A joint initiative of Ludwig-Maximilians University’s Center for Economic Studies and the Ifo Institute for Economic Research
Area Conference on
Behavioural
Economics
28 – 29 October 2011 CESifo Conference Centre, Munich
Mechanism Design and Intentions
Nick Netzer and Felix Bierbrauer
CESifo GmbH
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81679 Munich
Germany
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Mechanism Design and Intentions∗
Felix Bierbrauer
University of Cologne
Nick Netzer
University of Zurich
July 2011
Abstract
We introduce intentions-based social preferences into a Bayesian mechanism design framework. If social preferences are observable, any tension between material eciency, incentive
compatibility, and voluntary participation can be resolved. Hence, the classical impossibility
results that the conventional mechanism design literature has established are turned into
possibility results. We also investigate dierent possibilities how to incorporate kindness
sensations into assessments of welfare. For the case of unobservable social preferences, we
suggest a notion of psychological robustness. Psychologically robust mechanisms can be implemented without any need to acquire information about the intensity of social preferences.
We show that the mechanisms which have been the focus of the conventional mechanism
design literature need to be modied only slightly to achieve psychological robustness.
Keywords: Mechanism Design, Psychological Games, Social Preferences, Intentions,
Reciprocity, Revelation Principle.
JEL Classication: C70, C72, D02, D03, D82, D86.
∗
Email: [email protected] and [email protected]. We gratefully acknowledge helpful com-
ments by Florian Englmaier, Hans Peter Grüner, Martin Hellwig, Holger Herz, Benny Moldovanu, Armin Schmutzler, Roberto Weber and seminar participants at LMU Munich, MPI Bonn, and the Universities of Cologne,
Mannheim and Zurich. All errors are our own.
1
Introduction
People with intentions-based social preferences are willing to give up own material payos in
order to either reward behavior by others that they attribute to good intentions, or to punish
behavior that they attribute to bad intentions. After reporting on an experiment that provides
evidence for the prevalence of intentions-based social preferences, Falk et al. (2008) conclude
that ...people not only take the distributive consequences of an action but also the intention
it signals into account when judging the fairness of an action. This result casts serious doubt
on the consequentialist practice in standard economic theory that denes utility of an action
solely in terms of its consequences...
(p.
299).
This paper is an attempt to take up this
challenge and explore the implications of intentions-based social preferences for the theory of
mechanism design. We use the workhorse model of mechanism design theory, also known as the
independent private values model, and adapt the analysis of intentions-based social preferences
by Rabin (1993) to games of incomplete information, which leads to the solution concept of a
Bayes-Nash fairness equilibrium. We then provide a systematic study of the set of implementable
allocations. We also provide a systematic account of the possibility or impossibility to include
psychological payos into assessments of economic welfare.
Finally, we provide a systematic
analysis of mechanisms that are psychologically robust in the sense that they are operational for
varying degrees of social preferences.
A major methodological diculty that we have to address is that an assessment of intentions
implies that procedural questions matter. It does not only matter what players do, but also what
they could have done instead. If player
interpret
i's
i's
behavior neither as kind nor as unkind, and
motivated by concerns for reciprocity. If
been higher,
set of feasible strategies is a singleton, player
j
might interpret
i's
i
j 's
will
behavior will therefore not by
could have chosen a strategy so that
j 's
payo had
strategy as unkind and therefore be willing to harm player
Alternatively, if i's strategy has been to give up own material payo so as to increase
j
j
j 's
i.
payo,
might interpret this as kind and therefore be willing to reciprocate. This implies that the set
of attainable allocations will be aected by options that ultimately remain unchosen, so that
standard tools of mechanism design theory, such as the revelation principle, are not available.
1
While this complicates the formal analysis, there is also a benet that one can reap. Procedural
concerns oer a chance to make outcomes available that would be out of reach if individuals did
not care about intentions. Actions that are calibrated so that they remain unchosen but aect
the agents' perceptions of kindness become an important tool of mechanism design. For instance,
one of our results will be to show that every ecient social choice function can be implemented
under certain assumptions. This result would not be available with selsh individuals, and its
proof makes use of the possibility to engineer just the right intensity of kindness that is needed
for ecient outcomes to be obtained.
1
The behavioral relevance of unchosen actions has impressively been illustrated by the experiments of Andreoni
et al. (2002) or Falk and Fischbacher (2006), among others. Falk and Fischbacher (2006) report on how individuals
assess the kindness of proposals for the division of a cake of xed size. They show that this assessment depends
on the choice set that is available to the proposer. In particular, they compare the following two situations: (i)
the proposer gets 80 per cent of the cake and the only alternative option is that the proposer gets 20 per cent, (ii)
the proposer gets 80 per cent of the cake and there are two alternative options, a fty-fty division and 20 per
cent for the proposer. They show that the 80 per cent proposal is considered very unfair in the second situation,
and less unfair in the rst situation.
1
Our formal analysis has two main parts, which dier with respect to the information that is
available to the mechanism designer. In the rst part, we assume that individuals have private
information only about their material payos.
The process by which kindness sensations are
generated and traded o against material payos is assumed to be known to the mechanism
designer.
We refer to this case as mechanism design with known kindness generating process.
In the second part, we assume in addition that the designer does not know to what extent
individuals are willing to make trade-os between material and psychological payos, i.e., he
does not know the strength of the individuals' concern for intentions. This part of the analysis
comes under the heading of psychologically robust mechanism design.
Part I: Mechanism Design with Known Kindness Generating Process.
Our analysis
under the assumption that the kindness generating process is known begins with an examination
of the revelation principle.
We show that the revelation principle fails.
There exist social
choice functions that cannot be implemented by direct mechanisms with a truth-telling BayesNash fairness equilibrium, but that can be implemented by means of a non-direct mechanism.
With a direct mechanism, every available message is used in a truth-telling equilibrium. Put
dierently, this class of mechanism-equilibrium-pairs excludes unused actions from the analysis.
With intentions-based social preferences, this restricts the set of implementable social choice
functions, because unused actions are important for the interpretation of equilibrium play and
hence a valuable design tool.
We can show, by contrast, that a pseudo-revelation principle holds. Accordingly, it is without
loss of generality to focus on mechanisms so that a player's action set includes the set of possible
types, and which possess truth-telling equilibria. Hence, while the restriction that every action
must be used in equilibrium would involve a loss of generality, the restriction that every used
action is a truthfully communicated type is without loss of generality.
We then turn to the welfare implications of intentions-based social preferences. In particular,
we explore various approaches to the treatment of kindness sensation for assessments of economic
welfare. First, kindness sensations and psychological payos might be considered as relevant from
a behavioral but not from a welfare perspective. We are then left with the question what one
can say about the implementability of social choice functions that are, in a conventional sense,
ecient. We show in Theorem 1 that every ecient social choice function can be implemented by
an appropriately chosen non-direct mechanism. Our proof of this observation makes use of the
possibility to engineer kindness sensations in such a way that every individual's utility function
is turned into a utilitarian welfare function. The construction is akin to a Groves mechanism,
in that it aligns private and social interests.
It is dierent, however, because it is not based
on a clever choice of payments that individuals have to make in equilibrium, but on a clever
choice of payments that individuals refuse to make in equilibrium, and which are important for
the intentions they impute to their mutual behavior. This shows that intentions-based social
preferences can enlarge the set of implementable social choice functions in a drastic way. Every
ecient social choice function and not just those that also happen to be incentive-compatible
according to the conventional mechanism design approach can be implemented in Bayes-Nash
fairness equilibrium. Many applications of mechanism design theory also include participation
2
constraints, which make sure that the players prefer the outcome of the mechanism over a
status quo. The mechanism that we construct in order to prove Theorem 1 turns the players
into welfare-maximizers, which implies that they will prefer a mechanism giving rise to ecient
outcomes over any status quo that is not ecient.
Hence it not only eliminates any tension
between incentive compatibility and eciency, but also any tension between eciency, incentive
compatibility and voluntary participation.
We then turn to the possibility of treating the individuals' overall utility, which aggregates
material payos and kindness sensations, as the relevant criterion of economic welfare. We have
nothing to say about whether this utility function is, in an empirical sense, indeed a good measure
of individual well-being. Instead we explore the logical consequences of having a welfare objective
based on overall utility. We nd that the ambition to construct a welfare-maximizing mechanism
yields paradoxical results.
It is possible to construct mechanisms with the property that the
allocation becomes entirely unresponsive to the individuals' private information implying that
there is no real reason to let individuals communicate in a mechanism and still welfare goes out
of bounds because every player's kindness payo goes out of bounds. Our preferred interpretation
is that the existence of such kindness pumps, which are detached from any material consideration,
reveals fundamental problems behind the all-embracing welfare notion.
As an alternative, we therefore consider the following approach.
Suppose we x a social
choice function, i.e., we x the material outcome we want to achieve.
We then compare all
the dierent mechanisms that may be used to implement this social choice function, and ask
whether they can be ranked in the kindness dimension.
In particular, we ask whether there
exists a mechanism that implements the given social choice function with maximal kindness.
For environments with two players, Theorem 2 claries the conditions under which such a best
mechanism exists. A necessary and sucient condition is that the social choice function gives
rise to bilateral externalities, implying that the expected payo of player 1 varies with the type
of player 2 and vice versa. If there were no such externalities, unused actions could again be used
to let the players' kindness sensations grow arbitrarily. If there are externalities, the players'
kindness must remain bounded in equilibrium because, if they became too kind or unkind, their
concern for the other player's payo would eventually erode their willingness to act truthfully.
Part II: Psychologically Robust Mechanism Design.
The previous results relied on the
possibility to ne-tune out-of-equilibrium actions and payos so that players are either turned
into maximizers of the sum of material payos, or, for a given material outcome, into maximizers
of kindness. In the second part of the paper, we compare this benchmark to the problem that
arises if the mechanism designer does not know how strongly kindness sensations aect individual
behavior. We say that a mechanism has a psychologically robust equilibrium if there is a strategy
prole that constitutes a Bayes-Nash fairness equilibrium whatever the individuals' willingness
to trade-o material payos and kindness sensations.
One attractive feature of this solution concept is the following: If we obtain possibility results,
i.e., if we can characterize a social choice function as being implementable in a psychologically
robust way, then there is no need to worry about the details of multidimensional design in
environments in which players have private information both about their material payos and
3
their willingness to trade-o material payos and kindness sensations. In fact, our main results
are of this kind.
We show that many, if not most, of the social choice functions that have
attracted the attention of mechanism design theorists in the past, can indeed be implemented
in psychologically robust equilibrium.
As a rst result, we retrieve the revelation principle for the solution concept of a psychologically robust equilibrium. Our further analysis is then based on the following observation.
Suppose that, for some social choice function, the expected payo of any one player
depend on the type of any other player
j,
i
does not
so that each player is insured against the risk asso-
ciated with the randomness of each other player's type. If this insurance property holds, then
players cannot aect each other's payo by unilateral deviations from truth-telling in the direct
mechanism, so that kindness sensations become irrelevant and behavior is exclusively motivated
by own material payos. Consequently, a sucient condition for robust implementability of a
social choice function with the insurance property is the conventional requirement of incentive
compatibility.
Theorem 3 states that to any social choice function that is incentive compatible in the
conventional sense, there exists an equivalent version that has the insurance property and is
thus psychologically robust. Equivalence holds with respect to the allocation rule, the expected
payos of players (even in an ex interim sense), and the expected decit or surplus. By contrast,
the social choice functions may dier with respect to their respective decits or surpluses from an
ex post perspective. The proof is based on the observation that oering insurance is not in conict
with both the requirements of individual incentive compatibility and the requirement of, say,
budget balance in an average sense. Hence, for environments where budget balance is required
only in expectation, every social choice function that is implementable in an environment with
selsh individuals is also implementable in an environment with arbitrarily strong intentionsbased social preferences.
The theorem covers essentially any application of the independent
private values model that has been studied in the literature. In particular, it also covers the study
of optimal mechanisms that introduce participation constraints in addition to the requirement of
incentive compatibility, since ex interim payos are preserved by our construction of insurance.
Well-known examples include mechanisms for bilateral trade (Myerson and Satterthwaite 1983),
partnership dissolution problems (Cramton et al.
1987), or public goods provision (Hellwig
2003). The theorem implies that the mechanisms which are studied in these papers need to be
twisted only slightly if we want to make sure that they become psychologically robust.
A limitation of the theorem is that budget decits and surpluses may become unavoidable in
an ex post sense. One cannot insure individuals against the risk in the other individuals' types
and simultaneously balance the budget in each and every circumstance. Theorem 4 therefore
states sucient conditions under which psychological robustness is compatible with ex post
budget balance, and hence full material eciency of a social choice function. At the core of this
result lies the observation that the expected externality mechanism due to d'Aspremont and
Gerard-Varet (1979) and Arrow (1979) which satises ex post budget balance and implements
an ecient social choice function satises the insurance property under an assumption of
symmetry.
This follows from construction of this mechanism, which requires each agent to
compensate all others for the expected implications of a change in his type.
4
A comprehensive view on Parts I and II.
Our results on psychological robustness are
reassuring from the perspective of conventional mechanism design theory. Even if individuals
are inclined to respond to the behavior of others in a reciprocal way, this will in many cases
not upset implementability of the outcomes that have been the focus of this literature.
Our
analysis shows that, for many applications of interest, there is a way to design mechanisms so
that the transmission channel for reciprocal behavior is simply shut down. If it is shut down,
then individuals are, by design, acting as selsh payo maximizers and incentive compatibility
in the traditional sense is all that is necessary to ensure the implementability of a social choice
function.
By contrast, our analysis under the assumption of a known kindness generating process shows
the potential of exploiting the reciprocity channel, rather than shutting it down. Every ecient
social choice function becomes implementable. In addition, there is no longer a tension between
eciency and voluntary participation. Moreover, the question whether there exists a best mechanism for a given social choice function becomes meaningful.
With an analysis that is based
exclusively on outcomes-based preferences, it would be impossible to even ask this question.
The remainder of the paper is organized as follows.
discussion of the related literature.
The next section gives a more detailed
Section 3 states the mechanism design problem and in-
troduces the solution concept of a Bayes-Nash fairness equilibrium.
Section 4 deals with the
analysis of mechanism design when the kindness generating process is known. It also contains
several examples of public goods provision problems that illustrate the main results in Section
4. Section 5 contains our analysis of psychologically robust mechanism design. The last section
contains concluding remarks, in particular on avenues for further research. Several proofs are
relegated to the appendix.
2
Related Literature
Our work is related to several strands in the literature, (i) a literature which tries to model and
to empirically identify interdependent preferences, (ii) a literature that studies the implications
of interdependent preferences for various applications, and (iii) the theory of mechanism design.
Interdependent Preferences.
Models of interdependent or social preferences are usually
2
distinguished according to whether they are outcome-based or intentions-based.
Prominent
examples for the rst class are Fehr and Schmidt (1999) and Bolton and Ockenfels (2000), while
Rabin (1993) and Dufwenberg and Kirchsteiger (2004) belong to the second class of models.
3 An
extensive experimental literature examples include Andreoni et al. (2002), Falk et al. (2003)
and Falk et al. (2008) has concluded that behavior is most likely inuenced by both types of
considerations.
2
3
4 The theoretical models proposed by Levine (1998), Charness and Rabin (2002),
See Sobel (2005) for an excellent survey, with a focus on reciprocity.
Theories of procedural fairness (Bolton et al. 2005, Trautmann 2009, Krawczyk 2011) might be considered
as a third category, where probabilistic outcomes enter social comparisons in ways not compatible with either
outcome- or intentions-based models.
4
See Stanca (2010) for an instance where only outcomes matter, and McCabe et al. (2003) for evidence in
favor of purely intentions-based models. Oerman (2002) distinguishes between positive and negative intentions
5
Falk and Fischbacher (2006) and Cox et al. (2007) combine outcomes and intentions as joint
motivations for social behavior.
In this paper, we consider intentions-based social preferences only. We do this for a methodological reason.
The distinguishing feature of intentions-based preferences is their procedural
nature, i.e., sensations of kindness are endogenous to the game form.
This is a challenge for
mechanism design theory, which is concerned with nding optimal game forms. With outcomebased social preferences, this methodological issue would not arise. The validity of the revelation
principle, for instance, would not be in question. To keep the exposition straight, we therefore
refrain from also modelling outcome-based social preferences. That said, enriching our framework so that also outcome-based social preferences come into play would be straightforward, at
5
least conceptually.
The formal framework for modelling intentions is provided by psychological game theory, as
6 To
introduced by Geanakoplos et al. (1989), which allows payos to depend directly on beliefs.
the best of our knowledge, the literature does not yet contain a general treatment of intentionsbased social preferences for games of incomplete information: Rabin (1993) focusses on normal
form games and Dufwenberg and Kirchsteiger (2004) consider extensive form games, but both
7 Our mechanism design approach requires a general
contributions assume complete information.
theory of intentions for Bayesian games, and we will outline such a theory in Section 3.2.
Applications.
Social preferences have been modelled theoretically and investigated empiri-
cally within a wide range of applications to contracts, rms, and other relevant elds.
Most theoretical studies rely on outcome-based concepts, and here mostly on the inequality aversion models of Fehr and Schmidt (1999) or Bolton and Ockenfels (2000). For instance,
Engelmaier and Wambach (2010) derive optimal contracts under moral hazard and inequality
aversion, and Bartling (2011) studies team incentives in a framework that allows for both inequality aversion or pure status preferences.
We refer the reader to these papers for further
references on moral hazard models with outcome-based social preferences. Another strand of
the outcome-based literature investigates how competing rms can screen agents that dier in
their unobservable productivities or social preferences, resulting in equilibrium skill heterogeneity (Cabrales et al. 2007, Cabrales and Calvó-Armengol 2008) or dierent corporate cultures
(Kosfeld and von Siemens 2011). Furthermore, the literature on auctions has investigated genand nds stronger evidence for the latter.
5
Frey et al. (2004) provide a general discussion of procedural preferences and their potentially important role
for the design of institutions.
6
The framework of Geanakoplos et al. (1989) has been further developed by Battigalli and Dufwenberg (2009).
Besides intentions-based social preferences, psychological game theory has been used, among others, to study
optimal AIDS policies in the presence of fear (Caplin and Eliaz 2003), preferences over the timing of information
in strategic settings (Caplin and Leahy 2004), feelings of guilt (Battigalli and Dufwenberg 2007), adherence to
social norms (Li 2008), and framing eects (Dufwenberg et al. 2011a).
7
Segal and Sobel (2007) generalize the model of Rabin (1993) and provide an axiomatic foundation. Sebald
(2010) extends the model of Dufwenberg and Kirchsteiger (2004) by introducing objective randomization devices,
but still under the assumption of perfect observability.
The related contributions by Sebald (2007), Aldashev
et al. (2009) and Aldashev et al. (2010) model dierent applications where such randomization devices, called
procedures, play an important role, and they present experimental evidence in favor of the model. Von Siemens
(2009, online appendix) contains a model of intentions for a two-stage bargaining game with incomplete information about the second-mover's social type. Imperfect information about the other players' social types is also
the driving force for reciprocity in the model of Levine (1998), which does not build on the psychological game
framework considered here.
6
eral structures of outcome-based externalities (Jehiel et al. 1996), as well as spiteful preferences
(e.g., Morgan et al. 2003) and their role for the empirical phenomenon of overbidding.
Intentions-based social preferences have been utilized far less often.
Applications exist to
moral hazard problems (Englmaier and Leider 2008, Netzer and Schmutzler 2010), the puzzle
of wage rigidity (Dufwenberg and Kirchsteiger 2000), and the hold-up problem (von Siemens
2009). Nishimura et al. (2011) investigate, in an auctions model with two players and complete
information, the impact of intentions-driven reciprocal responses to spiteful behavior.
They
compare two dierent auction formats and can explain both over- and underbidding.
The experimental literature has long since emphasized the benecial role of reciprocity for
the design of incentive contracts (Fehr et al. 1997, Fehr and Falk 2002), and recent studies have
revealed that principals in laboratory settings do in fact adapt their contract oers accordingly
8 On the other hand, reciprocity also involves a negative
(Fehr et al. 2007, Cabrales et al. 2010).
side, which can make contracting more dicult. In Hart and Moore (2008), for instance, ex post
aggravation by at least one party is inevitable with exible contracts, which triggers negative
9 Experimental evidence by Fehr et al. (2011a)
reciprocal reactions and yields ex post ineciency.
supports this prediction. It is therefore not ex ante clear how intentions-based social preferences
will aect the set of implementable social choice functions.
Mechanism Design Theory.
Our basic mechanism design framework, the independent pri-
vate values model with quasilinear utilities, corresponds to standard textbook treatments (MasColell et al. 1995, chapter 23). Our analysis addresses the key question of this literature, which
is the characterization of conditions under which ecient outcomes can or cannot be reached.
Several authors have investigated mechanism design problems under behaviorally motivated
assumptions.
10
In an environment with symmetric information, Eliaz (2002) studies imple-
