Uniform Order and Foundations of
Dominant-Strategy Mechanisms∗
Yi-Chun Chen†
Jiangtao Li‡
September 30, 2014
Abstract
A major question in mechanism design is whether Bayesian mechanisms may have
advantages over dominant-strategy mechanisms. Following Chung and Ely (2007)
who study maxmin and Bayesian foundations for dominant-strategy mechanisms in
the single unit auction setting, we revisit these foundations in general social choice
environments with quasi-linear preferences and private values. Based on a graphtheoretic formulation of incentive constraints, we propose a condition called uniform
order that, under regularity and nonsingularity, ensures foundations of dominantstrategy mechanisms. We show that uniform order is satisfied in several classical
environments including single unit auction, public good, bilateral trade as well as
some nonlinear utility environments and multi-dimensional environments. When
uniform order is not satisfied, we show that, by means of a bilateral trade model with
ex ante unidentified traders, maxmin/ Bayesian foundations might not exist.
∗ We
are grateful to Jeffrey Ely for bringing us to this topic and Stephen Morris for encouragement. We
also would like to thank Tilman Börgers, Eduardo Faingold, Alex Gershkov, Takashi Kunimoto, Qingmin
Liu, Georg Nöldeke, Satoru Takahashi, Olivier Tercieux, Takuro Yamashita for helpful discussions, and
conference participants at the 14th SAET Conference on Current Trends in Economics for their valuable
comments. All errors are our own.
† Department of Economics, National University of Singapore, [email protected]
‡ Department of Economics, National University of Singapore, [email protected]
1
1
Introduction
Wilson (1987) criticizes applied game theory’s reliance on common-knowledge assumptions. This has motivated the search for detail-free mechanisms in the mechanism design
world. In practice, this has meant to use simple mechanisms such as dominant-strategy
mechanisms. A dominant-strategy mechanism does not rely on any assumptions of agents’
beliefs and is robust to changes in agents’ beliefs. However, dominant-strategy mechanisms constitute just one special class of detail-free mechanisms and a fundamental issue
is to justify the leap from detail-free mechanisms in general to dominant-strategy mechanisms in particular.1 In an important paper, Chung and Ely (2007) study the foundations
to justify the use of dominant-strategy mechanisms in the single unit private-value auction
setting. In this paper, we revisit this question in general social choice environments with
quasi-linear preferences and private values.
Throughout this paper, we suppose that the mechanism designer has an estimate of
the distribution of the agents’ payoff types, but he does not have any reliable information
about the agents’ beliefs (including their beliefs about one another’s payoff types, their
beliefs about these beliefs, etc.), as these are difficult to observe. Furthermore, the designer
does not want to make restrictive assumptions about such beliefs. We say that the designer is a maxmin decision maker if he ranks mechanisms according to their worst-case
performance, that is, the minimum expected utility where the minimum is taken over
all possible agents’ beliefs. The use of dominant-strategy mechanisms has a maxmin
foundation if the mechanism designer as a maxmin decision maker finds it optimal to use
a dominant-strategy mechanism.
The notion of maxmin foundation is best illustrated graphically. Consider Figure
1 excerpted from Chung and Ely (2007), where the horizontal axis represents different
assumptions about (distributions of) agents’ beliefs, the vertical axis represents the mechanism designer’s expected utility and different lines represent different mechanisms.
Dominant-strategy mechanisms (including the optimal dominant-strategy mechanism)
always secure a fixed expected utility for the mechanism designer, being independent of
agents’ beliefs. Therefore, the optimal dominant-strategy mechanism is represented by the
1 While
this issue arises for mechanism designers with different objectives (for example, efficiency/
welfare), this paper is mainly concerned with revenue maximization. The mechanism designer’s utility is
taken to be the total transfer from the agents.
2
straight line. Graphically, maximin foundation means that the graph of an arbitrary detailfree mechanism must dip below (or intersect) the graph of the optimal dominant-strategy
mechanism at some point.
Figure 1: The graph of any detail-free mechanism dips below (or intersects) the graph of
the optimal dominant-strategy mechanism at some point.
A closely related notion is Bayesian foundation. We say that the use of dominantstrategy mechanisms has a Bayesian foundation if there is a particular assumption about
(the distribution of) agents’ beliefs, against which the optimal dominant-strategy mechanism achieves the highest expected utility among all detail-free mechanisms. If there exists
such an assumption, then the worse case expected utility of an arbitrary detail-free mechanism cannot exceed its expected utility against this particular assumption, which in turn
cannot exceed the (fixed) expected utility of the optimal dominant-strategy mechanism.
Therefore, Bayesian foundation is a stronger notion than maxmin foundation.
In the context of an expected revenue maximizing auctioneer, Chung and Ely (2007)
show that a regularity condition on the distribution of the bidders’ valuations ensures the
maxmin/ Bayesian foundations for dominant-strategy mechanisms. It is well known that
the revenue maximizing dominant-strategy auction can be found by solving a "relaxed"
problem which, roughly speaking, corresponds to an assumption that only the "downward"
incentive constraints for bidders are binding. More importantly, the set of binding incentive
constraints is robust to changes in other bidders’ reports. As it turns out, this embedded
3
structure in the single unit auction setting is the key for Chung and Ely’s results.
Based on a graph-theoretic formulation of incentive constraints, we formulate such
a structure and examine its applicability in many important economic environments.
Consider the optimal dominant-strategy mechanism ( p∗ , t∗ ). Fix other agents’ reports v−i ,
v
we define an order i −i on agent i’s payoff types. Such order is closely related with the set
of binding incentive constraints.2 Say there is uniform order if for each agent i, the order
on agent i’s payoff types is independent of other agents’ report v−i . We show that under
regularity and nonsingularity, uniform order ensures the maxmin/ Bayesian foundations
for dominant-strategy mechanisms.3
Uniform order is naturally satisfied in environments with linear utilities and onedimensional types. This fits many classical applications of mechanism design, including
single unit auction (e.g., Myerson (1981)), public good (e.g., Mailath and Postlewaite (1990))
and standard bilateral trade (e.g., Myerson and Satterthwaite (1983)). In this case, the
payoff types are completely ordered via a single path. Uniform order can also be satisfied
in nonlinear utility environments and multi-dimensional environments. For multi-unit
auctions with capacitated bidders (a representative multi-dimensional environment), the
agent’s payoff types are located on different paths and are only partially ordered.
Lastly, we show that when uniform order is not satisfied, maxmin/ Bayesian foundations might not exist. We consider a bilateral trade model with ex ante unidentified traders.
In this important economic environment, we construct an example whereby uniform order
is not satisfied and we explicitly construct one Bayesian mechanism that does strictly better
than the optimal dominant-strategy mechanism, regardless of the assumption about (the
distribution of) agents’ beliefs. In other words, there is no maxmin/ Bayesian foundation.4
The rest of the paper is organized as follows. The reminder of this introduction
discusses some related literature. Section 2 presents the notations, concepts and the model.
2 Such interpretation of incentive constraints is not uncommon and has been used before by Rochet (1987),
Border and Sobel (1987), Rochet and Stole (2003), Heydenreich, Müller, Uetz, and Vohra (2009) and Vohra
(2011).
3 Chung and Ely (2007) have shown that, when regularity is not satisfied, the use of dominant-strategy
mechanisms may not have a Bayesian foundation. We discuss the singular case in Section 6.2.
4 (Chung and Ely, 2007, Proposition 2) present an example whereby their regularity condition is violated
and for each assumption about the agents’ beliefs, there is a Bayesian mechanism that does strictly better
than the optimal dominat-strategy mechanism. Thus, while the example shows that a Bayesian foundation
does not exist, it leaves open the existence of a maxmin foundation in a single unit auction setting.
4
Section 3 formulates the notion of uniform order and presents our main result. Section
4.1 applies the result in a standard social choice environment with linear utilities and onedimensional types, Section 4.2 extends the discussion to a nonlinear utility environment
and Section 4.3 extends the discussion to a multi-dimensional environment. In Section
5, we consider a prominent bilateral trade model with ex ante unidentified traders and
illustrate that when uniform order is not satisfied, the maxmin/ Bayesian foundations
might not exist. Section 6 concludes with several discussions.
1.1
Related literature
Several papers have sought to offer a foundation for dominant-strategy mechanisms. In
a seminal paper, Bergemann and Morris (2005) examine the implementability of a given
social choice correspondence (SCC) and ask when ex post implementation (equivalent
to dominant-strategy implementation in private value settings) is equivalent to interim
(or Bayesian) implementation for all possible type spaces. They focus exclusively on
mechanisms in which the outcome can depend only on payoff relevant data and these
mechanisms are naturally suited to study efficient design. In contrast, Chung and Ely
(2007) and this paper study revenue maximization for the mechanism designer and the
optimal mechanism will almost always depend not just on the payoff types, but also on
payoff-irrelevant data, such as beliefs and higher order beliefs.
