The Emergence of Trust∗
Matthias Blonski†
Daniel Probst‡
this draft: 23:5:2004 (first draft Summer 99)
Abstract
We analyze the phenomenon of gradual increase of stakes in bilateral social
interactions in the context of a repeated game with two-sided incomplete information. The underlying stage-game resembles properties of a ”Continuous Prisoner’s
Dilemma” with a continuous action space plus the possibility of stealing the opponent’s investment. The continuous investment variable corresponds to cooperation
and is interpreted as individual level of trust while the defective action is interpreted as break of trust which terminates the partnership. Players can be of two
different types/discount factors.
We analyze the structure of efficient equilibria and characterize the Pareto frontier of types’ payoffs. Efficient equilibria are shown to display a typical pattern
consisting of three phases. First, in the Communication Phase information about
the respective opponent’s type is transmitted, after which in the Adjustment Phase
stakes grow over time until in the Established Phase partners behave stationary as
under complete information. The efficient adjustment of stakes is governed by the
impatience of the low type.
Furthermore, we demonstrate that in some equilibria all players do strictly better than in a corresponding game with complete information.
JEL Classification: C72, D82, D83
Keywords: trust, gradualism, repeated games, incomplete information
∗
We are grateful for insightful comments and advice from Christian Ewerhart, David Levine, Larry
Samuelson, Joel Sobel and Joel Watson and two anonymous referees. The comments of seminar participants at the Universities of Berkeley, Bonn, Mannheim, San Diego, Stanford, UCLA and Wien as well as
participants of the 2000 World Congress of the Game Theory Society, ESEM 2001 and the 2001 Meeting
of the Verein für Socialpolitik are appreciated.
†
Department of Economics, Goethe University, Frankfurt am Main, 60054 Frankfurt a.M., Germany;
telephone: +49-(0)69-798-23858; e-Mail: [email protected]
‡
BASF, Germany e-Mail: [email protected]
1
. . . What makes many agreements enforceable is only the recognition of
future opportunities for agreement that will be eliminated if mutual trust is
not created and maintained, and whose value outweighs the momentary gain
from cheating in the present instance.
. . . Neither (party) may be willing to trust the other’s prudence (or the
other’s confidence in the first’s prudence, and so forth) on a large issue. But,
if a number of preparatory bargains can be struck on a small scale, each may
be willing to risk a small investment to create a tradition of trust.
(Thomas Schelling, 1980, The Strategy of Conflict, p.45)
1
Introduction
How should business parties build up trust? There is a common and widely accepted
perception that the mutual formation of trust is a key for successful business relationships
and that it is a complex and genuinely dynamic process. Thomas Schelling suggests a
number of preparatory bargains on a small scale to create a tradition of trust. In this
article we analyze the structure of this dynamic formation process. More precisely, we
study how many bargains on which scale should be struck under various conditions in
order to successfully build up a relationship being able to perform on large issues.
A salient part of any explanation how to build up trust is mutual uncertainty about
the other partie’s true motives and characteristics. Our intuition tells us – as Schelling
implicitly does – that foregone immediate larger business opportunities are the price to be
paid in order to collect information and learn about prospective business partners. Hence,
we are interested in more details of this information transmission process. How should we
communicate how trustworthy we are? Clearly, networks and modern legal institutions
offer various possibilities to alleviate or circumvent the trust formation problem. However,
we are convinced that a deeper understanding of a basic structure is paramount for the
analysis of more realistic social settings. Therefore, we concentrate in this article on the
pure bilateral problem of building up a relationship without any help by third parties or
institutions.
We analyze these questions in a repeated game framework with two-sided incomplete
information. Suppose that in each period a party can either cooperate by investing
a variable amount into a bilateral project or defect by stealing all tangible assets of his
partner. If both parties cooperate, the joint project generates a profit. On the other hand,
if a party decides to defect, no return is generated. The defector steals the current period
investment of his partner, and the relationship is terminated (simultaneous defection
results in no transfer). We quantify trust in a relationship with the stakes that can get
lost in case of betray.
In this article we analyze the linear payoff case which dissolves into two generic linear
specifications for the cooperation profit function1 . According to our motivation joint
profits should be maximized if both parties invest maximally and any partie’s individual
payoff should increase with the partner’s investment. Within a linear framework this
leaves us with one further degree of freedom. Should individual payoffs decrease or
increase with a partie’s own investment? The first case is known as linear public goods
1
This paper reveals that the linear case is already ambitious. Although we believe that many of our
results hold more generally non-linear specifications turn out to be much more challenging and are left
to future research
2
game and we call the latter case linear or continuous trust game. Since we assess both
cases as most relevant we study both of them. Both stage game structures share the
common feature with the Prisoner’s Dilemma that the unique stage game equilibrium is
non-cooperation. The continuous trust game, however, needs further motivation since it
is less known and since it contains an additional and interesting twist that turns out to
be very important for the structure of the results in the repeated game formulation. In
contrast to the Prisoner’s Dilemma and the linear public goods game non-cooperation
or quit Q is not a dominant strategy in the linear contribution game because the best
response to a low cooperative investment of your partner may be to cooperate on the
maximal investment level. In this case you can steal only very little but given your
partner cooperates you can raise your cooperation profits by investing more yourself2 .
We believe this interdependence in which high investments can lead to high returns,
but also result in high losses when a partner misbehaves, to be a central feature of many
real-world economic and social situations: joint ventures, trading relationships, employeremployee relationships, social partnerships (co-authorships, marriage, friendships) and so
forth.
Our central and unifying result is that not only do there exist equilibria with gradually increasing investment levels. Most importantly, such equilibria whenever they exist
they are uniquely efficient and display a characteristic pattern. First, parties communicate their type, we call this Communication Phase. Then, both parties adjust their
investments up to a final stable level. We call this Adjustment or Enhancement Phase.
Finally, partners behave stationary just as they should behave efficiently in the absence
of uncertainty, which we label Established Phase.
The phenomenon of gradually increasing stakes has entered the recent economic literature under labels such as “gradualism” or “starting small”. Most closely related to our
approach and results are the articles of Watson (1996; 1999; 2002) as well as Ghosh and
Ray (1996). Postponing a more detailed discussion of the related literature to Section 7,
we note at this point that in contrast to these articles we embed our model in the standard setting of repeated games (albeit with incomplete information regarding discount
factors). We therefore allow agents to select their possibly differing investment levels
endogenously in each period, which is not the case in other approaches.
This possibility of asymmetric investment levels leads us to some surprising additional
results regarding the information transmission process. Contrary to our first intuition it
is not generally true that incomplete information in this context necessarily worsens the
expected payoff of any type of agent. We show that for some parameters, there exist
efficient equilibria in which each possible type’s expected payoff strictly dominates his
payoff in a benchmark setting of complete information.3
We proceed with a more detailed discussion of our main results. As already mentioned,
we model the uncertainty about opponent’s motives by assuming double sided incomplete
information regarding discount factors. Specifically, we assume the discount factors to
be independently drawn by nature from a binary set 0 < δl < δh < 1 according to a
commonly known distribution (p, 1 − p). Our preferred interpretation of this setup is
as a prototypical model for general relationships in a society whereby two individuals
are randomly matched to interact while each knows only his own type. In line with
2
Probably all co-authors are from time to time most familiar with this phenomenon
In the benchmark setting assume that a type is randomly matched and observes the type of the
opponent he is matched against.
3
3
this interpretation we focus on symmetric, pure strategy4 equilibria of the corresponding
incomplete information game. We interpret equilibria as conventions of interaction in a
society. Therefore, from a normative viewpoint, our prime interest will regard efficient
equilibria.5 We call an equilibrium interim efficient if there exists no other equilibrium in
which all types do as least as well and some type attains strictly higher expected payoffs.
We also compare to the weaker notion ex-ante efficiency which we define regarding a
player’s expected payoff, i.e., before he observes his own type.
As a benchmark we first analyze the game with complete information, i.e. players
observe their opponent’s type. Under the assumption that cooperative returns and losses
are each linear in own investment and bounded, one can partition the discount parameter
space into two non-trivial Regions I and II.6 In Region I only high type matches can
potentially cooperate (low types are too impatient for cooperation to be a viable option
in an equilibrium). Region II, in contrast, allows for equilibria in which mixed high-low
matchings can cooperate too.7
Returning to the incomplete information case we observe that in the continuous trust
game all interim efficient Region I equilibria (with positive investment in some period)
are of two different types which we call Q−gradual equilibria and C−gradual equilibria.
In contrast, we show that in the linear contribution game only Q−gradual equilibria
exist. Both Q−gradual equilibria and C−gradual equilibria have a prominent feature
in common: the behavior of a high type when matched against another high type. In
this case the amount invested increases over time until it hits the maximal level where it
stays thereafter. The rate of investment growth turns out to be mainly determined by the
impatience of the low type. The low type is exactly indifferent between his equilibrium
behavior and imitating a high type in hope of cheating on a high type opponent in
some later period. Q−gradual equilibria and C−gradual equilibria, however, differ in
the manner in which the separation of types occurs. In Q−gradual equilibria a low type
quits in the first period, while in a C−gradual equilibrium low types cooperate in the
first period at the highest level and then quit in the second.
At first sight, one may be irritated by the fact that, in an efficient equilibrium, high
type matches only increase investment gradually, even though they are sure of their
opponent’s type along the equilibrium path because separation occurs in the first period.
We argue (see Remark 5) that such behavior is consistent with both renegotiation and
refinement considerations.
Q−gradual equilibria reflect the intuitive effect of incomplete information one would
expect in such a setting: High types have an incentive to discover their opponent’s type
in order to benefit from cooperation with a high type opponent. Low types are enticed
to reveal their identity by the offer of a rent/transfer in the first period, i.e., low types
4
If one were to either drop symmetry or allow mixed strategies, our results will still hold. We
conjecture that this is also true for asymmetric, mixed strategy equilibria.
5
Alternatively, one could motivate our focus on efficient equilibria as the standard procedure of
characterization of the set of feasible equilibrium payoffs of the repeated game.
6
Trivial regions are those for which both discount factors are so low that no equilibria with positive
investment exist or, alternatively, for which both discount factors are sufficiently high such that all
possible type combinations can cooperate at the highest level. Although a binary type space clearly
simplifies reality we think that this very qualitative structure of the parameter space is very intuitive
and calls for a careful investigation.
7
In the continuous trust game in such equilibria low types invest more than high types (in each
period). Low types do not steal as it is more worthwhile to obtain returns on the own large investment
than to steal the opponent’s small amount. High types do not quit because they are patient.
4
steal the first period investment of a high type if they meet one. Furthermore there is a
social loss due to the fact that high-high matches can only increase investment gradually
to deter low types from posing as high types and stealing later on. Q−gradual equilibria
can be indexed by the first period investment a of a high type. While clearly a low type’s
expected payoff is strictly increasing8 in a, the high type’s expected payoff turns out to be
concave in a. This characterizes the Pareto frontier of Q−gradual equilibria in region I.
Compared to expected (efficient) complete information payoffs, the low type does strictly
better9 , while the high type does strictly worse due to the low type’s stealing and the
slow increase in high-high matchings.
C−gradual equilibria only exist in the continuous trust game and are of a different
nature. In the first period low types invest the maximal amount possible and quit in the
second period. As before, high types start off with a small investment which gradually
grows in case they are matched with a high type opponent. When they are matched
with a low type they also quit in the second period. One may ask why low types do
not quit and steal in the first period? Low types face a tradeoff in the first period. If
they cooperate they obtain a reasonably high payoff, namely the cooperative return on
the maximal possible investment. The amount they obtain from stealing depends on
the type of their opponent. With probability p they meet a high type opponent from
whom they can steal only little, whereas with probability 1 − p they can steal the largest
amount possible from a low type opponent. We show that for a nontrivial parameter
region this tradeoff points in favor of cooperation, i.e., C−gradual equilibria exist. Given
that in C−gradual equilibria a low type obtains the return from the maximal investment
for one period, interim efficient equilibria are obtained by maximizing the high type’s
expected payoff subject to low/high type non-deviation conditions. C−gradual equilibria
are therefore generically unique. In contrast to both Q−gradual equilibria and efficient
complete information equilibria, low types obtain strictly positive payoffs in a C−gradual
equilibrium against both low and high type opponents (they therefore do strictly better
than in any complete information equilibrium). Similarly, high types also obtain positive
payoffs against low type opponents. Compared to efficient complete information equilibria, high types do strictly worse against high type opponents due to the inefficiency
of increasing investment only gradually, while they do strictly better against low type
opponents. We show that for some parameter regions the latter effect compensates the
former, i.e., there exist C−gradual equilibria which dominate all complete information
equilibria with respect to interim efficiency. This surprising result indicates that the “high
type pays low type information rent” story is not the complete picture. The possibility of
differing types investing differing amounts in one period allow for the separation of types
in a way which is beneficial to all types, which is in contrast to results in the previous,
related literature.10
While both Q− and C−gradual equilibria need not exist11 , at least one of them
8
With probability p a low type is matched with a high type from whom he can steal a.