mentability under the assumption that some players behave in an unpredictable faulty way.
In the same framework, Tumennasan (2010) investigates quantal response equilibria and Renou
and Schlag (2011) apply minimax regret equilibrium as solution concept. Matsushima (2008a,b)
assume that agents dislike lying, in frameworks with symmetric and private information, respectively. These papers are also related to our robustness approach in Section 5, because they share
the goal of designing mechanisms which do not rely on details of the environment (Matsushima
2005).
Another strand of literature is concerned with the design of mechanisms that exhibit
good learning and stability properties. Recent examples include the papers by Mathevet (2010)
and Cabrales and Serrano (2011), which also contain further references.
The possibility that institutions aect individual preferences has received some attention
(Bowles 2008). We add to this literature because in our model kindness sensations and hence
the willingness to trade-o own and others' payos depend on the mechanism.
Bowles and
Hwang (2008) investigate a public goods game where attitudes towards voluntary contribution
may interact with the level of a subsidy, yielding the possibility of crowding in or out.
A
common pool game where the population shares of dierent social types are aected by extrinsic
8
Cabrales et al. (2010) report that concerns about strategic uncertainty play an even greater role for contract
oers than social preferences.
9
Netzer and Schmutzler (2010) show that negative intentions must arise in any equilibrium of a two player
model based on Rabin (1993) whenever one of the players is materialistic.
10
See also the survey in McFadden (2009).
7
incentives is analyzed theoretically and experimentally by Rodriguez-Sickert et al. (2008). The
contributions by Bar-Gill and Fershtman (2004, 2005) and Heifetz et al. (2007) are all based
on the assumption that preferences including outcome-based social preferences in the rst
two papers are aected by the institutional framework due to an underlying evolutionary
adaptation process.
Finally, some attempts have been made to include outcome-based social preferences into
11 Desiraju and Sappington (2007) allow for inequality
more general mechanism design setups.
aversion in a model where a prot-maximizing principal faces two agents with private information about their production costs. Inequality aversion has no impact on optimal employment
contracts when the agents are symmetric, but modies their structure otherwise (see also our
discussion in Section 5.4).
A related model, with additional private information about social
preferences, is presented by von Siemens (2011), who studies the optimality of excluding agents
from the rm and hence the social reference group. Kucuksenel (2011) introduces altruism in
an environment similar to ours. Standard tools such as the revelation principle remain available
in this context. Increasing degrees of altruism help to achieve eciency, because individual utilities already internalize social goals.
12 With intentions-based social preferences, internalization
through preferences is not given by implication. Instead, it is the task of the (non-direct) mechanism to endogenously generate the right levels of kindness for the purpose of internalization.
3
Fairness Equilibria and Mechanism Design
3.1 Environment, Social Choice Functions, Mechanisms
An environment
E = [I, A, (Θi , πi )i∈I , p]
nite set of agents denoted by
type
is dened by the following characteristics: There is a
I = {1, . . . , n}
θi which belongs to a nite set Θi ⊂ R.
a material payo given by
πi (a, θ),
where
and a set of feasible allocations
If an allocation
θ = (θ1 , . . . , θn )
p(θ)
is the probability of type vector
θ.
independence we have
p(θ−i |θi ) = p(θ−i )
p(θ−i |θi )
i
has a
is a vector that lists the individuals'
p
with support
We use expressions like
marginal distributions for subsets of agents, and
Player
a ∈ A is chosen then player i realizes
types. Types are random, described by a probability distribution
so that
A.
p(θi )
or
Θ=
Qn
p(θ−i )
i=1 Θi ,
for the
for conditional distributions. Under
for all players and types.
We focus on the conventional textbook environment with quasilinear preferences and independent private values, by making the following assumptions.
13 First, types are independently
distributed and privately observed by the agents. Second, an allocation
species for each individual a consumption level
transfer
ti .
of a private or public good and a monetary
πi (a, θ) = vi (qi , θi ) + ti .
n
R species all possible consumption
Finally, material payos take the quasilinear form
The set of feasible allocations is
A = Q×T.
proles. Costs associated with a prole
11
qi
a = (q1 , . . . , qn , t1 , . . . , tn )
Here,
Q⊆
(q1 , ..., qn ) ∈ Q
can, without loss of generality, be cap-
Gaspart (2003) follows a dierent approach to modelling fairness. He captures fairness of a mechanism by
the requirement that all players can induce the same outcomes by deviations from equilibrium behavior.
12
Glazer and Rubinstein (1998) study the aggregation of information across experts.
There, the optimal
decision cannot be achieved when all experts care solely about the social goal, but can be achieved when egoistic
motives exist.
13
That said, some of our results, for instance those on solution concepts in the next subsection and in Section
5, hold more generally and require neither independence nor quasilinearity.
8
vi (qi , θi ), with an arbitrary cost sharing default when
n
R then describes the possible transfers to the agents. Depending on the
tured directly through the payo functions
necessary. The set
T ⊆
application, they might have to satisfy dierent constraints. If not mentioned otherwise, we rule
out subsidies from outside, that is, we assume
A social choice function
types. We also write
f=
f :Θ→A
to be Pareto ecient in
T = T̄ ,
A mechanism
T̄ = {(t1 , ..., tn ) ∈ Rn |
Pn
i=1 ti
≤ 0}.
species an allocation as a function of the individuals'
A,
f
f
i=1 ti (θ)
Pn
=0
f
requires, in the most general approach, the allocation
for every type prole
θ ∈ Θ.
In our quasilinear framework with
Pn
i=1 vi (qi , θi ) must be maximized by
this is equivalent to saying that
and that budget balance
where
(q1f , . . . , qnf , tf1 , . . . , tfn ) when referring to the dierent components of
separately. Material eciency of an SCF
f (θ)
T = T̄
must be satised, for every prole
q1f (θ), ..., qnf (θ)
θ ∈ Θ.
Φ = [M1 , . . . , Mn , g] contains a message set Mi for each player i and an outcome
g : M → A which species an allocation for each message prole m = (m1 , . . . , mn ) ∈
Qn
M = i=1 Mi . We also write g = (q1g , . . . , qng , tg1 , . . . , tgn ). Players privately observe their types
function
and hence can condition their message on their type. Therefore, a pure strategy for player
a mechanism
Φ
is a function
si : Θi → Mi .
The set of all pure strategies of player
d
by Si . Throughout, we focus on pure strategies. The mechanism Φ
the direct mechanism for
i
i
in
is denoted
= [Θ1 , . . . , Θn , f ]
is called
f.
s = (s1 , . . . , sn )
We will shortly dene what it means that a strategy prole
is a Bayes-Nash
equilibrium (BNE) or a Bayes-Nash fairness equilibrium (BNFE). We then obtain the following
denition of an implementable social choice function: A social choice function
f
in BNE/ BNFE if there is a mechanism with a BNE/ BNFE strategy prole
θ, f (θ) = g(s∗ (θ)),
where
is implementable
s∗
so that, for all
s∗ (θ) = (s∗1 (θ1 ) . . . , s∗n (θn )).
3.2 Solution Concepts
Bayes-Nash Equilibrium.
We are primarily interested in a characterization of social choice
functions that are implementable in BNFE. However, as a benchmark, we rst introduce the more
familiar solution concept of a BNE. To facilitate the comparison to BNFE we state the denition
in a way that emphasizes the role of the players' beliefs about their opponents' strategies. Since
we focus on pure strategy equilibria, we can, without loss of generality, assume that player
beliefs about player
j 's
k 's
strategy. Given an environment
payo from following strategy
Πi (si , (sbij )j6=i ) =
Sj . In the following,
bb
sijk player i's belief about j 's belief
strategy put unit mass on a particular strategy in
b
we denote this strategy by sij . Analogously, we denote by
about
i's
X
si ,
E
and a mechanism
Φ,
player
i's
expected material
given his beliefs, then equals
p(θ)πi (g(si (θi ), (sbij (θj ))j6=i ), θ) .
θ∈Θ
As throughout the section, this expression is still dened for an arbitrary prior and arbitrary
payo functions
πi ,
as none of the concepts considered here rely on quasilinear payos or inde-
pendence of types.
Denition 1.
A BNE is a prole
s∗
such that, for all players
∗
b
(1) si ∈ arg maxsi ∈Si Πi (si , (sij )j6=i ), and
b
∗
(2) sij = sj for all j 6= i.
9
i ∈ I,
Denition 1 is based on expected payos from an ex ante perspective, that is, from the
perspective of a player who has not yet discovered his type, but plans to behave in a typecontingent way and who evaluates a plan according to the associated expected material payo.
There is a second, equivalent denition, which evaluates actions (as opposed to strategies) from
an interim perspective, where each player has learned his own type, but has not yet observed the
other players' types. We denote the interim expected payo that type
with message
mi
θi
of player
i
associates
by
X
Πi (mi , (sbij )j6=i |θi ) =
p(θ−i |θi )πi (g(mi , (sbij (θj ))j6=i ), (θi , θ−i )).
θ−i ∈Θ−i
Denition 2.
s∗
A BNE is a prole
∗
(1) si (θi ) ∈ arg maxmi ∈Mi
b
∗
(2) sij = sj for all j 6= i.
such that, for all players
Πi (mi , (sbij )j6=i |θi ) for all types
i ∈ I,
θi ∈ Θi ,
and
It is straightforward to see that Denitions 1 and 2 are equivalent as soon as every type of every
player occurs with strictly positive probability.
Bayes-Nash Fairness Equilibrium.
In an attempt to model intentions-based social prefer-
ences, Rabin (1993) has introduced the notion of a fairness equilibrium for normal form games
of complete information. In the following we adopt this solution concept to normal form games
of incomplete information. In particular, we follow Rabin in that we enrich the individuals' payo functions by psychological components which capture the desire to reward kind and punish
14
unkind behavior.
Given an environment
E
and a mechanism
Φ,
we assume that player
i's
expected utility is
given by
b
Ui (si , (sbij , (sbb
ijk )k6=j )j6=i ) = Πi (si , (sij )j6=i ) +
X
yij κij (si , (sbik )k6=i ) λiji (sbij , (sbb
ijk )k6=j ) .
j6=i
Expected utility has two components.
One source of expected utility is the material payo
Πi (si , (sbij )j6=i ). In addition, player i's interaction with any other player
j
gives rise to sensations
of kindness (or unkindness). This is captured by
yij κij (si , (sbik )k6=i ) λiji (sbij , (sbb
ijk )k6=j ) .
In this expression,
yij
is an exogenous parameter, interpreted as the weight that kindness in
the relationship between players
i
and
j
has in player
i's
expected utility function. The term
kij (si , (sbik )k6=i ), which is dened in more detail below, is a measure of how kind player
to treat player
j.
implications of
i's
wants
It is a measure of intended as opposed to actual kindness since the
behavior for the well-being of player
other players, about which player
i
j
also depend on the behavior of the
b
holds beliefs (sik )k6=i . Whether i's intended kindness enters
the expected utility function with a positive or a negative sign depends on
14
i
i's
beliefs about
In contrast to Rabin (1993), we will omit a normalization of kindness terms for simplicity, in line with
Dufwenberg and Kirchsteiger (2004). See also footnote 19.
10
j , λiji (sbij , (sbb
ijk )k6=j ).
the kindness intended by
kindness, player
i's
Note that in forming beliefs about
beliefs about the beliefs of player
j
j 's
intended
matter.
b
Following the literature, we model kij (si , (sik )k6=i ) as the dierence between
j 's
actual ex-
pected payo and an equitable reference payo,
κij (si , (sbik )k6=i ) = Πj (si , (sbik )k6=i ) − πjei ((sbik )k6=i ) .
Alternative modelling choices for equitable payos have been explored in the literature.
We
follow Rabin (1993), but provide a brief discussion of conceivable alternatives below. Specically,
πjei ((sbik )k6=i )
where
1
=
2
!
max
si ∈Eij ((sbik )k6=i )
Πj (si , (sbik )k6=i )
+
min
si ∈Eij ((sbik )k6=i )
Eij ((sbik )k6=i ) is the set of bilaterally Pareto ecient strategies.15
Πj (si , (sbik )k6=i )
,
The restriction to Pareto
ecient strategies makes sure that kindness cannot be inuenced by adding irrelevant strategies
that yield Pareto inecient outcomes. Given this denition of
i's
intended kindness towards
j,
λiji
is
we analogously dene
b
bb
λiji (sbij , (sbb
ijk )k6=j ) = κji (sij , (sijk )k6=j )
as player
i's
belief about how kind player
obtained from
κji
j
intends to be to himself.
simply by replacing all arguments by player
i's
The expression
corresponding beliefs. Armed
with these denitions we can now state our denition of a Bayes-Nash fairness equilibrium,
which amounts to the denition by Rabin (1993) applied to expected payos.
Denition 3.
A BNFE is a prole
s∗
such that, for all
i ∈ I,
∗
b
bb
(1) si ∈ arg maxsi ∈Si Ui (si , (sij , (sijk )j6=k )j6=i ),
b
∗
(2) sij = sj for all j 6= i, and
b
bb
(3) sijk = sjk for all j 6= i and k 6= j .
Denition 3 is an ex ante denition in the spirit of Denition 1. In fact, it becomes equivalent
to the denition of BNE whenever
yij = 0
for all
i, j ∈ I , j 6= i,
so that concerns for reciprocity
disappear. The BNFE denition does not generally permit an equivalent interim version. The
reason is that the action
no longer inuence player
si (θi )
i's
that player
i
chooses upon receiving the information
payo only in the corresponding information set.
b
depends on the term Πj (si , (sik )k6=i ), which is player
the complete strategy
si .
θi
does
Formally,
κij
j 's expected payo and as such depends on
Hence, we cannot write an individual's ex ante optimality condition
as a collection of mutually independent interim optimality conditions.
Discussion of Alternative Modelling Choices.
We have dened kindness based on ex
ante expected payos. A second possibility would be to dene kindness conditional on players'
types, i.e., based on updated interim payos. A player's ex ante kindness would then correspond
15
A strategy
Πi (s0i , (sbik )k6=i )
si belongs to Eij ((sbik )k6=i ) if and only if there is no alternative strategy s0i ∈ Si so that
≥ Πi (si , (sbik )k6=i ) and Πj (s0i , (sbik )k6=i ) ≥ Πj (si , (sbik )k6=i ), with at least one of those inequali-
ties being strict.
11
to the expectation of the interim kindness of his dierent types.
We introduce this concept
formally in Appendix A.1, where we also derive conditions under which the two approaches are
equivalent. Under these conditions, the above BNFE denition again possesses an equivalent
interim version.
The literature agrees to dene an equitable payo as a value in between the largest and
the smallest material payo that one player can give to another when varying the own strategy,
where attention is restricted to a set of strategies that are ecient, for the reasons outlined above.
Dierent authors invoke dierent eciency concepts, however. Rabin (1993) denes eciency
conditional on the opponents' strategies.
Hence, a strategy is ecient if it induces a Pareto
ecient material payo prole, given the xed strategies of the opponents.
Dufwenberg and
Kirchsteiger (2004) dene eciency unconditionally: a strategy is ecient whenever it induces
a Pareto ecient material payo prole for some possible strategies of the opponents.
More
precisely, a strategy is inecient only if it is Pareto dominated by some other strategy for all
opponent strategies.
Given our focus on games in normal form, we follow Rabin (1993) and apply the conditional
16 When moving to
eciency concept. However, Rabin (1993) considers two-player games only.
more than two players, the eciency concept can be applied bilaterally or population-wide. As
much of the earlier literature, we model social preferences as purely bilateral: In the assessment of
the kindness in his relation to player
j , player i does not care about how j
treats a third player
k.
In our formalism, this also shows up in the denition of equitable payos: From the perspective
of player
i,
outcome for
the payo deserved by player
j,
conditional on
i
j
is a weighted average of the best and the worst
choosing a strategy in the bilateral eciency set
Eij ((sbik )k6=i ).
Our reason for using the bilateral eciency concept is twofold. First, this enables us to generalize
some of Rabin's (1993) results in a straightforward way. Second, as argued above, it ts well
with the bilateral notion of reciprocity, and we avoid the reintroduction of more complicated
interpersonal eects. Trivially, the dierence between bilateral and population-wide eciency
notions disappears for all our applications with only two players.
4
Mechanism Design with Known Kindness Generating Process
In the following we will study the problem of mechanism design from the perspective of a designer
who knows how kindness sensations are generated. In particular, we consider a designer who
knows how individuals compute equitable payos,
o material payos and mutual kindness,
yij ,
πjei ,
etc.
to what extent they are willing to trade-
We are interested in two main questions:
First, are standard tools of mechanism design theory still available in a model with intentions
based social preferences? Particularly, we are interested in understanding whether it still involves
no loss of generality to restrict attention to direct mechanisms and to truth-telling equilibria.
Second, we discuss alternative notions of eciency with a focus on the question to what extent
kindness sensations can enter eciency considerations as an own source of individual well-being.
We present various possibility and impossibility results.
16
The appendix in Charness and Rabin (2002) contains a model of normal form games with more than two
players, but their approach to modelling reciprocity diers from Rabin (1993). The same holds for the models of
Levine (1998) and Falk and Fischbacher (2006).
12
In models in which individuals care only about their material payos, the revelation principle is the main tool for the analysis of mechanism design problems. Accordingly, it is without
loss of generality to focus on outcomes that can be obtained as the truth-telling equilibrium
of a direct mechanism.
This approach makes it possible to separate outcomes from the pro-
cedures according to which they are obtained: implementability (or incentive compatibility)
becomes a property of a social choice function that can be investigated without recourse to specic allocation mechanisms or institutions. We show that this is no longer true if psychological
considerations are introduced into the model (Propositions 2 and 3). The reason is that individuals do no longer care exclusively about the outcomes that are obtained in equilibrium, but also
about alternatives that might have been chosen instead. Consequently, with the solution concept
of a BNFE, procedures matter in the following two senses: First, the revelation principle does
not hold so that the question whether a social choice function is implementable can no longer
be answered by looking only at direct mechanisms. Second, as will be studied in more detail in
Section 4.3 below, dierent mechanisms which implement the same social choice function dier
in the kindness sensations that they generate. We can therefore ask which mechanism performs
best in the kindness-dimension.
4.1 Failure of the Revelation Principle
We start the discussion of why the revelation principle fails for the solution concept of a BNFE
by recalling why it holds for the solution concept of BNE.
Proposition 1
.
social choice function
implements
f,
Φ = [M1 , . . . , Mn , g] implements the
∗
d
in some BNE s . Then the direct mechanism Φ = [Θ1 , . . . , Θn , f ] also
(Myerson 1979)
f
in a BNE
sT
where
Suppose a mechanism
sTi (θi ) = θi
for all
i∈I
and
θi ∈ Θi .
The proposition immediately follows from the observation that if, under the direct mechanism
Φd , some individual
i had an incentive to deviate from truth-telling sTi , then the same individual
would have an incentive to deviate from
is a BNE in
s∗
Φ.
s∗i
in
Φ,
thereby contradicting the assumption that
Alternatively, one can interpret
Φd
and
sT
as being obtained from
by relabelling messages in line with the types who use them in
work for BNFE. When moving from an arbitrary mechanism
Φ
s∗ .
mi ∈
Mi with s∗i (θi )
6= mi
for all
θi ∈ Θi .
and
This proof does not
to the direct mechanism
we generally omit (i.e., do not relabel but simply leave out) messages from
∗
in s , i.e., messages
Φ
s∗
Φ
Φd ,
that were unused
This is no problem for BNE:
the removal of unused actions cannot destroy the BNE property. Such messages can, however,
be important for the kindness terms in a BNFE, and omitting them can change the BNFEstructure. In the following, we will illustrate this by means of a public goods application, where
BNFE implementation of ecient provision with equal cost sharing is impossible in a direct
mechanism, but is possible using a suitably dened non-direct mechanism. We will also continue
to use Example
Example 1.
1
throughout the paper, to illustrate several of our results.
There are two individuals,
indivisible public good has to be taken.
q2 = q
Θ2 =
with
q ∈ {0, 1},
and
T = T̄ .
I = {1, 2},
and a decision about the provision of an
Formally, we have
Q = {(0, 0), (1, 1)},
Θ1 =
1
θ2 . We assume that high
The individuals' type spaces are
{θ20 , θ21 }, where the taste parameters satisfy
13
θ10
<
θ11 and
θ20
<
q1 =
0
1
{θ1 , θ1 } and
so that
and low taste parameters are equally likely for both players. Net utility from the public good is
given by
vi (1, θi ) = θi − c,
where
c
is the per capita cost of public goods provision. If the public
good is not provided then payos are given by
vi (0, θi ) = 0,
We seek to implement a social choice function
f∗
i∈I
for all
and
θi ∈ Θi .
which prescribes that the public good pro-
vision is ecient,
∗
q f (θ1 , θ2 ) = 1
θ1 + θ2 > 2c,
if and only if
and that there are no transfers,
∗
∗
tf1 (θ1 , θ2 ) = tf2 (θ1 , θ2 ) = 0
for all
(θ1 , θ2 ) ∈ Θ,
so that there is equal cost sharing. For concreteness, we assume that the parameters are such
that the public good should be provided if at least one individual values it highly, and should not
be provided otherwise. Formally,
min{θ10 + θ21 , θ11 + θ20 } > 2c > θ10 + θ20 .
We also assume that individual
1
benets (in material terms) from public good provision even if
his taste parameter is low, while individual
2 benets only with a high taste parameter.
Formally,
min{θ10 , θ21 } > c > θ20 .
Since we have only two players, we can simplify notation and write
weight of player
i
yi
for the reciprocity
j 6= i.
toward player
The following Proposition shows that, whatever the individuals' reciprocity concerns, the
social choice function
f∗
in Example 1 cannot be implemented as a truth-telling BNFE of a
direct mechanism. This impossibility result covers the case where
yi = 0, i = 1, 2,
and hence
the impossibility of implementation in a BNE.
Proposition 2.
Consider Example 1 and the direct mechanism for
[0, ∞[2 , the strategy prole
sT
=
f ∗.
For every
(y1 , y2 ) ∈
(sT1 , sT2 ) is not a BNFE.
Sketch. A formal of proof of the Proposition is in the Appendix. To see the logic of the argument,
suppose rst that
y1 = y2 = 0 so that the players care only for their material payos.
Then truth-
telling is not an equilibrium, because player 1 wants to ensure that the public good is provided
even if his taste is low, by assumption
θ10 > c.
Hence, given that player 2 behaves truthfully
T
(s2 = s2 ), the best response of player 1 is to exaggerate, i.e. to always communicate a high
H
taste parameter to the mechanism. For brevity, denote this strategy by s1 in the following.
Now suppose that
If player
1
plays
sT1 ,
y1 > 0 and/ or y2 > 0, and hypothesize that truth-telling is an equilibrium.
the best player
2
can do for player
1
H
is to exaggerate (s2 ), which makes
sure that the public good is always provided. It is easily veried that all strategies dierent from
sT2
and
sH
2
are inecient because they harm both players
T
Hence, by choosing s2 , player
player
2
2
gives player
is strictly unkind toward player
1.
1
and
2,
compared to truth-telling.
the lowest ecient payo, which implies that
Player
14
1
1
therefore nds it optimal to choose
sH
1
irrespective of the size of
y1 ≥ 0,
because
s1 = sH
1
simultaneously maximizes
T
minimizes Π2 (s1 , s2 ). This is contrary to the assumption that
Π1 (s1 , sT2 )
and
(sT1 , sT2 ) is a BNFE.
Proposition 2 shows that ecient public goods provision with equal cost sharing is out of
reach if only direct mechanisms are considered.
We will now show that, whenever there are
mutual reciprocity concerns, we can nd a non-direct mechanism
0
possible. More specically, consider the mechanism Φ
Φ0
in which implementation is
[M10 , M20 , g 0 ] in which individual
=
i
has
0
0 1
the extended message set Mi = {θi , θi , di }. The outcome of the mechanism is, for every pair of
0
0
g0
messages (m1 , m2 ) ∈ M1 × M2 , a decision on public goods provision q (m1 , m2 ) ∈ {0, 1} and a
0
g0
g0
g0
g0 g
pair of transfers t1 (m1 , m2 ) and t2 (m1 , m2 ). The following table gives the triplet (q , t1 , t2 )
for every possible pair of messages.
m2
m1
θ10
θ11
d1
θ20
θ21
d2
(0, 0, 0)
(1, 0, 0)
(1, 0, 0)
(1, 0, 0)
(1, 0, 0)
(1, 1 , −δ2 )
(1, 0, 0)
(1, −δ1 , 2 )
(1, 0, 0)
Table 1: The non-direct mechanism
We assume
0 < 2 ≤ δ1
0 < 1 ≤ δ2 ,
and
mechanism works like a direct mechanism
{θ10 , θ11 } × {θ20 , θ21 }.
Φ
Φ0
so that all transfers are actually feasible.
with outcome function
This
f ∗ as long as messages
g=
d1 , the consequence is the same
1
1
as when announcing the high type θ1 , except that when individual 2 sends θ2 , individual 1
are in
If individual 1 chooses message
receives an additional payment
1 ,
δ2 .
the outcome is the same as when announcing
Analogously, if 2 sends
d2 ,
whereas individual 2 has to make an additional payment of
θ21 ,
except that
1
when individual 1 sends θ1 , 2's received transfer is increased by some number 2 , and individual
δ1 .
(d1 , d2 ) will be irrelevant for our purpose.
0
0
0
0
With the non-direct mechanism Φ , player i's set of pure strategies equals Si = Mi × Mi . A
1 suers a loss of
generic element
s0i
of
The outcome after messages
Si0
is a tuple in which the rst entry is the message chosen in case of having
a low taste parameter, and the second entry is the message chosen in case of having a high taste
parameter. Note that, for both players, the strategy set of the direct mechanism,
Si = Θi × Θi ,
is a subset of the extended strategy set
Si0 , and that the outcome under the strategy pair (sT1 , sT2 )
is the outcome stipulated by the SCF
f ∗,
ecient outcome. The use of the messages
an inecient outcome (when
Proposition 3.
(y1 , y2 ) ∈
and
2
Sketch.
2 < δ1
or
for every
d1
or
1 < δ2
d2