Nevertheless, we highlight the following similarity. Under a separability condition,
Bergemann and Morris (2005) show that if a SCC cannot be implemented ex post, it
cannot be interim implemented for all type spaces. In other words, Bayesian mechanisms
cannot do better (in terms of implementing the SCC) than the ex post incentive compatible
mechanisms at some type space. In this sense, the notion of maxmin foundation is
reminiscent of the problem in their setting, but the mechanism designer in our setting has
a different objective - revenue maximization. The main contribution of this paper is, for
expected revenue maximizing mechanism designers, we identify a sufficient condition for
the maxmin/ Bayesian foundations of dominant-strategy mechanisms.
Another contribution of this paper is to add to the family of results where Bayesian
mechanism robustly improves over dominant-strategy mechanisms. Bergemann and
Morris (2005) points out that in non-separable environments, dominant-strategy implementability may be a stronger requirement than Bayesian implementability on all type
5
spaces. In particular, in the prominent quasilinear environment with budget balance, once
there are more than two agents and at least one agent has at least three types, a SCC can
be interim implementable on all type spaces without being ex post implementable.5 In
this paper, we highlight the role of uniform order in the foundation of dominant-strategy
mechanisms and illustrate in Section 5 that, in environments where uniform order is not
satisfied, a single Bayesian mechanism can robustly achieve higher expected revenue than
the optimal dominant-strategy mechanism.
This paper also joins a recently growing literature exploring mechanism design with
worst case objectives. This includes the work of Hurwicz and Shapiro (1978), Frankel (2014),
Garrett (forthcoming), Carroll (forthcoming), Carroll (2014) and work such as Yamashita
(2014) that studies the mechanism design problem of guaranteeing desirable performances
whenever agents are rational in the sense of not playing weakly dominated strategies.
Wolitzky (2014) studies mechanism design where agents have maxmin preferences. Börgers
(2013) criticizes the notion of the maxmin foundation, the discussion of which is relegated
to the last section.
Lastly, a line of literature focuses on the equivalence of Bayesian and dominantstrategy implementation. Manelli and Vincent (2010) shows that in the standard context of
single unit private value auctions, for any Bayesian incentive compatible auction, there
exists an equivalent dominant-strategy incentive compatible auction that yields the same
interim expected utilities for all agents. Gershkov, Goeree, Kushnir, Moldovanu, and Shi
(2013) extend this equivalence result to social choice environments with linear utilities
and independent, one-dimensional, private types. Goeree and Kushnir (2013) provides a
simpler proof by showing that the support functions of the set of the interim values under
Bayesian and dominant-strategy incentive compatibility are identical. These results are
valuable, as they imply that w.l.o.g. the mechanism designer could restrict attention to
dominant-strategy mechanisms. The key difference between our paper and these results
lies in that in our setting, the mechanism designer does not make any assumptions about
the agents’ beliefs.6
5 More
literature along this direction include Meyer-Ter-Vehn and Morris (2011) and Yamashita (2014).
our result holds even in nonlinear utility environments and multi-dimensional environments.
6 Moreover,
See Section 4.2 and Section 4.3.
6
2
Preliminaries
2.1
The environment
We consider an environment with a finite set I = {1, 2, ..., I } of risk neutral agents and a
finite set K = {1, 2, ..., K } of social alternatives. The set of possible payoff types of agent
i is a finite set Vi ⊂ RK .7 We denote a payoff type profile by v = (v1 , v2 , ..., v I ). The set
of all possible payoff type profiles is V ≡ V1 × V2 × · · · × VI . We write v−i for a payoff
type profile of agent i’s opponents, i.e., v−i ∈ V−i = × j6=i Vj . If Y is a measurable set, then
∆Y is the set of all probability measures on Y. If Y is a metric space, then we treat it as a
measurable space with its Borel σ-algebra.
2.2
Types
Agents’ information is modelled by a type space. A type space, denoted Ω = (Ωi , f i , gi )i∈ N
is defined by a measurable space of types Ωi for each agent, and a pair of measurable
mappings f i : Ωi → Vi , defining the payoff type of each type, and gi : Ωi → ∆(Ω−i ),
defining each type’s belief about the types of the other agents.
A type space encodes in a parsimonious way the beliefs and all higher-order beliefs
of the agents. One simple kind of type space is the naive type space generated by a payoff
type distribution π ∈ ∆ (V ) . In the naive type space, each agent believes that all agents’
payoff types are drawn from the distribution π, and this is common knowledge. Formally,
a naive type space associated with π ∈ ∆ (V ) is a type space Ωπ = (Ωi , f i , gi )i∈ N such that
Ωi = Vi , f i (vi ) = vi , and gi (vi )[v−i ] = π (v−i |vi ) for every vi and v−i .
The naive type space is used almost without exception in auction theory and mechanism design. The cost of this parsimonious model is that it implicitly embeds some
strong assumption about the agents’ beliefs, and these assumptions are not innocuous.
For example, if the agents’ payoff types are independent under π, then in the naive type
space, the agents’ beliefs are common knowledge. On the other hand, for a generic π, it is
common knowledge that there is a one-to-one correspondence between payoff types and
beliefs. The spirit of the Wilson Doctrine is to avoid making such assumptions.
7v
i
∈ Vi ⊂ RK represents agent i’s gross utility vector under all K alternatives. Sometimes we may use
different ways to represent agent i’s payoff types vi , when it is more convenient to do so.
7
To implement the Wilson Doctrine, the common approach is to maintain the naive
type space, but try to diminish its adverse effect by imposing stronger solution concepts.
To provide foundations for this methodology, we have to return to the fundamentals.
Formally, weaker assumptions about the agents’ beliefs are captured by larger type spaces.
Indeed, we can remove these assumptions altogether by allowing for every conceivable
hierarchy of higher-order beliefs. By the results of Mertens and Zamir (1985), there exists a
universal type space, Ω∗ = (Ωi∗ , f i∗ , gi∗ )i∈ N , with the property that, for every payoff type
vi and every infinite hierarchy of beliefs ĥi , there is a type of player i, ωi , with payoff type
vi and whose hierarchy is ĥi . Moreover, each Ωi∗ is a compact topological space.
2.3
Mechanisms
A mechanism consists of a set of messages Mi for each agent i, a decision rule p : M → ∆K,
and payment functions ti : M → R, one for each agent i. Each agent i selects a message
from Mi , and based on the resulting profile of messages m, the decision rule p specifies the
outcome from ∆K (lotteries are allowed) and the payment function ti specifies the transfer
of agent i to the mechanism designer. Agent i obtains utility p · vi − ti , for decision rule
p ∈ ∆K and transfer ti .
The mechanism defines a game form, which together with the type space constitutes
a game of incomplete information. The mechanism design problem is to fix a solution
concept and search for the mechanism that delivers the maximum expected utility for the
mechanism designer in some outcome consistent with the solution concept. To implement
the Wilson Doctrine and minimize the role of assumptions built into the naive type space,
the common approach is to adopt a strong solution concept which does not rely on these
assumptions. In practice, the often-used solution concept for this purpose is dominantstrategy equilibrium.
The revelation principle holds, and we can restrict attention to direct mechanisms.
Definition 1. A direct-relevation mechanism Γ for a type space Ω is dominant-strategy incentive
compatible (dsIC) if for each agent i and type profile ω ∈ Ω,
p(ω ) · f i (ωi ) − ti (ω ) ≥ 0, and
p(ω ) · f i (ωi ) − ti (ω ) ≥ p(ωi0 , ω−i ) · f i (ωi ) − ti (ωi0 , ω−i ),
for any alternative type ωi0 ∈ Ωi .
8
Definition 2. A dominant-strategy mechanism is a dsIC direct-revelation mechanism for the naive
type space Ωπ . We denote by Φ the class of all dominant-strategy mechanisms.
To provide a foundation for using the optimal dominant-strategy mechanism, we
shall compare it to the route of completely eliminating common-knowledge assumptions
about beliefs. We maintain the standard solution concept of Bayesian equilibrium, but
now we enlarge the type space all the way to the universal type space.
Definition 3. A direct-revelation mechanism Γ for type space Ω = (Ωi , f i , gi ) is Bayesian
incentive compatible (BIC) if for each agent i and type ωi ∈ Ωi ,
Z
Ω −i
Z
Ω −i
( p(ω ) · f i (ωi ) − ti (ω )) gi (ωi )dω−i ≥ 0, and
( p(ω ) · f i (ωi ) − ti (ω )) gi (ωi )dω−i ≥
Z
Ω −i
p(ωi0 , ω−i ) · f i (ωi ) − ti (ωi0 , ω−i ) gi (ωi )dω−i
for any alternative type ωi0 ∈ Ωi .
Definition 4. Let Ψ be the class of all BIC direct-revelation mechanism for the universal type
space. We say that such a mechanism is detail free.