With complete information a low type’s payoff is always 0
10
There are somewhat less related fields in which incomplete information has been shown to be beneficial. In the reputation literature (see, for example, Kreps et al. (1982)), the addition of uncertainty
can lead to previously infeasible cooperative outcomes. However, we would argue that our case is more
salient in that we do not add uncertainty regarding an artificial, exogenously chosen type to a given
economic setting. Instead we motivate a specific economic setting with given types, and then introduce
incomplete information while keeping the type space fixed.
11
This contrasts Watson (1996), where equilibria related to Q−gradual equilibria always exist because
differing types are assumed to invest the same amount, i.e., deviations to higher investment levels are
9
5
exists as soon as there is any equilibrium in which at least one type-matching invests
a positive amount in some period. As Q−gradual equilibria can be shown to dominate
C−gradual equilibria with respect to interim efficiency — and vice versa for a different
parameter region —, they together as well as the trivial “All Quit” equilibrium completely
characterize the Pareto frontier of Region I equilibria.
In Region II also low types can potentially cooperate with high types for prolonged
periods of time. Only low-low matches cannot. We observe that both Q−gradual equilibria and C−gradual equilibria still remain equilibria in Region II. Since these equilibria
do not take advantage of mixed matching cooperation they tend not to be efficient. However, there are additional equilibria which do take advantage of mixed cooperation. We
call them CII −equilibria. The structure of CII −equilibria is more involved for several
reasons. First, in contrast to Region I we cannot exclude pooling phases. Further, since
only low-low matchings leave the game after separation the following behavior is characterized by three potentially different investment paths, those of a high type meeting
another high type or a low type and that of a low type meeting a high type. These paths
are interrelated by a complex variety of non-deviation conditions. Nevertheless, we can
show that the previously discovered CAE−pattern of C ommunication Phase followed by
Adjustment and E stablished Phases still prevails as efficient mode of behavior. Since with
linear payoff functions all potentially binding non-deviation conditions are weak linear
inequalities the Pareto frontier of the payoff space in region II is simple to describe. The
dimensionality of a detailed description of efficient behavior in region II grows rapidly
in the efficient duration of the Adjustment Phase. To get a better intuition for this we
further elaborate on the specific rapid enhancement equilibrium. This equilibrium with
immediate adjustment (Adjustment Phase of length 0) is of particular interest since it is
efficient if it exists. Further, it only depends on two investment parameters, namely the
separation investments, after which players switch to the stationary efficient. Therefore
it is the simplest CII −equilibrium and can be studied by graphical analysis.
The paper is organized as follows: in Section 2 we present the formal model. Section
3 presents the benchmark case of complete information, characterizing the structure and
payoffs of efficient equilibria. Some preparatory results are presented in Section 4 before
a complete characterization of efficient Region I equilibria is given in Section 5. Section 6
characterizes Region II equilibria and provides a geometrical approach. Related literature
is discussed in Section 7. Section 8 closes with the conclusions.
2
The Model
Stage game We first define the stage game Γ := N := {1, 2}, (Ai), (vi ), where N is
the set of players, Ai := [0, 1] ∪ {Q} is player i’s action set, and payoff vi is defined as


fi (ai , aj ) if ai , aj ∈ [0, 1],




−(1 + )ai if ai ∈ [0, 1], aj = Q,
vi (ai , aj ) :=

aj if ai = Q, aj ∈ [0, 1],




0 if ai = aj = Q.
where i, j ∈ N and i = j (which will always be assumed for indices i, j). The set of action
profiles is denoted by A := A1 × A2 .
not feasible and the separation of type can only occur by someone quitting.
6
As mentioned in the introduction the structure of the stage game Γ is closely related
to the Prisoners’ Dilemma. However, we consider a continuous action space. By playing
a cooperative action ai ∈ [0, 1] – denoted by C – each player individually chooses his
level of trust measured by the potential loss if the partner defects12 . The project of the
relationship only pays off fi (ai , aj ) when both players cooperate.
Instead of investing, player i can quit – denoted by Q – and steal the other player’s
stake aj . In order to allow for a social inefficiency of stealing, we assume that the victim
j incurs losses of size (1 + )aj , which potentially exceed his investment (for > 0). We
recapitulate with the following “collapsed” normal form matrix:
C
Q
C
f1 (a1 , a2 ), f2 (a1 , a2 )
−(1 + )a1 , a1
Q
a2 , −(1 + )a2
0, 0
In this article we restrict attention to symmetric linear return functions f . While it
seems intuitively plausible that many of our results also hold for concave return functions,
such an analysis turns out to be technically extremely demanding13 . Within the linear
formulation we consider two salient normalized specifications differing by the individual
incentive to raise the level of the relationship which encompass all relevant features of
linear return functions.
Case 1: fi (ai , aj ) = −λai + aj for i = 1, 2 and λ ∈ (0, 1). This is a normalized
parametrization of the linear contribution game or linear public goods game. Here
each player has the incentive not to contribute (free ride) while maximal contribution ai = aj = 1 is efficient. Due to the normalization, contributing 0 is payoffequivalent to playing Q.14
Case 2: Let fi (ai , aj ) = (1 − λ)ai for i = 1, 2 and λ ∈ (0, 1). We call this the continuous
trust game 15 . Contrasting the linear contribution game, in this specification a
player’s return only depends on his own stake when both players cooperate. We
envisage a project which solely pays off when both players cooperate, and whereby
returns are split proportionally to the investment levels. Here, a player would always
like to invest if he were sure the opponent wouldn’t steal. An important difference
to the standard Prisoners’ Dilemma is that the action Q is not strictly dominant.
While (Q, Q) is the unique stage game equilibrium, maximal cooperation at level
1, and not Q, is a best response to low (i.e., below 1 − λ) levels of cooperation.
This latter feature turns out to have surprising and interesting implications for the
structure of information transmission under incomplete information.
12
Somewhat loosely, we use in this article ’investment’ and ’the level of the relationship or’ or what
can be stolen synonymously for variable ai .
13
Our linear specification can be seen as a piecewise linear approximation of a concave function since
we assume an upper bound on investments which amounts to the assumption that additional investment
from this upper bound does not result in additional returns.
14
In this case we will from here implicitly identify the strategies 0 and Q to avoid unsubstantial case
differentiations.
15
An earlier version of this paper analyzed this specification only.
7
Repeated game We define the repeated game Γ∞ of incomplete information as follows:
Players can be from a set of two different types T := {l, h} — interpreted as low and
high types. Types differ in their discount factor δi ∈ {δl , δh }, where 0 < δl ≤ δh < 1.
At the beginning of the game nature draws players’ types independently according to
Pr(δh ) = p, Pr(δl ) = 1 − p. Unless indicated otherwise each player’s type is private
information.
For technical convenience we assume action Q or investment ai = 0 to end the game.16
Let ω 0 := ∅ and define nonterminal histories as ω t = (a0 , a1 , . . . , at−1 ) where at ∈ A and
until then no player quits or invests 0, that is aτi = Q, 0 for all τ = 1, . . . , t − 1 and
t
i = 1, 2. Let Ωt := {ω t} and Ω := ∪∞
t=0 Ω . Terminal histories are either infinite sequences
of (stage game) outcomes in which Q was never played, or finite sequences of outcomes
in which one (or both) players choose Q only in the last period. A strategy for player i is
then given by si := Ω ∪ T → Ai . s := (s1 , s2 ) ∈ S := S1 × S2 . Type contingent strategies
will be denoted si (h) and si (l).
A tuple of strategies s induces an outcome o(s) consisting of four (one for each
type-combination) terminal histories which we will also call outcome paths. Specifically
o(s) := {ohh (s), ohl (s), olh (s), oll (s)} , where ohh (s) corresponds to the terminal history
induced by reducing the strategies to s(h) = (si (h), sj (h)); and similarly the other expressions. For a history ω define by #ω the (possibly infinite) index of the last period
of ω; therefore #ohh (s) + 1 denotes the length of the outcome path/terminal history induced when both players play according to s = (si , sj ) reduced to the behavior of a pair
of h-type. Furthermore index ohh (s) with t = 0, . . . , #ohh (s) such that othh (s) ∈ A.
The preferences of player i over outcomes are given by utility
ui (s) := p
2
#ohh (s)
δht vi (othh (s))
#ohl (s)
+ p(1 − p)
t=0
δht vi (othl (s))+
t=0
#olh (s)
+ p(1 − p)
δlt vi (otlh (s)) + p2
t=0
#oll (s)
δlt vi (otll (s))
t=0
Given that utility depends only on the outcome induced by strategies we will switch
between the notation ui (s) and ui(o) where o is the outcome induced by s.
We use Bayesian Nash equilibrium as our equilibrium concept. We call an outcome
o an equilibrium outcome if there exist strategies s∗ and beliefs β ∗ such that (s∗ , β ∗ ) is a
Bayesian Nash equilibrium with induced outcome o = o(s∗ ).
Following the interpretation given in the introduction, namely a random matching
problem in a population consisting of a proportion p of high types, we restrict attention
to symmetric equilibria of Γ∞ . We therefore define type-contingent utilities for n ∈ T as
follows:
#onh
#onl
t
t
un (o(s)) := un (s) := p
δn vn (onh ) + (1 − p)
δnt vn (otnl ).
t=0
t=0
Our prime interest regards equilibria which are undominated with respect to type contingent payoffs:
Definition 1 An equilibrium (s∗ , β ∗ ) and its outcome o(s∗ ) are called interim efficient or
(weakly) interim undominated if there exists no equilibrium outcome õ(s̃) with uh (õ) ≥
16
This is without loss of generality; all results would go through in the standard repeated game setting.
By this assumption we just get rid of various cases that do not differ in substance.
8
uh (o) and ul (õ) ≥ ul (o) where at least one of the inequalities is strict. We will use
“efficient” and “undominated” synonymously.
2
A weaker concept which we will be using is:
Definition 2 An equilibrium (s∗ , β ∗ ) and its outcome o(s∗ ) are called ex-ante efficient
or ex-ante undominated if there exists no equilibrium outcome õ(s̃) with puh (õ) + (1 −
p)ul (õ) > puh (o) + (1 − p)ul (o).
2
Ex-ante and interim efficiency reflect player’s preferences over different equilibria before and after learning their own type. Obviously, every interim inefficient equilibrium is
ex-ante inefficient. Most results in this paper will refer to the more intuitive notion of
interim efficiency.
3
Complete Information
As a benchmark for the influence of incomplete information, and to familiarize ourselves
with the basic mechanics of our model, we start with a complete information, matching
version of our game17 . Assume as before a population of players, of which proportion
p have discount factor δh (and 1 − p have δl < δh ). Players are matched randomly in
pairs, however, in this section, they observe their opponent’s type. The main goal is
to characterize efficient equilibria. We note that there are no general characterization
results concerning repeated game equilibrium correspondences as a function of discount
parameters.
A first simple observation for equilibria of the repeated game is:
Lemma 1 In any finite equilibrium outcome path both players obtain payoff 0. Any
infinite equilibrium outcome path where some player’s payoff is not 0 consists of an infinite
number of periods with strictly positive investment.
2
Proof In the converse of both cases there exists a last period T where no player plays
Q or invests 0. This cannot be an equilibrium since in this period players can always
improve either by stealing or by investing more or less.
As a consequence we can partition the λ, δl , δh parameter space into different regions,
corresponding to whether indefinite cooperation (at a positive level) is viable or not. As
we will see in the next section, this also determines the possible structure of equilibria
in the incomplete information case. In any period t in an equilibrium it must hold that
stealing is worse than continuing along the equilibrium path, i.e.,
fi (ati , atj ) + δi Vit+1 ≥ atj ,
(1a)
fj (ati , atj ) + δj Vjt+1 ≥ ati ,
(1b)
where Vit+1 and Vjt+1 denote the continuation payoffs along the corresponding outcome
paths from period t + 1 on.
A seemingly obvious but nontrivial observation is that efficient outcome paths are
stationary.