(θ1 , θ2 ).
Moreover,
that satisfy
0 < 2 ≤ δ1
and
δ1 > 0
can only lead to redistribution and possibly
and
δ2 > 0
Φ0
Again, a complete proof of the Proposition is relegated to the Appendix.
and
1 = δ2 .
15
The logic
0 < 2 ≤ δ1
and 0 < 1 ≤ δ2
T
T
(s1 , s2 ) we can, in order
While we seek to implement a budget balanced SCF, the complete mechanism
2 = δ1
1
0 < 1 ≤ δ2 .
T T
so that (s1 , s2 ) is a BNFE. Starting from a hypothetical equilibrium
when
in Table 1. For every
T T
so that (s1 , s2 ) is a BNFE, for all
is as follows: We seek to verify that there there exist numbers
17
still induces an
17
).
Consider Example 1 and the non-direct mechanism
]0, ∞[2 there exist numbers
(sT1 , sT2 )
Φ0
is only budget balanced
to derive player 2's equitable payo, solve for the set of ecient strategies of player 1 if he
believes player 2 to behave according to
sT2 .
As under a direct mechanism, the best that player
1 can do for player 2 is to play understatement,
0 0
sL
1 = (θ1 , θ1 ).
The worst outcome for player
2 is obtained if player 1 maximizes his material payo. Under the non-direct mechanism this
requires that player 1 chooses
(d1 , d1 ),
rather than
sH
1 .
Using these observations to solve for
= 14 δ2 in the hypothetical truth-telling equilibrium.
T
By not using the action d1 , player 1's strategy s1 becomes strictly kind.
1
b
bb
0
A symmetric reasoning for player 2 yields λ121 (s12 , s121 ) = (δ1 + c − θ1 ) in the hypothetical
8
b
bb
player 1's kindness, we obtain λ212 (s21 , s212 )
δ1 , player 1 now also feels that he is treated kindly by player
T
2 if the latter plays s2 , because player 2 could have increased his material payo by using the
equilibrium. For suciently large
message
d2 .
δ1
Given these observations we can now calibrate the numbers
parameters
y1
and
y2
and
δ2
to the fairness-
to turn every player's utility-maximization problem into a problem of
welfare-maximization. To see how this works, consider player 2's problem. If player 1 chooses
sT1 ,
player 2 chooses
s2
in order to maximize
T
Π2 (sT1 , s2 ) + y2 λ212 (sb21 , sbb
212 )Π1 (s1 , s2 ).
Now let
λ212 (sb21 , sbb
212 ) =
1
y2 , or, equivalently,
δ2 =
4
y2
> 0.
Then the problem becomes: choose
s2 in order to maximize expected utilitarian welfare Π2 (sT1 , s2 ) + Π1 (sT1 , s2 ). By construction,
sT2 is a solution to this problem. Similarly, if player 2 chooses sT2 , player 1 chooses s1 in order
to maximize
T
Π1 (s1 , sT2 ) + y1 λ121 (sb12 , sbb
121 )Π2 (s1 , s2 ).
Now let
choose
λ121 (sb12 , sbb
121 ) =
s1
1
y1 , or, equivalently,
δ1 = θ10 − c +
8
y1
in order to maximize expected utilitarian welfare
> 0.
Then the problem becomes:
Π1 (s1 , sT2 ) + Π2 (s1 , sT2 ).
Again,
sT1
is a solution.
Proposition 2 and 3 together imply the failure of the revelation principle for BNFE. With
reciprocity, ecient public goods provision with equal cost-sharing can be achieved, but not
with a direct mechanism. This illustrates that, with intentions-based social preferences, we can
no longer separate outcomes or social choice functions from the procedures or game forms which
can be used for implementation.
4.2 The Pseudo-Revelation Principle
The non-direct mechanism
Φ0
that can be used to implement
f∗
in BNFE resembles a direct
mechanism. This is not a coincidence. In the following, we will show that if implementation of
a social choice function is possible at all, then it is also possible in a class of mechanisms that
we will refer to as a pseudo-direct mechanisms.
A mechanism
Θi ⊆ Mi ,
for all
Φ
is called a pseudo-direct mechanism for a social choice function
i∈I
, and
g(m) = f (m)
sets include the type sets and the SCF
f
for all
m ∈ Θ,
f
whenever
i.e., whenever the individual message
is realized in the event that all messages are possible
0
types. Mechanism Φ in Table 1 is an example of a pseudo-direct mechanism for
16
f ∗.
Observe that
the strategy sets of a pseudo-direct mechanism include the strategy sets of the corresponding
direct mechanism. Specically, truth-telling
a pseudo-direct mechanism
a BNFE of
Φ
sTi
is a viable strategy for every player. We say that
f
truthfully implements
in BNFE if the truth-telling prole
sT
is
Φ.
We seek to prove a pseudo-revelation principle for BNFE, i.e., the claim that a social choice
function
f
is implementable in BNFE if and only there exists a pseudo-direct mechanism that
truthfully implements
f
in BNFE. It is more convenient, however, to rst establish the strate-
gic equivalence of arbitrary mechanisms and pseudo-direct mechanisms. The pseudo-revelation
principle for BNFE as well as pseudo-revelation principles for the equilibrium concepts to be
considered later (utility-ecient implementation, psychologically robust implementation) will
follow immediately from this result.
Our denition of strategic equivalence relies on the following construction: We start from
an arbitrary mechanism
Φ = (M1 , ..., Mn , g)
of some type. The prole
s̃
θ ∈ Θ.
(Φ, s̃)
Based on the pair
induces the social choice function
θi ∈ Θi
Formally, for every
i∈
Mi .
usually an equilibrium
given by
f (θ) = g(s̃(θ))
for all
hi :
Mi0
→
Mi
that is used by
Any unused action from
Mi
be the message sets of
s̃
is relabelled
is kept unchanged.
Φ0 (Φ, s̃).
Next, we
Mi that maps actions from Mi0 back into actions
Formally,
(
hi (m0i )
=
Observe that
hi
s̃i (m0i )
if
m0i ∈ Θi ,
m0i
if
m0i ∈ Mi \ s̃i (Θi ) .
is surjective, i.e. for every
We can now complete the denition of
all
any action from
= Θi ∪ (Mi \s̃i (Θi ))
dene for every player a function
from
i ∈ I,
that uses it.
I , let Mi0
f
s̃,
we will now construct an equivalent pseudo-direct mechanism
Φ0 (Φ, s̃) as follows. For every player
according to the type
and some strategy prole
mi ∈ Mi
Φ0 (Φ, s̃)
there exists
m0i ∈ Mi0
with
by dening the outcome function
hi (m0i ) = mi .
g0
so that, for
m0 = (m01 , . . . , m0N ),
g 0 (m0 ) = g(h(m0 )),
where
(1)
h(m0 ) = (h1 (m01 ), . . . , hn (m0n )).
In words, announcing a type
same consequences as choosing the action
in
Φ0 (Φ, s̃)
s̃i (θi )
in
Φ,
θi ∈ Θi
f,
because
Φ0 (Φ, s̃)
and choosing an action from
has the same consequences as choosing that same action in
is in fact a pseudo-direct mechanism for
in
Φ.
has the
Mi \s̃i (Θi )
Observe that
Φ0 (Φ, s̃)
g 0 (sT (θ)) = g 0 (θ) = g(s̃(θ)) = f (θ)
for all
θ ∈ Θ.
Strategic equivalence of
Φ
and
Φ0 (Φ, s̃)
holds provided that, for any player, the outcomes
that he can induce by varying his action, are the same for the game induced by
Φ
and the game
0
induced by Φ . More formally, these sets of achievable outcomes are dened as follows: For any
i∈I
and
m−i ∈ M−i ,
let
Gi (m−i ) = {a ∈ A | ∃mi ∈ Mi
be the set of outcomes that player
i
can induce in
17
so that
Φ
g(mi , m−i ) = a}
if the opponents behave according to
m−i .
Analogously,
G0i (m0−i ) = {a ∈ A | ∃m0i ∈ Mi0
are the outcomes player
Proposition 4.
for every player
it holds that
i
can induce in
The mechanisms
i∈I
Φ
Φ0 (Φ, s̃),
Φ0 (Φ, s̃)
and
and any two proles
so that
g 0 (m0i , m0−i ) = a}
given any prole
0 .
m0−i ∈ M−i
are strategically equivalent, in the sense that,
m−i ∈ M−i
and
0
m0−i ∈ M−i
Let
a ∈ G0i (m0−i ).
m−i = h−i (m0−i ),
with
Gi (m−i ) = G0i (m0−i ).
Proof. We rst show that
G0i (m0−i ) ⊂ Gi (h−i (m0−i )).
Hence, there exists
m0i so that
g 0 (m0i , m0−i ) = a. By (1), this implies that g(hi (m0i ), h−i (m0−i )) = a, and hence
a ∈ Gi (h−i (m0−i )). We now show that Gi (h−i (m0−i )) ⊂ G0i (m0−i ). Let a ∈ Gi (h−i (m0−i )). Hence,
0
there exists mi ∈ Mi so that g(mi , h−i (m−i )) = a. Since the function h is surjective there exists
m0i with hi (m0i ) = mi . Then (1) implies that g 0 (m0i , m0−i ) = a. Hence, a ∈ G0i (m0−i ).
There are several immediate corollaries from Proposition 4. For instance, if we start from
an arbitrary mechanism
Φ
with BNE
s∗
that implements an SCF
f,
the above construction
0
∗
yields a pseudo-direct mechanism Φ (Φ, s ) in which truth-telling induces
BNE as well.
in
Φ0 (Φ, s∗ )
f
and is in fact a
This latter conclusion follows from the fact that unilateral deviations from
can achieve exactly the same outcomes as unilateral deviations from
s∗
in
Φ.
sT
The
equivalence of achievable deviation outcomes also implies that the kindness terms associated to
s∗
and all unilateral deviations in
Φ are identical to those of sT
and all corresponding deviations
0
∗
in Φ (Φ, s ). Hence a pseudo-revelation principle holds for implementation in BNFE.
Proposition 5.
Suppose a mechanism
Φ
implements the social choice function
0
Then there exists a pseudo-direct mechanism Φ that truthfully implements
f
f
in BNFE.
in BNFE.
4.3 Eciency and Kindness
The solution concept of a BNFE relies on two sources of utility, material payos and kindness
sensations. This raises the question how to treat these from a welfare perspective. This question can be formulated using the notions of decision utility and experienced utility which are
frequently used by behavioral economists (Kahneman et al. 1997). Our whole analysis is based
on the assumption that the desire to reward kind and to punish unkind behavior matters for
the individuals' behavior. Hence, behavior is as if individuals were maximizing the utility function
Ui .
That is,
Ui
is a decision utility function, i.e., a tool for making predictions about the
behavior of individuals. This leaves open the question whether sensations of kindness should be
counted as a source of well-being in addition to the individuals' material payos.
If we disregard kindness sensations, we are left with a conventional notion of eciency that
is based on material payos only. The only interesting question that remains is then how the
behavioral implications of kindness sensations aect the possibility to implement material payoecient outcomes. We study this question in subsection 4.3.1, and we return to it in the context
of psychological robustness in Section 5.
We then take an alternative perspective and entertain the possibility that kindness sensations
are an own source of well-being or experienced utility. We thus follow the path outlined by Rabin
(1993) who argues that welfare economics should be concerned not only with the ecient
18
allocation of material goods, but also with designing institutions such that people are happy
about the way they interact with others (p. 1283). We explore two routes for how this can be
done. We will rst show (in Section 4.3.2) that a welfare objective which puts material payos
and kindness sensations on an equal footing leads to paradoxical results: It is possible to achieve
unbounded utility levels by an appropriate choice of a mechanism. Importantly, this does not
require material eciency, in contrast to models with outcome-based social preferences, where,
at least under plausible conditions, material eciency is necessary and might even be sucient
for overall utility eciency (Benjamin 2010). Moreover, achieving unbounded utility levels does
not even require that the implemented allocation is responsive to the agents' types. Kindness
pumps can thus be constructed without recourse to any substantial material allocation problem.
As an alternative, in Section 4.3.3 we will therefore suggest the notion of utility ecient
implementation of a given SCF. That is, we consider the class of mechanisms which all implement the same social choice function, and we ask which of those mechanisms performs best in
the kindness dimension. We show that a best mechanism implementing a given ecient social
choice function may indeed exist.
4.3.1 Material Eciency
The introduction of psychological utility enlarges the set of implementable social choice functions,
because unused actions become an additional design instrument. Therefore, before we turn to
a possibility result, we want to illustrate by means of an example that not every SCF can
be implemented in BNFE. Let
I = {1, 2}
Θi = {1, 2, 3}
and
types are equally likely. Allocations are given by
vi (qi , θi ) = qi θi .
are
We assume, however, that
implementing transfers. Now consider the SCF
f
for both
(q1 , q2 , t1 , t2 ) ∈ Q × T̄
Q = {(0, 0)},
i = 1, 2,
where all
and material payos
so we are only interested in
given in the following table, that contains the
f f
transfers (t1 , t2 ) for every realization of types:
θ2
θ1
1
2
3
1
2
3
(−3, −3)
(0, 0)
(0, 0)
(0, 0)
(−2, −2)
(0, 0)
(0, 0)
(0, 0)
(−1, 0)
This SCF is not materially ecient, because it does not satisfy budget balance. It generates a
strictly positive surplus whenever the players' types coincide.
According to the pseudo-revelation principle, if
f
is implementable in BNFE, then also in
a pseudo-direct mechanism with truth-telling being a BNFE. In any such mechanism, feasible
strategies for player
1
include the strategy of always announcing type
2,
denoted
sM
1 ,
and
H
the strategy of always announcing type 3, denoted s1 . This implies, however, that there are
T for player 1 which are of opposite direction: sH makes both
coexisting deviations from s
1
M
players better o and thus requires strict unkindness for not being used, and s1 leaves the
deviator unaected but makes the opponent worse o, and will thus be used whenever there is
strict unkindness. Formally, for truth-telling
λ121
(sT )
< 0
we must have
sT1
being weakly preferred to
T
in the hypothetical equilibrium s , while for
λ121 (sT ) ≥ 0.
sH
1
we would need
sT1 being weakly preferred to
sM
1
Hence no matter how unused actions are used to manipulate
19
λ121 (sT ),
these deviations cannot simultaneously be made unattractive and the SCF cannot be
implemented in BNFE.
The example crucially relies on the SCF being materially inecient.
For the case of two
players, consider an ecient SCF and a pseudo-direct mechanism with truth-telling
pothetical equilibrium. In this equilibrium candidate, player
1
sT
as hy-
maximizes
Π1 (s1 , sT2 ) + y1 λ121 (sT )Π2 (s1 , sT2 )
by choice of
had
λ121
(sT )
s1 ∈ S1 ,
= 1/y1 ,
where the term
λ121 (sT )
is treated as xed.
Now, if
y1 > 0
and if we
then the individual's problem would become the problem of maximizing
T
T
welfare Π1 (s1 , s2 ) + Π2 (s1 , s2 ). Since the social choice function to be implemented is materially
ecient, i.e., it maximizes the sum of payos for every prole
θ ∈ Θ,
truth-telling is clearly
a solution to this problem, because it ensures that the maximal payo sum results for every
possible realization of types. Hence we need to ask whether or not we can manipulate equitable
payos and therefore equilibrium kindness in the desired way to achieve
appropriately designed unused actions.
λiji (sT ) = 1/yi ,
using
This is akin to a Groves mechanism, where transfers
between individuals are designed so as to align individual interests with the objective of surplusmaximization.
Here, out-of-equilibrium-payments are used for that purpose.
The same line
of argument has already been used to prove Proposition 3, and it provides the basis for the
following result, which shows that eciency is in fact sucient for an SCF to be implementable
in BNFE. We denote by
Theorem 1.
Suppose
y = (yij )i,j∈I,i6=j
y ∈ ]0, ∞[n(n−1) .
the prole of individual reciprocity weights.
If a social choice function
f
is materially ecient, then
it is implementable in BNFE.
Proof. See Appendix A.4.
The proof of the theorem is constructive: it shows how to add unused actions to achieve
the desired equilibrium kindness values.
This problem is not straightforward, as additional
messages can have a non-trivial impact on the set of bilaterally Pareto-ecient strategies, and
they must yield bilaterally Pareto-ecient outcomes themselves. The number of unused actions
our construction requires for each player pair depends on the problem: starting from the direct
mechanism, if we need to increase the kindness of player
actions for player
j 's
i
i
toward
j,
the number of additional
is equal to the number of dierent types he announces when minimizing
payo in the direct mechanism. In some cases, a single unused action per player might be
sucient (as in the pseudo-direct mechanism from Table
1), but in general more than one unused
actions can be asked for. When we want to decrease kindness toward
unused actions for player
i
j,
as he announces types when maximizing
we need to add as many
j 's
payo in the direct
mechanism. In any case, it is always possible to let the transfers associated with unused actions
satisfy budget balance, so that budget balance holds not only on but also o the equilibrium.
The relevance of Theorem 1 is that it shows any materially ecient SCF
mentable.
f
to be imple-
With the solution concept of a BNE, by contrast, a surplus-maximizing choice of
(q1f , ..., qnf ) requires specic transfers
(tf1 , ..., tfn ).
individuals are granted information rents.
In particular, transfers have to be such that
Consequently, an ecient social choice function is
20
implementable only if it has certain distributional characteristics. The construction in Theorem
1 removes this restriction.
From a purely materialistic perspective, the existence of kindness
sensations allows for a separation of eciency and distributional goals. We can solve for the set
of ecient SCFs and, in a second step, choose the one that is preferred for distributive reasons.
The argument in the proof of Theorem 1 can be adapted so that, in addition, any tension
between eciency and the requirement of voluntary interim participation in the mechanism
18 The argument is as follows. The proof of Theorem 1 starts with a direct mechanism
disappears.
a mechanism so that
Mi = Θ i ,
for all
i
and then adds unused actions that are calibrated
in such a way that every agent's objective function is turned into a utilitarian welfare function.
To include the requirement that interim participation in the mechanism is voluntary, we can as
well start out from a direct mechanism with veto rights where
Miv = Θi ∪ {veto},
for all
i
with the understanding that if at least one player chooses the action veto then a status quo
outcome is implemented. We can then add actions to the message sets
Miv
in exactly the same
way as in the proof of Theorem 1 so as to align individual preferences with the objective of
welfare maximization. Consequently, if the outcome of the mechanism is ecient and the status
quo is not, then all players will refrain from exercising their veto rights and communicate their
types truthfully to the mechanism.
Theorem 1 can hence be interpreted as a universal possibility result for social choice functions
that are in a material sense ecient. Not only does the theorem imply that every ecient social
choice function is implementable, it also implies that the famous impossibility results á la Myerson and Satterthwaite (1983) on the incompatibility of eciency and voluntary participation
are turned into possibility results.
4.3.2 Utility Eciency
We now seek to dene a notion of Pareto eciency which is based on the interpretation of the
entire utility function
Ui
as a measure of individual well-being.
Developing this notion faces
the following diculty: In the conventional approach one denes ecient social choice functions
taking only the players' material payos and the economy's resource constraint into account.
A key question then is whether there exists a mechanism that implements this social choice
function as the equilibrium outcome of a Bayesian game (in BNE or possibly in BNFE). This
route is not available if we seek to dene a notion of eciency based on the players' utilities,
because the latter endogenously depend on the mechanism that is used and the equilibrium that
is played. Therefore, we cannot simply dene a utility-ecient social choice function. Instead,
we dene eciency for mechanism-equilibrium pairs.
Denition 4.
(1)
s∗
∗
A mechanism-equilibrium-pair (Φ, s ) is utility-ecient if
is a BNFE of
Φ,
and
0 0
(2) there is no pair (Φ , s ) of a mechanism
Φ0
with BNFE
s0
whose equilibrium utilities Pareto
∗
dominate those of (Φ, s ).
We once more consider the public goods application in Example 1 and the pseudo-direct
mechanism
18
Φ0
from Table 1, to make the point that this eciency notion is problematic.
Well-known examples where participation constraints are in conict with eciency are Myerson and Sat-
terthwaite (1983) and Mailath and Postlewaite (1990).
21
Proposition 6.
(1) For every
Consider Example 1 and the pseudo-direct mechanism
(y1 , y2 ) ∈
]0, ∞[2 there exist numbers
0 < 2 ≤ δ1
L H
0
strategy prole (s1 , s2 ) is a BNFE of Φ whenever δ2 ≥ δ .
L H
(2) In (s1 , s2 ) it holds that Ui → ∞ as δ2 → ∞, for both i
and
Φ0
in Table 1.
0 < 1 ≤ δ
so that the
= 1, 2.
H
(sL
1 , s2 )
is a BNFE if δ2 is chosen suciently
H
large. Given that player 2 behaves according to s2 , the public good is provided anyway and
Proof.
Step 1.
We rst seek to verify that
player 1 can aect the outcome only by the extent to which he makes use of the redistributive
action
d1 .
Strategy
s1 = (d1 , d1 )
simultaneously maximizes player
1's
and minimizes player
H H
2's payos, yielding Π2 ((d1 , d1 ), sH
2 ) = Π2 (s1 , s2 ) − δ2 . Making no use of d1 at all leaves a
e1 H
1
H H
H H
maximal payo of Π2 (s1 , s2 ) to player 2. Hence we obtain π2 (s2 ) = Π2 (s1 , s2 ) − δ2 ,
2
e1 H
e1 H
1
H
H
b
bb
L
H
implying λ212 (s21 , s212 ) = Π2 (s1 , s2 )−π2 (s2 ) = Π2 (s1 , s2 )−π2 (s2 ) = δ2 in the equilibrium
2
candidate.
sL
1,
d2 because every strategy
1
s2 that uses d2 is outcome equivalent to a strategy that uses θ2 instead. The ecient strategies
T
H
of player 2 are now truth-telling, s2 , which maximizes the own material payo, and s2 , which
1 0
1 1
bb
b
maximizes player 1's material payo. We then obtain λ121 (s12 , s121 ) = (θ1 − c) + (θ1 − c) > 0
8
8
Given that player 1 behaves according to
we can ignore action
in the equilibrium candidate, which is independent of
In the equilibrium candidate, player
2
δ2 .
maximizes
b
bb
L
Π2 (sL
1 , s2 ) + y2 λ212 (s21 , s212 )Π1 (s1 , s2 ).
1
1
2 δ2 as shown above, there exists δ such that player 2
1
L
H
chooses s2 as to maximize Π1 (s1 , s2 ) whenever δ2 ≥ δ , hence s2 = s2 . Player 1, in turn,
H
b
bb
H
b
bb
maximizes Π1 (s1 , s2 ) + y1 λ121 (s12 , s121 )Π2 (s1 , s2 ). Since y1 λ121 (s12 , s121 ) > 0 as shown above,
L
player 1 is willing to choose s1 = s1 rather than any strategy that uses action d2 whenever
H
δ2 ≥ 1 /(y1 λ121 ) =: δ 2 . Hence when δ2 ≥ δ = max{δ 1 , δ 2 , 1 }, the prole (sL
1 , s2 ) is a BNFE.
L H
Step 2. We now show that utility in equilibrium grows without limit as δ2 → ∞. In (s1 , s2 ),
L H
L H
the players' utility levels Ui , i = 1, 2, are given by material payos, Π1 (s1 , s2 ) or Π2 (s1 , s2 ),
b
bb
b
bb
b
bb
respectively, plus yi λ121 (s12 , s121 )λ212 (s21 , s212 ). Since yi λ121 (s12 , s121 ) > 0 is independent of δ2 ,
1
b
bb
and λ212 (s21 , s212 ) = δ2 , we have that Ui → ∞ as δ2 → ∞, for both i = 1, 2.
2
Since
y2 > 0
and
In the BNFE
d1 ,
λ212 (sb21 , sbb
212 ) =
H
(sL
1 , s2 ), player 1 does not make use of the egoistic but socially wasteful action
because he wants to treat player
2
kindly.
type because he wants to be kind to player
1
Player
2,
in turn, always announces the high
and make sure that the public good is always
L H
provided. The outcome of (s1 , s2 ) is hence not materially ecient but exhibits over-provision
of the public good, independent of the players' types. Psychological utility in the equilibrium, on
the other hand, can be inated innitely by increasing the destructive potential
δ2
of player
1.
An immediate implication of this observation is that in the environment of Example 1 a utilityecient mechanism-equilibrium-pair does not exist. That is, to any mechanism-equilibrium-pair,
we can use the construction in the proof of Proposition 6 to nd a pair that gives more utility
19
to all players.
Corollary 1.
19
In Example 1, a utility-ecient mechanism-equilibrium-pair does not exist.
If we exogenously bound psychological utility components (as in Rabin 1993), our arguments still imply an
openness-problem and the associated issues of non-existence. The fact that kindness can be generated without a
meaningful underlying allocation problem remains valid as well.
22
4.3.3 Implementation with Maximal Kindness
Our previous results suggest that a notion of Pareto eciency that treats material payos and
kindness sensations as two equally welcome sources of individual well-being leads to paradoxical
results. We will now explore a dierent route for introducing kindness considerations into welfare
assessments. In the following we x an SCF and then ask whether or not we can implement it
in BNFE, as done in Section 4.3.1. If we can, we look for the mechanism and the BNFE which
implements it with maximal utility levels. That is, kindness sensations play a subordinate role
in our welfare analysis. They are a good thing, but only to the extent that they do not upset
the material outcomes of a given implementation exercise.
As the previous section has shown, the problem to implement an SCF with a maximal degree
of kindness might not be well-dened in some cases, such as when we consider inecient overprovision of the public good with equal cost-sharing.
In general, however, rst imposing the
SCF to be implemented does act as a constraint and can yield bounds on utility levels.
Denition 5.
∗
A mechanism-equilibrium-pair (Φ, s ) implements an SCF
∗
(1) s is a BNFE of
Φ
which implements
f,
0 0
(2) there is no pair (Φ , s ) of a mechanism
f
utility-eciently if
and
Φ0
with BNFE
s0
which also implements
f,
but with
∗
equilibrium utilities that Pareto dominate those of (Φ, s ).
As a starting point, we characterize the mechanism which, for Example 1, implements the
social choice function
Proposition 7.
]0, ∞[2 . Let
f∗
(ecient provision and equal cost-sharing) utility-eciently.
Consider Example 1 and social choice function
Assume that
(y1 , y2 ) ∈
Φ0 be the extended mechanism in Table 1 with
δ1 = −
and any
f ∗.
1 , 2
with
8 θ11 − c
+ θ10 − c > 0
y1 θ20 − c
0 < 2 ≤ δ1 , 0 < 1 ≤ δ2 .
and
Then
δ2 = −
(Φ0 , sT )
4 θ20 − c
> 0,
y2 θ10 − c
utility-eciently implements
f ∗.
Proof. See Appendix A.5.
Proposition 7 shows that utility-ecient implementation of a given materially ecient SCF
can be meaningful in the sense that utilities are not necessarily unbounded.
Second, in the
context of Example 1, it shows how to construct a mechanism which achieves utility-ecient
implementation. In particular, utility ecient implementation is viable within the simple pseudodirect mechanism
Φ0
in Table 1.
The proof of the proposition reveals why utilities must be bounded under certain conditions.
First, observe that a pseudo-revelation principle applies to utility-ecient implementation. This
follows directly from Proposition 4: Whenever some mechanism implements
f∗
in BNFE, then
∗
there exists a pseudo-direct mechanism that truthfully implements f in BNFE with identical
utilities.
We can therefore restrict attention to pseudo-direct mechanisms and truth-telling
when searching for utility-ecient mechanism-equilibrium-pairs.
∗
mechanism for f the strategies
sL
i and
Now, in any pseudo-direct
sH
i are viable for both players
i = 1, 2.
In our example,
these strategies can be used to strictly increase or strictly decrease the opponent's payo. The
23
condition that neither player wants to deviate to any of these strategies then immediately implies