2.4
The mechanism designer as a maxmin decision maker
The designer has an estimate of the distribution of the agents’ payoff types, π. Following
Chung and Ely (2007), we assume that π has full-support. An assumption µ about the
distribution of the payoff types and beliefs of the agents is consistent with this estimate
if the induced marginal distribution on V is π. Let M(π ) denote the compact subset
of such assumptions. The mechanism designer derives utility from the transfers from
the agents.8 For any mechanism Γ, the µ−expected utility/ revenue of Γ is defined as
R
Rµ (Γ) = Ω∗ ∑i∈I ti (ω )dµ(ω ).
We do not assume that the designer has confidence in the naive type space as his
model of agents’ beliefs. Instead, the designer considers other assumptions within the set
8 We
see no difficulty in extending our results to cover slightly more general objective function as long as
the mechanism designer’s objective function is increasing w.r.t. the transfers from the agents. A prominent
such case would be sometimes, the mechanism designer also has an interest in the selected alternative.
For example, consider the public good setting, if the selected alternative is to provide the public good, the
mechanism designer incurs the cost of providing the good.
9
M (π ). The designer as a maxmin decision maker solves the maxmin problem of
sup
inf
Γ∈Ψ µ∈ M(π )
R µ ( Γ ).
If the mechanism designer chooses to use a dominant-strategy mechanism, his maximum revenue would be Π D (π ) = supΓ∈Φ Rπ (Γ), where Rπ (Γ) = ∑v∈V π (v) ∑i∈I ti (v)
for any dominant-strategy mechanism Γ ∈ Φ. We can formulate the optimal dominant
strategy mechanism design problem as follows:
max
∑ π ( v ) ∑ ti ( v )
p(·),t(·) v∈V
(B.1)
i ∈I
subject to ∀i ∈ I , ∀vi , vi0 ∈ Vi , ∀v−i ∈ V−i ,
p(v) · vi − ti (v) ≥ 0,
p(v) · vi − ti (v) ≥ p(vi0 , v−i ) · vi − ti (vi0 , v−i ).
h DIRvi i
D
E
0
DIC vi →vi
Lemma 1. The maximization problem ( B.1) has a solution.
Proof. By h DIRvi i, ti (v) is bounded from above. Since the designer can guarantee a fixed
revenue by choosing a constant decision rule p ∈ ∆K and payment ti (v) = minvi ∈Vi p · vi ,
we can assume w.l.o.g. ti (v) is bounded from below. Moreover, the set of p(·) and t(·)
that satisfy the constraints is a closed set. Hence, the constraint set is closed and bounded.
Since the objective function is a continuous function of t (·) , the maximization problem
admits a solution.
Say that a decision rule p is dsIC if there exists transfer scheme t such that the
D
E
0
mechanism ( p, t) satisfies the constraints h DIRvi i and DIC vi →vi . We omit the proof of
the following standard lemma. For a detailed explanation and proof of the lemma, please
see Rochet (1987).
Lemma 2. A necessary and sufficient condition for a decision rule p to be dsIC is the following
cyclical monotonicity condition: ∀i = 1, ..., I, ∀v−i ∈ V−i and every sequence of length k ∈ N of
payoff types of agent i, (vi,1 , vi,2 , ..., vi,k ) with vi,k = vi,1 , we have
k −1
∑ [ p(vi,κ , v−i ) · vi,κ+1 − p(vi,κ , v−i ) · vi,κ ] ≤ 0.
κ =1
10
3
3.1
Main result
Uniform order
To formally define uniform order, we first present an order-based interpretation of incentive
constraints.
Definition 5. Consider the optimal dominant-strategy mechanism ( p∗ , t∗ ). Fix other agents’
v
reports v−i , we define an order i −i on agent i’s payoff types using the following recursive
procedure:
Step 1) Denote by ViT = {vi ∈ Vi : p∗ (v) · vi − ti∗ (v) = 0}. That is, ViT is the collection of
payoff types such that agent i is indifferent between reporting truthfully and not participating. Let
Vi0 = ViT .
Step 2) Choose some vi ∈ Vi − Vi0 such that p∗ (v) · vi − ti∗ (v) = p∗ (vi− , v−i ) · vi −
ti∗ (vi− , v−i ) for some vi− ∈ Vi0 . That is, agent i is indifferent between reporting truthfully and
v
reporting vi− when his payoff type is vi . Write vi i −i vi− and update Vi0 = Vi0 ∪ {vi }.
0
Step 3) Repeat Step 2) until Vi = Vi .
Remark 1. (a) We show in a series of lemmas that such recursive procedure is well defined. Lemma
3 says that ViT 6= ∅. Hence, Step 1) is well defined. Lemma 4 says that if Vi0 ⊇ ViT and Vi − Vi0
v
6= ∅, there exists vi ∈ Vi − Vi0 such that vi i −i vi− for some vi− ∈ Vi0 . Hence, Step 2) is well
defined. Finally, since each agent has finitely many payoff types, the procedure ends after finitely
many rounds.
(b) Graphically, fix any order, the ordering of payoff types comprises a "tree" - a connected
graph without cycles such that, there is a unique path from any payoff type to some vi ∈ ViT .
Furthermore, if vi0 belongs to the path from some payoff type to some vi ∈ ViT , the truncation of the
path from vi0 to vi defines the path from vi0 to vi .
(c) Agent i with payoff type vi could be indifferent between reporting many distinct payoff
types. Hence, the order may not be unique. Having said that, if we impose a genericity condition,
there is a unique order from some vi ∈ ViT to any payoff type. Please refer to (Rochet and Stole,
2003, Page 163).
Lemma 3. Fix other agents’ reports v−i ,
p∗ (v) · vi − ti∗ (v) = 0 for some vi ∈ Vi .
11
Proof. Suppose not. p∗ (v) · vi − ti∗ (v) > 0 for all vi ∈ Vi . Let
λ = min { p∗ (v) · vi − ti∗ (v)} > 0.
vi ∈Vi
The designer could increase ti∗ (vi , v−i ) by λ for all vi ∈ Vi and achieve a higher expected
revenue. We reach a contradiction.
Lemma 4. Fix other agents’ reports v−i . If Vi0 ⊇ ViT and Vi − Vi0 6= ∅, there exists vi ∈ Vi − Vi0
such that
p∗ (v) · vi − ti∗ (v) = p∗ (vi− , v−i ) · vi − ti∗ (vi− , v−i )
for some vi− ∈ Vi0 .
Proof. Suppose not.
p∗ (vi , v−i ) · vi − ti∗ (vi , v−i ) > p∗ (vi0 , v−i ) · vi − ti∗ (vi0 , v−i ) for vi ∈ Vi − Vi0 , vi0 ∈ Vi0 .
Furthermore, inequalities (1) - (2) hold,
p∗ (vi , v−i ) · vi − ti∗ (vi , v−i ) > 0 for vi ∈ Vi − Vi0 ;
(1)
p∗ (vi , v−i ) · vi − ti∗ (vi , v−i ) ≥ p∗ (vi0 , v−i ) · vi − ti∗ (vi0 , v−i ) for vi , vi0 ∈ Vi − Vi0 .
(2)
If (1) is violated, p∗ (v) · vi − ti∗ (v) = 0 for some vi ∈ Vi − Vi0 , we have a contradiction. (2)
is simply the incentive constraint for vi , vi0 ∈ Vi − Vi0 .
Let
a =
b =
min { p∗ (vi , v−i ) · vi − ti∗ (vi , v−i )} > 0,
vi ∈Vi −Vi0
min
vi ∈Vi −Vi0 ,vi0 ∈Vi0
{ p∗ (vi , v−i ) · vi − ti∗ (vi , v−i ) − [ p∗ (vi0 , v−i ) · vi − ti∗ (vi0 , v−i )]} > 0,
λ = min{ a, b} > 0.
The designer could increase ti∗ (vi , v−i ) by λ for all vi ∈ Vi − Vi0 . The expected revenue is
higher and no constraints are violated. We reach a contradiction.
Definition 6. There is uniform order if for each agent i ∈ I , there exists an order i that holds
for all v−i .
Remark 2. With uniform order, we drop the superscript and denote the uniform order of agent i
by i and its transitive closure by i+ . For notational convenience, write vi0 i+ vi if vi0 i+ vi or
vi0 = vi and write ViI = {vi ∈ Vi : there is no vi0 such that vi0 i vi }.
12
With uniform order, the rent of any payoff type is easily calculated and this further
gives the maximum expected revenue for the mechanism designer if the mechanism
designer insists on a dominant-strategy mechanism. We record this as the following
proposition.
Proposition 1. With uniform order i , the maximum expected revenue the mechanism designer
could achieve if he insists on a dominant-strategy mechanism is

∑ ∑
∑
π ( vi , v −i )  p ∗ ( v ) · vi −
i ∈I vi ∈Vi v−i ∈V−i
∑+
vi0 i

p∗ ((vi0 )− , v−i ) · [vi0 − (vi0 )− ] .