17
In contrast to the Folk Theorem literature in evolutionary game theory as for example Hirshleifer
and Martinez-Coll (1988) we need to concentrate on the λ, δl , δh −parameter space where cooperation
among the different matchings is viable.
9
Lemma 2 All efficient outcome paths are stationary.
2
Proof Along efficient equilibrium outcome paths (1a,1b) must either hold in equality
or the investment bound of 1 is reached. Writing the non-deviation conditions for period
t − 1 as:
t−1
t t
2 t+1
fi (at−1
= at−1
i , aj ) + δi fi (ai , aj ) + δi Vi
j ,
t−1
t t
2 t+1
= at−1
fj (at−1
i , aj ) + δj fj (ai , aj ) + δj Vj
i .
Substituting the V t+1 from (1a,1b) we obtain:
1 t−1
t−1
ai − fi (at−1
i , aj )
δj
1 t−1
t−1
aj − fj (at−1
atj =
i , aj )
δi
ati =
(2a)
(2b)
For the case of linear f , these equations define a linear dynamical system with fixed point
(0, 0). As we are restricting the system to the unit square we obtain either convergence
to (0,0) or into a point on the boundary of the unit square.
In a stationary equilibrium at level (ai , aj ), continuation payoffs in each period are
given by (fi (ai , aj )/(1 − δi )), fj (ai , aj )/(1 − δj )). Inserting such continuation payoffs into
(1a,1b) we obtain the following necessary equilibrium conditions:
fi (ai , aj )
≥ aj ,
1 − δi
fj (ai , aj )
≥ ai .
1 − δj
(3a)
(3b)
We define the set A(δi , δj ) as the set of points in the unit square for which inequalities
(3a,3b) hold.
Lemma 3 In the linear contribution game, the set of equilibrium payoffs for the repeated
game with discount factors (δi , δj ) is equal to A(δi , δj ).
For the continuous trust game all points in A(δi , δj ) for which ai ≥ 1−δi and aj ≥ 1−δj
can be implemented as equilibria of the repeated game.
2
Proof The statement follows from standard results in the theory of repeated games.
Any point in A(δi , δj ) can be implemented as an equilibrium payoff18 of the repeated
contribution game. Players could, for example, play well known “grim-trigger” strategies,
sticking to the a point in A(δi , δj ) and punishing off-equilibrium behavior by quitting. The
resulting equilibrium is also subgame perfect.
For the continuous trust game, as mentioned in the model description, a deviation to
1 may also be a best response. The above inequalities ensure that such deviations are
not beneficial in an equilibrium with “grim-trigger” strategies.
We can therefore proceed by analyzing the Pareto-frontier of A(δi , δj ) as a function of
the discount factors. The next lemma shows that parameter λ describes two important
regions of discount factors:
18
We gloss over the fact that one actually should renormalize the repeated game payoffs by a factor
of 1 − δ to compare directly with stage game payoffs.
10
δ2
δ2
1
1
II
I
λ
λ
II
II
I
II
I
I
δ1
1
λ
(a) Contribution Game
λ
(b) Continuous
Game
0
0
1
δ1
Trust
Figure 1: In the lower left, white region no cooperation is possible, while in the upper
right, white region all type combinations can cooperate. In Region I only high types
amongst themselves have the option of cooperating, while in Region II all type combinations except low-low can do so.
Lemma 4 If discount rates are such that λ > δh > δl then the unique equilibrium payoff
of the repeated game is (0, 0). Alternatively if λ ≤ δl < δh then the unique efficient
equilibrium has both players cooperating at the maximal level of 1 in every period.
2
Proof Note that if λ > δh > δl then A(δi , δj ) contains only (0, 0) (resp. is empty for the
case of the continuous trust game in which case (Q, Q) is the unique equilibrium). To see
the second point, note that (1, 1) ∈ A(δi , δj ) whenever λ < δi , δj .
Obviously the more interesting case arises when δl < λ ≤ δh . This ensures that while
low types cannot cooperate with one another, high types can. But how about mixed pairs
consisting of a high and a low type?
Restricting attention for the moment to δl < λ ≤ δh we define Region I to be the set
of (δl , δh ) such that A(δl , δh ) is empty (i.e. mixed pairs cannot cooperate), while Region
II is the set of (δl , δh ) such that A(δl , δh ) is nonempty.
For the contribution game Region I is described by δl δh > λ2 and Region II by δl δh ≤
2
λ . In the case of the continuous trust game Region I is described by (1 − δl )(1 − δh ) >
(1 − λ)2 and Region II by (1 − δl )(1 − δh ) ≤ (1 − λ)2 These regions are graphed in Figure
1.
Therefore, if mixed pairs cannot cooperate, i.e. in Region I, efficient equilibria can be
characterized as follows:
Lemma 5 Assuming δl < λ ≤ δh and restricting attention to Region I, all efficient
equilibria have all type combinations except hh quitting in the first period, while high
types meeting high types cooperate at the maximal level 1 in every period.
2
It remains to analyze efficient equilibria in Region II. We start with the contribution
game:
Lemma 6 Assuming δl < λ ≤ δh and restricting attention to Region II in the contribution game, efficient equilibrium outcome paths are characterized as follows:
• high types amongst themselves cooperate at the highest level in every period,
• low types amongst themselves quit immediately,
11
• if a high type meets a low type, then the high type cooperates at level 1 forever, while
the low type cooperates at some level in
λ δl
,
.
2
δh λ
Lemma 7 Assuming δl < λ ≤ δh and restricting attention to Region II in the continuous
trust game, efficient equilibrium outcome paths are characterized as follows:
• high types amongst themselves cooperate at the highest level in every period,
• low types amongst themselves quit immediately,
• if a high type meets a low type, then the low type cooperates at level 1 forever, while
the high type cooperates at level
1−λ
.
2
1 − δl
Lemmas 6 and 7 follow straightforwardly from Lemma 3 and the analysis of the Pareto
frontier of A(δi , δj ).
We note that for high-low matchings in the contribution game we obtain a Pareto
frontier of equilibria in which the patient type invests maximally. In contrast in the
continuous trust game we obtain a single point in which the low type player invests
more than the high type and therefore obtains higher stage game payoffs in an efficient
equilibrium (however, the low type still has a strictly lower repeated game payoff than
the high type).
To wrap up, we point again to Figure 1 which displays the partitioning of the discountparameter-space into regions with differing efficient equilibria.
In Region I the payoffs attained in efficient equilibria will play a role as a benchmark
for the incomplete information setting. Recalling the population random matching setting
we obtain
uIl = 0,
p(1 − λ)
uIh = p
.
1 − δh
4
(4)
(5)
Incomplete Information: Preparatory Results
We now turn to incomplete information, i.e. nature selects types independently at the
start of the game according to (p, 1 − p) (with p being the probability of a high type).
Players observe their own types perfectly and obtain no information about their opponent.
Our main goal is to characterize interim efficient equilibrium payoffs and the corresponding equilibrium outcome paths. We focus on the interesting parameter region
δl < λ ≤ δh (in the other regions efficient equilibria induce the same outcome paths as in
the complete information case).
Remark 1 An important point to note is that given any equilibrium, setting all offequilibrium behavior to Q retains the equilibrium property (recall that (Q, Q) is the only
12
equilibrium of the stage game). It therefore suffices to describe an equilibrium’s outcome
path, implicitly assuming all off-equilibrium behavior to be Q, while taking beliefs to be
consistent with behavior.19 In the following we will take this route and describe outcome
paths when talking about (efficient) equilibria.20
2
As usual we call an equilibrium with strategies s separating (with separation period
T ) if
otll (s) = otlh (s) = othl (s) = othh (s) for all t < T and
oTll (s) = oTlh (s) = oThl (s) = oThh (s),
i.e. both types l and h behave identically up to period T − 1 and differ (are separated) in
period T (recall we restrict attention to symmetric equilibria which allows this simplifying
language).21 We will also call periods 0, . . . , T − 1 the pooling phase. An equilibrium with
no separation will be called pooling equilibrium.22
For an outcome path to be an equilibrium one generally has to check numerous nondeviation conditions. To play Q off-equilibrium greatly simplifies this task as only deviations in one period need to be verified. Note furthermore that playing either Q or
1 are best possible deviations in any period. Hence, these are the only deviations one
has to check for since others are dominated by these. Therefore the set of potentially
characterizing deviations is i) sticking to equilibrium outcome behavior for a (possibly
zero) number of periods and then deviating to Q or 1, and ii) imitating the other type
for a number of periods and then deviating to Q or 1.
From i) it trivially follows that:
Lemma 8 Consider a separating equilibrium with separation period T . Then all outcome paths restricted to periods after T are equilibrium outcome paths of the complete
information game.
2
This tells us that the range of equilibrium behavior we can expect to observe may
differ in parameter regions I and II. We therefore discuss the structure in separate Sections
5 and 6.
Before doing this, we proceed with observations which hold independently of the
region. Equilibria in which all players obtain payoff 0 are called 0-equilibria – i.e. players
only invest 0 or play Q. First, we observe that there are no pooling equilibria exept
0-equilibria.
Proposition 1 In any equilibrium — except 0-equilibria — separation occurs in finite
time.
2
The intuition for this is that during a pooling phase in which all types invest the same
strictly positive amount, the only way to keep a low type from quitting is by offering him
more to steal later on. However, as feasible investments are bounded from above, this is
19
As (Q, Q) is a subgame perfect equilibrium independently of discount factors, off-equilibrium beliefs
will not play a role.
20
In Remark 5 we also discuss the appropriateness of thereby induced off equilibrium beliefs and
possible renegotiation issues.
21
Note that separation will not necessarily be reflected in beliefs, if it occurs when one player plays Q
(the game ends and there is no future period for the opponent to update his beliefs).
22
The outcome path of a pooling equilibrium can be of finite or infinite length.
13
not feasible infinitely often. The technicalities of the proof are relegated to page 32 in
the Appendix.
The next result shows that even if equilibria with a pooling phase are relevant in
characterizing the Pareto frontier, there will always exist undominated equilibria with an
empty pooling phase:
Proposition 2 For any equilibrium E in which separation of types occurs in period T > 0
there exists another equilibrium Ẽ in which separation of types occurs in period 0, and
which is strictly preferred by the low type.
2
The proof is relegated to page 33 in the Appendix.
As suggested by Lemma 8 we proceed with analyzing parameter Regions I an II in
turn.
5
Region I
For this section we restrict the discount parameters to Region I, that is, after separation
or in a complete information setting only cooperation amongst h types is viable.
From Proposition 1 we know that separation of types occurs in finite time. This can
occur in two distinct ways: i) the l type plays Q, or ii) both types cooperate at distinctive
levels.23 For the low type to separate by a cooperative action can only be an equilibrium
in the linear trust game since there his own investment is beneficial to him. In the linear
contribution game a low type could improve by quitting the game. As this differentiation
will be important to our results, we define:
Definition 3 An equilibrium in which separation occurs by the l type playing Q will
be called Q−equilibrium or just Q-equilibrium. In contrast, an equilibrium in which
separation occurs by both types playing distinct levels of cooperation will be called
C−equilibrium or C-equilibrium.
2
The goal of this section is to give a complete characterization of interim efficient
(in the sense of definition 1) Q−or C−equilibria. The main results will be presented
in the following subsections 5.1 and 5.2, respectively. It turns out that the structure
of Q−equilibria is the same for the linear contribution game and the continuous trust
game. However, C−equilibria do only exist for the continuous trust game since in the
linear contribution game maximal investment is never a best response in the stage game.
There is the salient unifying feature of gradualism or starting small in both corresponding
sorts of efficient equilibria: namely the path played after separation when two h types
are matched. Both efficient Q− and C−equilibria have qualitatively similar hh outcome
paths, which can be described as follows:
Definition 4 A gradual path α(a0 , a1 ) (for Region I) — indexed by a parameters a0 , a1 ∈
[0, 1] — is a sequence of actions:
2
τ (a1 )
λ
λ
λ
α(a0 , a1 ) := a0 , a1 , a1
, . . . , a1
, 1, 1, . . . ,
(6)
, a1
δl
δl
δl
23
Note that there cannot exist a separating equilibrium in which the h type separates by playing Q
in T for the following reason: If h exits the game in T it follows from Lemma 8 that the low types play
Q in the next period. Therefore, as l’s continuation payoff is 0, it cannot be a best response for l to
cooperate in T .