upper and lower bounds on kindness, and hence on equilibrium utilities in any mechanism that
implements
f ∗.
It is the possibility for both players to increase and decrease the opponent's
payo by announcing types non-truthfully, which implies that kindness cannot grow without
bounds.
In the context of two players, we can formalize this insight through the concept of
bilateral externalities.
In the following, we will make use of an expectations operator
E
to
simplify notation. For instance, we write
h
i
h
i
X
Eθi vi (qif (θi , θj ), θi ) + tfi (θi , θj ) =
p(θi ) vi (qif (θi , θj ), θi ) + tfi (θi , θj )
θi ∈Θi
for the expected payo of agent
i
under SCF
independence, we can use the marginal
Denition 6.
n = 2.
Let
when agent
and
j 6=
j 's
type is xed to
θj ,
where, due to
instead of the conditional distribution
Given an environment
i = 1, 2
externalities if, for each
p(θi )
f
E,
a social choice function
i, there exist types θj0 , θj00
∈ Θj
f
p(θi |θj ).
exhibits bilateral
such that
h
i
h
i
Eθi vi (qif (θi , θj0 ), θi ) + tfi (θi , θj0 ) 6= Eθi vi (qif (θi , θj00 ), θi ) + tfi (θi , θj00 ) .
Bilateral externalities are a joint property of a social choice function
part of the environment
E.
f
(2)
and the prior
p,
as
When bilateral externalities do exist, both agents are not indierent
with respect to the other's type, because their own expected payo diers for at least two of
these types. The ecient social choice function
1's
expected payos are larger for
are larger for
θ1 =
θ10 than for
in Example
1
satises this property: Player
θ20 , while player
θ21 than for
θ2 =
2's expected payos
1
θ1 . Inecient over-provision of the public good with equal
θ2 =
θ1 =
f∗
cost-sharing, as examined in the previous subsection, does not satisfy the property, because
types have no impact at all on the chosen allocation. Consequently, in the latter case players
do not care about the other's type announcement in a pseudo-direct mechanism, but they do so
in the rst case. This is the reason why one SCF can be implemented utility-eciently, while a
20
utility-ecient mechanism does not exist for the other.
Theorem 2.
Let
n=2
and
(y1 , y2 ) ∈ ]0, ∞[2 ,
and let
f
be a materially ecient SCF. Then,
there exists a mechanism that utility-eciently implements
f
if and only if
f
exhibits bilateral
externalities.
Proof. See Appendix A.6.
4.4 Kindness and Coercion
The problem to implement a given SCF with maximal kindness is, at least for some SCFs, welldened in a mathematical sense. While the proofs in the previous section are constructive, they
are probably not appealing from an intuitive perspective. We will therefore use a more colorful
example to illustrate how unused actions might play important roles in real-world mechanisms.
20
Utility-ecient implementation as opposed to simple implementation in BNFE as examined by Theorem
1 does not generally allow to adhere to budget balance o the equilibrium path. While budget balance was
possible in the special case of Proposition 7, the proof of Theorem 2 requires to construct unused messages with
an associated budget surplus in some cases.
24
More specically, we will study another example of public goods provision, and show that
the introduction of veto-rights so that, by exercising his veto right, each individual can force
a status quo outcome with no public goods provision may increase all individuals' kindness
sensations without impeding ecient public goods provision.
Hence, in our framework, the
introduction of veto rights may be a good thing. People enjoy kindness sensations if they (i) are
not forced to contribute to public goods that they do not like, and (ii) nevertheless voluntarily
choose to contribute because they want to reciprocate the kindness of others.
This stands in stark contrast to the view of participation constraints which emerges with the
conventional solution concept of a Bayes-Nash equilibrium. With BNE as opposed to BNFE,
participation constraints are, if anything, bad, because they may render ecient public goods
21
provision impossible.
Example 2.
There are three individuals,
I = {1, 2, 3},
indivisible public good has to be taken. We have
with
q ∈ {0, 1}, and T = T̄ .
and a decision about the provision of an
Q = {(0, 0, 0), (1, 1, 1)},
so that
q1 = q2 = q3 = q
Each individual values the public good either at 1 or at 3,
for all i, with equal probability. The per capita cost of public goods provision equals
vi (1, 3) = 1
vi (1, 1) = −1,
and
whereas
vi (0, θi ) = 0
for all
θi ∈ {1, 3}.
Θi = {1, 3}
c = 2.
Hence,
Material payo eciency
requires that the public good is provided if at least two individuals value it highly, and not to
provide it otherwise.
We again seek to implement this rule without any additional transfers,
hence with equal cost-sharing.
We compare the performance of two mechanisms which we refer to as simple majority voting
and majority voting with veto rights. Under simple majority voting, each individual's message
set is given by
Mi = {no, yes}
and the public good is provided if at least two individuals vote
yes and is not provided otherwise. Since both the set of types and the set of messages is binary,
and the voting mechanism entails no transfers, simple majority voting is equivalent to a direct
mechanism for the SCF we seek to implement.
Under majority voting with veto rights, each individual's message set is given by
{no, yes, veto}.
Mi0 =
The public good is provided if no individual chooses the action veto and at
least two individuals choose the action yes. Majority voting with veto rights is equivalent to a
pseudo-direct mechanism for our SCF.
A strategy for player
{no, yes}2 ,
where
vi1
i
in the game induced by simple majority voting is a tuple
is the vote that is cast by type 1 of individual
i,
and
vi3
(vi1 , vi3 ) ∈
is the vote that
is cast by type 3. Likewise, a strategy in the game induced by majority voting with veto rights
is an element of
(no, yes),
{veto, no, yes}2 .
A strategy of particular interest is sincere voting
(vi1 , vi3 ) =
which can readily be interpreted as truth-telling. For both voting mechanisms, material
payo eciency is reached if and only if all players vote sincerely.
The following Proposition establishes a robust possibility result: Whatever the intensity of
the players' kindness sensations, sincere voting by all players is an equilibrium of the game
induced by simple majority voting.
21
In an independent private values model, ecient public goods provision is possible if there are no participation
constraints, see d'Aspremont and Gerard-Varet (1979), but impossible if voluntary participation is required, see
Güth and Hellwig (1986) or Mailath and Postlewaite (1990). Our view of participation decisions is more in line
with the ndings of Frey and Stutzer (2004), according to which greater political participation rights lead to
increased life-satisfaction.
25
Proposition 8.
Consider Example 3. For all values of
y ∈ [0, ∞[6 ,
sincere voting is a BNFE
under simple majority voting.
Proof. In the following we hypothesize (i) that players 2 and 3 vote sincerely, and (ii) that all
players believe all other players to vote sincerely, and (iii) that all players have correct beliefs
about the beliefs of the other players. We show that this implies that it is a best response for
player 1 to also vote sincerely.
Step 1. We leave it to the reader to verify that, given that players 2 and 3 vote sincerely, the
expected material payo of players 2 and 3 is given by
1
Π2 (s1 , (no, yes), (no, yes)) = Π3 (s1 , (no, yes), (no, yes)) = ,
4
for all
s1 ∈ {no, yes}2 .
Since player 1 cannot aect the expected material payo of the other
players, this trivially implies that the kindness of player 1 towards players 2 and 3 must satisfy
κ12 = κ13 = 0
in the hypothetical equilibrium. By symmetry, and since beliefs are correct in
equilibrium, this also implies that
λ121 = λ131 = 0
in the hypothetical equilibrium.
Step 2. Given that all terms involving kindness sensations are equal to zero, player 1 chooses
s1
in order to maximize
s1 = (no, yes)
Π1 (s1 , (no, yes), (no, yes)).
We leave it to the reader to verify that
is the unique solution to this problem.
Simple majority voting makes it possible to reach material eciency. Moreover, this comes
without (positive or negative) sensations of kindness. The reason is the following: If one player
unilaterally changes his strategy, this does not aect the other players' expected payos. Consequently, no player has the possibility to be kind or unkind to the other players. Therefore, the
only remaining concern is the own payo, which is maximized by sincere voting.
The following Proposition shows that majority voting with veto rights may yield the same
outcome as simple majority voting, and moreover, generate positive kindness. These sensations
are induced because players refrain from exercising their veto power.
If a player has a low
valuation of the public good and chooses the action no instead of the action veto, he takes
the risk of ending up with a payo of
−1,
of making sure that he gets a payo of
in case the other two players both vote yes, instead
0.
The other players will interpret this behavior as
kind: The player in question harms himself so as to make it possible for them to benet from
public goods provision. The player's willingness to sacrice the own payo is motivated by the
desire to reciprocate that the other players would also refrain from exercising their veto rights
if they were in his situation. Consequently, in circumstances where both majority voting with
veto rights and simple majority voting work in the sense of generating the materially ecient
outcome majority voting with veto rights is preferable because it comes with the extra benet
of positive kindness sensations.
Proposition 9.
Consider Example 3.
Sincere voting is a BNFE under majority voting with
veto rights if and only if, for all players
i ∈ I,
P
j6=i yij
≥ 16.
Whenever sincere voting is
a BNFE, then the equilibrium utilities Pareto dominate those of sincere voting under simple
majority voting.
26
Proof. We hypothesize (i) that players 2 and 3 vote sincerely, and (ii) that all players believe all
other players to vote sincerely, and (iii) that all players have correct beliefs about the beliefs of
the other players. We show that, under these assumptions, it is a best response for player 1 to
vote sincerely if and only if
P
j6=1 y1j
≥ 16.
A symmetric reasoning applies to the other players.
Step 1. We leave it to the reader to verify that, given that players 2 and 3 vote sincerely, player 1
has two ecient strategies,
(no, yes) and (veto, yes).
If player 1 chooses
(no, yes) the associated
expected material payos are
1
4
Π1 ((no, yes), (no, yes), (no, yes)) =
and
1
Π2 ((no, yes), (no, yes), (no, yes)) = Π3 ((no, yes), (no, yes), (no, yes)) = .
4
If he chooses
(veto, yes),
the payos are
Π1 ((veto, yes), (no, yes), (no, yes)) =
and
3
8
1
Π2 ((veto, yes), (no, yes), (no, yes)) = Π3 ((veto, yes), (no, yes), (no, yes)) = .
8
Given that players 2 and 3 vote sincerely, their equitable payos are therefore
1/8)/2 = 3/16.
π2e1 = π3e1 = (1/4+
Consequently, in a hypothetical equilibrium where all players vote sincerely, the
κ12 = κ13 = 1/4 − 3/16 = 1/16.
kindness of player 1 towards players 2 and 3 equals
symmetry, and since beliefs are correct in equilibrium, this also implies that
By
λ121 = λ131 = 1/16
in the hypothetical equilibrium.
Step 2. Given that player 1 expects the other players to be kind, he will choose from his set of
ecient strategies, as he has no incentive to sacrice own payo in order to harm others. Hence
he will either choose
s1 = (veto, yes)
or
s1 = (no, yes),
depending on which of the two yields a
larger value of
Π1 (s1 , (no, yes), (no, yes)) +
1
(y12 Π2 (s1 , (no, yes), (no, yes)) + y13 Π3 (s1 , (no, yes), (no, yes))) .
16
It is straightforward to verify that the optimal choice is
s1 = (no, yes) if and only if y12 +y13 ≥ 16.
Step 3. To prove the last statement in the proposition, observe that, whenever sincere voting
is an equilibrium, all players have strictly positive kindness sensations, since, for all
κij = λiji = 1/16.
i
and
j,
By contrast, in a sincere voting equilibrium under simple majority voting
(recall the proof of Proposition 8) we have
κij = λiji = 0
for all
i
and
j.
Majority voting with veto rights can outperform simple majority voting in the kindness
dimension, whenever both yield the same material outcome.
In that case, the explicit intro-
duction of veto rights into the game helps to solve, paradoxically, an apparent participation
problem. Sincere voting in the simple majority voting mechanism induces zero kindness, so that
equilibrium utilities coincide with equilibrium material payos. Treating interim participation
constraints in the conventional way, one would have to conclude that voluntary participation
fails because individuals with type
θi = 1
prefer to veto the mechanism. Once veto rights are
27
included as part of the game, however, their existence enables the appearance of an equilibrium
in which they remain unused, in which utilities are increased and voluntary participation is
assured. This can be recast as an application of the Lucas Critique to the case of endogenous
preferences, as pointed out by Bowles and Reyes (2009). In our example, the introduction of
veto rights changes the players' preferences towards more social behavior, and fundamentally
alters their participation decisions.
However, majority voting with veto rights can yield ecient outcomes only if kindness sensations carry enough weight in the players' utility functions. Simple majority voting, by contrast,
works whatever those weights are. It is therefore the more robust procedure. In the following
section, we will make this distinction more precise and study psychologically robust implementation, that is, implementation which works if (i) the players' behavior may be driven by kindness
sensations, but (ii) the designer of the mechanism remains ignorant with respect to the strength
of this behavioral force.
5
Psychologically Robust Mechanism Design
In the preceding we have studied mechanism design problems under the assumption that the designer knows the kindness generating process and its behavioral implications. More specically,
the designer was assumed to know how individuals compute equitable payos, how deviations
from those equitable payos translate into sensations of kindness or unkindness, and nally how
much own material payo individuals are willing to sacrice in order to reciprocate the kindness
of other players. In the following, we seek to complement this analysis by asking what a mechanism designer can accomplish who lacks this degree of psychological sophistication. That is, we
try to characterize social choice functions that admit a psychologically robust implementation,
which means that they can be implemented by a mechanism that yields the intended outcome,
whatever the players' inclination to trade-o material payos and reciprocal kindness. Hence
we model robustness with respect to one specic of the above-mentioned aspects, the individual
weights
y = (yij )i,j∈I,i6=j .
Denition 7.
for all
We will comment on other aspects of robustness as we go along.
A strategy prole
s∗
is a psychologically robust equilibrium (PRE) if it is a BNFE
y ∈ [0, ∞[n(n−1) .
The solution concept of PRE is interesting for two reasons: First, as opposed to BNFE, the
concept of PRE is a renement of BNE, since we require
all
s∗
to be a BNFE even when
yij = 0 for
i, j ∈ I , in which case BNFE coincides with BNE. Hence we can start from established results
on Bayesian implementation and investigate their robustness in a standard renement sense. In
particular, we will focus on the robustness of well-known possibility results for material-payo
ecient implementation such as the result by d'Aspremont and Gerard-Varet (1979) according
22 Second, individual heterogeneity in social
to which ecient public goods provision is possible.
preferences is well-documented (Fehr and Schmidt 1999, Engelmann and Strobel 2004, Falk et
22
Trivially, impossibilities for implementation in BNE such as the impossibility to achieve simultaneously
eciency and voluntary participation which has been established for a bilateral trade application by Myerson
and Satterthwaite (1983) and for a public goods environment by Mailath and Postlewaite (1990) remain
impossibilities if we use the more demanding solution concept of PRE.
28
al.
2008, Dohmen et al.
2009).
Private information of individuals about their marginal rate
of substitution between material payos and kindness sensations, in addition to their private
information about material payos, gives rise to a two-dimensional mechanism design problem.
Such problems are notoriously dicult to solve. A robust possibility to reach material payo
eciency implies that there is no need to look at a complicated two-dimensional mechanism
design problem. Instead, there will be an easy solution which makes it possible for the designer
to remain entirely ignorant about the intensity of kindness sensations.
Bergemann and Morris (2005) have also looked at robust mechanism design, but with a
robustness requirement that applies to the specication of the individuals' probabilistic beliefs
about the types and the beliefs of other players.
Bergemann and Morris call a social choice
function robustly implementable whenever, for every specication of belief types, there is a
mechanism that implements the social choice function. That is, while dierent type spaces may
warrant the use of dierent mechanisms, robustness holds provided that, for every type space,
an appropriately calibrated mechanism can be found.
If we followed a similar approach, we
would dene psychological robustness of a social choice function
every collection
y
there exists a mechanism
Φ(y)
f
that implements
by the requirement that for
f
in BNFE.
Our possibility results in Propositions 3 and 7 and Theorems 1 and 2 are all robust in this
weak sense.
Proposition 3, for instance, established that, for every strictly positive
y
we can
0
construct a pseudo-direct mechanism Φ (y) so that truth-telling is a BNFE that implements
ecient public goods provision with equal cost sharing. Dierent
y -proles,
however, warrant
the use of dierent pseudo-direct mechanisms. We use our stronger notion of robustness, which
also has a predecessor in the mechanism design literature that is concerned with probabilistic
beliefs (see Ledyard 1978), because we are interested in outcomes that can be achieved even if
the designer does not know how strongly the individuals' behavior may be inuenced by kindness sensations. Trivially, everything that turns out to be possible with this strong robustness
requirement remains possible with a weaker requirement in the spirit of Bergemann and Morris.
5.1 The Revelation Principle
From a practical point of view, the solution concept of PRE has another advantage. If we insist
on robustness, the revelation principle comes back:
Proposition 10.
Then
f
Suppose a mechanism
Φ
implements the social choice function
f
in PRE.
is truthfully implementable in PRE in the corresponding direct mechanism.
Appendix A.7 contains a characterization of psychologically robust equilibria and the proof
of the revelation principle. It is worth being pointed out that the revelation principle does not
immediately follow from the fact that PRE is a renement of BNE. Any SCF
f
that can be
implemented in BNE can be truthfully implemented in BNE in the direct mechanism, but to
23
achieve robustness, a non-direct mechanism could still be necessary.
23
Our PRE characterization, which generalizes a result by Rabin (1993), states that a Bayes-Nash equilibrium
i and j , the strategy chosen by i, s∗i , minimizes
is psychologically robust if and only if, for every pair of players
j 's
expected payo, whenever kindness of
j
towards
i
is not zero. This follows from two observations: First, in
a Bayes-Nash equilibrium kindness is non-positive, since players only care about their own payo. Second, if
strategy was not minimizing
j 's
expected payo but
j
i's
was being strictly unkind to i, then for some high value of
29
5.2 A Sucient Condition for Robustness: The Insurance Property
In the following we establish that a social choice function with a property that we refer to as
the insurance property is implementable in PRE whenever it is implementable in BNE. From
player
i's
perspective, the types of all other players are random quantities.
function is such that player
i
is (pairwise) insured against this randomness, then each other
player is unilaterally unable to aect player
player and player
If a social choice
i is therefore equal to zero.
i's
payo, and the kindness between any other
Consequently, i's only concern is the maximization
of his expected material payo, which implies that the social choice function in question is
implementable in PRE provided that it is implementable in BNE.
We will present formal versions of these statements below. In subsequent sections, we will
then make heavy use of the insurance property. It will enable us to show that, in our independent private values environment, we can implement essentially any social choice function in a
psychologically robust way, provided that is implementable in BNE and provided that we require
budget balance only in expected terms. Hence, in these cases the requirement of psychological
robustness is not more demanding than the conventional notion of implementability in BNE. We
will then show that, in symmetric environments, material payo-ecient social choice functions
can be implemented even if we insist on ex post budget balance. Finally, we will use the insurance property to show that social choice functions that can be made the outcome of a screening
procedure or those that admit a decentralization via a price system are psychologically robust.
Formally, the insurance property is dened in terms of conditional expected payos as follows.
Denition 8.
if, for all
i
and
Given an environment
E,
f
a social choice function
has the insurance property
j 6= i,
Eθ−j [vi (qif (θj0 , θ−j ), θi ) + tfi (θj0 , θ−j )] = Eθ−j [vi (qif (θj00 , θ−j ), θi ) + tfi (θj00 , θ−j )]
for any pair
θj0 , θj00
of possible realizations of
j 's
type.
If the insurance property holds, then the expected payo of any individual
on the type of any single other individual
insures individual
i
j.
(3)
i does not depend
Put dierently, the social choice function perfectly
against the randomness of
j 's
type.
When we consider the case of only
two individuals, the insurance property is virtually the opposite of the bilateral externalities
property from Denition 6.
24 In the general case with an arbitrary number of players, insurance
is required only bilaterally, so that simultaneously changing types of two or more players can
have an impact on player i's expected payo. Also observe that the insurance property is again
a joint condition on
f
and
p,
with the prior
p
being part of the xed environment
E.
The insurance property relates to an observation about other-regarding preferences already
made by Levine (1998), Fehr and Schmidt (1999) or Bolton and Ockenfels (2000). There are situations, presumably competitive ones, where in equilibrium players do not have the possibility
yij
it would become attractive for
i
to deviate from
s∗i
so as to punish player
j
more severely for his unkindness,
which would contradict the robustness of the equilibrium under consideration.
The revelation principle then
follows from the fact that an action remains unused only if it is not needed to minimize the other players' payos,
so that a removal of unused actions does not alter the equilibrium structure.
24
It is possible that an SCF neither exhibits bilateral externalities nor satises the insurance property, whenever
one player is indierent with respect to the other's type, but the other player is not.
30
to aect others' payos. Other-regarding preferences will then become behaviorally irrelevant
25 We are utilizing essentially
and individuals will behave as if they were egoistic maximizers.
this insight in our mechanism design exercise, by designing institutions that, due to the insurance property, are robust to psychological considerations.
The following proposition provides
the corresponding characterization of the insurance property in game-theoretic terms: Given
that all players except
j
tell the truth in the direct mechanism for
f,
player
i's
payo does not
26
depend on the strategy chosen by player j .
Proposition 11.
A social choice function
f
has the insurance property if and only if in the
corresponding direct mechanism we have that, for all
i
and
j 6= i,
Πi (s0j , sT−j ) = Πi (s00j , sT−j )
for any pair
s0j , s00j
(4)
j.
of possible strategies of player
Proof. Step 1. We rst show that (4) implies (3). If (4) holds then it must be true that
θ0
θ00
Πi (sj j , sT−j ) = Πi (sj j , sT−j ) ,
where
θj0
and
θj00
are arbitrary types from
θ0
Θj , sj j
is the strategy where player
θ00
θj0 , whatever his true type, and sj j is the strategy where
j
j
always announces
always announces
θj00 .
This condition
is equivalent to
X
h
i
h
i X
p(θ−j ) vi (qif (θj00 , θ−j ), θi ) + tfi (θj00 , θ−j ) ,
p(θ−j ) vi (qif (θj0 , θ−j ), θi ) + tfi (θj0 , θ−j ) =
Θ−j
Θ−j
so that condition (3) holds.
Step 2. We now show that (3) implies (4). For an arbitrary strategy
θj ∈ Θj ,
sj
and an arbitrary type
dene
Λ(θj |sj ) = {θj0 ∈ Θj | sj (θj0 ) = θj } ,
and observe that
P
i
f
f
0)
p(θ
p(θ
)[v
(q
(θ
,
θ
),
θ
)
+
t
(θ
,
θ
)]
0
−j
i i
j −j
i
j
θj ∈Θj
i j −j
hPθj ∈Λ(θj |sj )
Θ−j
i
P
f
f
0
=
p(θ
)
E
[v
(q
(θ
,
θ
),
θ
)
+
t
(θ
,
θ
)]
,
j −j
i
θ−j i i
j
θj ∈Θj
θ0 ∈Λ(θj |sj )
i j −j
Πi (sj , sT−j ) =
P
hP
j
25
See Dufwenberg et al. (2011b) and the discussion in Sobel (2005) for a general treatment of other-regarding
preferences in a general equilibrium framework, and specically for the role of separability of preferences for the
above claim.
26
In their concluding remarks, Baliga and Sjöström (2011) conjecture that mechanisms in which players can
inuence their opponents' payos without own sacrice ...may have little hope of practical success if agents
are inclined to manipulate each others' payos due to feelings of spite or kindness.
From the perspective of
our results, the following qualications are appropriate. First, with an observable kindness generating process,
giving players the opportunity to aect each others' payos becomes, quite to the contrary, an important design
instrument. With unobservable social preferences, the insurance property is in fact sucient for robustness. Note
that the insurance property does not contain any requirement about the impact of deviations on
Renou and Schlag (2011) make a related observation, albeit in a very dierent context.
own
payos.
They show that the
canonical Maskin mechanism is robust to their concept of a minimax regret equilibrium, because players cannot
inuence the maximal regret that other players can impose on them.
31
Now, if (3) holds then there is a number
ρ
so that
Eθ−j [vi (qif (θj , θ−j ), θi ) + tfi (θj , θ−j )] = ρ
for all
θj ∈ Θj .
Hence