(3)
vi
Proof. We first calculate the rent of the payoff types as follows:
Ui (v) = p∗ (v) · vi − ti∗ (v) = 0 for vi ∈ ViT ;
Ui (v) = p∗ (v) · vi − ti∗ (v)
= p∗ (vi− , v−i ) · vi − ti∗ (vi− , v−i )
= p∗ (vi− , v−i ) · vi− − ti∗ (vi− , v−i ) + p∗ (vi− , v−i ) · (vi − vi− )
= Ui (vi− , v−i ) + p∗ (vi− , v−i ) · (vi − vi− ) for vi ∈ Vi − ViT .
By induction,
Ui (v) =
∑+
p∗ ((vi0 )− , v−i ) · [vi0 − (vi0 )− ].9
vi0 i vi
Therefore,
ti∗ (v) = p∗ (v) · vi − Ui (v)
= p ∗ ( v ) · vi −
∑+
p∗ ((vi0 )− , v−i ) · [vi0 − (vi0 )− ].
vi0 i vi
It follows that
∑ π (v) ∑ ti∗ (v)
v ∈V
i ∈I


=
∑ ∑
∑
π ( vi , v −i )  p ∗ ( v ) · vi −
i ∈I vi ∈Vi v−i ∈V−i
9 For
∑+
vi0 i vi
simplicity of exposition, write vi − = vi for vi ∈ ViT .
13
p∗ ((vi0 )− , v−i ) · [vi0 − (vi0 )− ] .
Definition 7. Say π is regular if p∗ solves the reduced objective function under p∗ . That is,


p∗ ∈ arg max ∑
∑
∑
π ( vi , v −i )  p ( v ) · vi −
i ∈I vi ∈Vi v−i ∈V−i
∑+
p((vi0 )− , v−i ) · [vi0 − (vi0 )− ] .
vi0 i vi
To the best of our knowledge, there is no formal definition of regularity in the general
environments. Our definition is not in terms of primitive condition, but it does reduce to
familiar regularity condition in environments such as single unit auction. Furthermore, our
regularity is defined in such a way that is consistent with its usage in the literature. Loosely
speaking, to ensure that the cyclical monotonicity constraint is automatically satisfied for
the maximizer of the "relaxed" problem.
Definition 8. Let σvi = π (v−i |vi ) be the conditional distribution over the payoff types of agents
j 6= i, conditional on agent i having payoff type vi . Say π is nonsingular if the collection of vectors
{σvi }vi ∈Vi is linearly independent.
Theorem 1. If π is regular, nonsingular and there is the uniform order, then the use of dominantstrategy mechanisms has a Bayesian/ maxmin foundation. That is,
sup
inf
Γ∈Ψ µ∈ M(π )
R µ ( Γ ) = Π D ( π ).
The proof is relegated to the Appendix.
An illustrating example10
3.2
Imagine that you and your colleagues are buying a coffee maker to be kept at work. While
everyone is in favor, different people might have different preferences as to which model
to buy. We consider the problem of selecting a public good among mutually exclusive
choices. While all agents are in favor of providing a public good, different agents might
have different preferences as to which public good to provide.
We consider 2 agents and 2 public goods. Each agent is either indifferent between
the two public goods, or strictly prefers a particular one. Each agent is described by a
pair v1 = (1, 1), v2 = (2, 1) or v3 = (1, 2), where the first number denotes the agent’s
10 In
this illustrating example and the applications in Section 4, to show that there is uniform order, we
show a stronger condition - the order on agent’s payoff types is independent of other agents’ reports v−i , for
all decision rule p.
14
valuation for public good 1 and the second number denotes his valuation for public good 2.
The valuations of the public goods are privately known to the agents and the mechanism
designer chooses a mechanism that maximizes the expected profit. The costs of the public
goods are normalized to be 0.
Each agent submits his payoff type and the mechanism designer chooses decision
rule p(v) = ( p1 (v), 1 − p1 (v)) ∈ [0, 1]2 , where p1 (v) is the probability of providing public
good 1 and 1 − p1 (v) is the probability of providing public good 2. Consider the optimal
dominant-strategy mechanism design problem. Fix p and v−i . To ensure that agent i has
the incentive to report truthfully and voluntarily participate, the mechanism design must
take care of the following constraints:
p(vm , v−i ) · vm − ti (vm , v−i ) ≥ p(vl , v−i ) · vm − ti (vl , v−i ) and
p(vm , v−i ) · vm − ti (vm , v−i ) ≥ 0, ∀l, m = 1, 2, 3.
We show that for the optimal transfer scheme t∗ ,
p(v1 , v−i ) · v1 − ti∗ (v1 , v−i ) = 0;
p(v2 , v−i ) · v2 − ti∗ (v2 , v−i ) = p(v1 , v−i ) · v2 − ti∗ (v1 , v−i );
p(v3 , v−i ) · v3 − ti∗ (v3 , v−i ) = p(v1 , v−i ) · v3 − ti∗ (v1 , v−i ).
First, it must be the case that p(v1 , v−i ) · v1 − ti∗ (v1 , v−i ) = 0. Suppose not,
p(v1 , v−i ) · v1 − ti∗ (v1 , v−i ) > 0 and
p(v j , v−i ) · v j − ti∗ (v j , v−i ) ≥ p(v1 , v−i ) · v j − ti∗ (v1 , v−i )
≥ p(v1 , v−i ) · v1 − ti∗ (v1 , v−i )
> 0 for j = 2, 3.
Therefore, the mechanism designer could increase ti (v j , v−i ), j = 1, 2, 3 by the same amount
slightly and achieve a higher expected revenue.
Second, since
p(v1 , v−i ) · v1 − ti∗ (v1 , v−i ) ≥ p(v3 , v−i ) · v1 − ti∗ (v3 , v−i ) and
p(v3 , v−i ) · v3 − ti∗ (v3 , v−i ) ≥ p(v1 , v−i ) · v3 − ti∗ (v1 , v−i ),
we have
p(v3 , v−i ) · (v3 − v1 ) ≥ p(v1 , v−i ) · (v3 − v1 ), or equivalently
p ( v1 , v −i ) · ( v2 − v1 ) ≥ p ( v3 , v −i ) · ( v2 − v1 ).
15
Therefore,
p(v1 , v−i ) · v2 − ti∗ (v1 , v−i ) ≥ p(v1 , v−i ) · v1 − ti∗ (v1 , v−i ) = 0 and
p(v1 , v−i ) · v2 − ti∗ (v1 , v−i ) = p(v1 , v−i ) · v1 − ti∗ (v1 , v−i ) + p(v1 , v−i ) · (v2 − v1 )
≥ p(v3 , v−i ) · v1 − ti∗ (v3 , v−i ) + p(v3 , v−i ) · (v2 − v1 )
= p ( v3 , v −i ) · v2 − t i ( v3 , v −i ).
Then we have
p(v2 , v−i ) · v2 − ti∗ (v2 , v−i ) = p(v1 , v−i ) · v2 − ti∗ (v1 , v−i ).
Similarly,
p(v3 , v−i ) · v3 − ti∗ (v3 , v−i ) = p(v1 , v−i ) · v3 − ti∗ (v1 , v−i ).
(1,2)
(1,1)
(2,1)
Figure 2: The set of binding incentive constraints is the same regardless of the decision
rule and the other agent’s report. The uniform order: (1, 2) i (1, 1) and (2, 1) i (1, 1)
4
Applications
4.1
One-dimensional environments with linear utilities
In this and the next subsections, we show that uniform order is satisfied in several classical environments. We first consider a standard social choice environment with onedimensional payoff types and linear utilities.11 This fits many classical applications of
mechanism design, including single unit auction (e.g., Myerson (1981)), public good
11 Hence,
our set-up covers the environment studied in (Gershkov, Goeree, Kushnir, Moldovanu, and Shi,
2013, Page 199, Section 2).
16
(e.g., Mailath and Postlewaite (1990)) and standard bilateral trade (e.g., Myerson and
Satterthwaite (1983)). There is a finite set I = {1, 2, ..., I } of risk neutral agents and a finite
set K = {1, 2, ..., K } of social alternatives. Agent i’s gross utility in alternative k equals
uik (vi ) = aik vi , where vi ∈ R is agent i’s payoff type, aik ∈ R are constants and aik ≥ 0. Agent
i obtains utility
p ( v ) · Ai vi − ti ( v )
for decision rule p ∈ ∆K and transfer ti , where Ai = ( a1i , a2i , ..., aiK ). For notational simplicity,
we assume that each agent has M possible payoff types and that the set Vi is the same for
each agent i : Vi = {v1 , v2 , ..., v M }, where vm > vm−1 for m = 2, ..., M.
Proposition 2. There is uniform order: v M i v M−1 i ... i v1 .
The proof is standard and relegated to the Appendix.