14
τ (a1 )
a
1
1
a1
a
1
a
λ
δl
2
λ
δl
a0
t
0
0
1
2
3
4
5
6
7
Figure 2: An example of a gradual cooperation path. Starting off with investments a0 and
a1 , in the following the investment level is increased by the factor δλl until period τ (a1 ),
after which the maximal amount of 1 is invested thereafter.
where

t
 max t ∈ {0, 1, . . .} | a1 λ < 1
for a1 < 1,
δl
1
τ (a ) :=

0
for a1 = 1.
(7)
2
While this might look somewhat complicated, the intuition is reasonably simple. Noting the graphic visualization of such a path in Figure 2 we subdivide a gradual cooperation
path into three phases:
Communication Phase: consists solely of period 0 in which amount a0 is invested. As
we will see later, in this phase information about types is transmitted.
Adjustment Phase: comprises periods 1 to τ (a1 ) (and therefore may be empty for
a1 > δλl ). During periods 1 to τ (a1 ), the investment level is increased by a factor
of δλl . The term τ (a1 ) is the maximal number of periods one can increase the
investment by a factor of δλl before hitting the maximal stake 1.
Established Phase: encompasses all periods from τ (a1 )+1 onwards where the maximal
amount 1 is invested.
The fraction δλl which determines the growth speed of the path in the Adjustment
Phase will subsequently crop up frequently (as well as its reciprocal value). It has the
following interpretation. Imagine for a moment type l playing along a gradual path (and
his opponent too). Assume furthermore that we are in the Adjustment Phase and l is
pondering quitting in t or t + 1 (both in the Adjustment Phase). λ/δl is the exact growth
rate such that an l type is indifferent between (i) stealing in t, and (ii) cooperating in t
and stealing in t + 1, i.e., if at+1 = δλl at then:
λ
at = (1 − λ)at + δl at .
δl
15
(8)
5.1
5.1.1
Separation by Quitting
Characterization
We now use the definition of a gradual cooperation path given in the previous section to
define a special class of Q−equilibria:
Definition 5 A Q-gradual-equilibrium with starting value a — denoted by QGE(a) — is
a Bayesian Nash equilibrium in which the outcome path of a pair of high types corresponds
24
to a gradual cooperation path, while low types
quit
in the first period. The parameters
are: a0 = a, a1 = δλl γa, and τa := τ (a1 ) = τ δλl γa where
γ :=
pλ + (1 − p)(1 + )
.
pλ
(9)
2
In order to present the central result of this section as early as possible we relegate a
detailed description of the properties (existence etc.) of Q−gradual equilibria to Propositions 3 and the following.
A point, however, which we cannot delay is the interpretation of γ. As will be shown
later Q−gradual equilibria have the property that l is indifferent between playing Q or
imitating h for t − 1 periods and then playing Q (for all periods t ≤ τa ). If this statement
is correct for t = 0 and t = 1, then the growth rate discussion preceding (8) shows that it
is also true for t = 2, . . . , τa . The difference between period 1 and the following periods
is: When l decides to deviate from playing Q in 0, to imitating h in 0 and playing Q
in 1 there is a 1 − p chance that he will meet another l-type in which case he will loose
(1 + )a0h . The γ parameter corrects for this feature, that is the indifference growth rate
in the separation period is γ δλl while later on it is δλl .
We now present the main result of this section:
Theorem 1 Every interim efficient Q−equilibrium is a Q−gradual equilibrium. Every
Q−equilibrium is weakly dominated by a some QGE(a).
2
This implies that whenever Q−equilibria exist, a Q−gradual equilibrium exists also.
A more detailed characterization of existence will follow in Proposition 3.
For the proof of Theorem 1 we proceed in two steps: i) an interim efficient Q−equilibrium
after separation resembles a Q−gradual equilibrium (Lemmas 9 and 10), and ii) in all
efficient Q−equilibria separation occurs in the first period (Lemma 11).
For step i) we first note that in an efficient Q−equilibrium the path of play of an hh
match must hit the top stake 1 at some stage (and stay there forever).
Lemma 9 In any undominated Q−equilibrium there exists τ such that in the event of
a hh match, both players invest the maximal amount of 1 in all periods after τ , i.e.,
othh = (1, 1) for all t > τ .
2
The proof can be found on page 33 in the Appendix.
Now we can complete step i) by arguing that any equilibrium with a pooling phase
is dominated by another equilibrium with the same pooling phase followed by behavior
corresponding to a Q−gradual equilibrium.
24
We remind the reader that for convenience sake, all off-equilibrium behavior is set to Q and beliefs
are consistent with actions (see Remark 1).
16
Lemma 10 Every Q−equilibrium E in which separation takes place in period T is weakly
dominated by an equilibrium Ẽ in which behavior starting from the separation period is
defined by a QGE(a).
2
The proof can be found on page 34 in the Appendix.
It remains to show that equilibria with pooling phases are dominated, i.e. step ii)
above. Lemmas 9 and 10 leave us with equilibria which consist of a pooling phase followed
by a gradual path. The following crucial lemma shows that unraveling the pooling phase,
i.e. having the l quit earlier while keeping him indifferent, is beneficial to the h-type.
Lemma 11 In any undominated Q−equilibrium, separation takes place in period 0.
2
The proof can be found on page 34 in the Appendix and concludes the proof of
Theorem 1.
5.1.2
Existence
The expected payoff of a low-type player in a QGE(a) is given by
uQ
l (a) := pa,
whereas the high-type obtains:
uQ
h (a)
t
τa λ
δhτa +1
:= p(1 − λ) a + γa
+
δh
δl
1 − δh
t=1
(10)
− (1 − p)(1 + )a.
(11)
As mentioned previously, the relevant non-deviation conditions which have to be fulfilled in an equilibrium are i) sticking to equilibrium outcome behavior for a (possibly
zero) number of periods and then deviating to Q or 1, and ii) imitating the opposite type
for a number of periods and then deviating to Q or 1.
Denote by a the maximum of the non-deviation conditions that bound initial invest¯
ment a from below (more precisely defined in the proof of Proposition 3 in the Appendix)
and let
the solution of uQ
h (a) − pa = 0 if there exists a solution in [0, 1],
ā :=
(12)
1
otherwise.
Proposition 3 For any a ∈ [a, ā] there exists a gradual cooperation equilibrium with
¯
starting value a.
2
The proof is on page 35 in the Appendix. We proceed with a characterization of the
Pareto frontier restricted to efficient Q−equilibria. Define
a∗ := max argmax uQ
h (a).
(13)
a∈[a,ā]
¯
Proposition 4 Restricting attention to Q−equilibria, the Pareto frontier in the (ul ,uh )
space is a strongly decreasing, piecewise linear function described by the payoffs attained
in a gradual cooperation equilibrium with a ∈ [a∗ , ā].
2
17
Proof From the definition in (10), ul (a) is increasing in a. From the proof of Proposition
3 it follows that uh (a) is a concave, piecewise linear function which increases for a ∈ [0, a∗ ]
and decreases for a ∈ [a∗ , 1]. Therefore the Pareto frontier is described by a ∈ [a∗ , ā]. It is furthermore possible to give a more precise characterization of the parameter
region for which the optimal τ is small, i.e., when equilibria entail high type matches
investing the maximal amount early on.
Proposition 5 Let
δl < (1 − λ)δh ,
(14)
and E be an interim efficient Q-separating equilibrium. Then E is a Q−gradual equilibrium with τ = 0, (i.e., from period 1 onwards hh-matches play 1).
2
Proof We know from the characterization Theorem 1 that E is a Q−gradual equilibrium. It remains to show that τ = 0. Take a Q−equilibrium and let Vlt−1 be the
best possible continuation payoff of a low type in period t − 1, where t − 1 is after the
separation period (i.e. assume l were to imitate h through periods 0 to t − 2 and let
him play a best response thereafter). Assume l’s best response to be Q in period t,
i.e., Vlt−1 = (1 − λ)at−1
+ δl ath . This implies that (while keeping l’s incentive and Vlt−1
h
constant) one can change investments at−1
and ath by
h
∆ath = −
1 − λ t−1
∆ah .
δl
(15)
Such a change is strictly preferred by h if:
(1 − λ)∆at−1
+ δh (1 − λ)∆ath > 0.
h
(16)
if δl > (1 − λ)δh
Combining (15) with (16) we observe that h prefers a positive ∆at−1
h
and a negative ∆at−1
if
δ
<
(1
−
λ)δ
.
Noting
that
a
negative
(positive)
∆at−1
implies
l
h
h
h
t
— via (15) — a positive (negative) ∆ah , we see that if (14) holds, h prefers to transfer
cooperation returns from earlier periods (after separation) to later ones. This would not
lead to paths of gradual increase of investment, but to U-shaped paths. However, we have
proved in Theorem 1 that Q−gradual equilibria are undominated. The only Q−gradual
equilibrium for which previously described Pareto-improvements are not feasible, are the
ones with τ = 0.
Remark 2 It is interesting to recapitulate the seemingly contradictory ideas in the preceding proof. We optimize h’s path while keeping l indifferent to his equilibrium payoff,
i.e., his best possible deviation payoff is equal to the equilibrium payoff obtained by quitting in period 0. Under these conditions, the h faces following trade-off i) getting to the
maximal investment level as early as possible versus ii) maximizing the earliest cooperation benefits possible. In case i) we obtain a path resembling an U while for ii) we obtain
a gradual path. Depending on whether (14) holds or not, h prefers i) to ii) or vice versa.
This area is visualized in Figure 3.
Having proved that gradual paths are a feature of undominated equilibria, a parameter
region in which i) is preferred by h can only imply that such a change is not feasible. This
is the case when both U-paths and gradual paths are degenerate and therefore coincide,
that is, when τ = 0.
2
18
δ2
1
λ
λ
0
1
δ1
Figure 3: When δl < (1 − λ)δh — the darker colored parts of Region I —, then optimal
QGE entail high types jumping immediately to the investment level of 1 in the second
period.
5.2
Separation by Cooperation
We have already seen that for the low type to separate by a cooperative action cannot be
an equilibrium for the linear contribution game. Consequently, for this case the Region
I characterization is complete. In this subsection we consider the continuous trust game
with fi (ai , aj ) = (1 − λ) ai for i = 1, 2. We shall investigate C−equilibria, i.e., equilibria
in which separation occurs by different levels of cooperation. It will be shown that they
too resemble gradual cooperation paths.
5.2.1
Characterization
We start by noting an important property of C−equilibria which holds independently of
efficiency:
Lemma 12 In every C−equilibrium separation occurs in period 0. Furthermore in every
C−equilibrium the l-type’s payoff is
uC
l = 1 − λ.
(17)
2
Proof Assume separation occurs in T > 0. Since the l type separates by a cooperative
action aTl > 0 and his continuation payoff is 0, it must be that aTl = 1. Pooling in T − 1
implies alT −1 = ahT −1 = aT −1 . In an equilibrium it must hold that l has no incentive to
play Q or 1 in period T − 1. This yields two conditions:
(1 − λ)aT −1 + δl (1 − λ) ≥ aT −1 ,
(1 − λ)aT −1 + δl (1 − λ) ≥ 1 − λ.
Both conditions together can only be satisfied at the same time if δl (1−λ)
≥ aT −1 ≥ 1 − δl .
λ
These inequalities imply δl ≥ λ which contradicts our assumption that λ > δl .
As with respect to the second point, it follows straightforwardly that if a cooperative
action is a best response in period 0 for the low type, and his continuation payoff is zero,
then he will cooperate at the maximum level of 1. His expected payoff is therefore 1 − λ.
Note that this contrasts Q−equilibria, for which equilibria with pooling phases exist
but happen to be inefficient.
19
Before proceeding with the characterization (and existence) of optimal C−equilibria
we point out a few striking properties of equilibrium payoffs compared to the complete
information case.
Remark 3 Unsurprisingly, as in the case of Q−equilibria, the low type’s C−equilibrium
I
payoff exceeds that of the complete information case, i.e., uC
l > ul = 0. However, in
contrast to Q−equilibria, a low type not only achieves a positive payoff when matched
with a high type, but also in a matching with another low type. Furthermore, again in
contrast to Q−equilibria, a high type’s payoff against a low type opponent is also higher
than for complete information (although his payoff against high types, as we will see,
turns out to be lower). We will show and discuss later, that there exist parameters, such
that in C−equilibria both high and low types do strictly better than in the corresponding
complete information case.
Straightforward intuition about the effects of the introduction of incomplete information can be subsumed as follows. High types have an interest in discovering their
opponent’s type. Therefore, they have to pay a rent to low types to extract this information. This effect, which clearly shows up in the case of Q−equilibria, no longer holds
for C−equilibria. The structure of the stage-game allows for information revelation in a
way which is beneficial to all types.