Πi (sj , sT−j ) = ρ
X
sj
p(θj0 ) = ρ .

θj ∈Θj
Since our choice of

X
θj0 ∈Λ(θj |sj )
was arbitrary, this shows that, for all
s0j
and
s00j ,
Πi (s0j , sT−j ) = Πi (s00j , sT−j ) = ρ
and hence (4) holds.
The insurance property and implementability in BNE are jointly sucient for implementability of a social choice function in PRE:
Proposition 12.
If a social choice function is implementable in BNE and has the insurance
property, then it is implementable in PRE.
Proof. Let
f
be a social choice function that is implementable in BNE and that has the insurance
property. We show that the direct mechanism truthfully implements
mechanism and given that
and
j 6=
f
f
in PRE. Given the direct
has the insurance property, Proposition 11 implies that for all
i
i, there exists a number ρi (sT−j ) so that
Πi (sj , sT−j ) = ρi (sT−j )
for any strategy
sj ∈ Sj .
Now consider the truth-telling BNE
sT
and suppose all rst- and
ej T
T
T
second-order beliefs to be correct. Then, trivially, πi (s−j ) = ρi (s−j ) and κji (sj , s−j ) = 0 for
T
all sj , and thus λiji (s ) = 0. Consequently, truth-telling is a best response of player i, for all
T
T
T
parameter values (yij )j6=i , if si is a maximizer of Πi (si , s−i ). This holds because s is a BNE,
T
so that s is also a PRE.
Before moving on to our possibility results for implementation in PRE, we would like to pick
up on our earlier discussion of a mechanism designer who is not sophisticated enough to utilize
all details of the kindness generating process. The insurance property has the appealing feature
that it implies robustness not only for varying values of
y,
as captured by the concept of PRE,
but also in other dimensions. For instance, we could speculate that the designer does not know
the details of how equitable payos are computed. Possible variations might include dierent
denitions of the set of ecient strategies, varying weights placed on maximal and minimal
payos, or the possibility that equitable payos depend on how much own payo a player must
sacrice to inuence the opponent's payo. The insurance property implies that no player can
unilaterally aect the payo of any other player through deviations from truth-telling, so that
psychological considerations will disappear from his optimization problem in all these cases,
yielding robustness to an even greater extent than examined here.
32
5.3 A Possibility Result Based on Expected Budget Balance
Consider an environment where both surplus and decit of a mechanism are in principle possible,
that is, where
T = Rn
is the unrestricted set of transfers. In such environments, we often place
constraints on the expected surplus or decit of a social choice function. For instance, admissible
social choice functions
f = (q1f , . . . , qnf , tf1 , . . . , tfn )
might not be allowed to yield a decit in
expectation, and hence must satisfy
Eθ
" n
X
#
tfi (θ)
≤ 0.
i=1
The following theorem shows that, to any social choice function that is implementable in BNE,
there exists another one that has the insurance property and is essentially equivalent otherwise.
27
In particular, it entails the same expected transfers and utilities.
Theorem 3.
Let
f
be an SCF that is implementable in BNE. Then there exists an SCF
f¯ that
has the following properties:
¯
(b) Expected transfers are the same as
.
f : qif (θ)h= qif (θ) fori all i ∈hI and θ ∈ Θ
Pn f i
Pn f¯
under f : Eθ
i=1 ti (θ) .
i=1 ti (θ) = Eθ
(c) Interim payos of every individual
i∈I
(a) The decision rule is the same as under
¯
and type
θi ∈ Θi
are the same as under
f:
¯
Eθ−i [vi (qif (θi , θ−i ), θi ) + tfi (θi , θ−i )] = Eθ−i [vi (qif (θi , θ−i ), θi ) + tfi (θi , θ−i )].
(d)
(e)
f¯ is implementable in BNE.
f¯ has the insurance property.
Proof. Step 1. For any arbitrary SCF
f = (q1f , . . . , qnf , tf1 , . . . , tfn ),
we dene the following ex-
pressions:
Tif (θi ) = Eθ−i [tfi (θi , θ−i )]
are the expected transfers to
i
conditional on type
θi ,
and
Vif (θi ) = Eθ−i [vi (qif (θi , θ−i ), θi )]
are, analogously,
i's
conditional expected payos net of transfers.
f
f f
f
Step 2. Now, starting from f = (q1 , . . . , qn , t1 , . . . , tn ) as given in the theorem, we construct a
f¯
f¯
payment scheme (t1 , . . . , tn ) as follows. For every i ∈ I , θi ∈ Θi , and θ−i ∈ Θ−i , we let
¯
tfi (θi , θ−i ) = Vif (θi ) + Tif (θi ) − vi (qif (θi , θ−i ), θi ).
Now consider
payment rule.
¯
¯
f¯ = (q1f , . . . , qnf , tf1 , . . . , tfn ), which has the same decision
We claim that f¯ satises, for all i ∈ I and θi ∈ Θi ,
¯
Tif (θi ) = Tif (θi ),
27
rule as
f
but the new
(5)
Börgers and Norman (2009) investigate a related question, asking under which conditions an otherwise
equivalent but ex post budget balanced SCF exists for a given SCF. In contrast, we are interested in insurance
against other players' type realizations, not insurance against mechanism decits. Mathevet (2010) constructs
transfers to obtain an otherwise identical but supermodular SCF.
33
that is, the expected payment to every type of every individual is the same under
f
and
f¯.
In
fact, it holds that
¯
¯
Tif (θi ) = Eθ−i [tfi (θi , θ−i )]
= Eθ−i [Vif (θi ) + Tif (θi ) − vi (qif (θi , θ−i ), θi )]
= Vif (θi ) + Tif (θi ) − Eθ−i [vi (qif (θi , θ−i ), θi )]
= Tif (θi ).
Step 3. We now verify that
(a)
¯
¯
f¯ = (q1f , . . . , qnf , tf1 , . . . , tfn )
is satised by construction. Property
(b)
satises properties
(a)
-
(e).
Property
follows from (5) above, after an application of the
law of iterated expectations:
Eθ
i
Pn P
f¯
f¯
t
(θ)
=
i=1 i
i=1
θi ∈Θi p(θi )Ti (θi )
Pn P
f
=
i=1
i i )Ti (θi )
hP θi ∈Θi p(θ
f
n
= Eθ
i=1 ti (θ) .
hP
n
Properties
(d).
(a)
and (5) together also immediately imply property
The revelation principle for BNE implies that
f¯ is
¯
We next turn to property
implementable in BNE if and only if the
following incentive compatibility constraints are satised: for all
¯
(c).
¯
i∈I
and
θi , θi0 ∈ Θi ,
¯
Eθ−i [vi (qif (θi , θ−i ), θi )] + Tif (θi ) ≥ Eθ−i [vi (qif (θi0 , θ−i ), θi )] + Tif (θi0 ) .
Because of property
(a)
and (5), this inequality can be equivalently written as
Eθ−i [vi (qif (θi , θ−i ), θi )] + Tif (θi ) ≥ Eθ−i [vi (qif (θi0 , θ−i ), θi )] + Tif (θi0 ),
which is satised because
the insurance property
f
(e).
is implementable in BNE. We complete the proof by establishing
From the denition of
¯
¯
tfi (θi , θ−i )
and property
(a)
it follows that
¯
vi (qif (θi , θ−i ), θi ) + tfi (θi , θ−i ) = Vif (θi ) + Tif (θi )
for all
θi
and
θ−i .
Hence, for any
¯
j 6= i,
¯
Eθ−j [vi (qif (θj , θ−j ), θi ) + tfi (θj , θ−j )] = Eθ−j [Vif (θi ) + Tif (θi )] = Eθi [Vif (θi ) + Tif (θi )]
is independent of
θj ,
which is the insurance property.
Theorem 3 implies that, when budget balance (or surplus) is required only in expected terms,
we can implement any decision rule in PRE that would be implementable in BNE, i.e., if people
cared only about their own material payos. Implementability in a world inhabited by selsh
homines oeconomicii is then not only necessary but also sucient for implementability in PRE.
For instance, d'Aspremont and Gerard-Varet (1979) establish the possibility to implement any
ecient decision rule in BNE, together with the stronger requirement of ex post budget balance.
Theorem 3 then implies that we can implement any such decision rule in PRE, if we are allowed
to replace ex post budget balance by the weaker requirement of budget balance in expectation.
34
Theorem 3 implies an additional robustness property. It tells us that the transfers
needed for robust implementation are equivalent to
(tf1 , ..., tfn )
expected revenues and interim payos are the same.
¯
¯
(tf1 , ..., tfn )
from important perspectives:
For instance, if we are faced with in-
terim participation constraints in addition to implementability in BNE, i.e. minimal values of
Eθ−i [vi (qif (θi , θ−i ), θi ) + tfi (θi , θ−i )]
ises these constraints implies that
for dierent players
f¯ satises
i
and types
θi ,
then the fact that
f
sat-
them as well. This observation implies that the
following well-known results for dierent applications of the independent private values model
with participation constraints are still true if we use the more demanding solution concept of a
PRE:
•
Second-price auction: A second-price auction is a special case of the above environment,
where
f
is such that a private good is assigned to an individual with maximal valuation,
who then has to pay the second highest valuation.
As is well known, truth-telling is a
BNE of the corresponding direct mechanism (in fact, even in dominant strategies), and all
types of all individuals are willing to participate (when interim outside options all yield a
payo of zero). Moreover, the second price auction runs an expected budget surplus. For
this setting, Theorem 3 implies that there exists a modied version of the second-price
auction that shares all these properties and is psychologically robust.
•
Partnership dissolution: The problem to dissolve a partnership eciently, which has been
studied by Cramton et al. (1987), is also a special case of our setup. In this application,
shares of an object have to be assigned to a number of agents (the partners) who have
private information about their valuation of those shares and who have pre-specied property rights. Cramton et al. (1987) require that participation constraints are satised, or
equivalently, that an agreement is reached with unanimity, and obtain possibility results
for an ecient allocation of shares. Theorem 3 implies that these results can be generalized
to a setting where the partners are inclined to reward kind and punish unkind behavior of
other partners.
•
Public goods provision: Various authors have studied the provision of excludable and nonexcludable public goods in the presence of participation constraints (Güth and Hellwig
1986, Hellwig 2003, Norman 2004). Again, Theorem 3 implies that these results generalize
to a setting where individuals are willing to react in a reciprocal way to the other agents'
contributions to a public good.
Whether or not expected (as opposed to ex post) budget balance is a reasonable requirement,
will depend on the application. When it comes to public goods provision, insisting on ex post
budget balance is appropriate if there is no external source of funds that may help to cover
the provision costs.
By contrast, an auctioneer who runs several independent auctions may
be willing to accept losses on some, provided that overall there is positive expected surplus. A
focus on expected budget balance can also be justied if the number of individuals is large. With
many individuals, the probability that the mechanism runs a surplus or a decit larger than
for an arbitrary
ε,
ε > 0, converges to zero if and only if expected budget balance holds, due to the
law of large numbers. Bierbrauer (2011) proves this fact in the context of redistributive income
taxation.
35
5.4 A Possibility Result Based on Ex Post Budget Balance
Let us return to an environment where
T = T̄ ,
so that mechanisms can never run a decit.
Overall eciency of an SCF then requires, in addition to the decision rule being ecient (or
surplus-maximizing), the transfers to satisfy ex post budget balance. Clearly, ex post budget
balance is a stronger requirement than budget balance in expectation, which was examined in
the preceding subsection.
To provide sucient conditions for psychological robustness under ex post budget balance,
we study the expected externality mechanism of d'Aspremont and Gerard-Varet (1979) or Arrow
(1979), following the exposition in Mas-Colell et al. (1995, chapter 23). The expected externality
mechanism is a direct mechanism
Φ = [Θ1 , ..., Θn , f ]
where the social choice function
f
is
f
f
constructed as follows. First, the decision rule (q1 , ..., qn ) must be chosen ecient, i.e., for all
θ ∈ Θ,
(q1f (θ), . . . , qnf (θ))
The associated transfers
∈ arg
n
X
max
(q1 ,...,qn )∈Q
(tf1 , ..., tfn )
vi (qi , θi ).
i=1
are then constructed by


X
tfi (θi , θ−i ) = Eθ−i 
vj (qjf (θi , θ−i ), θj ) + hi (θ−i ),
(6)
j6=i
where
hi (θ−i )
is dened as as
hi (θ−i ) = −
1
n−1
X


X
Eθ−j 
vh (qhf (θj , θ−j ), θh ) .
j6=i
(7)
h6=j
One can show that ex post budget balance,
f
i∈I ti (θ)
P
= 0 for all θ ∈ Θ, in fact holds.
Moreover,
d'Aspremont and Gerard-Varet (1979) have shown that truth-telling is a BNE in the expected
externality mechanism.
In the following we show that this mechanism, or more precisely its SCF
property if
n = 2 or if environment and decision rule are symmetric.
Proposition 12 then implies
that the expected externality mechanism implements
f
Denition 9.
Given an environment
and a decision rule
holds if, for all
i
and
E
with
n≥3
in PRE.
(q1f , ..., qnf ),
(8)
θi ∈ Θi .
Symmetry requires that all opponents of
which type
symmetry
j 6= i, k 6= i,
h
i
h
i
Eθ−i vj (qjf (θi , θ−i ), θj ) = Eθ−i vk (qkf (θi , θ−i ), θk )
for all types
f , has the insurance
θi ∈ Θi
is realized.
i
obtain an identical expected payo, no matter
This should not be confused with the insurance property.
Symmetry does not require the opponents' payos to be independent of
θi ,
but rather that the
opponents' payos are always identical and are thus aected equally by player i's type. Further,
36
it is a condition on payos net of transfers. Symmetry is satised whenever the environment
iid
is symmetric, which requires identical payo functions for all players and
types, and the
f
f
decision rule (q1 , ..., qn ) treats all individuals symmetrically.
Theorem 4.
Consider an expected externality mechanism
the insurance property if
n=2
Φ = [Θ1 , ..., Θn , f ].
The SCF
has
or if symmetry holds.
i, j ∈ I , j 6= i,
Proof. Step 1. We rst show that for all players


1
n−1
Πj (si , (sTk )k6=i ) = Eθ vj (qjf (si (θi ), θ−i ), θj ) −
where
f
ξji is independent of si .
X
vh (qhf (si (θi ), θ−i ), θh ) + ξji ,
(9)
h6=i
To see that this is true, note that, given the denition of
(tf1 , ..., tfn )
in the expected externality mechanism, we have
Πj (si , (sTk )k6=i )
=
h
i
h
Eθ vj (qjf (si (θi ), θ−i ), θj )
i
Eθ tfj (si (θi ), θ−i )
+
i
= Eθ vj (qjf (si (θi ), θ−i ), θj )



X
vh (qhf (θj , θ−j ), θh ) + Eθ [hj (si (θi ), θ−ij )] ,
+Eθ Eθ−j 
h
h6=j
where
θ−ij
denotes the type prole of all players except
corresponds to the rst term in (9).
subsumed into
ξji .
i and j .
The rst term in this expression
The second term is independent of
Now consider the third term.
si
and can thus be
Again using the denition of the expected
externality mechanism we obtain
hj (si (θi ), θ−ij ) =







X
X
X
1
f
f




vh (qh (si (θi ), θ−i ), θh ) +
Eθ−l
vh (qh (θl , θ−l ), θh )
.
Eθ
−

n − 1  −i
h6=i
l6=j,i
The second term is again independent of
si
h6=l
and can be subsumed into
ξji .
The rst term, in
turn, becomes the second term in (9) after taking the expectation with respect to
Step 2. First, assume that there are only two players (n
= 2).
θ.
The term in squared brackets in
(9) then cancels out, which implies that
Πj (s0i , (sTk )k6=i ) = Πj (s00i , (sTk )k6=i )
for any pair of strategies
s0i
and
s00i
of player
i.
Proposition 11 then implies that the insurance
property is satised. Now, suppose that symmetry holds. Under this assumption it is also true
that the term in squared brackets in (9) vanishes. Again, this implies that the insurance property
holds.
The intuition for this result is as follows: The expected externality mechanism derives its
name from the fact that each player pays for the expected impact that his strategy choice has
on the other players' payos. If there are just two players, this implies that player 1 is perfectly
insured against the randomness in player 2's type, or equivalently, against changes of player 2's
37
strategy. Under symmetry, the argument generalizes to more than two players: If each player's
externalities are evenly distributed among all other players, then, once more, the insurance
28
property can be veried.
To illustrate that the insurance property will generally not be satised without symmetry,
in section A.8 of the Appendix we give a simple three player example of an ecient but nonsymmetric decision rule for which the the expected externality mechanism does not satisfy
the insurance property and is not psychologically robust. This observation sheds a new light
on symmetry properties.
From a normative perspective, requiring an SCF to be symmetric
captures common ideas about fairness, so that notions related to symmetry appear repeatedly
as axioms in social choice theory, most prominently as anonymity (Mas-Colell et al. 1995, p.
791). In our approach, symmetry plays an important role from a positive perspective: symmetric
decision rules can be implemented in the expected externality mechanism even if players have
intentions-based social preferences.
5.5 Extension: Robustness of Screening Mechanisms and Price Systems
As an extension, we use our formalism to assess the psychological robustness of social choice
functions that are of particular interest from the perspective of general equilibrium theory, public
nance and contract theory. In general equilibrium theory and public nance, one typically looks
at social choice functions that admit a decentralization via a (possibly non-linear) price system,
that may be shaped by the government's tax policy. A prominent topic in contract theory is the
design of optimal screening or incentive schemes. Classical applications include the regulation of
monopolistic rms, the study of market outcomes in insurance markets with private information
about risks, or prot-maximizing price-discrimination.
In the following we will rst introduce a more general setup and provide sucient conditions
for implementability of certain social choice functions in PRE. In a second step, we argue that
various well-known models can be viewed as a special case of this general setup.
We consider an environment
E
as described in Section 3.1, but we now allow for allocations
a = (a1 , . . . , an ) ∈ A = A1 × . . . × An
that specify for each individual
ai ∈ Ai , which can be multidimensional.
i
a consumption bundle
The set of possible types of individual
i is given by Θi =
{θi1 , . . . , θimi }, and types are independently distributed. Payo functions are given by
without imposing quasi-linearity. In this framework, a social choice function
species for each individual
i
f
a bundle ai (θ)
∈ Ai
f =
πi (ai , θi ),
f
(a1 , . . . , afn )
for each vector of types.
We now focus on social choice functions that are simple in the following sense: For each
individual
i
there exist bundles
i
ā1i , . . . , ām
i ∈ Ai
so that
afi (θik , θ−i ) = āki ,
for any
k = 1, ..., mi ,
independently of
θ−i .
The term simple is borrowed from Dierker and
Haller (1990). A simple social choice function does not make use of the possibility to make the
28
For an example with two-players, the property that each player's payo is independent of the other's an-
nouncement has also been observed by Mathevet (2010, p. 414). Desiraju and Sappington (2007) show that, in
their model with two agents and privately observed cost parameters, transfers can be structured so as to avoid
ex post
inequality
and hence render inequality aversion irrelevant, whenever the agents are ex ante identical.
38
outcome of individual
for
i
is a function of
i
i's
dependent on the types of the other individuals. Rather, the outcome
type only.
has the insurance property.
29 This immediately implies that any simple choice function
The property of simplicity is in fact much stronger than the in-
surance property. Hence, with an appeal to Proposition 12, a simple social choice function is
implementable in PRE if and only if it is implementable in BNE. Implementability in BNE in
turn holds if and only if the classical incentive compatibility constraints are satised: For each
individual
i
and all
k, l ∈ {1, ..., mi },
πi (āki , θik ) ≥ πi (āli , θik ).
(10)
The following proposition summarizes these observations.
Proposition 13. A simple SCF is implementable in PRE if and only if it is incentive compatible.
There are many applications where one is interested in simple social choice functions that
satisfy the incentive compatibility constraints in (10). Examples include the the study of insurance markets with adverse selection á la Rothschild and Stiglitz (1976), the study of optimal
monopoly regulation in the tradition of Baron and Myerson (1982), mechanism design approaches
to the Mirrlees (1971)-problem of optimal income taxation, such as Stiglitz (1982), or mechanism
design approaches to problems of non-linear pricing such as Mussa and Rosen (1978).
Many applications are special cases, with
i∈
Ai = Ā
and
I , identical independent probabilities pk of having type
identical payo functions
πi (ai , θi ) = π(ai , θi ).
Θi = Θ̄ = {θ1 , . . . , θm }
θi =
for all
θk for all individuals, and also
The denition of a simple SCF
f
would then
1
m
usually contain the requirement that there exist individual-independent bundles ā , . . . , ā
∈ Ā
so that
afi (θk , θ−i ) = āk
for all
i
and
k = 1, ..., m.
Hammond (1979) provides an alternative characterization of such
simple and incentive compatible social choice functions.
Accordingly, a simple social choice
function is incentive compatible if and only if it can be decentralized by means of a (possibly
non-linear) budget set.
This requires that there exists some set
B ⊆ Ā
such that, for all
k = 1, ..., m,
āk ∈ arg max π(b, θk ) .
(11)
b∈B
Proposition 13 can then be reformulated as saying that a simple SCF is implementable in PRE
if and only if it can be decentralized by means of a budget set, or a price system, respectively.
In many applications one is actually interested in simple social choice functions that admit
a decentralization via a price system.
29
This is true for any application of general equilibrium
Simple SCFs raise the question of what the appropriate resource constraint looks like.
With independent
private values, all individuals might turn out to be of the same type, for instance. For simple SCFs, this makes
ex post resource constraints inappropriate. As a consequence, applications that work with simple SCFs require
feasibility in expectation, i.e., expected consumption levels must not exceed the economy's resources. As noted
earlier, for large economies it can be possible to reinterpret expected consumption as deterministic aggregate
consumption.
39
theory. It is also true for any model of public nance which rests on the assumption that the
tax system shapes an individual's budget set, and that, given those budget sets, individuals
solve consumer choice problems.
For instance, this would be true for any model of taxation
using the framework of Ramsey (1927). It is also true for the original formulation of the optimal
income tax problem by Mirrlees (1971).
Proposition 13 shows that all these applications are
concerned with social choice functions that are psychologically robust. Whatever the inclination
of individuals to reward kind and to punish unkind behavior of others, a competitive equilibrium
allocation or an allocation that is induced by some tax system is implementable provided that
it is implementable in a model with selsh individuals.
Dufwenberg et al. (2011b) arrive at
the similar conclusion of behavioral irrelevance of other-regarding preferences in competitive
equilibrium under complete information, for a comprehensive model of social preferences that
may depend both on outcomes and consumption opportunities.
6
Conclusions
In this paper, we have enriched the independent private values framework of Bayesian mechanism
design with intentions-based social preferences. Starting from our results, we can identify a large
range of interesting questions to be addressed by future research. First, one might want to go
beyond the independent private values case, or examine a framework with symmetric information
(Maskin 1999). Second, modelling the mechanism designer as a player, to whose behavior the
agents attribute good or bad intentions, can be a relevant exercise for applications such as
auction design. Finally, working out the details of psychologically robust mechanisms for many
of the relevant applications strikes us as important and promising.
In addition, the following issues deserve closer scrutiny.
As is well-known, the focus on
normal form mechanisms is not restrictive in the classical mechanism design framework where
uniqueness of the equilibrium is not required, because any equilibrium in an extensive form
mechanism remains an equilibrium in the corresponding normal form.
With intentions-based
social preferences, whether or not this is still true remains an open question. A major obstacle
to answering this question is the fact that a general theory of intentions in extensive form games
with incomplete information is still lacking. Related to this point, the question of equilibrium
uniqueness is of course equally relevant in our framework as it is for the classical approach, but
further developments in the area of psychological game theory might be necessary rst.
Finally, several of our results lend themselves to experimental testing. First and foremost,
this concerns the role of unused actions as a design instrument. Second, our analysis has explored, theoretically, the possibility to rank dierent mechanisms that implement the same
material outcome in the kindness dimension.
This raises the question whether dierences in
kindness perceptions across outcome-equivalent mechanisms can also be identied empirically.
Our results on psychological robustness are another candidate for testing.
For instance, in a
recent experimental study Fehr et al. (2011b) report on the behavioral non-robustness of the
Moore-Repullo mechanism for subgame-perfect implementation under symmetric information.
In fact, this mechanism does not satisfy what would be an appropriate modication of our
insurance property for the Moore-Repullo framework.
40
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46
A
Appendix
A.1 BNFE with Updated Kindness
Consider the second possibility, where we dene kindness conditional on players' types. Let
X
Πj (mi , (sbik )k6=i |θi ) =
p(θ−i |θi )πj (g(mi , (sbik (θk ))k6=i ), (θi , θ−i ))
θ−i ∈Θ−i
i of type θi expects to give player
b
given i's beliefs (sik )k6=i about his opponents' strategies. Dene
denote the payo that player
j
by choice of action
mi ,
and
θi .
The
upi
b
b
b
κup
ij (mi , (sik )k6=i |θi ) = Πj (mi , (sik )k6=i |θi ) − πj ((sik )k6=i |θi ),
which in principle allows the equitable payo
πjupi
to depend on player
current formulation is analogous to the kindness term
ference that all payos are now conditional on player
κij
i's
mi = si (θi )
chosen upon receiving the information
information
dened in Section 3, with the dif-
type. As a result, the kindness term
b
κup
ij (mi , (sik )k6=i |θi ) no longer depends on the complete strategy
action
i's
si
of player
i,
but only on the
θi .
λiji is simply player i's belief about κji . Then,
up
b
starting from κji (mj , (sjk )k6=j |θj ), we can replace all arguments by player i's beliefs and take
As explained in Section 3, the kindness term
player
i's
expectation with respect to
θj ,
to obtain
b
bb
λup
iji (sij , (sijk )k6=j |θi ) =
h
i
X
upj
bb
p(θ−i |θi ) Πi (sbij (θj ), (sbb
)
|θ
)
−
π
((s
)
|θ
)
.
j
j
k6
=
j
k6
=
j
ijk
ijk
i
θ−i ∈Θ−i
The belief of player
i
about the (believed) kindness of
setup, depend on player
θi
i's
j
toward himself will, in the most general
information in a non-trivial way.
might give him additional information about player
j 's
simplify the expression by summing over
θj
θj ,
i's
type
type, which in turn is important to
calculate the expected kindness when dierent types of player
since the term in squared brackets depends only on
This is because player
j
are dierently kind. Note that,
not on the complete prole
θ−i ,
we could
only, with the appropriate probabilities.
Utility functions are now given by
Uiup (si , (sbij , (sbb
ijk )k6=j )j6=i ) =
Πi (si , (sbij )j6=i ) +