4.2
Nonlinear utility
In this subsection, we adopt the set-up of (Maskin and Riley, 1989, Chapter 14, pp312-335)
and consider the problem of finding the revenue maximizing multi-unit auction. The
seller has Q units to sell. There is a finite set I = {1, 2, ..., I } of risk neutral buyers where
buyer i has a demand curve p = p( x, vi ), where vi is buyer i’s payoff type.12 That is, the
seller has uncertainty about the demand curve of the buyers. For notational simplicity,
we assume that each buyer has M possible payoff types and that the set Vi is the same for
each buyer i : Vi = {v1 , v2 , ..., v M }, where vm > vm−1 for m = 2, ..., M. A buyer of type vm
has preferences represented by the utility function
qi (vm ,v−i )
∑
p( x, vm ) − ti (vm , v−i ),
x =1
where qi (vm , v−i ) is the number of units purchased from the seller and ti (vm , v−i ) is the
total spending on these units. Throughout, we shall assume that higher levels of v are
associated with higher demand. To be precise, p( x, v) is increasing in v.
Proposition 3. This is uniform order: v M i v M−1 i ... i v1 .
The proof is similar to the proof of Proposition 2 and omitted.
12 Buyer’s
demand curve need not be linear.
17
Multi-dimensional environments13
4.3
In this subsection, we consider multi-unit auction with capacity-constrained bidders.14
Consider the problem of finding the revenue maximizing auction when bidders have
constant marginal valuations as well as capacity constraints. Both the marginal values
and capacity constraints are private information to the bidder. We denote a bidder’s type
by ( a, b), where a is the maximum amount he is willing to pay for each unit and b is the
largest number of units he seeks. That is, units beyond the bth unit are worthless. Let the
range of a be R = {1, 2, ..., R} and the range of b be K = {1, 2, ..., K }. The seller has Q units
to sell.
A crucial assumption is that no bidder can inflate his capacity but can shade it down.
In other words, the auctioneer can verify, partially, the claims made by a bidder. Although
this assumption seems odd in the selling context, it is natural in a procurement setting.
Consider a procurement auction where the auctioneer wishes to procure Q units from
bidders with constant marginal costs and limited capacity. No bidder will inflate his
capacity when bidding because of the huge penalties associated with not being able to
fulfill the order. Equivalently, we may suppose that the designer can verify that claims
that exceed capacity are false.
Following (Vohra, 2011, pp150-159), we can verify that uniform order is satisfied in
such an environment.
Proposition 4. There is uniform order: (i, j) i (i − 1, j) i ... i (1, j) i (1, j − 1) i ... i
(1, 1).
The proof is similar to (Vohra, 2011, pp150-159) and omitted.
13 Solving
for the optimal mechanism in a multi-dimensional environment is in general a daunting task.
See Rochet and Stole (2003) for a survey of multi-dimensional screening.
14 (Vohra, 2011, pp150-159) studies the optimal Bayesian mechanism in such an environment, assuming
that types are independent.
18
3,3
2,3
1,3
3,2
2,2
1,2
3,1
2,1
1,1
Figure 3: (i, j) i (i − 1, j) i ... i (1, j) i (1, j − 1) i ... i (1, 1).
5
Bilateral trade with ex ante unidentified traders
In this section, we consider a bilateral trade model with ex ante unidentified traders.15 In
the context of this important economic environment, this example illustrates that, when
the uniform order condition is not satisfied, maxmin/ Bayesian foundations might not
exist. Section 5.1 presents the basics of the model, Section 5.2 calculates the maximum
expected revenue that could be achieved by a dominant-strategy mechanism, and Section
5.3 constructs a Bayesian mechanism that achieves a strictly higher expected revenue.
5.1
Setup
We consider the problem of designing a rule to determine the terms of trade between
two agents. Each agent is endowed with
1
2
unit of a good to be traded and has private
information about his valuation for the good. Agent 1’s valuation for the good could be
either 1 or 3. Agent 2’s valuation for the good could be either 0 or 2. The mechanism
designer has the following estimate of the distribution of the agents’ valuations:
15 See
Cramton, Gibbons, and Klemperer (1987) and Lu and Robert (2001) for the motivation and complete
exposition of such bilateral trade models.
19
v2 = 0
v2 = 2
v1 = 1
v1 = 3
4
10
1
10
1
10
4
10
.
(4)
In this context, each agent may be either the buyer or the seller, depending on the
realization of the privately observed information and the choice of the mechanism. In
other words, whether an agent is the buyer or the seller is endogenously determined
by the agents’ reported valuations and cannot be identified prior to trade. The buyer’s
utility from purchasing p unit of the good and paying a transfer t B is pv B − t B and the
seller’s utility from selling p unit of the good and paying a transfer tS is − pvS − tS , where
0 ≤ p ≤ 12 . The mechanism designer chooses a mechanism that maximizes the expected
profit.
5.2
Optimal dominant-strategy mechanism
We show that the maximum expected profit the designer can generate from a dominantstrategy mechanism is 25 . To see this, we focus on the following case that an agent is
assigned to be the buyer if and only if the agent reports a higher valuation, where B (resp.
S) means that agent 1 is assigned to be the buyer (resp. seller).
v1 = 1
v1 = 3
v2 = 0
B
B
v2 = 2
S
B
(5)
We use p (v1 , v2 ) to denote the expected trading amount when agent 1 reports v1 and
agent 2 reports v2 . For example, in the assignment in (5), p (1, 0) denotes the expected
amount of good that agent 1 buys from agent 2 while p (1, 2) denotes the expected amount
of good that agent 1 sells to agent 2. We can formulate the optimal dominant-strategy
mechanism design problem as follows:
4
1
[t1 (1, 0) + t2 (1, 0)] +
[t1 (3, 0) + t2 (3, 0)]
10
10
1
4
+ [t1 (1, 2) + t2 (1, 2)] +
[t1 (3, 2) + t2 (3, 2)]
10
10
max
20
by choosing
1
p(1, 0), p(3, 0), p(1, 2), p(3, 2) ∈ 0,
, t i ( v1 , v2 ) ∈ R
2
subject to the following constraints:
1p(1, 0) − t1 (1, 0) ≥ max {0, 1p(3, 0) − t1 (3, 0)} ;
3p(3, 0) − t1 (3, 0) ≥ max {0, 3p(1, 0) − t1 (1, 0)} ;
−1p(1, 2) − t1 (1, 2) ≥ max {0, 1p(3, 2) − t1 (3, 2)} ;
3p(3, 2) − t1 (3, 2) ≥ max {0, −3p(1, 2) − t1 (1, 2)} ;
0p(1, 0) − t2 (1, 0) ≥ max {0, 0p(1, 2) − t2 (1, 2)} ;
2p(1, 2) − t2 (1, 2) ≥ max {0, −2p(1, 0) − t2 (1, 0)} ;
−0p(3, 0) − t2 (3, 0) ≥ max {0, −0p(3, 2) − t2 (3, 2)} ;
−2p(3, 2) − t2 (3, 2) ≥ max {0, −2p(3, 0) − t2 (3, 0)} .
Following standard procedure, the above maximization problem is equivalent to:
max
2
3
1
2
p(1, 0) + p(3, 0) + p(1, 2) + p(3, 2)
10
10
10
10
subject to monotonicity constraints:
p(3, 0) − p(1, 0) ≥ 0;
(6)
p(3, 2) + p(1, 2) ≥ 0;
(7)
p(1, 2) + p(1, 0) ≥ 0;
(8)
p(3, 0) − p(3, 2) ≥ 0.
(9)
Obviously, p(0, 1) = p(3, 2) = p(0, 2) = p(3, 1) = 1/2 solves the unconstrained
problem. The monotonicity constraints in (6)-(9) are automatically satisfied. The maximal
expected profit of the designer is 25 .
Remark 3. We also need to consider other possible assignment rules and calculate the maximum
expected revenue for each case. Similar calculations show that the maximal revenue for all other
cases are strictly less than 25 .
Therefore, the optimal dominant strategy mechanism Γ is as follows, where the first
number in each cell indicates the amount of good agent 1 buys from agent 2, the second
number is the transfer from agent 1 and the third number is the transfer from agent 2.
21
v1 = 1
v2 = 0
v2 = 2
1 1
2, 2, 0
− 12 , − 21 , 1
v1 = 3
1
2,
1
2,
1
2 , −1
3
2 , −1
(10)
Remark 4. The designer’s probability distribution in (4) is regular in the sense that p(0, 1) =
p(3, 2) = p(0, 2) = p(3, 1) = 1/2 solves the unconstrained problem.
5.3
No maxmin/ Bayesian foundation
We show that there is no maxmin foundation. That is, the mechanism designer could
employ a Bayesian mechanism and achieves a strictly higher expected revenue than he
does using the optimal dominant-strategy mechanism. To do this, we first explicitly
identify one mechanism and proceed by verifying i) the mechanism is BIC for the universal
type space; and ii) this mechanism achieves a strictly higher expected revenue than
2
5
regardless of the designer’s assumptions about bidders’ beliefs. Note that this implies there
is no Bayesian foundation. Following Chung and Ely (2007), we use a (resp. b) to denote
the first-order belief of a low-valuation (resp. high-valuation) type of agent 1 that agent
2 has low valuation; we use c (resp. d) to denote the first-order belief of a low-valuation
(resp. high-valuation) type of agent 2 that agent 1 has low valuation.