2
We now define C−gradual equilibria, of which we will show that they dominate other
C−equilibria.
Definition 6 A C-gradual-equilibrium — denoted by CGE(a0 , a1 ) — is a Bayesian Nash
equilibrium in which the outcome path of high types corresponds to a gradual cooperation
path α(a0 , a1 ), while low types play 1 in the first period and quit thereafter (high types
meeting low types also quit in the second period).
2
Theorem 2 Every interim efficient C−equilibrium is a C−gradual-equilibrium.
2
Proof The theorem follows if we can show that, to be efficient, the path of play of high
types from period 1 onwards must resemble a gradual cooperation path.
Given that the low type already obtains 1 − λ in equilibrium the only potentially
binding deviation constraint for the low type is that he imitates the high type for a
number of periods and then quits (for the case that he met a h-type opponent in 0).
Denote such a constraint CIClt where t is the period he intends to quit when he meets a
high type (implicitly assuming that he quits in 1 after having met another l-type).25
We first note that the equivalent to Lemma 9 also holds for efficient C−equilibria, i.e.,
in any undominated C−equilibrium there exists τ such that in the event of a hh match,
both players invest the maximal amount of 1 in all periods after τ , i.e., othh = (1, 1) for
all t > τ .
Therefore , it remains to show that in an undominated equilibrium CIClt are binding
for all t = 2, . . . , τ . Clearly CIClτ must be binding otherwise we could raise aτh , thereby
increasing h’s expected payoff. We distinguish between two parameter regions (known
from Proposition 5) i) δl ≥ (1 − λ)δh and ii) δl < (1 − λ)δh .
25
When we say that CIClt is binding we mean that the low type’s 1 − λ equilibrium payoff exactly
equals the expected payoff obtained by the low type when he imitates the high type for t − 1 periods
and quits in t.
20
Case i): δl ≥ (1 − λ)δh . Assume that there exists a period t , 1 < t < τ such that CIClt
does not bind. Assume furthermore, without loss of generality, that CIClt +1 binds.
Raising ath and simultaneously lowering ath +1 , while ensuring
∆ath = −
δl
∆ath +1 ,
(1 − λ)
(18)
keeps CIClt +1 binding (i.e., does not induce the low type to deviate from his equilibrium
behavior). Such a change is preferred by the h-type if:
(1 − λ)∆ath + δh (1 − λ)∆aht +1 ≥ 0.
(19)
which holds if δl ≥ (1 − λ)δh which is true by assumption.
Case ii): δl < (1 − λ)δh . For this case we will take a slightly different track. We
we argue that it
define an equilibrium Ẽ := CGE(ã0 , ã1 ), for which dominates any other
p−λ
pδl
0
equilibrium — denoted by E. We set ã = min p , 1 − (1−λ) and ã1 = 1 (which
implies that in Ẽ hh matches always play (1, 1) except in period 0). For Ẽ to be an
equilibrium it must be that the low type has neither an incentive a) to quit in period 0,
nor b) to imitate the high type in period 0 and then quit:
1 − λ ≥ pã0 + 1 − p
⇔
1 − λ ≥ (1 − λ)ã0 + pδl
⇔
p−λ
,
p
pδl
ã0 ≤ 1 −
.
1−λ
ã0 ≤
(20a)
(20b)
Note that by definition Ẽ fulfills these requirements, while equivalent conditions have to
hold E as well. Clearly Ẽ will dominate any E for which a0 < ã0 . As, due to (20a),
pδl
a0 > ã0 cannot hold when ã0 = p−λ
, the only case we need to check is a0 > ã0 = 1 − 1−λ
,
p
while a1 < ã1 = 1. The deviation condition corresponding to (20b) for E is:
(1 − λ) ≥ (1 − λ)a0 + pδl a1 ⇔ a0 ≤ 1 −
If E were preferred to Ẽ then:
0
(1 − λ) ah − 1 −
pδl
(1 − λ)
pδl 1
a.
1−λ
(21)
− pδh (1 −
a1h )
> 0,
(22)
which, together with (21), reduces to
δl
− δh
p(1 − a )
1−λ
1
> 0.
(23)
This, however, contradicts our assumption δl < (1 − λ)δh , which completes the proof.
5.2.2
Existence
In our definition of C−gradual equilibrium there remains a degree of freedom in the choice
of a0 and a1 . The above conditions bound a0 , a1 and depending on parameter values
C−gradual equilibrium — and therefore also C−equilibria — may not exist. However,
when they exist, efficient C−gradual equilibrium are generically unique.
21
Proposition 6 Efficient C−gradual equilibrium (amongst the set of C− equilibria) are
generically unique.
2
The proof can be found on page 37 in the Appendix. It therefore trivially follows:
Corollary 1 Restricting attention to C−equilibria, the Pareto frontier in ul , uh space is
a unique point.
2
The parameter values for which C−equilibria exist are those for which all non deviation conditions (42) through (46) given in the proof of Proposition 6 hold.
5.3
Efficient Region I Equilibria
Having analyzed C−separation and Q−separation separately, we now combine both
types. Combining Theorems 1 and 2 we obtain:
Corollary 2 All region I equilibria which entail positive investment in some period by
some type are weakly Pareto dominated (with respect to interim efficiency) by either a
C−gradual-equilibrium or a Q−gradual-equilibrium.
2
The next proposition shows that, indeed, both types of equilibria are relevant in
describing the Pareto frontier in Region I.
Proposition 7 The Pareto frontier of Region I equilibria contains both C−gradualequilibrium and Q−gradual-equilibrium.
2
Proof The simplest proof is to note that for large enough Q−gradual-equilibrium
no longer exist. C−gradual-equilibrium are then trivially better. On the other hand
C−gradual-equilibrium do not exist for λ > p, which is not necessarily an impediment
to the existence of Q−gradual-equilibrium.
In Remark 3 on page 20 we commented on the existence of equilibria which dominate
efficient complete information equilibria:
Proposition 8 There exist parameters such that an incomplete information equilibrium
strictly dominates the efficient complete information equilibrium with respect to interim
efficiency.
2
1
19
Proof Let λ = 18 , p = 34 , δl = 20
, δh = 100
. The efficient C−gradual-equilibrium entails
5
0
1
a = 6 , a = 1. Then, recalling (4) and (5), we obtain
δh
1
C
I
uh − uh = (1 − λ) a + p
−
1 − δh 1 − δh
= (1 − λ)(a − p)
7
=
> 0,
96
and
I
uC
l − ul = 1 − λ =
22
7
> 0.
8
Remark 4 The central properties of the C−gradual-equilibrium underlying the previous
proof where already discussed in Remark 3. While an l-type’s complete information payoff
is 0, he obtains a positive payoff in a C−gradual-equilibrium both when matched with
an l-type opponent or an h-type. This is also true for the h-type. However, compared
with the complete information case the h-type faces a trade-off: a) against an l-opponent
he earns more (cooperation at level a0 for one period), while b) against an h-opponent
he has to wait longer until cooperation at the highest level of 1. As the example in
the proof demonstrates, for some parameters this trade-off turns in favor of incomplete
information.
2
Remark 5 As mentioned in the introduction, it may seem somewhat unconvincing that
in both Q−gradual-equilibrium and C−gradual-equilibrium high type matches insist on
gradually raising investment levels, although they are sure from period 1 onwards that
their respective opponent is also a high type. If the focus is on efficiency, would not
renegotiation in periods 1 and later break our equilibria? This is not the case. Indeed
we constructed our equilibria by setting all off-equilibrium behavior to Q and having
beliefs consistent with behavior (i.e., probability 1 on high types for later nodes), which
may give rise to renegotiation concerns. However, even if we were to explicitly model
renegotiation procedures, nothing would hinder us from setting beliefs such that all renegotiation attempts would be interpreted as coming from low types. Such type switches
are not precluded by the specification of Bayesian or sequential equilibrium, and the
support restrictions of stronger refinements like perfect sequential equilibrium are highly
contentious.26
Furthermore, even alluding to heuristically motivated refinements like the intuitive
criterion would be to no avail. If any renegotiation would lead to a faster rise of investment
than in our equilibria, low types would also have the strict incentive to pose as a high type
and propose such renegotiation. Therefore, beliefs that renegotiation attempts stem from
low types would be perfectly reasonable (and all renegotiation attempts would fail). 2
6
Region II
The remaining terra incognita of our parameter space is Region II. We learned in Lemmas
6,7 of Section 3 that in the complete information case these parameter constellations
allow not only cooperation in hh-matches, but also in mixed hl-pairings. As in region I
we expect that there exist equilibria where after separation behavior converges in finite
time τ to the corresponding efficient complete information behavior. Clearly, in that case
separation has to be C−separation. The adjustment process after separation in period T
in such a symmetric equilibrium is described by the 3 investment levels hh, hl, lh in each
period T + 1, . . . , T + τ after separation, where hh and hl denote the investment paths of
h after meeting another h or l and lh correspondingly denotes l’s investment path after
meeting an h. Remember that l meeting another l cannot cooperate in region II and
therefore must quit the game immediately after separation. Since we have 3 adjustment
paths after separation (instead of one in region I) which are potentially interrelated by
various equilibrium non-deviation conditions region II tends to be more complicated and
less intuitive. Nevertheless, the salient feature of starting small or gradualism of efficient
outcome paths survives in a generalized way as is formulated precisely in the following
definitions.
26
For a discussion of such issues see Nöldeke and van Damme (1990).
23
Definition 7 A gradual path or CAE-path27 α(τ, a∗ ) := (αhh , αll , αhl , αlh ) for region II
with adjustment time τ is the 4-tuple of investment paths given by
αhh := a0h , a1hh , a2hh , . . . , aτhh , 1, 1, . . . ,
αhl := a0h , a1hl , a2hl , . . . , aτhl , a∗hl , a∗hl , . . . ,
αlh := a0l , a1lh , a2lh , . . . , aτlh , a∗lh , a∗lh , . . . ,
αll := a0l , Q .
As before we call t = 0 the Communication Phase described by C−separating investments a0h = a0l . The Adjustment Phase of length τ are the periods t = 1, . . . , τ described
by ((a1hh , a1hl , a1lh ) , . . . , (aτhh , aτhl , aτlh )). Remaining periods t = τ + 1, τ + 2, . . . are called the
Established Phase. Investments in the Established phase abbreviated by a∗ := (1, a∗hl , a∗lh )
are those defined in Lemmas 6,7 of Section 3, i.e. by efficient play in the corresponding complete information game28 . An equilibrium with a gradual or CAE-outcome path
α(τ, a∗ ) := (αhh , αll , αhl , αlh ) is called CII −gradual equilibrium for region II with adjustment time τ .
2
The CII in the terminology CII −gradual equilibrium indicates the separation mode
again by cooperation on different levels and distinguishes these equilibria from the C−gradual
equilibria and the Q−gradual equilibria which also may exist in region II but – as we will
see– tend to be dominated by CII −gradual equilibria with their extended cooperation
possibilities for mixed hl−matchings.
The central result of this section is the following theorem stating that gradualism29 is
also efficient in region II.
Theorem 3 Every interim efficient equilibrium that is not a Q−gradual or a C−gradual
equilibrium in region II resembles a CII −gradual equilibrium starting from the separation
period.
2
Note that this is a weaker claim with respect to pooling phases compared to region
I. We cannot exclude the existence of a pooling phase in region II as we did in region I.
But with proposition 2 in section 4 we know that for every CII −gradual equilibrium with
pooling phase there always exists an interim efficient CII −gradual equilibrium without
pooling phase that is strictly preferred by l. Since we know from proposition 1 in section
4 that separation occurs in finite time it remains to show for the proof that finitely many
periods after separation efficient behavior converges to the efficient stationary behavior
of the corresponding complete information game. This is done in the following lemma30 .
Lemma 13 In any efficient region II equilibrium that is not a Q−equilibrium or a
C−equilibrium as in region I there exists τ such that in hh−matchings both players invest
1 and in all hl−matchings the l−type invests a∗lh and the h−type invests a∗hl in all periods
after τ , i.e. othh = (1, 1) and othl = (a∗hl , a∗lh ) for all t > τ where a∗hl , a∗lh are those investment levels defined by stationary efficient play in complete information given in Lemmas
6,7 of Section 3.
2
27
CAE = Communication Phase, Adjustment Phase and Established Phase.
Remember that while in the continuous trust game this was unique in the contribution game we
have a continuum of such efficient complete information equilibria. For this definition pick any of these
and keep it fixed within the Established Phase.