X

p(θi ) 
yij κup (si (θi ), (sbik )k6=i |θi )λup (sbij , (sbb
ijk )k6=j |θi ) .
X
ij
θi ∈Θi
iji
j6=i
The corresponding ex ante denition of BNFE would then be exactly like Denition 3, using the
payo functions
Uiup
instead of
Ui .
It is now fairly easy to see that the present formulation admits an equivalent interim BNFE
denition. To move from an ex ante to an interim denition, we rst decompose material payos
Πi (si , (sbij )j6=i ) =
P
θi ∈Θi
p(θi )Πi (si (θi ), (sbij )j6=i |θi )
as usual.
The same decomposition for the
up
kindness terms is already given in the above denition of Ui .
47
Hence we obtain an interim
BNFE denition that requires maximization of
Πi (mi , (sbij )j6=i |θi ) +
X
up b
b
bb
yij κup
ij (mi , (sik )k6=i |θi )λiji (sij , (sijk )k6=j |θi )
j6=i
for each player
i∈I
and each type
θi ∈ Θi ,
given correct beliefs.
We now address the question how the updated kindness concept diers from the one in
Section 3. The following properties are crucial.
Assumption 1.
(i) For all
i, j ∈ I , i 6= j ,
X
and all proles
(sbik )k6=i ∈
Q
k6=i Sk , it holds that
p(θi )πjupi ((sbik )k6=i |θi ) = πjei ((sbik )k6=i ).
θi ∈Θi
(ii) The players' types are independent.
Part (i) species a systematic relation between the equitable payos used in the two concepts.
It requires that the equitable payos
πjei
used in Section
3
are identical to the expected value
of the updated equitable payos introduced above. Part (ii) is the standard condition, imposed
throughout the main part of the paper, that the individual types are statistically independent.
Based on Assumption 1, we can show the following result.
Proposition 14.
Under Assumption 1, the functions
Uiup
and
Ui
are identical.
Proof. First, we can rewrite
κij (si , (sbik )k6=i ) =

X
X
p(θi ) 
θi ∈Θi

p(θ−i |θi )πj (g(si (θi ), (sbik (θk ))k6=i ), (θi , θ−i )) − πjei ((sbik )k6=i ),
θ−i ∈Θ−i
which, using assumption 1(i), becomes
X
h
i
p(θi ) Πj (si (θi ), (sbik )k6=i |θi ) − πjupi ((sbik )k6=i |θi ) .
θi ∈Θi
Hence we have
κij (si , (sbik )k6=i ) =
X
b
p(θi )κup
ij (si (θi ), (sik )k6=i |θi ).
θi ∈Θi
Next, given independence 1(ii) and again assumption 1(i), we can rewrite
b
bb
λup
iji (sij , (sijk )k6=j |θi ) =