Proposition 5. There is no maxmin (Bayesian) foundation.
Proof. Consider the following mechanism Γ0 :
a ∈ [0, 1]
b ∈ [0, 1]
c ∈ [0, 12 )
0, 0, 1
c ∈ [ 12 , 1]
1 1
2, 2, 0
− 12 , − 21 , 1
1 3
2 , 2 , −1
1 1
2, 2, 0
1 3
2 , 2 , −1
d ∈ [0, 1]
.
To see that Γ0 is BIC for the universal type space, note that
i) truth telling continues to be a dominant strategy for agent 1;
ii) high type of agent 2 has a dominant strategy to tell the truth;
iii) c ∈ [0, 21 ) will not announce d ∈ [0, 1] as utility is unchanged;
iv) c ∈ [ 12 , 1] will not announce d ∈ [0, 1] as expected utility is lower; and
22
v) lastly, a low type of agent 2 will announce c ∈ [ 12 , 1] if and only if c ≥ 12 .
To see that Γ0 is expected revenue improving, note that Γ0 achieves at least
enue everywhere and hence expected revenue is at least
1
2
1
2
rev-
regardless of the designer’s
assumptions about bidders’ beliefs.
6
6.1
Discussion
Foundations of ex post incentive compatible mechanisms
Our paper focuses exclusively on the private value setting and a natural question to ask
is under regularity, whether uniform order provides foundations for ex post incentive
compatible mechanisms in the interdependent value setting. In a recent work by Yamashita
and Zhu (2014), they focus on the so-called "digital-goods" auctions and show that if each
agent has a stable preference ordering over all his payoff types, regardless of what payoff
type profile the other agents have, the use of ex post incentive compatible mechanisms
has a maxmin/ Bayesian foundation. We conjecture that our formulation of uniform order
has a natural extension in the interdependent setting and if properly formulated, ensures
maxmin/ Bayesian foundations of ex post incentive compatible mechanisms.
6.2
The case of singular π
For our result to cover singular cases, we conjecture that we need a stronger version of
uniform order - the order on agent’s payoff types needs to be robust to changes in other
agents’ reports v−i , the decision rule p and even the agent’s estimate of the distribution
of agents’ payoff types π. This possibly could help us find a sequence of nonsingular
distributions to approach any π satisfying uniform order and regularity. We can then
apply the limiting argument as in Chung and Ely (2007) and show the foundations of
dominant-strategy mechanisms. We note that all applications studied in this paper also
satisfy this stronger version of uniform order.
6.3
On the notion of maxmin foundation
Börgers (2013) argues that the maxmin foundation requires little of an optimal mechanism.
For every dominant-strategy mechanism, Börgers constructs another mechanism which
23
never yields lower revenue and yields strictly higher revenue sometimes. The construction
of this new mechanism is particularly simple and is built on the possibility of side bets.
Among other assumptions, Börgers requires at least three agents, and the mechanism
involves agents entering into side bets with each other on a third agent’s type. The new
mechanism achieves strictly higher revenue because the mechanism designer charges a
small fee for each bet.
In this perspective, we view the maxmin foundation as the minimal requirement that
the optimal mechanism needs to satisfy. This makes our example stronger, as when uniform
optimal path condition is violated, dominant-strategy mechanism does not even satisfy
the requirement of maxmin foundation. Lastly, if we restrict attention to the consistent
beliefs type spaces as defined in Morris (1994), the use of dominant-strategy mechanisms
would still have Bayesian/ maxmin foundations. And yet, betting opportunities as argued
in Börgers (2013) would no longer be desirable for the agents.
A
A.1
Appendix
Proof of Proposition 2
We can formulate the optimal dominant-strategy mechanism design problem as follows:
max
∑ π ( v ) ∑ ti ( v )
p(·),t(·) v∈V
i ∈I
subject to
∀i ∈ I , ∀m, l = 1, ..., M, ∀v−i ∈ V−i ,
p(vm , v−i ) · Ai vm − ti (vm , v−i ) ≥ 0,
p ( v m , v −i ) · A i v m − t i ( v m , v −i ) ≥ p ( v l , v −i ) · A i v m − t i ( v l , v −i ).
Lemma 5. Any dsIC allocation rule must be monotonic. That is, for m ≥ l, we have p(vm , v−i ) ·
Ai ≥ p ( v l , v −i ) · Ai .
Proof. Consider two types vm and vl with m ≥ l. Incentive compatibility requires
p(vm , v−i ) · Ai vm − ti (vm , v−i ) ≥ p(vl , v−i ) · Ai vm − ti (vl , v−i ) and
p ( v m , v −i ) · A i v l − t i ( v m , v −i ) ≤ p ( v l , v −i ) · A i v l − t i ( v l , v −i ).
24
Subtracting these two inequalities, we obtain
p ( v m , v −i ) · Ai ( v m − v l ) ≥ p ( v l , v −i ) · Ai ( v m − v l ) ⇔
p ( v m , v −i ) · Ai ≥ p ( v l , v −i ) · Ai .
Lemma 6. All constraints are implied by the following:
p(vm , v−i ) · Ai vm − ti (vm , v−i ) ≥ p(vm−1 , v−i ) · Ai vm − ti (vm−1 , v−i ) for m = 2, 3, ..., M;
p(vm , v−i ) · Ai vm − ti (vm , v−i ) ≥ p(vm+1 , v−i ) · Ai vm − ti (vm+1 , v−i ) for m = 1, 2, ..., M − 1;
p(v1 , v−i ) · Ai v1 − ti (v1 , v−i ) ≥ 0.
Proof. Constraints p(vm , v−i ) · Ai vm − ti (vm , v−i ) ≥ 0, m = 2, 3, ..., M are implied by p(v1 , v−i ) ·
Ai v1 − ti (v1 , v−i ) ≥ 0.
p ( v m , v −i ) · Ai v m − ti ( v m , v −i )
≥ p ( v1 , v −i ) · A i v m − t i ( v1 , v −i )
≥ p ( v1 , v −i ) · A i v1 − t i ( v1 , v −i )
≥ 0.
Downward constraints p(vm , v−i ) · Ai vm − ti (vm , v−i ) ≥ p(vl , v−i ) · Ai vm − ti (vl , v−i )
for m > l can be implied by the adjacent downward constraints. We show that the
following pair of inequalities:
p ( v m , v − i ) · A i v m − t i ( v m , v − i ) ≥ p ( v m −1 , v − i ) · A i v m − t i ( v m −1 , v − i )
p ( v m −1 , v − i ) · A i v m −1 − t i ( v m −1 , v − i ) ≥ p ( v m −2 , v − i ) · A i v m −1 − t i ( v m −2 , v − i )
imply
p ( v m , v − i ) · A i v m − t i ( v m , v − i ) ≥ p ( v m −2 , v − i ) · A i v m − t i ( v m −2 , v − i ).
The rest will follow by induction. Indeed,
p ( v m , v −i ) · Ai v m − ti ( v m , v −i )
≥ p ( v m −1 , v − i ) · A i v m − t i ( v m −1 , v − i )
= p ( v m −1 , v − i ) · A i v m −1 − t i ( v m −1 , v − i ) + p ( v m −1 , v − i ) · A i ( v m − v m −1 )
≥ p ( v m −2 , v − i ) · A i v m −1 − t i ( v m −2 , v − i ) + p ( v m −2 , v − i ) · A i ( v m − v m −1 )
= p ( v m −2 , v − i ) · A i v m − t i ( v m −2 , v − i ),
25
where the second last line follows from Lemma 5.
Similar arguments show that upward constraints can be implied by the adjacent
upward constraints.
Lemma 7. If the adjacent downward constraint binds
p ( v m , v − i ) · A i v m − t i ( v m , v − i ) = p ( v m −1 , v − i ) · A i v m − t i ( v m −1 , v − i ),
then the corresponding upward constraint
p ( v m −1 , v − i ) · A i v m −1 − t i ( v m −1 , v − i ) ≥ p ( v m , v − i ) · A i v m −1 − t i ( v m , v − i )
is satisfied. Similarly, if the adjacent upward constraint binds, the corresponding downward
constraint is satisfied.
Proof. Suppose
p ( v m , v − i ) · A i v m − t i ( v m , v − i ) = p ( v m −1 , v − i ) · A i v m − t i ( v m −1 , v − i ),
then
p ( v m −1 , v − i ) · A i v m −1 − t i ( v m −1 , v − i )
= p ( v m −1 , v − i ) · A i v m − t i ( v m −1 , v − i ) − p ( v m −1 , v − i ) · A i ( v m − v m −1 )
= p ( v m , v − i ) · A i v m − t i ( v m , v − i ) − p ( v m −1 , v − i ) · A i ( v m − v m −1 )
= p(vm , v−i ) · Ai vm−1 − ti (vm , v−i ) + [ p(vm , v−i ) − p(vm−1 , v−i )] · Ai (vm − vm−1 )
≥ p ( v m , v − i ) · A i v m −1 − t i ( v m , v − i ),
where the last line follows from Lemma 5.