29
We interpret gradualism in region II in a less narrow sense than in region I because our definition of
a region II gradual path is less explicit about the details of the corresponding Adjustment Phase.
30
Remember that lemma 9 for region 1 stated a similar result.
28
24
Proof Assume separation occurs in period T . Clearly, ll−matchings cannot cooperate
after separation. For the hh−outcome path of an efficient equilibrium there exists some
τhh such that othh = (1, 1) for all t > τhh for the same reason as in region I (see proof
of lemma 9). It remains to show that also the mixed outcome path at some point hits
the ”upper bound” defined by the efficient stationary investment levels for the complete
information case. First, note that #ohl = #ohh = ∞ since otherwise there must be a
last period where someone invests after separation (which cannot be an equilibrium). An
outcome path ohl with othl = (a∗hl , a∗lh ) for some t > T but not thereafter cannot be outcome
path of an efficient equilibrium since any h−investment above a∗hl or l−investment above
a∗lh violates the equilibrium condition that nobody wants to steal and raising investment
levels to (a∗hl , a∗lh ) after period t would raise equilibrium payoffs. The only remaining
possibility would be that there is an efficient equilibrium outcome path where either h
or l do never hit their upper bound. The non-deviation conditions binding l’s path from
above is that l imitating h should have no incentive to steal l’s investment for any t ≥ T .
This condition must be binding infinitely many times since otherwise equilibrium payoffs
could be improved by raising investment after the last period where this condition was
binding. Let us assume the condition binds in periods t1 , t2 with t2 > t1 > T . Denote
the relevant segment of the low type path in the mixed matching by aTl , aTl +1 , . . . . This
implies
t2 −t1
λ
t2
al ≥
atl 1 .
δl
But, as investment is bounded by 1 there must be a last such period which is a contradiction. Hence there must be period τl after which l invests a∗lh in the mixed matching. A similar argument defines τh for the h−path in the mixed matching. Now define
τ = max {τhh , τl , τh } which completes the proof.
We call τ ”efficient adjustment time” because in an efficient equilibrium in τ periods
after separation players behave as efficient players in the full information game – i.e. they
play the stationary efficient equilibrium.
So far we did not provide a description of an efficient Adjustment Phase. In order
to further characterize region II-equilibrium outcome paths recall the set of relevant
(potentially binding) equilibrium non-deviation conditions31 .
• LQT:
l cannot improve by playing Q in period T .
• LQt̃:
l cannot improve by playing Q in periods T + 1 < t̃ ≤ τ .
l cannot improve by imitating h and play Q in periods T + 1 ≤ th , tl ≤
• ILQth tl :
τ after meeting an h or l, respectively.
• IH∞:
h does not want to imitate l forever. Note that this condition together
with δh > δl implies ul (ατ ) < uh (ατ ).
• HQ0:
h cannot improve by playing Q in period 0 is dominanted by LQ0 since
pa0h + (1 − p) a0l ≤ ul (ατ ) < uh (ατ )
h cannot improve and play Q in periods T + 1 ≤ t̂h , t̂l ≤ τ after
• HQt̂h , t̂l :
meeting an h or l, respectively.
31
Conditions for t > τ are satisfied by construction of τ .
25
• IHQt̂:
h cannot improve by imitating l and play Q in periods T + 1 ≤ t̂ ≤ τ .
• All previous conditions with the deviation to play C (to invest 1) instead of playing
Q. As we know from region I all these conditions are only relevant for the trust
game and for existence since they define lower bounds for investments. As before,
they have no role in the characterization of efficient equilibrium outcome paths if
these exist.
Generally this rather large number of potentially binding non-deviation conditions
exactly defines efficient behavior in the Adjustment Phase. In order not to get lost between numerous case distinctions we proceed to describe efficient behavior in region II
geometrically. First, we provide a general description and second, to get a better intuition we study some special cases. Let a(τ, a∗ ) = (a0h , a0l , a1hh , a1hl , a1lh , . . . , aτhh , aτhl , aτlh ) ∈
A (τ, a∗ ) ⊂ [0, 1]2+3τ where
A (τ, a∗ ) = {a(τ, a∗ ) |all non-deviation conditions are satisfied } ⊂ [0, 1]2+3τ
is defined as the set of investment parameters in the Communication and Adjustment
Phases such that all non-deviation conditions are satisfied. CII −gradual equilibrium
payoffs are given by
τ
τ +1
δ
uh (a(τ, a∗ )) = p (1 − λ) a0h +
δht athh + h
+
1 − δh
t=1
τ
τ +1
∗
∗
δ
(θahl + ηalh )
δht θathl + ηatlh + h
,
+ (1 − p) θa0h + ηa0l +
1 − δh
t=1
τ
τ +1
∗
∗
δ
(θa
+
ηa
)
lh
hl
δlt θatlh + ηathl + l
+
ul (a(τ, a∗ )) = p θa0l + ηa0h +
1
−
δ
l
t=1
+ (1 − p) (1 − λ) a0l ,
where θ = −λ, η = 1 for the linear contribution game and θ = 1 − λ, η = 0 for the
continuous trust game. Both payoffs contain the two expessions reflecting continuation
payoffs starting from the separation period conditional on meeting an h−type (with
probability p) or an l−type (with probability 1 − p). For all matchings exept ll payoffs
further reflect the three phases CAE of the relationship. Note that these payoffs are linear
in all investment parameters. Hence, since all non-deviation conditions are linear weak
inequalities A (τ ) is a closed convex linear 2 + 3τ −dimensional polygon and a subset of
[0, 1]2+3τ ⊂ R2+3τ . Therefore the Pareto-frontier as in region I consists of subsets of the
boundaries of these polygons A (τ ) for τ = 0, 1, . . .. More precisely, let
 =
A (τ, a∗ )
τ =0,1,...
then define
P FII =
∗
there is no b ∈ Â with un (b) > un (a(τ, a ))
.
a(τ, a∗ ) ∈ Â for some type n = h, l and ≥ for both n = h, l
Clearly, P FII = ∅ if there exists some CII −equilibrium.
While Q−gradual equilibria and C−gradual equilibria as defined for region I do not
take advantage of of hl−cooperation between high/low types, they still remain equilibria
in Region II:
26
Lemma 14 Both QGE and CGE remain equilibria in Region II, if they exist.
2
Proof The non-deviation conditions for QGCE and CGCE do not depend on the
Region II parameter restriction.
Of course CII −equilibria are only efficient if they are not dominated by some Q−gradual
equilibrium or C−gradual equilibrium. In the remainder of this section we study the
special case τ = 0 where players separate and then immediately switch to the efficient
complete information stationary mode of behavior. This case is of particular interest for
two reasons. First, whenever this case exists CII −equilibria are efficient and therefore
substantial part of the general structure. Second, since this case can be described by
just two parameters a0l , a0h we can use it for a graphical analysis which is helpful for a
better understanding of region II in general. To avoid redundancies we will perform this
anylysis of special cases only for the continuous trust game. Analogous considerations
hold for the linear contribution game.
Definition 8 A rapid enhancement equilibrium for Region II in the continuous trust
game — parameterized by (al , ah ) — is an equilibrium in which an h-type’s actions are
given by:
(ah , 1, 1, . . .)
when meeting an h-type,
αh =
1−λ 1−λ
,
, . . .) when meeting an l-type,
(ah , 1−δ
l 1−δl
while an l-type’s actions are given by:
(al , 1, 1, . . .) when meeting an h-type,
αl =
when meeting an l-type.
(al , Q)
2
Proposition 9 There exist parameters in Region II such that a rapid enhancement equilibrium exists and is efficient.
2
Proof We show this by giving a numerical example. Let δh = .9, δl = .3, λ = .4 be in
9
and al = 1. Following 3
Region II and p = .4. As initial investment levels we set ah = 35
critical non-deviation-conditions have to be are satisfied:
1. LQ0 : l does not want to quit in t = 0:
pδl
≥ pah + (1 − p)al
(1 − λ) al +
1 − δl
2. LIQ1,1 : l does not like to pretend to be h and then quit in t = 1 for both opponent’s
types:
pδl
δl (1 − λ)
≥ p ((1 − λ) ah + δl ) + (1 − p) (1 − λ) al +
(1 − λ) al +
1 − δl
(1 − δl )
3. HIQ∞,∞ has no incentive to imitate l forever:
(1 − p) (1 − λ)
δh
pδh
≥ (1 − λ) al +
p+
(1 − λ) ah +
1 − δh
(1 − δl )
1 − δh
27
It is easy to check that all other non-deviation-conditions are either trivially satisfied or
dominated by these. For example, h does not like to quit in t = 0 by combination of
the first and third conditions. An easy calculation shows that indeed for δh = .9, δl =
9
.3, λ = .4, p = .4, ah = 35
, al = 1 all conditions are satisfied. In order to show that
this CII -equilibrium-outcome-path is efficient we need to show that it is not dominated
by a Q- or a C-equilibrium nor by another CII -equilibrium. First, with respect to the
Q-equilibrium just check that for the given parameters
δh
(1 − p) (1 − λ)
pδh
CII
> uQ
uh = (1 − λ) ah +
p+
>
h.
1 − δh
(1 − δl )
1 − δh
pδl (1−λ)
II
= uC
.
Second any C-equilibrium is always worse for l since uC
l = 1 − λ < 1 − λ + 1−δl
l
Third, observe that for the given parameter values condition 1 is binding and does not
depend on al since λ = p. Therefore al = 1 is maximal and a higher ah cannot be an
equilibrium. Therefore the given outcome path is supported by an efficient equilibrium
which completes the proof.
As in the previous section, it is interesting to compare efficient incomplete information
equilibria with the complete information benchmark. In contrast to Region I we can no
longer expect interim dominance of incomplete information equilibria in Region II. To
see this, note that in an efficient equilibrium in the complete information setting of
the continuous trust game an h-type will cooperate at the highest possible level with
both l-type and h-type opponents. Adding incomplete information can therefore only be
detrimental to h-type.
Nevertheless, low types can gain more as they loose by the introduction of incomplete
information. This is due to the fact that in a C II -equilibrium separation works by cooperation on different levels. Therefore, pairs of l-types can cooperate before and during
the separation, which is not possible in the complete information case. Depending on
parameters, one can show that this effect can offset the losses of h-types in the ex-ante
sense.
Proposition 10 There exist parameters in region II such that an efficient C II -equilibrium
strictly dominates the efficient complete information equilibrium with respect to ex-ante
efficiency.
2
Proof Consider the same numerical example δh = .9, δl = .3, λ = .4, p = .4, ah =
9
, al = 1 as in the proof of the previous Proposition 9. We calculate the according
35
equilibrium utility levels:
pδl
C II
ul = (1 − λ) al +
= 0.702857,
1 − δl
δh
(1 − p)(1 − λ)
CII
p+
uh = (1 − λ) ah +
= 5.09143.
1 − δh
(1 − δl )
Similarly for the efficient complete information utilities we obtain:
uII
l = 0.342857,
uII
h = 5.48571.
28
II
II
C
As we expect uC
> uII
< uII
l (pairs of low types can cooperate one period) and uh
h
l
(under complete information h cooperates with both types at the maximal level). Finally
we check
UC
U
II
II
II
= puC
+ (1 − p)uC
h
l
=
puII
h
+ (1 −
p)uII
l
II
= 2.45829,
= 2.4,
II
and hence U C − U II = 0.0582857 > 0, which completes the proof.
For which parameters λ, p, δl , δh in region II exist CII −equilibria with τ = 0?
A sufficient condition is
A (0, a∗ ) = a0l , a0h |all non-deviation conditions are satisfied = ∅
It is easy to see that only the corresponding non-deviation conditions
pδl
0
0
0
LQ0 : pah + (1 − p) al ≤ (1 − λ) al +
1 − δl
p − λ 0 (1 − λ) δl
al +
,
⇔ a0h ≤
p
1 − δl
pδl
0
0
ILQ11 : (1 − λ) ah + δl ≤ (1 − λ) al +
1 − δl
pδ
δ
l
l
,
−
⇔ a0h ≤ a0l +
1 − δl 1 − λ
pδh
pδh
(1 − p) δh 1 − λ
0
0
+
IH∞ : (1 − λ) al +
≤ (1 − λ) ah +
1 − δh
1 − δh
1 − δh 1 − δl
(1 − p) δh 1 − λ
⇔ a0h ≥ a0l −
1 − δh 1 − δl
are potentially binding. Looking at these conditions we obtain two cases:
1. If p ≥ λ then P FII ⊂ A (0, a∗ ) is the unique point
p (1 − λ) [δl (1 − δh ) + δh (1 − p)]
0
al = min 1,
λ (1 − δl ) (1 − δh )


 (1−λ)[(p−λ)δh (1−p)+pδl (1−δh )] , 
λ(1−δl )(1−δh )
a0h = min
 (1−δl )(1−δh )−δh (1−p)(1−λ) 
(1−δl )(1−δh )
A possible constellation is illustrated in figure 1.