X
X
upj
bb

p(θj ) 
p(θ−j |θj )πi (g(sbij (θj ), (sbb
ijk (θk ))k6=j ), (θj , θ−j )) − πi ((sijk )k6=j |θj )
θj ∈Θj
=
X
θ−j ∈Θ−j
e
j
bb
b
bb
p(θ)πi (g(sbij (θj ), (sbb
ijk (θk ))k6=j ), θ) − πi ((sijk )k6=j ) = λiji (sij , (sijk )k6=j ).
θ∈Θ
48
Hence we have
κij (si , (sbik )k6=i )λiji (sbij , (sbb
ijk )k6=j ) =
up b
b
bb
p(θi )κup
ij (si (θi ), (sik )k6=i |θi )λiji (sij , (sijk )k6=j |θi ),
X
θi ∈Θi
which immediately implies that
Ui
and
Uiup
are identical.
The idea behind Proposition 14 is the following. If assumption 1(i) is satised, we can write
the kindness
kij (.)
as player
which depends on the action
i's
expectation of the updated kindness terms
si (θi ) ∈ Mi
only. Thus we can interpret player
up
kij
(.|θi ),
i's
each of
kindness as his
expectation of how kind his own possible types will be at the interim stage. The term
then reects player
i's
expectation of the kindness of player
j 's
λiji (.)
dierent interim types. If types
are independent (assumption 1(ii)), the own type contains no information about the opponents'
types. This implies that, upon learning the own type, player
i
will not change his belief about
up
player j 's kindness. Hence λiji (.|θi ) is in fact independent of θi , and identical to λiji (.). Taken
up
together, the functions Ui and Ui
are identical, which implies that it is irrelevant which of them
is used for the denition of BNFE.
A.2 Proof of Proposition 2
In the direct mechanism, the set of pure strategies for individual
sTi
denotes truth-telling,
sH
i
L −T
i is Si = {sTi , sH
i , si , si }, where
prescribes to announce the high type
θi1
whatever the true type,
−T
sL
is a strategy which requires to lie always,
i requires to always anonounce a low type, and si
−T 0
−T 1
1
0
∗
f∗
i.e., si (θi ) = θi and si (θi ) = θi . Material payos are given by πi (f (θ̂), θ) = q (θ̂) (θi − c),
where
θ̂ denotes the prole of announced types.
For any strategy pair
(s1 , s2 ) we then obtain the
Π1 (s1 , s2 ) and Π2 (s1 , s2 ) as described in Section 3. We seek to show
T
T
2
that (s1 , s2 ) is not a BNFE. We proceed by contradiction. Hence x some (y1 , y2 ) ∈ [0, ∞[
T T
and suppose that (s1 , s2 ) is a BNFE. In the hypothetical equilibrium beliefs are correct, which
b
bb
T
b
bb
T
implies that s12 = s212 = s2 and s21 = s121 = s1 .
expected material payos
Ecient strategies of player 1.
Solving for the ecient strategies of player 1, given that
T
player 2 plays s2 , requires, in a rst step, to look at how the two players' payos are aected as
player 1 varies his strategy. Straightforward computations yield
Π1 (sT1 , sT2 ) =
1 1
1 1
1 0
1 0
θ1 − c +
θ1 − c , Π2 (sT1 , sT2 ) =
θ2 − c +
θ2 − c ,
4
2
4
2
1 1
1 1
1 0
1 0
T
θ1 − c +
θ1 − c , Π2 (sH
θ2 − c +
θ2 − c ,
1 , s2 ) =
2
2
2
2
1 1
1 0
1 1
T
T
Π1 (sL
θ1 − c +
θ1 − c , Π2 (sL
θ2 − c ,
1 , s2 ) =
1 , s2 ) =
4
4
2
1 1
1 1
1 0
1 0
T
T
Π1 (s−T
θ1 − c +
θ1 − c , Π2 (s−T
θ2 − c +
θ2 − c .
1 , s2 ) =
1 , s2 ) =
2
4
4
2
T
Π1 (sH
1 , s2 ) =
Inspection of these expressions reveals that strategy
makes player
1
better o and leaves player
2
s−T
1
is not ecient, because a switch to
sT1
unaected. All other strategies are ecient since
T
T T
H T
Π1 (sL
1 , s2 ) < Π1 (s1 , s2 ) < Π1 (s1 , s2 )
49
and
T
T T
H T
Π2 (sL
1 , s2 ) > Π2 (s1 , s2 ) > Π2 (s1 , s2 ).
The equitable payo for player
2
in the equilibrium
(sT1 , sT2 )
is therefore
1 1
1
1 0
T
H T
π2e1 (sT2 ) = (Π2 (sL
θ2 − c +
θ2 − c = Π2 (sT1 , sT2 ).
1 , s2 ) + Π2 (s1 , s2 )) =
2
4
2
(12)
Ecient strategies of player 2. Analogously we solve for the ecient strategies of player 2
given that player 1 behaves according to
Π1 (sT1 , sT2 ) =
sT1 .
An inspection of
1 1
1 1
1 0
1 0
θ1 − c +
θ1 − c , Π2 (sT1 , sT2 ) =
θ2 − c +
θ2 − c ,
4
2
4
2
1 1
1 1
1 0
1 0
θ1 − c +
θ1 − c , Π2 (sT1 , sH
θ2 − c +
θ2 − c ,
2 )=
2
2
2
2
1 0
1 1
1 1
θ1 − c , Π2 (sT1 , sL
θ2 − c +
θ2 − c ,
Π1 (sT1 , sL
2) =
2) =
2
4
4
1 1
1 1
1 0
1 0
Π1 (sT1 , s−T
θ1 − c +
θ1 − c , Π2 (sT1 , s−T
θ2 − c +
θ2 − c
2 )=
2 )=
4
2
2
4
Π1 (sT1 , sH
2 )=
makes it possible to verify that neither strategy
s−T
2
nor strategy
sL
2
are ecient.
The other
strategies are ecient since
Π1 (sT1 , sT2 ) < Π1 (sT1 , sH
2 )
and
Π2 (sT1 , sT2 ) > Π2 (sT1 , sH
2 ).
The equitable payo for player
1
in
(sT1 , sT2 )
is therefore equal to
1 1
1
3 0
T T
π1e2 (sT1 ) = (Π1 (sT1 , sH
θ2 − c +
θ2 − c .
2 ) + Π1 (s1 , s2 )) =
2
8
2
Best response of player 2. Using (12) we obtain that
e1 T
T T
λ212 (sb21 , sbb
212 ) = Π2 (s1 , s2 ) − π2 (s2 ) = 0
in the hypothetical equilibrium
material payo
Π2 (sT1 , s2 ).
(sT1 , sT2 ).
Hence, player 2 chooses
s2
to maximize only the
Our assumptions imply that the best response of player 2 is then
sT2 .
Best response of player 1. Analogously, we obtain
e2 T
T T
λ121 (sb12 , sbb
121 ) = Π1 (s1 , s2 ) − π1 (s1 ) < 0
in the hypothetical equilibrium. Hence, since
T
minimizes Π2 (s1 , s2 ), player
T T
that (s1 , s2 ) is a BNFE.
s1 = sH
1
simultaneously maximizes
1's best response to sT2 is
50
Π1 (s1 , sT2 )
and
sH
1 , which contradicts the assumption
A.3 Proof of Proposition 3
We seek to verify that
(sT1 , sT2 )
is a BNFE for appropriately chosen values of
2 . In this hypothetical equilibrium beliefs are correct, which implies sb12
T
sb21 = sbb
121 = s1 .
=
δ1 , δ2 , 1
sbb
212
=
and
sT2 and
Ecient strategies of player 1. We solve for the ecient strategies of player 1, given that
player 2 behaves according to
sT2 .
By varying his strategy, player 1 can induce the following
expected payo pairs:
1
1
T
T
H T
Π1 ((d1 , d1 ), sT2 ) = Π1 (sH
1 , s2 ) + 1 , Π2 ((d1 , d1 ), s2 ) = Π2 (s1 , s2 ) − δ2 ,
2
2
1
1
Π1 ((θ10 , d1 ), sT2 ) = Π1 (sT1 , sT2 ) + 1 , Π2 ((θ10 , d1 ), sT2 ) = Π2 (sT1 , sT2 ) − δ2 ,
4
4
1
1
−T T
T
0
T
Π1 ((d1 , θ10 ), sT2 ) = Π1 (s−T
1 , s2 ) + 1 , Π2 ((d1 , θ1 ), s2 ) = Π2 (s1 , s2 ) − δ2 ,
4
4
1
T
Π1 ((d1 , θ11 ), sT2 ) = Π1 ((θ11 , d1 ), sT2 ) = Π1 (sH
1 , s2 ) + 1 ,
4
1
T
Π2 ((d1 , θ11 ), sT2 ) = Π2 ((θ11 , d1 ), sT2 ) = Π2 (sH
1 , s2 ) − δ2 .
4
Given our assumptions on parameters, it is easily veried that, among the ecient strategies,
sL
1
yields the highest and
(d1 , d1 )
yields the smallest payo for player
2.
The equitable payo of
player 2 is therefore equal to
π2e1 (sT2 ) =
=
1
L T
T
2 (Π2 (s1 , s2 ) + Π2 ((d1 , d1 ), s2 ))
Π2 (sT1 , sT2 ) − 14 δ2 .
1
T
H T
= 21 (Π2 (sL
1 , s2 ) + Π2 (s1 , s2 )) − 4 δ2
Consequently, in the hypothetical equilibrium,
1
e1 T
T T
λ212 (sb21 , sbb
212 ) = Π2 (s1 , s2 ) − π2 (s2 ) = δ2 .
4
Ecient strategies of player 2. Analogously, holding
sT1
xed, we obtain
1
1
T
T H
Π1 (sT1 , (d2 , d2 )) = Π1 (sT1 , sH
2 ) − δ1 , Π2 (s1 , (d2 , d2 )) = Π2 (s1 , s2 ) + 2 ,
2
2
1
1
Π1 (sT1 , (θ20 , d2 )) = Π1 (sT1 , sT2 ) − δ1 , Π2 (sT1 , (θ20 , d2 )) = Π2 (sT1 , sT2 ) + 2 ,
4
4
1
1
T
0
T −T
Π1 (sT1 , (d2 , θ20 )) = Π1 (sT1 , s−T
2 ) − δ1 , Π2 (s1 , (d2 , θ2 )) = Π2 (s1 , s2 ) + 2 ,
4
4
1
Π1 (sT1 , (d2 , θ21 )) = Π1 (sT1 , (θ21 , d2 )) = Π1 (sT1 , sH
2 ) − δ1 ,
4
51
1
Π2 (sT1 , (d2 , θ21 )) = Π2 (sT1 , (θ21 , d2 )) = Π2 (sT1 , sH
2 ) + 2 .
4
It is now again straightforward to nd the worst and the best payo for player
2's
sT1
ecient strategies, holding
1
among player
xed, and we obtain
1
1
1
T
0
T H
T T
π1e2 (sT1 ) = (Π1 (sT1 , sH
2 ) + Π1 (s1 , (θ2 , d2 ))) = (Π1 (s1 , s2 ) + Π1 (s1 , s2 )) − δ1 .
2
2
8
Consequently, in the hypothetical equilibrium,
e2 T
1
1
T T
T T
T H
λ121 (sb12 , sbb
121 ) = Π1 (s1 , s2 ) − π1 (s1 ) = 2 (Π1 (s1 , s2 ) − Π1 (s1 , s2 )) + 8 δ1
1
8 (δ1
=
+ c − θ10 ).
sT1 ,
player 2 chooses s2 in order to maximize
1
T
b
bb
T
Π2 (s1 , s2 ) + y2 λ212 (s21 , s212 )Π1 (s1 , s2 ). Now let λ212 (sb21 , sbb
212 ) = y2 , or, equivalently, δ2 =
4
y2 > 0. Then the problem becomes: choose s2 in order to maximize expected utilitarian
T
T
T
welfare Π2 (s1 , s2 ) + Π1 (s1 , s2 ). By construction, s2 solves this problem whenever 2 is such
Best response of player 2. If player 1 chooses
that
0 < 2 ≤ δ1 .
Best response of player 1. If player 2 chooses
Π1 (s1 , sT2 )
θ10 − c + y81
+
T
y1 λ121 (sb12 , sbb
121 )Π2 (s1 , s2 ).
> 0.
sT2 ,
b
bb
Now let λ121 (s12 , s121 )
Then the problem becomes: choose
T
welfare Π1 (s1 , s2 )
+
Π2 (s1 , sT2 ). By construction,
s1
s1 in order to maximize
1
y1 , or, equivalently, δ1 =
player 1 chooses
=
in order to maximize expected utilitarian
sT1 solves this problem when
1
is such that
0 < 1 ≤ δ2 .
A.4 Proof of Theorem 1
Idea and structure of proof. We rst deal with the case of 2 agents,
n = 2.
Below, we explain
how to generalize the argument for an arbitrary number of players.
Fix an ecient SCF
M1 = Θ1 , M2 = Θ2
and
f
Φ = [M1 , M2 , g]
and consider the direct mechanism
g = f . Si
is the set of player
i's
T
are player i's ex ante expected material payos. Let s
with its associated kindness values
T
implies that s is a BNFE in
messages for each player
holds in
Φ̃
j
when
Φ
λ121
and
(sT )
that is,
Πi : S1 × S2 → R
T
T
(s1 , s2 ) be the truth-telling prole,
pure strategies, and
=
λ212 (sT ).
= 1/y1
f,
As argued before, eciency of
f
(sT )
λ212
= 1/y2 . Now, in general
T
this condition on kindness values will not be satised by s in the direct mechanism. We will
therefore show how to extend
Φ
λ121 (sT )
for
and
to a pseudo-direct mechanism
for both players, still given the truth-telling prole
T is a BNFE in
that s
Φ̃,
Φ̃,
T
to adjust λiji (s ) to the desired value
sT ,
by introducing additional
1/yi .
Once
eciency of
λiji (sT ) = 1/yi
f
again implies
so we do not need to care about the problem that the additional
messages must remain unused. We rst study the problem of adding messages for player
1
to
manipulate
λ212 (sT ) = Π2 (sT ) − π2e1 (sT ).
It can be increased by decreasing the equitable payo
π2e1 (sT ),
and decreased by increasing
π2e1 (sT ) can in fact be adjusted
T
to any arbitrary value by an appropriate mechanism extension. Let E1 (s2 ) denote the set of
π2e1 (sT ). We will show that, starting from the direct mechanism,
52
conditionally ecient strategies of player
1
π2e1 (sT2 ) =
2
1
in the direct mechanism, and remember that
#
"
min
s1 ∈E1 (sT
2)
Π2 (s1 , sT2 ) +
max
s1 ∈E1 (sT
2)
Π2 (s1 , sT2 )
denes the equitable payo in the direct mechanism.
Decreasing
player
2's
π2e1 (sT2 ).
smin
∈ arg mins1 ∈E1 (sT ) Π2 (s1 , sT2 )
1
Let
2
be a strategy that minimizes
payo among ecient strategies in the direct mechanism
min , and
the range of s1
r=
|smin
1 (Θ1 )| is its cardinality. Let
Φ.
Then
smin
1 (Θ1 ) ⊆ Θ1
σ : {d1 , d2 , ..., dr } →
is
smin
1 (Θi ) be an
{d1 , d2 , ..., dr } a distinct
−1
is denoted σ
. We now construct the extended
arbitrary bijective function, which assigns to every element of the set
min (Θ ). The inverse of
type from the range s1
i
mechanism
Φ̃ = [M̃1 , M̃2 , g̃]
i.e. we add
r
from
Φ
σ
as follows. Let
M̃1 = Θ1 ∪ {d1 , d2 , ..., dr }
M̃2 = Θ2 ,
1 and keep player 2's message set unchanged. Hence we
˜
S2 = S2 . We construct g̃ from g by keeping g̃(m) = g(m)
new messages for player
have new strategy sets
S̃1 ⊃ S1
and
m ∈ Θ1 × Θ2 . When m = (dj , θ2 ) for some
g̃
(q1 (m), q2g̃ (m), tg̃1 (m), tg̃2 (m)) by
whenever
g̃(m) =
and
j ∈ {1, ..., r}
and
θ2 ∈ Θ2 ,
we dene
q2g̃ (dj , θ2 ) = q2g (σ(dj ), θ2 )
q1g̃ (dj , θ2 ) = q1g (σ(dj ), θ2 ),
and
tg̃1 (dj , θ2 ) = tg1 (σ(dj ), θ2 ) + ,
for some pre-specied
as announcing type
player
1.
and
σ(dj ) ∈
Φ̃,
Πi
with
0 < ≤ δ.
smin
1 (Θ1 )
⊆ Θ1 ,
Hence announcing
because payos in
to the new strategy sets. Let
conditional on player
2
dj
has the same consequences
2
to
Φ̃
by
except for additional transfers from player
With slight abuse of notation, we denote player
Πi : S̃1 × S̃2 → R as well,
extend
δ
tg̃2 (dj , θ2 ) = tg2 (σ(dj ), θ2 ) − δ
i's
ex ante expected payos in
Φ and Φ̃ coincide on S1 × S2 ,
Ẽ1 (sT2 )
so that we can simply
be the set of ecient strategies for player
1
in
still telling the truth.
We rst prove that the maximization part in the denition of
π2e1 (sT2 )
remains unaected by
the mechanism extension.
Lemma 1.
For any arbitrary
, δ
with
max
s1 ∈Ẽ1 (sT
2)
0 < ≤ δ,
Π2 (s1 , sT2 ) =
it holds that
max
s1 ∈E1 (sT
2)
Π2 (s1 , sT2 ).
Proof. Step 1. We rst claim that, for the maximization in
max
s1 ∈E1 (sT
2)
The inequality
S1 .
Φ,
we can replace
E1 (sT2 )
by
S1 ,
i.e.,
Π2 (s1 , sT2 ) = max Π2 (s1 , sT2 ).
s1 ∈S1
maxs1 ∈E1 (sT ) Π2 (s1 , sT2 ) ≤ maxs1 ∈S1 Π2 (s1 , sT2 ) immediately follows from E1 (sT2 ) ⊆
2
To obtain a contradiction, assume
maxs1 ∈E1 (sT ) Π2 (s1 , sT2 ) < maxs1 ∈S1 Π2 (s1 , sT2 ),
2
smax
1
and let
∈ arg maxs1 ∈S1 Π2 (s1 , sT2 ), which implies smax
∈
/ E1 (sT2 ). Then there exists, by niteness of
1
0
T
max , sT ) >
S1 , a strategy s01 ∈ E1 (sT2 ) that Pareto dominates smax
and satises Π2 (s1 , s2 ) ≥ Π2 (s1
1
2
53
maxs1 ∈E1 (sT ) Π2 (s1 , sT2 ),
2
a contradiction.
The same argument holds for the extended mechanism
when maximizing
Π2 .
Φ̃,
i.e. we can replace
Ẽ1 (sT2 )
by
S̃1
Thus the lemma follows when we have established
max Π2 (s1 , sT2 ) = max Π2 (s1 , sT2 ).
s1 ∈S1
s1 ∈S̃1
Step 2. The inequality
maxs1 ∈S̃1 Π2 (s1 , sT2 ) ≥ maxs1 ∈S1 Π2 (s1 , sT2 )
the other inequality, observe that for every
egy
s1 ∈ S1
s̃1 ∈ S̃1 \S1
by replacing the announcement of any
s1 (θ1 ) = s̃1 (θ1 )
whenever
T
It follows that Π2 (s̃1 , s2 )
follows from
dj ∈ {d1 , d2 , ..., dr }
σ(dj ).
by
<
and s1 (θ1 ) = σ(s̃1 (θ1 )) whenever s̃1 (θ1 ) ∈ {d1 , d2 , ..., dr }.
T
Π2 (s1 , s2 ), so that maxs1 ∈S̃1 Π2 (s1 , sT2 ) ≤ maxs1 ∈S1 Π2 (s1 , sT2 ). For any arbitrary
, δ
min
s1 ∈Ẽ1 (sT
2)
with
0 < ≤ δ,
π2e1 (sT2 ).
We show that the
Π2 (s1 , sT2 ) =
min
s1 ∈E1 (sT
2)
Π2 (s1 , sT2 ) − δ.
from above, and construct the associated
by replacing every announcement of a type
−1 (θ̄ ). Formally,
new message σ
1
s̃1 (θ1 ) =
δ.
it holds that
smin
∈ E1 (sT2 ) ⊆ S1
1
Proof. Step 1. Consider strategy
s̃1 ∈ S̃1
Formally,
s̃1 (θ1 ) ∈ Θ1 ,
minimum can be decreased to any arbitrary value, by an appropriate choice of
strategy
θ̄1 ∈ smin
1 (Θ1 )
σ −1 (smin
1 (θ1 )) for all
θ1 ∈ Θ1 .
by the associated
Then it follows that
T
T
min T
Π1 (s̃1 , sT2 ) = Π1 (smin
1 , s2 ) + , Π2 (s̃1 , s2 ) = Π2 (s1 , s2 ) − δ.
Step 2. We claim that
s̃1 ∈ Ẽ1 (sT2 ),
min
s1 ∈Ẽ1 (sT
2)
(13)
which then implies
Π2 (s1 , sT2 ) ≤
To obtain a contradiction, suppose
min
s1 ∈E1 (sT
2)
s̃1 ∈
/ Ẽ1 (sT2 ),
Π2 (s1 , sT2 ) − δ.
so that there exists
s̃01 ∈ S̃1
such that
Π1 (s̃01 , sT2 ) ≥ Π1 (s̃1 , sT2 ), Π2 (s̃01 , sT2 ) ≥ Π2 (s̃1 , sT2 ),
(14)
with at least one of the inequalities being strict. Starting from
the announcement of any
s̃01 (θ1 )
∈
Θ1 , and s01 (θ1 )
=
dj ∈ {d1 , d2 , ..., dr }
σ(s̃01 (θ1 )) whenever
by
σ(dj ).
s̃01 (θ1 )
s̃01 , construct s01 ∈ S1
0
Formally, s1 (θ1 )
∈ {d1 , d2 , ..., dr }.
=
x ∈ [0, 1]
is the probability of announcements from
by replacing
s̃01 (θ1 ) whenever
We obtain
Π1 (s01 , sT2 ) = Π1 (s̃01 , sT2 ) − x, Π2 (s01 , sT2 ) = Π2 (s̃01 , sT2 ) + xδ,
where
For
we can construct an associated strat-
We now examine the minimization part in the denition of
Lemma 2.
S1 ⊂ S̃1 .
{d1 , d2 , ..., dr }
(15)
under
s̃01 .
Conditions
(13), (14) and (15) together imply
T
0
T
min T
Π1 (s01 , sT2 ) ≥ Π1 (smin
1 , s2 ) + (1 − x), Π2 (s1 , s2 ) ≥ Π2 (s1 , s2 ) − (1 − x)δ,
where at least one inequality is strict. If
x = 1,
this contradicts
54
smin
∈ E1 (sT2 ).
1
(16)
Hence assume
x < 1,
T
Π2 (smin
1 , s2 )
on
T
Π1 (s01 , sT2 ) > Π1 (smin
1 , s2 ).
so that
E1 (sT2 ).
must hold. Now, if
Otherwise, if
s001 ∈ S1
such strategy
From
s01 ∈ E1 (sT2 ),
s01 ∈
/ E1 (sT2 ),
smin
∈ E1 (sT2 )
1
it then follows that
this contradicts that
there exist strategies in
must, however, still satisfy
S1
smin
1
Π2 (s01 , sT2 ) <
Π2 (s1 , sT2 )
minimizes
that Pareto dominate
T
Π2 (s001 , sT2 ) < Π2 (smin
1 , s2 ),
s01 .
Any
since otherwise it
min . Finiteness of S then implies that there exists
would also Pareto dominate s1
1
00
T
min
T
with Π2 (s1 , s2 ) < Π2 (s1
, s2 ), which is the nal contradiction.
s001 ∈ E1 (sT2 )
Step 3. The lemma follows when we can also establish the opposite inequality
min
s1 ∈Ẽ1 (sT
2)
Π2 (s1 , sT2 ) ≥
min
s1 ∈E1 (sT
2)
Π2 (s1 , sT2 ) − δ.
To obtain a contradiction, assume
min
s1 ∈Ẽ1 (sT
2)
Π2 (s1 , sT2 ) <
min
s1 ∈E1 (sT
2)
where the right hand side equals
s̃min
1
Eciency of
Π2 (s1 , sT2 ) − δ,
(17)
Π2 (s̃1 , sT2 ) according to (13).
Let
s̃min
∈ arg mins1 ∈Ẽ1 (sT ) Π2 (s1 , sT2 ).
1
2
then requires
T
T
Π1 (s̃min
1 , s2 ) > Π1 (s̃1 , s2 ).
Let
0
smin
∈ S1
1
{d1 , d2 , ..., dr }
by
be obtained from
σ(dj ).
(18)
s̃min
1
by again replacing the announcement of any
dj ∈
This implies
0
0
T
min
T
Π1 (smin
, sT2 ) = Π1 (s̃min
, sT2 ) = Π2 (s̃min
1
1 , s2 ) − y, Π2 (s1
1 , s2 ) + yδ,
where
(19)
y ∈ [0, 1] is the probability of announcements from {d1 , d2 , ..., dr } under s̃min
1 .
Conditions
(13), (17), (18) and (19) together then imply
0
0
T
min
T
Π1 (smin
, sT2 ) > Π1 (smin
, sT2 ) < Π2 (smin
1
1 , s2 ), Π2 (s1
1 , s2 ).
Now, if
0
smin
∈ E1 (sT2 )
1
this contradicts that
smin
1
minimizes
Π2 (s1 , sT2 )
on
E1 (sT2 ).
Otherwise,
we obtain a contradiction exactly as in step 2.
Lemmas 1 and 2 together imply that, starting from the direct mechanism, we can decrease
the equitable payo arbitrarily, with the help of the examined mechanism extension.
Increasing
π2e1 (sT2 ).
The mechanism extension used to increase the equitable payo works
max ∈ arg max
analogously. Let s1
Π2 (s1 , sT2 ) and let r = |smax
(Θ1 )| be the cardinality of
1
s1 ∈E1 (sT
2)
max
max
the range of s1
. Fix any bijection σ : {d1 , d2 , ..., dr } → s1
(Θ1 ) and extend Φ to Φ̃ by letting
M̃1 = Θ1 ∪ {d1 , d2 , ..., dr }
any
m = (dj , θ2 ),
let
and
g̃(m)
M̃2 = Θ2 .
The function
g̃
again coincides with
be given by
q1g̃ (dj , θ2 ) = q1g (σ(dj ), θ2 ),
55
q2g̃ (dj , θ2 ) = q2g (σ(dj ), θ2 )
g
on
Θ1 × Θ2 .
For
and
tg̃1 (dj , θ2 ) = tg1 (σ(dj ), θ2 ) − δ,
with
0 < ≤ δ.
We have
Here, using messages from
S̃1 ⊃ S1
on the extension
and
S̃2 = S2 , we let Πi
S̃1 × S̃2 ,
sT2 . Observe that, if
{d1 , d2 , ..., dr } redistributes from player 1 to player 2.
again denote player i's ex ante expected payos dened
Ẽ1 (sT2 )
and we write
s1 ∈
/
tg̃2 (dj , θ2 ) = tg2 (σ(dj ), θ2 ) + E1 (sT2 ) for some
for the ecient strategies in
s 1 ∈ S1 ,
s1 ∈
/
then
Φ̃,
conditional on
Ẽ1 (sT2 ) holds as well, because
enlarging the strategy set cannot make a previously inecient strategy ecient.
We rst prove that the minimization part in the denition of
π2e1 (sT2 )
remains unaected by
the mechanism extension.
Lemma 3.
For any arbitrary
, δ
min
s1 ∈Ẽ1 (sT
2)
Proof. Step 1. Let
0 < ≤ δ,
with
Π2 (s1 , sT2 ) =
it holds that
min
s1 ∈E1 (sT
2)
smin
∈ arg mins1 ∈E1 (sT ) Π2 (s1 , sT2 ).
1
2
Π2 (s1 , sT2 ).
We claim that
smin
∈ Ẽ1 (sT2 ), which then
1
implies
min
s1 ∈Ẽ1 (sT
2)
Π2 (s1 , sT2 ) ≤
min
s1 ∈E1 (sT
2)
smin
∈
/ Ẽ1 (sT2 ).
1
To obtain a contradiction, assume
Π2 (s1 , sT2 ).
This implies that there exists
s̃1 ∈ S̃1 \S1
with
T
T
min T
Π1 (s̃1 , sT2 ) ≥ Π1 (smin
1 , s2 ), Π2 (s̃1 , s2 ) ≥ Π2 (s1 , s2 ),
with at least one of the inequalities being strict. Let
by replacing the announcement of any
s01 ∈ S1
dj ∈ {d1 , d2 , ..., dr }
by
(20)
be the strategy obtained from
σ(dj ).
It follows that
Π1 (s01 , sT2 ) > Π1 (s̃1 , sT2 ), Π2 (s01 , sT2 ) < Π2 (s̃1 , sT2 ).
From (20) and (21), together with
smin
∈ E1 (sT2 ),
1
s̃1
(21)
it then follows that
T
0
T
min T
Π1 (s01 , sT2 ) > Π1 (smin
1 , s2 ), Π2 (s1 , s2 ) < Π2 (s1 , s2 ).
This is a contradiction to
smin
1
minimizing
Π2 (s1 , sT2 )
on
E1 (sT2 ),
with the same argument as in
the proof of Lemma 2.
Step 2. To establish the other inequality
min
s1 ∈Ẽ1 (sT
2)
Π2 (s1 , sT2 ) ≥
min
Π2 (s1 , sT2 ),
min
Π2 (s1 , sT2 ).
s1 ∈E1 (sT
2)
assume to the contrary that
min
s1 ∈Ẽ1 (sT
2)
Let
Π2 (s1 , sT2 ) <
s̃min
∈ arg mins1 ∈Ẽ1 (sT ) Π2 (s1 , sT2 ).
1
s̃min
1
2
∈
S̃1 \S1 . Similarly to above, let s01
Since
∈
s1 ∈E1 (sT
2)
s1 ∈
/ E1 (sT2 )
implies
s1 ∈
/ Ẽ1 (sT2 ),
we must have
S1 be the strategy obtained from s̃min
by replacing
1
56
the announcement of any
dj ∈ {d1 , d2 , ..., dr }
by
σ(dj ).
It follows that
T
0
T
min T
Π1 (s01 , sT2 ) > Π1 (s̃min
1 , s2 ), Π2 (s1 , s2 ) < Π2 (s̃1 , s2 ).
If
s01 ∈ Ẽ1 (sT2 ),
we have obtained a contradiction against
Otherwise, there exists
00 T
satises Π2 (s1 , s2 )
<
s001 ∈ Ẽ1 (sT2 )
that Pareto dominates
s̃min
∈ arg mins1 ∈Ẽ1 (sT ) Π2 (s1 , sT2 ).
1
s01
2
but, due to
s̃min
∈ Ẽ1 (sT2 ),
1
T
Π2 (s̃min
1 , s2 ), again a contradiction.
We now examine the maximization part in the denition of
π2e1 (sT2 ).
We show that the
maximum can be increased to any arbitrary value, by an appropriate choice of
Lemma 4.
For any arbitrary
, δ
max
s1 ∈Ẽ1 (sT
2)
with
0 < ≤ δ,
Π2 (s1 , sT2 ) =
Proof. As shown in the proof of Lemma 1, step
still
.
it holds that
max
s1 ∈E1 (sT
2)
Π2 (s1 , sT2 ) + .
1, the statement follows when we have established
max Π2 (s1 , sT2 ) = max Π2 (s1 , sT2 ) + ,
s1 ∈S1
s1 ∈S̃1
where the right hand side equals
Π2 (smax
, sT2 ) + .
1
Starting from
smax
,
1
construct the associated
max (Θ ) by the associated new
strategy s̃1 ∈ S̃1 by replacing every announcement of a type θ̄1 ∈ s1
1
−1 (θ̄ ). Formally, s̃ (θ ) = σ −1 (smax (θ )) for all θ ∈ Θ . It follows that Π (s̃ , sT ) =
message σ
1
1 1
1
1
1
2 1 2
1
T ) + , which establishes the rst inequality
Π2 (smax
,
s
1
2
max Π2 (s1 , sT2 ) ≥ max Π2 (s1 , sT2 ) + .
s1 ∈S1
s1 ∈S̃1
The opposite inequality holds as well, because for any
s 1 ∈ S1
s̃1 ∈ S̃1 \S1 we can construct the associated
dj ∈ {d1 , d2 , ..., dr }
by replacing the announcement of any
by
σ(dj ),
to obtain
Π2 (s̃1 , sT2 ) − Π2 (s1 , sT2 ) = x,
where
x ∈]0, 1]
is the probability of announcements from
{d1 , d2 , ..., dr }
under
s̃1 .
Lemmas 3 and 4 together imply that, starting from the direct mechanism, we can increase
the equitable payo arbitrarily, with the help of the examined mechanism extension.
Synthesis and generalization to an arbitrary number of players. The above construction can
be done equivalently for player
2.
For proles
from the extended mechanism, the outcome
need to address unilateral deviations from
m
where both players use an additional message
g̃(m)
sT .
can be specied arbitrarily, because we only
Hence we can achieve
T
players in Φ̃, which implies that s is a BNFE in
Φ̃,
λiji (sT ) = 1/yi
for both
which in turn implements the SCF
f.
If the number of players exceeds 2, the construction above can be done for each pair of
players separately. The actions
ij
{dij
1 , . . . , drij }
of player
i
that are added to manipulate
πjei (sT−i )
simply have to be chosen such that they do not aect the outcomes for all other players. Hence,
57
for every pair
i and j
we can make sure that
λiji (sT ) = 1/yij ,
so that all players become welfare
maximizers and our arguments about truth-telling apply unaltered.
A.5 Proof of Proposition 7
We prove the Proposition in two steps. First, we show that there exist upper bounds on utilities
for any
(Φ, s∗ )
that implements
f ∗.
Second, we show that
(Φ0 , sT ),
with parameters as given in
the proposition, reaches these bounds.
Step 1.
f∗
By Proposition 4, for any mechanism that implements
in BNFE, we can nd a
∗
pseudo-direct mechanism that truthfully implements f in BNFE with identical utilities (i.e., a
pseudo-revelation principle applies to utility-ecient implementation of an SCF). Hence consider
w.l.o.g. a pair
(Φ, sT ) where Φ is a pseudo-direct mechanism for f ∗
b
In this BNFE, we have s12
sbb
212
sT2 and
sb21
=
=
=
= sT1 .
H
Bounds for the kindness of player 2. For player 1, s1 and
and
sT
is the truthful BNFE.
sbb
121
sL
1
are viable strategies in
Φ.
Using the payos from the proof of Proposition 2, the condition for not wanting to deviate to
sL
1
(holding xed
sT2 )
can be rearranged to the (strictly positive) upper bound on kindness
max
λ121 (sb12 , sbb
121 ) ≤ λ121 = −
The condition for not wanting to deviate to
sH
1
yields the (strictly positive) lower bound
min
λ121 (sb12 , sbb
121 ) ≥ λ121 = −
where
1 θ11 − c
.
y1 θ20 − c
1 θ10 − c
,
y1 θ20 − c
max
λmin
121 < λ121 .
Bounds for the kindness of player 1. As for player
2, the condition for not wanting to deviate
H
to s2 yields the (strictly positive) upper bound
max
λ212 (sb21 , sbb
212 ) ≤ λ212 = −
The condition for not wanting to deviate to
sL
2
1 θ20 − c
.
y2 θ10 − c
yields the (strictly negative) lower bound
min
λ212 (sb21 , sbb
212 ) ≥ λ212 = −