Similar arguments show that if the adjacent upward constraint binds, the corresponding downward constraint is satisfied.
Lemma 8. At an optimal solution of the maximization problem, the downward constraints bind.
That is,
p(vm , v−i ) · Ai vm − ti∗ (vm , v−i ) = p(vm−1 , v−i ) · Ai vm − ti∗ (vm−1 , v−i ) for m = 2, 3, ..., M
p(v1 , v−i ) · Ai v1 − ti∗ (v1 , v−i ) = 0.
26
Proof. It follows from Lemma 6 that only the following constraints are relevant:
p ( v M , v −i ) · Ai v M − ti ( v M , v −i )
≥ p ( v M −1 , v − i ) · A i v M − t i ( v M −1 , v − i ).
p ( v m , v −i ) · Ai v m − ti ( v m , v −i )
≥ max{ p(vm−1 , v−i ) · Ai vm − ti (vm−1 , v−i ), p(vm+1 , v−i ) · Ai vm − ti (vm+1 , v−i )}, m = 2, ..., M − 1
p ( v1 , v −i ) · A i v1 − t i ( v1 , v −i )
≥ max{0, p(v2 , v−i ) · Ai v1 − ti (v2 , v−i )}.
At an optimal solution, it must be the case that
p(v M , v−i ) · Ai v M − ti∗ (v M , v−i ) = p(v M−1 , v−i ) · Ai v M − ti∗ (v M−1 , v−i ).
If not, we can increase ti∗ (v M , v−i ) slightly and achieve a higher expected revenue.
Furthermore,
p(v M−1 , v−i ) · Ai v M−1 − ti∗ (v M−1 , v−i ) = p(v M−2 , v−i ) · Ai v M−1 − ti∗ (v M−2 , v−i ).
If not, we can increase ti∗ (v M , v−i ) and ti∗ (v M−1 , v−i ) slightly by the same amount.
The constraint
p(v M , v−i ) · Ai v M − ti∗ (v M , v−i ) = p(v M−1 , v−i ) · Ai v M − ti∗ (v M−1 , v−i )
still holds and by Lemma 7, the constraint
p(v M−1 , v−i ) · Ai v M−1 − ti∗ (v M−1 , v−i ) ≥ p(v M , v−i ) · Ai v M−1 − ti∗ (v M , v−i )
is satisfied. Thus, we achieve a higher expected revenue and no constraint is violated.
By induction, we can show that at an optimal solution,
p(vm , v−i ) · Ai vm − ti∗ (vm , v−i ) = p(vm−1 , v−i ) · Ai vm − ti∗ (vm−1 , v−i ) for m = 2, 3, ..., M
p(v1 , v−i ) · Ai v1 − ti∗ (v1 , v−i ) = 0.
For any p, similar arguments show that the optimal transfer scheme t∗ must satisfy
the incentive constraints:
p(vm , v−i ) · Ai vm − ti∗ (vm , v−i ) = p(vm−1 , v−i ) · Ai vm − ti∗ (vm−1 , v−i ) for m = 2, 3, ..., M
p(v1 , v−i ) · Ai v1 − ti∗ (v1 , v−i ) = 0.
In other words, there is uniform order: v M i v M−1 i ... i v1 .
27
A.2
Proof of Theorem 1
The logic of the proof is summarized as follows: Step 1) explicitly constructs an assumption
about (the distribution of) bidders’ beliefs, against which we show in Step 2) and Step
3) that, the optimal dominant-strategy mechanism and the optimal Bayesian mechanism
deliver the same expected utility for the mechanism designer.
Step 1) Given π, write
∑
π vi =
π ( vi , v −i )
v−i ∈V−i
for the marginal probability of payoff type vi and write
Gi (vi ) =
0
∑+
π vi
∑+
π vi σ vi .
vi0 i vi
for the associated distribution function.
We define agent i’s beliefs as follows:
τ vi =
1
Gi (vi )
0
0
vi0 i vi
Note that the collection {τ vi }vi ∈Vi is linearly independent by the nonsingularity of π. Since
∑+
0
0
π vi σ vi =
vi0 i vi
∑
0
0
Gi (vi0 )τ vi + π vi σvi ,
vi i vi
we have the following equivalent recursive definition of τ vi :
τ vi = σvi for vi ∈ ViI ;


0
1 
τ vi =
Gi (vi0 )τ vi + π vi σvi  for vi ∈ Vi − ViI .
∑
Gi (vi ) v0 v
i
(11)
(12)
i i
Step 2) We argue that certain constraints in the Bayesian problem can be manipulated
or even ignored without cost to the mechanism designer in steps 2.1-2.3. To further simplify
v
the notations, let p̄vi · vi and t̄i i denote respectively the vectors ( p(vi , ·) · vi )v−i ∈V−i and
ti (vi , ·)v−i ∈V−i . Hence, the constraints in the Bayesian problem can be written as:
∀i ∈ I , ∀vi , vi0 ∈ Vi ,
v
τi i · ( p̄vi · vi − t̄vi ) ≥ 0,
vi
vi
vi
vi
vi0
vi0
τ · ( p̄ · vi − t̄ ) ≥ τ · ( p̄ · vi − t̄ ).
28
D
h IRvi i
E
0
IC vi →vi
Step 2.1) If vi and vi0 can not be ordered or vi0 i+ vi , the following incentive constraint
can be ignored:
0
0
τ vi · ( p̄vi · vi − t̄vi ) ≥ τ vi · ( p̄vi · vi − t̄vi )
(13)
Since the collection {τ vi }vi ∈Vi is linearly independent, there exists a lottery λ such
00
00
0
that τ vi · λ = 0 for vi00 + vi0 and τ vi · λ > 0 otherwise. Since σvi is a linear combination of
0
000
0
τ vi and τ vi for vi000 i vi0 , we have σvi · λ = 0. By adding (sufficiently large scale of) λ to
0
0
t̄vi , the revenue is the same since σvi · λ = 0 and no constraint is violated. Therefore, the
constraint (13) can be relaxed.
Step 2.2) for any mechanism ( p, t) that satisfies the remaining constraints, there exists
D
E
−
a mechanism ( p, t0 ) which satisfies the constraints h IRvi i for vi ∈ Vi and IC vi →vi for
vi ∈ Vi − ViT with equality and achieves at least as high a revenue. To prove this, fix any
D
E
−
mechanism ( p, t) that satisfies the remaining constraints. Suppose IC vi →vi holds with
strict inequality. Let Γ denote the matrix whose rows are the vectors {τ vi } for all vi ∈ Vi
−
−
and let {Γ−vi , σvi } be the matrix obtained by replacing τ vi by vector σvi .Note that the
matrix is full rank. We can thus solve the following equation for λ :
0
τ vi · λ = 0 for vi0 6= vi ;
−
σvi · λ = 1.
−
−
−
Note that because τ vi · λ = 0 < σvi · λ = 1, and because τ vi is a convex combination
v00
−
−
of τi i for vi00 i vi− and σvi , we have τ vi · λ < 0. We shall add the vector eλ to t̄vi . Because
0
τ vi · λ = 0 for vi0 6= vi , no constraints for types vi0 6= vi are affected. The only constraint that
could be violated is
−
−
τ vi · ( p̄vi · vi − t̄vi ) ≥ τ vi · ( p̄vi · vi − t̄vi ),
−
and this constraint was slack by assumption. Let Svi = τ vi · ( p̄vi · vi − t̄vi ) − τ vi · ( p̄vi · vi −
−
t̄vi ) > 0, and we can choose e = −Svi /(τ vi · λ) > 0 and the inequality becomes binding.
−
Since, σvi · eλ > 0, the auctioneer profits from this modification.
Step 2.3) each h IRvi i can be treated as equality without loss of generality. Define
v
Svi = τi i · ( p̄vi · vi − t̄vi ) ≥ 0 to be the slack in h IRvi i . Construct a lottery λ that satisfies
v
τi i · λ = Svi . By the full rank condition, such a lottery can be found. We will add λ to
each t̄vi . No constraints will be affected, but now each IR constraint holds with equality.
29
Revenue is at least the same. Indeed,
∑
∑
π vi σ vi · λ =
vi ∈Vi
vi ∈ViI
∑
=
∑
v
π vi τi i · λ +
[ Gi (vi )τ vi −
vi ∈Vi −ViI
∑
0
0
Gi (vi0 )τ vi ] · λ
vi i vi
vi
Gi (vi )τ · λ
vi ∈ViT
> 0.
Step 3) We can rewrite the Bayesian maximization problem as follows:
∑ π ( v ) ∑ ti ( v )
max
p(·),t(·) v∈V
∑ ∑
=
vi vi
i ∈I
vi
π σ · t̄
(14)
i ∈I vi ∈Vi
subject to
∀i ∈ I , ∀vi ∈ Vi ,
v
τi i · ( p̄vi · vi − t̄vi ) = 0,
v
−
v
−
τi i · ( p̄vi · vi − t̄vi ) = τi i · ( p̄vi · vi − t̄vi ).