2. If p < λ then P FII (0) is a set of points which is the upper boundary of the set
A (0, a∗ ). An example is illustrated in figure 2.
This pattern extends to the other cases where τ > 0 in dimension 2 + 3τ . In other
words, the Pareto frontier consists of subsets of the ”upper boundary” of the linear polygons A (τ, a∗ ) for τ = 0, 1, . . .which can be be unique points or a set of points depending
on exogenous parameters λ, p, δl , δh in region II.
29
0
0
ah
ah
1
1
Q
IL
LQ
11
0
PFII (0)
11
IH
4
A (0,a*)
IH
4
A (0,a*)
IL
Q
LQ 0
PFII (0)
0
1
1
0
0
al
(a) Case p ≥ λ
0
al
(b) Case p < λ
Figure 4: In figure (a) for p ≥ λ the Pareto frontier P FII (0) is a point. In figure (b) for
p < λ the Pareto frontier P FII (0) is a set.
7
Related Literature
Numerous authors before and after Schelling wrote in a verbal style about trust and its
necessity for doing business. Some of them even emphazised the worth of performing a
rigorous analysis. To keep this section short, however, we only refer to closely related
research with formal models or experiments and interpretations close to our approach.
Gradual increases of stakes in partnership situations have been modeled in the context
of random-matching games32 by Datta (1996) as well as Ghosh and Ray (1996) (see also
Kranton (1996)). In their models, players from a large population are matched to play a
potentially long term game (similar to ours). Players also have the option of terminating
the relationship, in which case they drop back into the pool of unmatched players where
they await a new matching. Due to the assumptions of large populations and the absence of information regarding player’s behavior in past matchings, contagion equilibria
as in Kandori (1992) do not work. Instead, both papers show that a gradual increase
of stakes can serve as an alternative disciplining device (for high types) by lowering the
attractiveness of defecting and starting a new relationship.33 Datta works in a complete
information setting (i.e., there are only high types) with an exogenous probability that
matchings break up. Ghosh/Ray assume an incomplete information setting. Their low
types players, however, are assumed to be myopic and therefore always quit in the first
period. Ghosh/Ray also apply a bilateral rationality constraint, which is similar in effect to our focusing on efficient equilibria. Another paper which combines search with
32
The study of random-matching games was initiated by Rosenthal (1972). Kandori (1992), amongst
others, extends the Folk Theorem to this context.
33
This is in contrast to our model, where the gradual buildup of investment serves as a disciplining
device for the low types to keep them from posing as high types.
30
incomplete information (albeit one-sided) to generate gradualism is Rauch and Watson
(2003).
Lockwood and Thomas (2002) show in a complete information version of a repeated
game with a general contribution game as a stage game, that the assumption of irreversibility of investments (in the sense that investments can only rise over time) also
leads to gradualism.
Sobel (1985) presents a loan model which is similar in flavor to our setting in that
stakes should change over time. The unique equilibrium has the lender starting off with
small loans. The principle theoretical difference to our work is the one-sidedness of
incomplete information.
Closest to our work is a series of papers by Watson (1996; 1999; 2002) the first of
which contributed to inspire the writing of this paper. Watson’s first article has been
subsequently subdivided into the two latter articles. These are updated, generalized but
substantially altered versions of the results in the first unpublished article, which was
in various respects closest to our framework. Watson’s work considers a game in which
players at each instant of time only choose whether to continue or to quit. The players
agree in advance upon a level function which describes the investment level as a function
of time. Watson focuses both on cases in which players can commit to level functions or
cases which are selected by a renegotiation condition. While we work in discrete time,
Watson’s model is in continuous time. Watson is more general in his latter two papers
by allowing for asymmetric probabilities to be a good type (including the symmetric
case and the one sided incomplete information case). Furthermore, Watson considers
mixed strategies (in his game strategies are simply stopping rules given by a probability
distribution for quitting over time). We only consider symmetric type probabilities and
only pure strategies.
There are, however, two fundamental differences between our results versus Watson’s.
Firstly, in Watson’s setup a gradual cooperation equilibrium regime exists even for very
pessimistic players (i.e., low prior probabilities for being a good type). This is not so
in our framework, as for very low starting values a high type (in a high-high matching)
has an incentive to deviate to a high level after separation has taken place (which is
not possible in Watson’s game, where deviations are restricted to quitting). Hence we
obtain parameter cases where only the noncooperative equilibrium exists (everybody quits
immediately). Secondly, in our framework asymmetric cooperation (players invest at two
differing levels) is supportable by an equilibrium (even with complete information). This
generates a richer structure of efficient equilibria than in Watson’s case. Conversely,
Watson’s more general stage game and the possibility of asymmetric type probabilities
generate other results which are beyond the scope of our current paper.
Also germane is the literature on repeated games with incomplete information as
presented for example in the seminal book by Aumann et al. (1995). Their book contains
a systematic investigation of the nature of information revelation for repeated games with
incomplete information. While most of the book is on repeated zero-sum games, the last
chapter turns to non-zero sum games with one sided incomplete information. We are not
aware of results (existence or characterization) on this level of generality for repeated
non-zero sum games with two-sided asymmetric information.
Since building and maintaining long-term relationships is an important issue if the
players are governments of countries or institutions our framework could also be usefully
applied on the impact of trust and reputation on investment levels in macro models. Noteworthy examples of this literature include Zak and Knack (2001), Marcet and Marimon
31
(1992) or Khan and Ravikumar (2002).
Most of the experimental literature on trust and reciprocity as for example Berg et al.
(1995), Smith (1998) or Houser and Kurzban (2002) deals with one-shot extensive-form
games and concentrates on other issues as the explanatory power of self-interest versus
altruism. We are not aware, however, of experimental studies within the narrow context of
our article — i.e. repeated interaction with variable stakes and double-sided uncertainty.
8
Conclusions
We have presented a model of a repeated game between two players with double sided
incomplete information regarding discount factors. Players invest variable stakes in a joint
project. We interpret the stakes as levels of trust since this is what can be stolen and at
the same time this measures the size of the damage if the partner steals. We explicitly
allow for situations where these levels can differ endogenously within a partnership. It
turns out that this possibility to differentiate is a very effective communication tool.
For the benchmark case of complete information we partition the parameter space
of discount factors into two non-trivial Regions in which either only high-high matches
can potentially cooperate (Region I), or only low-low matches cannot cooperate (Region
II). We characterize the structure and payoffs of interim and ex-ante efficient incomplete
information equilibria. Efficient behavior in all such equilibria follows a typical pattern.
First, in the Communication Phase players reveal their type. Then, in the Adjustment
Phase players adjust the level of the relationship according to the relevant equilibrium
non-deviation conditions which depend on the Region. In the Established Phase players
behave as they do without mutual uncertainty. The Communication Phase can differ in
the way in which low types reveal themselves. On the one hand one obtains equilibria
in which low types steal in the first period, which can be interpreted as obtaining a
rent from the high types in order to reveal information. On the other hand one obtains
equilibria in which low types separate by playing a high investment level in the first
period and quit thereafter (in Region I) or continue only with high types (in Region II).
Surprisingly, it can be shown that these equilibria may even dominate efficient equilibria
in a setting of complete information (irrespective of type). These effects do not crop up
in the immediately related literature, because it is usually either assumed that players
cannot choose different levels of investment endogenously in any period, or it is assumed
that quitting is always the unique best response in the stage game (which need not be
the case even if quit-quit is the unique stage game equilibrium).
Appendix
Proof (of Proposition 1) Assume a pooling equilibrium with more than one period
of play. Denote the actions of both types by at . We distinguish pooling with i) finite
strictly positive cooperation, and ii) infinite strictly positive cooperation.
In the case of i) there must be a last pooling period where players invest strictly
positively. This, however, cannot be an equilibrium since players can improve by playing
Q.
It remains to show that ii) implies a contradiction. We now have an infinite pooling
equilibrium (with strictly positive cooperation for an infinite number of periods). Let
a = sup at which is finite by the boundedness of the set of feasible investments and
32
strictly positive. In an equilibrium it must hold for the low type in any period t that
continuing to cooperate entails higher payoffs than to quit, i.e.,
(1 − λ)at + δl V ≥ at ,
where V is the continuation payoff of the low type from t + 1 onwards. V is bounded
1−λ a . Inserting this bound and rearranging we obtain:
above by 1−δ
l
a
a
(24)
δl t − λ + λδl 1 − t ≥ 0.
a
a
By definition of a we can find at arbitrarily close to a in which case a /at is arbitrarily
close to 1 and the bracketed term can be made arbitrarily small. Then it suffices to note
that δl < λ to see that there exists a period t such that (24) cannot hold.
Proof (of Proposition 2) Assume without loss of generality that all off-equilibrium
behavior of E is Q. Define Ẽ as a copy of E except that all players skip period T − 1.
By construction Ẽ is an equilibrium too. Let Vlt (resp. Ṽlt ) be the expected continuation
payoff of the l-type in the E (resp. Ẽ) equilibrium. As both equilibria induce the same
behavior in periods 0, . . . , T − 2, the claim then follows if:
ṼlT −1 − VlT −1 > 0.
(25)
Note that by construction ṼlT −1 = VlT and VlT −1 = (1 − λ)aT −1 + δl VlT . Substituting
these equations into (25) and rearranging we obtain:
(1 − δl )VlT > (1 − λ)aT −1 .
(26)
As E is an equilibrium, aT −1 must be such that a low type does not prefer stealing aT −1
to cooperation at level aT −1 plus the discounted continuation payoff of period T , i.e.
aT −1 ≤ δλl V T . Substituting this into (26) we obtain λ > δl , which holds by assumption.
Proof (of Lemma 9) Assume separation occurs in period T . We first note that #ohh =
∞. If this were not the case, i.e., if there is a period t > T in which ath = Q, then we
and increasing subsequent
could improve h’s equilibrium payoffs by setting ath = δλl at−1
h
λ
periods’ investments by δl until hitting 1 (and investment 1 thereafter).
Therefore, let #ohh = ∞ and assume conversely that there is an efficient equilibrium
such that ath < 1 infinitely often, i.e. for an infinite set
J ∗ := t ∈ {0, 1, 2, ...} ath < 1
of periods: #J ∗ = ∞. Then it must be the case that for infinitely many of these periods
t ∈ J ∗∗ ⊂ J ∗ (with #J ∗∗ = ∞) the l is indifferent between (i) his equilibrium payoff, and
(ii) imitating h up to period t − 1 and playing Q in t . If this were not the case h’s payoff
could be strictly increased by raising ath in some period after the last such period. For
any two such periods t , t ∈ J ∗∗ with t > t > T and for which an h imitating l-type is
indifferent between quitting in t or t , it must hold that:
t −t
λ
t
ah ≥
ath .
(27)
δl
However, the investment level is bounded above by 1. Since δλl > 1 there must be a last
such period which contradicts #J ∗∗ = ∞. This concludes the proof since there must be
a last period τ where aτh < 1.
33
Proof (of Lemma 10) Separation takes place in period T . Following Lemma 9 there
exists a first period τ such that ath = 1 for all t > τ . Furthermore the l-type is indifferent
between his equilibrium payoff (cooperating until T − 1 and quitting in T ) and imitating
the h-type’s play and quitting in τ . We define a new equilibrium Ẽ as a copy of E with
the following changes: in period T to τ the h-type plays a gradual cooperation path such
that ãτh = aτh . Note that this implies that ãth ≥ ath for all t ∈ [T, τ ]. We now have to
prove that payoffs are weakly higher in Ẽ. As E and Ẽ can differ at most in periods
between
including)T and τ − 1, we restrict attention to
subset of periods. Let
τ(and
τthis
−1 t t
τ −1 t t
τ −1 t t
−1 t t
Vl := t=T δl ah , Ṽl := t=T δl ãh , Vh := t=T δh ah , Ṽh := t=T δh ãh . By construction,
the l-type’s indifference implies:
paTh = p ((1 − λ)Vl + δlτ aτh ) − (1 − p)(1 + )aTh ,
T
τ τ
pãh = p (1 − λ)Ṽl + δl ah − (1 − p)(1 + )ãTh .