1 θ21 − c
.
y2 θ10 − c
Utility bounds. Given the previous results it follows that
max
Πi (sT ) + yi λmax
121 λ212
T
bound on player i's utility in any pair (Φ, s ) that implements
Step 2. Consider the extended mechanism
T
the hypothetical BNFE s we have
Choosing
δ1
and
δ2
Φ0 .
is an upper
f ∗.
Recall from the proof of Proposition 3 that in
λ121 (sb12 , sbb
121 )
= 18 (δ1 + c − θ10 )
as given in the proposition then implies that
and
1
λ212 (sb21 , sbb
212 ) = 4 δ2 .
max
λ121 (sb12 , sbb
121 ) = λ121
and
max
T
λ212 (sb21 , sbb
212 ) = λ212 , i.e. the upper bounds are reached. It remains to be shown that s is a
0
BNFE of Φ under these parameters. This is proven exactly as in the proof of Proposition 3,
except for the last two steps:
58
1
Best response of player 2. If player
Π2 (sT1 , s2 )
where
+
T
y2 λmax
212 Π1 (s1 , s2 )
(c − θ20 )/(θ10 − c) − 1 > 0
=
sT1 ,
chooses
Π2 (sT1 , s2 )
+
player
2
chooses
Π1 (sT1 , s2 )
+
s2
in order to maximize
c − θ20
− 1 Π1 (sT1 , s2 ),
θ10 − c
under our assumptions. By construction of
T
λmax
212 , s2
sH
2
and
yield the same value of this expression. From the payos derived in the proof of Proposition
3 it follows that any other strategy
Π2 (sT1 , s2 )
+
Π1 (sT1 , s2 ) and
s2 ∈ S20 \ {sT2 , sH
2 }
Π1 (sT1 , s2 ) whenever
2
Best response of player 1. If player
Π1 (s1 , sT2 )
where
+
T
y1 λmax
121 Π2 (s1 , s2 )
(θ11 − c)/(c − θ20 ) − 1 > 0
=
yields a weakly lower value of both
0 < 2 ≤ δ1 ,
T
chooses s2 , player
Π1 (s1 , sT2 )
+
so that
1
chooses
Π2 (s1 , sT2 )
+
sT2
is a best response.
s1
in order to maximize
θ11 − c
− 1 Π2 (s1 , sT2 ),
c − θ20
T
λmax
121 , s1
under our assumptions. By construction of
and
sL
1
yield the same value of this expression. From the payos derived in the proof of Proposition
3 it follows that any other strategy
Π1 (s1 , sT2 )
+
Π2 (s1 , sT2 ) and
s1 ∈ S10 \ {sT1 , sL
1}
Π2 (s1 , sT2 ) whenever
yields a weakly lower value of both
0 < 1 ≤ δ2 ,
so that
sT1
is a best response.
A.6 Proof of Theorem 2
We rst prove the if part: a utility-ecient mechanism exists when
f
exhibits bilateral exter-
nalities. To do so, we rst establish that kindness values must be bounded in that case. We then
show how these bounds can be reached. In the second step, we prove the only if part: without
bilateral externalities, kindness terms can be increased arbitrarily in equilibrium. Throughout,
we use notation and concepts from the proof of Theorem 1.
Step 1. Suppose
direct mechanism
S i , Ŝi
and
Si
where
j 6= i.
f
Φ
is materially ecient and exhibits bilateral externalities, and consider the
for
f,
with strategy sets
as follows. Let
Si
Si ,
for
i = 1, 2.
be the set of strategies
Bilateral externalities imply that
strategy of always announcing the type
θi
Si
si
that satisfy
Si
into three subsets
Πj (si , sTj ) > Πj (sTi , sTj ),
is nonempty. It contains, for instance, the
that maximizes
S i be the set of strategies si that satisfy Πj (si , sTj )
We partition
<
Eθj [vj (qjf (θj , θi ), θj ) + tfj (θj , θi )].
Πj (sTi , sTj ). Again,
Si
Let
is nonempty due to
T
T T
bilateral externalities. Finally, Ŝi is the set of strategies for which Πj (si , sj ) = Πj (si , sj ), so
T
that si ∈ Ŝi and Ŝi is also nonempty.
T
T
T
As argued in the proof of Theorem 1, si maximizes Πi (si , sj ) + Πj (si , sj ) among all si ∈ Si ,
T
T T
T
by material eciency of f . Hence we have Πi (si , sj ) < Πi (si , sj ) for all si ∈ S i and Πi (si , sj ) ≤
Πi (sTi , sTj ) for all si ∈ Ŝi . Furthermore, denoting by λiji (sT ) the kindness terms associated to the
T
T
T
T
truth-telling prole s in Φ, si is in fact a best response to sj for player i when λiji (s ) = 1/yi .
T T
T T
T
T
Consider any si ∈ S i . The condition Πi (si , sj )+yi λiji Πj (si , sj ) ≥ Πi (si , sj )+yi λiji Πj (si , sj ),
i.e., that a deviation from truth-telling to
λiji
si
is not attractive, can be rearranged to
1 Πi (sTi , sTj ) − Πi (si , sTj )
≤
,
yi Πj (si , sTj ) − Πj (sTi , sTj )
where the right hand side of the inequality is strictly positive by denition of
59
Si.
This denes
an upper bound
1 Πi (sTi , sTj ) − Πi (si , sTj )
yi Πj (si , sTj ) − Πj (sTi , sTj )
λmax
iji = min
si ∈S i
!
,
S i are unattracmax
λiji . From the previous arguments we must have 1/yi ≤ λmax
iji .
so that, in the direct mechanism, all deviations from truth-telling to strategies in
tive if and only if
λiji
(sT )
≤
The analogous argument for
Si
yields the lower bound
1 Πi (si , sTj ) − Πi (sTi , sTj )
yi Πj (sTi , sTj ) − Πj (si , sTj )
λmin
iji = max
si ∈S i
!
,
S i are unattractive if and only if
min
1/yi , but λiji can be positive or negative. DeviaT
can never be attractive. Altogether, truth-telling s
so that all deviations from truth-telling to strategies in
min
We must have λiji
≤
tions from truth-telling to strategies in
Ŝi
λiji
(sT )
≥
λmin
iji .
is a BNFE in the direct mechanism if and only if
T
max
λmin
iji ≤ λiji (s ) ≤ λiji
for
i = 1, 2
and
j 6= i.
Φ̃ with strategy sets S̃i , for i = 1, 2, satises Si ⊆ S̃i ,
max
λiji are also bounds on the kindness values of any truth-telling BNFE in
Since any pseudo-direct mechanism
min
the bounds λiji and
S i or S i are available in Φ̃ as well. By the
min
max are bounds for every mechanismpseudo-revelation principle, this implies that λiji and λiji
min
min max
max
equilibrium pair that implements f . Thus max{λ121 · λ212 , λ121 · λ212 } is an upper bound on
any pseudo-direct mechanism, because deviations to
the product of kindness terms for every mechanism-equilibrium pair that implements
max
Case 1a. Suppose λ121
·
which truthfully implements
λmax
212
f
f.
≥
· λmin
212 . We will construct a pseudo-direct mechanism
max are reached for both players, so
and in which the bounds λiji
that the mechanism implements
f
λmin
121
utility-eciently.
Using the construction given in the proof of Theorem 1, we can add messages to the direct
i
λjij (sT )
mechanism
Φ
for each player
shown that
sT
is a BNFE in the resulting pseudo-direct mechanism
to adjust
to the desired level
Φ̃.
λmax
jij .
It remains to be
It follows from the above
derivation of the kindness bounds that deviations from truth-telling to strategies
are not attractive. Hence we only need to show that no player
from
i
si ∈ Si ⊆ S̃i
wants to deviate to a strategy
S̃i \Si .
If no messages have been added for player
i in the construction of Φ̃, that is, if λjij (sT ) = λmax
jij
S̃i \Si is empty.
ei T
to decrease πj (s ), that is, if
already in the direct mechanism, this holds trivially because
If messages have been added for player
i
in the direct mechanism, then for any strategy
s0i
∈ Si
there exists an associated strategy
such that
Πi (s̃i , sTj ) = Πi (s0i , sTj ) + x
where
s̃i ∈ S̃i \Si
λjij (sT ) < λmax
jij
x ∈]0, 1]
and
is the probability of messages from
max
by the requirement to achieve λjij , and
Πj (s̃i , sTj ) = Πj (s0i , sTj ) − xδ,
{d1 , d2 , ..., dr }
is arbitrary with
under
0 < ≤ δ.
s̃i , δ > 0
is determined
Hence we have
max
T
0 T
max
0 T
Πi (s̃i , sTj ) + yi λmax
iji Πj (s̃i , sj ) = Πi (si , sj ) + yi λiji Πj (si , sj ) − x yi λiji δ − .
The last term in squared brackets is weakly positive since
60
yi λmax
≥ 1
iji
and
δ ≥ ,
so that all
strategies from
S̃i \Si are weakly less attractive than the associated strategies from Si .
from truth-telling to
S̃i \Si
Deviations
are therefore also not attractive.
If messages have been added for player
the direct mechanism, then for any
i
to increase
s̃i ∈ S̃i \Si
πjei (sT ),
that is, if
there exists an associated
λjij (sT ) > λmax
jij
s0i ∈ Si
in
such that
T
0 T
max
0 T
max
Πi (s̃i , sTj ) + yi λmax
iji Πj (s̃i , sj ) = Πi (si , sj ) + yi λiji Πj (si , sj ) − x δ − yi λiji ,
where
x ∈]0, 1], > 0
λmax
jij ,
and
δ
is arbitrary as
s̃i ∈ S̃i \Si we can choose δ large enough to make s̃i less attractive
0
than the associated si , so that, by niteness of S̃i , for large enough values of δ no deviation from
long as
δ ≥ .
is determined by the requirement to achieve
Hence for any
S̃i \Si is attractive.
T
Therefore, s is a BNFE in the pseudo-direct mechanism
truth-telling to
Φ̃,
which reaches the upper bound
λmax
121
· λmax
212 for psychological payos and thus implements f utility-eciently.
max
max
min
min
min
min
Case 1b. Suppose λ121 · λ212 < λ121 · λ212 , which requires λ121 < 0 and λ212 < 0. Suppose
T
min
further that, in the direct mechanism, λiji (s ) ≥ λiji holds for both players. Then we can use
ei T
the standard construction of Φ̃ to increase πj (s ) for both players and achieve the (negative)
lower bounds on kindness. Deviations to strategies
i = 1, 2,
λmin
iji .
by denition of
For any
si ∈ Si ⊆ S̃i
s̃i ∈ S̃i \Si ,
are again not attractive for any
there exists an associated strategy
s0i ∈ Si
such that
T
0 T
min
0 T
min
Πi (s̃i , sTj ) + yi λmin
iji Πj (s̃i , sj ) = Πi (si , sj ) + yi λiji Πj (si , sj ) − x δ − yi λiji ,
where
x ∈]0, 1]
and
0 < ≤ δ.
so that strategies from
S̃i \Si
The last term in squared brackets is positive, because
are less attractive than those from
Si ,
λmin
iji < 0,
which implies that
sT
is a
min
min
BNFE which reaches the upper bound λ121 · λ212 for psychological payos and thus implements
f
utility-eciently.
Case 1c.
The remaining case is characterized by
max
min
min
λmax
121 · λ212 < λ121 · λ212
and, for at
(sT )
least one player i, λjij
< λmin
jij holds in the direct mechanism. We claim that, in this case,
T
T
whenever s is a BNFE in any pseudo-direct mechanism for f , we must have λiji (s ) > 0 in this
max
max
min
min
equilibrium. The claim implies that λ121 · λ212 , and not λ121 · λ212 , is in fact an upper bound
on psychological payos in this case, which can then be reached as shown for case 1a above.
To establish the claim, observe again that
λmin
212 < 0.
λmin
iji < 0,
max
min
min
λmax
121 · λ212 < λ121 · λ212
requires
λmin
121 < 0
and
Πi (si , sTj ) < Πi (sTi , sTj )
for all si ∈ S i , so that all
T
strategies from S i yield Pareto inecient outcomes conditional on sj . This implies that, in the
If
direct mechanism
then we must have
Φ,
Πj (sTi , sTj ) =
because
sTi ∈ Ei (sTj )
min
si ∈Ei (sT
j )
Πj (si , sTj ),
clearly holds due to material eciency of
pseudo-direct mechanism
Φ̃,
f.
For
sT
we must achieve a kindness level of at least
to be a BNFE in a
λmin
jij ,
which requires
ei T
the equitable payo πj (s ) to be strictly smaller in Φ̃ than in the direct mechanism Φ. We
T
T
cannot have maxs ∈Ẽ (sT ) Πj (si , sj ) < maxs ∈E (sT ) Πj (si , sj ), because adding messages cannot
i
i j
i
i j
61
decrease the maximal payo for player
min
si ∈Ẽi (sT
j )
j
(see the proof of Lemma 1). Hence we must have
Πj (si , sTj ) <
i.e., there must exist a strategy
min
si ∈Ei (sT
j )
s̃i ∈ Ẽi (sTj )
Πj (si , sTj ) = Πj (sTi , sTj ),
such that
Πj (s̃i , sTj ) < Πj (sTi , sTj ),
and, by bilateral
T
T T
T
Pareto eciency, Πi (s̃i , sj ) > Πi (si , sj ). But a deviation from si to this strategy s̃i is clearly
T
protable whenever λiji (s ) ≤ 0, which establishes the claim and completes the proof of the if statement.
Step 2. Suppose
f
is materially ecient and does not exhibit bilateral externalities, so there
exists at least one player, say player
of
θ2 .
It follows that
Π1 (sT1 , s2 )
1, such that Eθ1 [v1 (q1f (θ1 , θ2 ), θ1 )+tf1 (θ1 , θ2 )] is independent
is independent of
s2 ∈ S2
in the direct mechanism
Proposition 11 for a more general statement of this fact). Material eciency of
T
that Π2 (s1 , s2 )
≤
Π2 (sT1 , sT2 ) for all
(see
then implies
s 2 ∈ S2 .
We now construct a pseudo-direct mechanism
Φ̃,
again as described in the proof of Theorem
(sT )
= 1/y1 is achieved, by adding messages for player 2. Eciency of f then
T
T
implies that s1 is a best response to s2 , irrespective of how unused messages are designed for
e1 T
player 1. Specically, we can now add unused messages to decrease π2 (s ) and hence increase
λ212 (sT ) arbitrarily, letting λ121 (sT ) · λ212 (sT ) = 1/y1 · λ212 (sT ) grow without bounds. We only
T
need to show that this is possible in a way such that s2 remains a best response for player 2.
T
T T
T
T T
Since any s2 ∈ S2 satises Π1 (s1 , s2 ) = Π1 (s1 , s2 ) and Π2 (s1 , s2 ) ≤ Π2 (s1 , s2 ), deviations
T
T
from s2 to any s2 ∈ S2 are never protable for player 2, irrespective of the size of λ212 (s ) > 0.
1, in which
λ121
f
Φ
Now consider strategies
to achieve
λ121
(sT )
s̃2 ∈ S̃2 \S2 , i.e., strategies that use messages which have been introduced
= 1/y1 .
The case where
has been decreased, strategies
s̃2 ∈ S̃2 \S2
S̃2 \S2 = ∅ is trivial.
If the equitable payo
are unprotable whenever
λ212 (sT ) ≥ 1/y2 ,
π1e2 (sT1 )
with the
e2 T
same argument as for case 1a above. When the equitable payo π1 (s1 ) has been increased, for
T
every value of λ212 (s ) we can choose δ for player 2 large enough to again make all deviations to
s̃2 ∈ S̃2 \S2
unprotable, as shown for case 1a above. Hence, letting
T
with λ212 (s ), we can ensure that
δ
grow to innity together
sT remains a BNFE.
A.7 Characterization of PRE and Proof of the Revelation Principle
The following lemma will prove helpful both for the characterization of psychologically robust
equilibria and for the proof of the revelation principle. It states that, in a BNE, kindness between
two players cannot be positive.
Lemma 5.
Let
s∗
30
be a BNE. Then it holds that
Proof. Consider any
i∈I
and
j 6= i.
30
Πi (si , (s∗k )k6=i ).
min
si ∈Eij ((s∗k )k6=i )
Πj (s∗i , (s∗k )k6=i ) ≤ πjei ((s∗k )k6=i )
∗
of BNE, si maximizes
for all
i, j ∈ I , j 6= i.
We claim that
Πj (s∗i , (s∗k )k6=i ) ≤
which implies
κij (s∗i , (s∗k )k6=i ) ≤ 0
Πj (si , (s∗k )k6=i ),
and thus
κij (s∗i , (s∗k )k6=i ) ≤ 0.
∗
Specically, Πi (s̃i , (sk )k6=i )
≤
By denition
Πi (s∗i , (s∗k )k6=i ) for all
See Netzer and Schmutzler (2010) for a similar result, in the context of dynamic games between one materi-
alistic and one reciprocal player.
62
s̃i ∈ Eij ((s∗k )k6=i ).
s̃i ∈
Bilateral eciency then implies
Πj (s∗i , (s∗k )k6=i ) ≤ Πj (s̃i , (s∗k )k6=i )
for all
Eij ((s∗k )k6=i ), which proves the claim.
In any BNE
s∗ ,
and hence in any PRE, every player is maximizing the own material payo.
This behavior will not be considered strictly kind by any opponent, as, with a conditional
and bilateral concept of eciency, positive kindness requires giving up payos for someone else's
benet. These observations provide the basis for the following characterization of psychologically
robust equilibria, which generalizes a result from Rabin (1993).
Proposition 15.
A BNE
s∗
is a PRE if and only if, for all
i, j ∈ I , j 6= i,
s∗i ∈ arg max λiji (s∗j , (s∗k )k6=j )Πj (si , (s∗k )k6=i ).
(22)
si ∈Si
Proof. Step 1. Suppose that
s∗
is a BNE and suppose that condition (22) holds. We seek to
∗
show that this implies that s is a PRE, i.e. that for every player i, the strategy
Πi (si , (s∗k )k6=i ) +
X
s∗i
is maximizing
yij λiji (s∗j , (s∗k )k6=j )Πj (si , (s∗k )k6=i ),
(23)
j6=i
independently of the size of the parameters
Πi (si , (s∗k )k6=j ). Condition (22) implies that
∗
second term of (23). Hence s is a PRE.
Step 2.
We now show that if
Suppose that
i 6= j .
s∗
s∗
(yij )j6=i .
Since
s∗
is a BNE,
s∗i
is a maximizer of
s∗i is also a maximizer of every summand in the
is a PRE (and hence a BNE), then condition (22) holds.
is a PRE. Lemma 5 then implies that
Condition (22) trivially holds for any pair
λiji (s∗j , (s∗k )k6=j ) ≤ 0 holds, for all i, j ∈ I ,
i, j
such that
λiji (s∗j , (s∗k )k6=j ) = 0.
Hence
∗ /
∗
∗
assume λiji (sj , (sk )k6=j ) < 0 but si ∈
Πi (s̃i , (s∗k )k6=i ) ≤ Πi (s∗i , (s∗k )k6=i ) and
where
yij > 0 and yik = 0 for all k 6=
yij >
arg minsi ∈si Πj (si , (s∗k )k6=i ). Then, there exists s̃i so that
κij (s̃i , (s∗k )k6=i ) < κij (s∗i , (s∗k )k6=i ). Consider a prole y
i, j . Player i has an incentive to deviate from s∗i to s̃i when
Πi (s∗i , (s∗k )k6=i ) − Πi (s̃i , (s∗k )k6=i )
,
λiji (s∗j , (s∗k )k6=j ) κij (s̃i , (s∗k )k6=i ) − κij (s∗i , (s∗k )k6=i )
which contradicts the assumption that
Proposition 15 says that a BNE
maximizes the expression
coecient
s∗
s∗
is a PRE.
is a PRE if and only if every player
λiji (s∗j , (s∗k )k6=j )Πj (si , (s∗k )k6=i )
for any opponent
λiji (s∗j , (s∗k )k6=j ) is a constant which does not depend on i's choice.
j.
i's
strategy
s∗i
Observe that the
By Lemma 5, it can
si ∈ Si
∗
∗
∗
∗
is a maximizer of λiji (sj , (sk )k6=j )Πj (si , (sk )k6=i ). In the latter case, si is a maximizer if and
∗
only it is a minimizer of Πj (si , (sk )k6=i ). Hence if, in any bilateral relation, a player experiences
∗
strictly negative kindness in a BNE s , then robustness requires that he minimizes the other's
either take a value of zero or a negative one. In the former case, trivially, any strategy
payo. In the context of two player normal form games, Rabin (1993) calls a strategy prole
mutual-min (p. 1290) when this is satised for both players. He proves that a mutual-min
Nash equilibrium is always a fairness equilibrium, and hence robust in our sense. Proposition
15 implies that the analogous result is true in our BNFE setting.
Furthermore, Proposition
15 applies to an arbitrary number of players, and it provides a condition that is necessary and
63
sucient for robustness.
31
Armed with Lemma 5 and Proposition 15 we can now prove the
revelation principle for the solution concept of a PRE.
Proof Proposition 10.
f.
First, we state the pseudo-revelation principle for PRE.
Φ = (M1 , ..., Mn , g)
Consider a mechanism
function
Step 1.
with a PRE
s∗
that implements a social choice
From Proposition 4 it follows that there exists a strategically equivalent pseudo-
direct mechanism
Φ0 = (M10 , ..., Mn0 , g 0 )
f
that truthfully implements
in PRE, i.e. in which
sT
is
0
∗
a PRE. Moreover, we can write Mi = Θi ∪ Mi− where Mi− = Mi \si (Θi ) are the unused actions
0
0
0
T
T
from Φ. The strategy sets in Φ are denoted Si . Lemma 5 implies that κij (si , (sk )k6=i ) ≤ 0
holds in the PRE
sT
Φ0 ,
of
for all
Step 2. The proof is completed if
i, j ∈ I , i 6= j .
Mi− = ∅ for all i ∈ I , so that Φ0
is a direct mechanism. Hence
0
0
00
00
00 00
assume Mi− 6= ∅ for some i, and let mi ∈ Mi− ⊂ Mi . Construct Φ = (M1 , ..., Mn , g ) from
Φ0 by letting Mi00 = Mi0 \{m0i } and keeping Mj00 = Mj0 for all j 6= i. Let g 00 be the restriction of
g 0 to M100 × ... × Mn00 . The strategy sets in Φ00 are Si00 ⊂ Si0 and Sj00 = Sj0 for all j 6= i. We have
T
00
only removed an unused action of player i, so s is still a BNE of Φ . Lemma 5 thus implies
that
κ00ij (sTi , (sTk )k6=i ) ≤ 0
still holds for all
j 6= i.
The kindness of all other players is completely
unaected by the removal.
Step 3. To prove that
For player
is still a PRE of
Φ00 , we need to show that condition (22) is still satised.
i this is immediate, since sTi ∈ Si00 ⊂ Si0
for all opponents
player
sT
j 6= i,
j 6= i,
for whom
and
λ00iji (sTi , (sTk )k6=i ) = λ0iji (sTi , (sTk )k6=i ) holds
i.e. the removal has left condition (22) unaected. Then consider any
Sj00 = Sj0 .
0
being violated in Φ , is that
The only way in which (22) could be violated in
λ0jij (sTi , (sTk )k6=i )
=0
0
held in Φ but
Φ00 ,
λ00jij (sTi , (sTk )k6=i )
while not
<0
holds in
Φ00 . We will show that this is impossible.
From the proof of Lemma 5 we know that
where
Thus
0 ((sT )
Eij
k k6=i )
Πj (sTi , (sTk )k6=i ) ≤ minsi ∈E 0
are the bilaterally ecient strategies in
λ0jij (sTi , (sTk )k6=i ) = 0
min
0 ((sT )
si ∈Eij
k k6=i )
Φ0 ,
T
ij ((sk )k6=i )
sT
due to
Πj (si , (sTk )k6=i ),
being a BNE in
Φ0 .
requires
Πj (si , (sTk )k6=i ) = Πj (sTi , (sTk )k6=i ) =
But the same two equalities must then hold in
the maximization, we can always replace
Φ00 ,
max
0 ((sT )
si ∈Eij
k k6=i )
implying
0 ((sT )
Eij
k k6=i )
by
Si0
λ00jij (sTi , (sTk )k6=i ) = 0
and
00 ((sT )
Eij
k k6=i )
T
T
changing the result. Thus since si maximizes Πj (si , (sk )k6=i ) on
00
have established the second equality for Φ . The rst equality for
the proof of Lemma 5, together with the fact that
sT
Πj (si , (sTk )k6=i ).
is a BNE in
by
as well. For
Si00 ,
without
Si0 , and sTi ∈ Si00 ⊂ Si0 , we
Φ00 then follows again from
Φ00 .
Hence
sT
still satises
00
condition (22) in Φ , and thus is a PRE.
Step 4.
Iterating steps 2 and 3, we can remove all unused actions until arriving at a direct
mechanism in which
sT
is a PRE.
31
Rabin (1993) also proves a robustness result for two player mutual-max Nash equilibria, where, phrased
s∗i ∈ arg maxsi ∈Si Πj (si , (s∗k )k6=i ) for both players. This result also follows from our
∗
∗
proposition, because the mutual-max property implies that λiji (sj , (sk )k6=j ) = 0 holds for both players.
in terms of our notation,
64
A.8 An Asymmetric Expected Externality Mechanism
The following example illustrates how lack of symmetry leads to a violation of the insurance
property and to non-robustness of the expected externality mechanism.
Example 3.
Consider the problem of sharing one unit of a private good among three players
I = {1, 2, 3}.
Each player's type is from the set
independent between players. Let
possibilities, so that
let
T = T̄
qi
Θi = {0, 1}.
[0, 1]3 |q
Q = {(q1 , q2 , q3 ) ∈
Both types are equally likely and
1 + q2 + q3
= 1}
be the set of sharing
denotes the share of the private good that is allocated to player
be the set of admissible transfers.
Preferences are given by
i,
and
vi (qi , θi ) = θi qi .
We
f f f
consider the expected externality mechanism for the ecient decision rule (q1 , q2 , q3 ) detailed
in Table 2, where each row corresponds to one possible type prole θ = (θ1 , θ2 , θ3 ) and contains
f f f
the associated shares. The transfers (t1 , t2 , t3 ) in Table 2 are those of the expected externality
mechanism.
θ1
θ2
θ3
q1f
q2f
q3f
tf1
tf2
tf3
0
0
0
1
0
0
0
0
0
0
0
1
0
0
1
1/16
1/16
-1/8
0
1
0
0
1
0
1/16
-1/8
1/16
0
1
1
0
1/2
1/2
1/8
-1/16
-1/16
1
0
0
1
0
0
-3/4
3/8
3/8
1
0
1
1
0
0
-11/16
7/16
1/4
1
1
0
1
0
0
-11/16
1/4
7/16
1
1
1
1
0
0
-5/8
5/16
5/16
Table 2: An asymmetric expected externality mechanism
The environment of Example 3 is symmetric, but the decision rule is not: it allocates the
good entirely to player
1
whenever
θ1 = 1 ,
symmetrically allocated between players
θi = 0
or when
2 and 3.
for all
i ∈ I.
Otherwise, the good is
While not being symmetric, it is still ecient:
it allocates a positive share of the private good only to those players with a maximal valuation.
We now obtain the following non-robustness result.
Proposition 16.
Consider Example 3. The SCF
f
in Table 2 violates the insurance property.
T
The truthful strategy prole s is not a PRE of the expected externality mechanism.
Proof. We can derive the players' payos in the expected externality mechanism, i.e., the direct
mechanism for
and
s−T
i .
f,
both for the truth-telling prole
sT
and for the unilateral deviations
L
sH
i , si
The derivations are tedious but straightforward, and the results are given in Table 3.
Proposition 11 now implies that
f
violates the insurance property, because player
can aect his opponents' payos by unilateral deviations.
Based on Table 3, we also obtain the bilateral eciency sets
T T
T L
T T
T H
E12 (sT2 , sT3 ) = {sT1 , sL
1 }, E13 (s2 , s3 ) = {s1 , s1 }, E21 (s1 , s3 ) = {s2 , s2 },
T T
T H
T T
T L
E23 (sT1 , sT3 ) = {sT2 , sL
2 }, E31 (s1 , s2 ) = {s3 , s3 }, E32 (s1 , s2 ) = {s3 , s3 } ,
65
2, for instance,
s1
sT1
sL
1
sH
1
s−T
1
sT1
sT1
sT1
sT1
sT1
sT1
s2
sT2
sT2
sT2
sT2
sL
2
sH
2
s−T
2
sT2
sT2
sT2
s3
sT3
sT3
sT3
sT3
sT3
sT3
sT3
sL
3
sH
3
s−T
3
Π1
Π2
Π3
3/16
11/32
11/32
3/16
11/32
11/32
-3/16
11/32
11/32
-3/16
11/32
11/32
5/32
7/32
3/8
7/32
9/32
5/16
3/16
5/32
11/32
5/32
3/8
7/32
7/32
5/16
9/32
3/16
11/32
5/32
Table 3: Expected payos
and the equitable payos
π2e1 (sT2 , sT3 ) = 11/32, π3e1 (sT2 , sT3 ) = 11/32, π1e2 (sT1 , sT3 ) = 13/64,
π3e2 (sT1 , sT3 ) = 23/64, π1e3 (sT1 , sT2 ) = 13/64, π2e3 (sT1 , sT2 ) = 23/64.
Based on these results, we obtain the equilibrium kindness values
κ12 (sT ) = κ13 (sT ) = 0, κ21 (sT ) = κ23 (sT ) = κ31 (sT ) = κ32 (sT ) = −1/64.
It is now immediate to see that
we have
λ323
(sT )
reduces player
2's
= κ23
(sT )
sT
< 0
payo. Hence
violates the condition given in Proposition 15. For instance,
but
sT
sT3 ∈
/ arg mins3 ∈S3 Π2 (s3 , (sTk )k=1,2 ),
is not a PRE.
66
because
sH
3
further