That is,
v
v
τi i · t̄vi = τi i · p̄vi · vi
v
τi i · t̄
vi−
= τivi · p̄
vi−
(15)
· vi .
(16)
It follows that for vi ∈ ViI ,
π vi σvi · t̄vi = π vi τ vi · t̄vi = π vi τ vi · p̄vi · vi = π vi σvi · p̄vi · vi ;
(17)
moreover, for vi ∈ Vi − ViI ,
π vi σvi · t̄vi = Gi (vi )τ vi · t̄vi −
0
∑
0
Gi (vi0 )τ vi · t̄vi
(18)
vi i vi
= Gi (vi )τivi · p̄vi · vi −
∑
vi0 i vi

= 
v0
Gi (vi0 )τi i · p̄vi · vi0
(19)

∑
0
0
Gi (vi0 )τ vi + π vi σvi  · p̄vi · vi −
vi i vi
= π vi σvi · p̄vi · vi −
∑
0
v0
Gi (vi0 )τi i · p̄vi · vi0
(20)
vi i vi
∑
0
0
Gi (vi0 )τ vi · p̄vi · (vi0 − vi )
vi i vi
30
(21)
where (17) follows from (11), (19) follows from (15) and (16), and (20) follows from (12).
Therefore
∑
π vi σvi · t̄vi
∑
π vi σvi · t̄vi +
∑
π vi σvi · t̄vi +
∑
π vi σvi · p̄vi · vi −
∑
π vi σvi · p̄vi · vi −
vi ∈Vi
=
vi ∈ViI
=
π vi σvi · t̄vi
∑
π vi σvi · p̄vi · vi −
vi ∈Vi −ViI
vi ∈ViI
=
∑
vi ∈Vi −ViI
vi ∈Vi
=
∑
vi ∈Vi −ViI vi i vi
∑
0
∑
0
Gi (vi0 )τ vi · p̄vi · (vi0 − vi )
(22)
0 −
∑ ∑+
π vi σvi · p̄(vi ) · [vi0 − (vi0 )− ]
(23)
vi ∈Vi v0 vi
i i


∑
0
Gi (vi0 )τ vi · p̄vi · (vi0 − vi )
vi ∈Vi −ViI vi i vi
vi ∈Vi
=
∑
0
π vi σvi · p̄vi · vi −
vi ∈Vi
∑+
vi0 i
0 −
π vi σvi · p̄(vi ) · [vi0 − (vi0 )− ]
(24)
vi
where (22) follows from (17) and (21), and (23) follows from the equality:16
∑
vi ∈Vi −ViI
=
∑
0
0
Gi (vi0 )τ vi · p̄vi · (vi0 − vi )
(25)
vi i vi
0 −
∑ ∑+
π vi σvi · p̄(vi ) · [vi0 − (vi0 )− ].
(26)
vi ∈Vi v0 vi
i i
It follows from (24) that the objective function of the Bayesian problem is
∑ ∑
π vi σvi · t̄vi
i ∈I vi ∈Vi

=
∑ ∑

π vi σvi · p̄vi · vi −
i ∈I vi ∈Vi
∑+
0 −
π vi σvi · p̄(vi ) · [vi0 − (vi0 )− ]
vi0 i vi

=
∑ ∑
∑

π ( vi , v −i )  p ( v ) · vi −
i ∈I vi ∈Vi v−i ∈V−i
∑+
p((vi0 )− , v−i ) · [vi0 − (vi0 )− ]
vi0 i vi
Regularity ensures that the optimal dominant-strategy mechanism and the optimal
Bayesian mechanism deliver the same expected utility for the mechanism designer.
16 Indeed,
the coefficient for any p̄vi · (vi0 − vi ) is the same. Indeed, fix any p̄vi · (vi0 − vi ), it is easy to see
0
that for (25), Gi (vi0 )τ vi is the coefficient. And for (23), it is the summation of π vi σvi for all vi i+ vi0 .
31
References
B ERGEMANN , D., AND S. M ORRIS (2005): “Robust Mechanism Design,” Econometrica, 73,
1771–1813.
B ORDER , K. C.,
AND
J. S OBEL (1987): “Samurai Accountant: A Theory of Auditing and
Plunder,” Review of Economic Studies, 54, 525–540.
B ÖRGERS , T. (2013): “(No) Foundations of Dominant-Strategy Mechanisms: A Comment
On Chung and Ely (2007),” mimeo. University of Michigan.
C ARROLL , G. (2014): “Informationally Robust Trade Under Adverse Selection,” mimeo.
Stanford University.
(forthcoming): “Robustness and Linear Contracts,” American Economic Review,
Forthcoming.
C HUNG , K.-S., AND J. C. E LY (2007): “Foundations of Dominant-Strategy Mechanisms,”
Review of Economic Studies, 74, 447–476.
C RAMTON , P., R. G IBBONS ,
AND
P. K LEMPERER (1987): “Dissolving a Partnership Effi-
ciently,” Econometrica, 55, 615–632.
F RANKEL , A. (2014): “Aligned Delegation,” American Economic Review, 104, 66–83.
G ARRETT, D. F. (forthcoming): “Robustness of simple menus of contracts in cost-based
procurement,” Games and Economic Behavior, Forthcoming.
G ERSHKOV, A., J. K. G OEREE , A. K USHNIR , B. M OLDOVANU ,
AND
X. S HI (2013): “On
the Equivalence of Bayesian and Dominant Strategy Implementation,” Econometrica, 81,
197–220.
G OEREE , J.,
AND
A. K USHNIR (2013): “A Geometric Approach to Mechanism Design,”
mimeo. ESEI Center for Market Design.
H EYDENREICH , B., R. M ÜLLER , M. U ETZ ,
AND
R. V OHRA (2009): “Characterization of
Revenue Equivalence,” Econometrica, 77, 307–316.
H URWICZ , L., AND L. S HAPIRO (1978): “Incentive Structures Maximizing Residual Gain
under Incomplete Information,” Bell Journal of Economics, 9, 180–191.
32
L U , H., AND J. R OBERT (2001): “Optimal Trading Mechanisms with Ex Ante Unidentified
Traders,” Journal of Economic Theory, 97, 50–80.
M AILATH , G., AND A. P OSTLEWAITE (1990): “Asymmetric Information Bargaining Problems with Many Agents,” Review of Economic Studies, 57, 351–367.
M ANELLI , A., AND D. V INCENT (2010): “Bayesian and Dominant-Strategy Implementation
in the Independent Private Values Model,” Econometrica, 78, 1905–1938.
M ASKIN , E.,
AND
J. R ILEY (1989): “Optimal Multi-unit Auctions,” in The Economics of
Missing Markets, Information, and Games, ed. by H. F., chap. 14, pp. 312–335. Oxford
University Press.
M ERTENS , J.-F., AND S. Z AMIR (1985): “Formulation of Bayesian Analysis for Games with
Incomplete Information,” International Journal of Game Theory, 14, 1–29.
M EYER -T ER -V EHN , M., AND S. M ORRIS (2011): “The robustness of robust implementation,”
Journal of Economic Theory, 146, 2093–2104.
M ORRIS , S. (1994): “Trade with Heterogeneous Prior Beliefs and Asymmetric Information,”
Econometrica, 62, 1327–1347.
M YERSON , R. (1981): “Optimal Auction Design,” Mathematics of Operations Research, 6,
58–71.
M YERSON , R., AND M. S ATTERTHWAITE (1983): “Efficient Mechanisms for Bilateral Trading,” Journal of Economic Theory, 29, 265–281.
R OCHET, J.-C. (1987): “A Necessary and Sufficient Condition for Rationalizability in a
Quasi-Linear Context,” Journal of Mathematical Economics, 16, 191–200.
R OCHET, J.-C., AND L. A. S TOLE (2003): “The Economics of Multidimensional Screening,”
in Advances in Economics and Econometrics: Theory and Applications: Eighth World Congress.
Cambridge University Press, Cambridge.
V OHRA , R. V. (2011): Mechanism Design: A Linear Programming Approach. Cambridge
University Press, Cambridge.
33
W ILSON , R. (1987): “Game-Theoretic Analyses of Trading Processes,” in Advances in
Economic Theory: Fifth World Congress. Cambridge University Press, Cambridge.
W OLITZKY, A. (2014): “Mechanism Design with Maxmin Agents: Theory and an Application to Bilateral Trade,” mimeo. Stanford University.
YAMASHITA , T. (2014): “Implementation in Weakly Undominated Strategies, with Applications to Auctions and Bilateral Trade,” mimeo, Toulouse School of Economics.
YAMASHITA , T.,
AND
S. Z HU (2014): “Foundations of Ex Post Incentive Compatible
Mechanisms,” mimeo, Toulouse School of Economics.
34