(28a)
(28b)
Subtracting (28a) from (28b) and noting that aτh = ãτh , rearranging leads to:
ãTh − aTh =
p(1 − λ)
Ṽl − Vl .
p + (1 − p)(1 + )
(29)
The high types payoffs from periods T, . . . , τ − 1 are given by:
Uh = p(1 − λ)Vh − (1 − p)(1 + )aTh ,
(30a)
Ũh = p(1 − λ)Ṽh − (1 − p)(1 + )ãTh .
(30b)
Subtracting (30a) from (30b), inserting (29) and dividing by p(1 − λ) we obtain:
(1 − p)(1 + ) Ũh − Uh = Ṽh − Vh −
Ṽl − Vl .
p(1 − λ)
p + (1 − p)(1 + )
(31)
We note that this expression is strictly positive due to following observations: The V ’s
are are strictly monotonic in the discount factors, therefore, as ãth ≥ ath , it must hold that
Ṽh − Vh > Ṽl − Vl > 0. Furthermore the fraction in the RHS of (31) is strictly below 1.
The only deviation condition which could be violated in Ẽ (given that E is an
equilibrium, which implies that Uh > Ul ) is that Ũl > Ũh . Simple calculation shows
Ũh − Uh ≥ Ũl − Ul .
Proof (of Lemma 11) Let E be an equilibrium in which separation takes place in
T > 0. We will construct an equilibrium Ẽ which is strictly preferred by h and weakly
preferred by l. Define Ẽ equivalently to E except that the low type quits in period T − 1
instead of T and the high type invests ãhT −1 and ãTh . Noting that ahT −1 = alT −1 , we define
uh := (1 − λ) ahT −1 + pδh aTh − (1 − p)(1 + )δh aTh ,
(32a)
ul := (1 − λ)ahT −1 + pδl aTh ,
(32b)
ũh := (1 − λ)p ãhT −1 + δh ãTh − (1 − p)(1 + )ãhT −1 ,
(33a)
and
ũl :=
pãhT −1 .
(33b)
34
We next set ãhT −1 := δλl aTh and ãTh := γ δλl ãhT −1 = γaTh . The non-deviation conditions of Ẽ
for these parameter values are very similar to those of E but one period earlier (replace
τ by τ − 1 and divide by δl or δh ).
As E is an equilibrium, stealing in T − 1 is no strict improvement for the l-type, i.e.,
T −1
ah ≤ (1 − λ)ahT −1 + δl paTh , or:
δl
ahT −1 ≤ p aTh .
(34)
λ
Inserting (34) plus the previous definitions into (32a), (32b), (33a), (33b) and rearranging
we obtain:
ũh − uh
(1 − p)(1 + )(δh − δl )
≥
> 0,
T
ah
λ
ũl − ul
≥ 0.
aTh
Proof (of Proposition 3) To be able to treat the linear contribution game and the
continuous trust game simultaneously we will use the notation fi (ai , aj ) = θai + ηaj .
This implies θ = −λ, η = 1 for the linear contribution game and θ = 1 − λ, η = 0 for the
continuous trust game. Existence of Q-gradual equilibrium is governed by whether the
non-deviation conditions can be simultaneously fulfilled. We state the relevant conditions
and start off the low type:
• l cannot improve by imitating h and playing Q in period t > 0: As already noted
in our discussion of gradual paths all such deviations are by construction weakly
dominated by equilibrium behavior.
• l cannot improve by investing 1 in period 0:
pa ≥ p(θ + ηa) − (1 − p)(1 + ) ⇔ a ≥
θ
1−p
−
(1 + ).
1 − η p (1 − η)
• l cannot improve by imitating h and playing 1 in period 1:
λ
pa ≥ p (1 − λ)a + δl (θ + η γa) − (1 − p)(1 + )a ⇔
δl
pθδl
a≥ .
p λ − ηγ
+
(1
−
p)(1
+
)
δl
(35)
(36)
(37)
• l cannot improve by imitating h and playing 1 in period t > 1: one can show that if
l wants to neither i) imitate h and play Q in period 1 (the first point above), nor ii)
imitate h and play 1 in period 1 (inequality (36)) then he does not want to imitate
h and deviate to 1 in some later period. As we will be using this result later, we
relegate the details of this claim to Lemma 15 later in this Appendix, page 37.
We proceed with the high type:
• h cannot improve by quitting in period 0:
QICh0 (a) := uQ
h (a) − pa ≥ 0.
(38)
Note, that this implies, that the h-type’s expected payoff is always higher than an
l-type’s.
35
• h cannot improve by quitting in a later period: After separation, the h-type’s
continuation payoffs from cooperation are always strictly higher than feasible Qpayoffs.34
• h cannot improve by playing 1 in period 0: This is implied by combining (35) and
(38).
• h cannot improve by investing 1 in period 1:
QICh1 (a) := uQ
h (a) − p(θ + ηa) + (1 − p)(1 + ) ≥ 0.
(39)
h cannot improve by quitting in period 0:
QICh0 (a) := uQ
h (a) − pa ≥ 0.
The monotonicity of continuation payoffs after period 1 guarantees that the corresponding non-deviation conditions for later periods do not bind. We now note that (35),
(36) and (39) all bound a from below. We denote the maximum of these expressions
with a. Now we show that QICh0 (a) is a continuous, piecewise linear function in a. Fur¯
thermore QICh0 (a) is weakly concave, attaining a strictly positive maximum on any set
a ∈ [0, x] where x > 0. To see this note first is that

t

 max t ∈ {0, 1, ...} | γλ a λ < 1
for γλ
a < 1,
δl
δl
δl
τa = τ (a1 ) =

 0
for γλ
a = 1.
δl
is a weakly decreasing, strictly positive step function with a countable number of jumps
and lim+
a→0 τa = ∞. Payoff uh (a) is therefore trivially linear for those a for which τa is
continuous. For those a for which τa is discontinuous it suffices to verify that uh (a) is
continuous. This proves the first statement.
To prove the second statement we first note that QICh0 (0) = 0 and proceed in two
steps: 1) δl = λδh and 2) δl = λδh .
1) δl = λδh : In this case, the LHS of condition (38) reduces to:
δhτa +1
p(1 − λ) a + γaτa +
− (1 − p)(1 + )a − pa.
1 − δh
which we differentiate with respect to a (for those a where τa is constant) to obtain:
p(1 − λ) + γτa − (1 − p)(1 + ) − p.
+
0
Recalling that lim+
a→0 τa = ∞ we see that lima→0 ∂QICh (a)/∂a = ∞. For similar reasons
the slope of successive linear segments of QICh0 (a) decreases in a.
2) δl = λδh : we rewrite the LHS of condition (38) as follows:
τ δh λ
δhτa +1
δh λ a
0
QICh (a) = p(1 − λ) a + aγ
+
1−
δl − δh λ
δl
1 − δh
− (1 − p)(1 + )a − pa.
34
This follows by construction of the Q−gradual equilibrium. Type l is indifferent between playing
the equilibrium and imitating h up to some t ≤ τ . Since h is strictly more patient he strictly prefers not
to quit.
36
Differentiating with respect to a (for those a where τa is constant), substituting γ by its
definition and rearranging we obtain:
∂QICh0 (a)
= p(1 − λ) − (1 − p)(1 + ) − p
∂a
τ δh λ a
(pλ + (1 − p)(1 + ))δh
1−
+
p(δl − δh λ)
δl
(40)
We analyze two subcases 2i) and 2ii) in turn: 2i) δl < λδh : As lim+
a→0 τa = ∞ the
0
∂QIC
(a)/∂a
=
∞. For similar
term on the second line of (40) goes to +∞, i.e. lim+
a→0
h
reasons the slope of successive linear segments of uh (a) decreases in a. 2ii) δl > λδh : the
expression with the τa exponent now converges to 0, and (40) can be written as follows:
δh
(pλ + (1 − p)(1 + ))
−1 .
δl − δh λ
0
This term is positive (as δh > δl ), therefore lim+
a→0 ∂QICh (a)/∂a > 0. As before, the
slope of successive linear segments of uh (a) decreases in a.
Combining these properties, it follows that QICh0 (a) is positive in a neighborhood of
a = 0, and furthermore weakly concave. This implies that QICh0 (a) attains a strictly
positive maximum on a ∈ [0, x] for any x > 0.
Consequently uQ
h (a) is a continuous, piecewise linear function which is differentiable
almost everywhere. Furthermore, uQ
h (a) is weakly concave, attaining a strictly positive
maximum on any set a ∈ [0, x] where x > 0. This concludes the proof.
Lemma 15 Consider Region I and an equilibrium in which some gradual path is played
amongst high types and separation takes place in period 0. If l wants neither i) to imitate
h and play Q in 1, nor ii) to imitate h and play 1 in period 1, then he does not want to
imitate h and deviate to 1 in some period t > 1.
2
Proof Assume without loss of generality that θ > 0 and l imitates h in 0 and he meets a
high type opponent. The continuation payoff of this branch is bounded below by θ + ηa1
which l gets when he plays 1. Could it be that he
1 to later periods?
delays playing
1
1
1λ
1
l)
This would be the case if θ + ηa ≤ (θ + η)a + δl θ + ηa δl or a ≥ θ(1−δ
. However,
θ+ηλ
l)
is strictly larger than θ + ηa1
straightforeward calculation shows that to steal a1 ≥ θ(1−δ
θ+ηλ
since δl < λ. Therefore, if Q is not a best response in period 1, then to invest 1 in period
1 is better than doing so later on.
Proof (of Proposition 6) We proceed with a more detailed analysis of the nature
of C-gradual equilibrium. An h-type’s payoff is given by:


τ (a1 ) τ (a1 )+1
δh λ t
δ
a0 + pa1
.
uC
+p h
(41)
h = (1 − λ)
δ
1
−
δ
l
h
t=1
By the constructive nature of the proof of Theorem 2 it follows that if C-separating
equilibria exist, then C-gradual equilibrium exist too. In any C-gradual equilibrium, the
following non-deviation conditions must hold:
37
• l cannot improve with Q in period 0:
1 − λ ≥ pa0 + 1 − p ⇔ a0 ≤
p−λ
.
p
(42)
Note that this is an upper bound for a0 . It also implies that p ≥ λ.
• l cannot improve by imitating h and playing Q in period 1:
1 − λ ≥ (1 − λ)a0 + pδl a1 ⇔ a0 ≤ 1 −
pδl 1
a.
1−λ
(43)
Recall that, by construction, corresponding non-deviation conditions must hold for
all periods later than 1.
• l cannot improve by imitating h and playing 1 in period 1:
1 − λ ≥ (1 − λ)(a0 + pδl ) ⇔ a0 ≤ 1 − pδl .
(44)
As in the case of Q-gradual equilibrium, the corresponding non-deviation conditions
also do not bind for later periods (see again previous Lemma 15).
• h cannot improve by playing 1 in period 0:
uC
h ≥ 1 − λ.
(45)
Note that together with (42) this implies that h does not want to play Q in period
0.
• h cannot improve by playing 1 in period 1:


1 )+1
τ (a1 ) τ
(a
t
δh λ
δ
.
δh (1 − λ) ≤ (1 − λ) a1
+ h
δ
1
−
δ
l
h
t=1
(46)
Again as in the case of Q-gradual equilibrium the monotonicity of continuation
payoffs after period 1 ensures that corresponding non-deviation conditions for later
periods do not bind (both for Q-deviations as well as deviations to 1).
Both (42) and (44) bound a0 from above. Call this bound ā0 . (46) bounds a1 from
below, denote a1 . Therefore a0 ∈ [0, ā] and a1 ∈ [a1 , 1]. In an efficient equilibrium —
¯
¯
0
1
0
1
given that uC
l is independent of a and a — we wish to maximize a and a . Therefore,
0
1
either i) (43) holds with equality, or ii) a and a hit a boundary. In case of ii) the
proposition trivially holds. In the case of i) we maximize uC
h subject to equality of (43).
given
by
equation
(41)
we obtain:
Substituting into the definition of uC
h


t
τ (a1 ) τ (a1 )+1
δh
δh λ  1
pδl

+p
,
(47)
a + (1 − λ)p
(1 − λ) 1 −
1−λ
δ
1
−
δ
l
h
t=1
which we maximize with respect to a1 (keeping the relevant constraints in mind). Depending on whether the expression in the large brackets is strictly positive or negative
(which is generically the case) we set a1 = 1 or a1 = a1 — with a0 determined by (43).35 ¯
35
Obviously one then has to check, given optimal a0 , a1 values, that (45) holds.
38
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