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WORKING PAPER SERIES
Working Paper
2014-049
Incentive compatible mechanisms in
multiprincipal multiagent games
Gwenaël Piaser
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IPAG Business School
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peer-reviewed and may not be reproduced without permission of the authors.
Incentive compatible mechanisms in
multiprincipal multiagent games∗
Gwenaël Piaser†
January 6, 2014
Abstract
It is argued that the revelation principle in multi-principal multi-agent games
cannot be generalized. In other words, one cannot restrict attention to incentive
compatible mechanisms, even if the concept of information is enlarged.
Keywords: Direct Mechanisms, Incentive compatible, Multiprincipals.
JEL Classification: D82.
Corresponding author
Gwenaël Piaser
184, bd Saint-Germain
75006 Paris
Tel: +33 (0)1 53 63 36 00
Fax: +33 (0)1 45 44 40 46
e-mail: [email protected]
∗I
thank Andrea Attar for his help.
Business School, Paris
† IPAG
1
1
Introduction
Contract theory is a well established branch of economics. Auctions, procurement,
regulation, insurance, corporate governance and many other fields have been revitalized
by the introduction of contract theory. A “contract problem” is defined as follows: We
distinguish two classes of players, the principals who play first and offer contracts. The
second class of players, the agents, observe the offers made by the principals; they
accept some of them and they may undertake some actions.
In this class of games, the issue of communication is interesting and has been one
the most studied features of contract theory. The class of single-principal games is
formally well understood. When the (unique) principal offers a contract to the agent and
centralizes communication, Myerson (1982) has shown that one can restrict attention to
contracts such that
• The principal asks (privately) each agent for his information, and sends to each
agent a private recommendation over his non-contractible actions;
• The contract must be such that the agents reveal their information (they are honest) and they follow the principal’s recommendation (they are obedient).
In other words, the restriction to incentive compatible direct mechanisms causes no loss
of generality.1 This result is a cornerstone of contract theory: it provides a simple and
elegant methodology for characterizing equilibria in single agent models.
Unfortunately, this elegant result does extend to multiprincipal games. Peck (1997),
Peters (2001) and Martimort and Stole (2002) have provided various counterexamples.2
For single-agent games, Peters (2001) and Martimort and Stole (2002) have shown,
without loss of generality, that we can assume principals compete through “menus”. In
1 This
result was discovered simultaneously by several researchers; see for example Gibbard (1973),
Myerson (1979), Dasgupta, Hammond, and Maskin (1979), or Harris and Townsend (1981). The result
of Myerson (1982) result can be considered the most general version. The relation between direct mechanisms incentive compatibility and the concept of correlated equilibrium, introduced by Aumann (1974),
is discussed by Myerson (1991).
2 In the presence of restrictive assumptions on players’ preferences, the restriction to direct mechanisms can be justified; see Peters (2004), Attar, Majumdar, Piaser, and Porteiro (2008), Attar, Piaser, and
Porteiro (2007) and Peters (2007).
2
other words, one can assume that principals offer a subset, or “menu”, of all possible
outcomes, and the (unique) agent chooses his favorite outcome in each menu. However,
as Peters (2001) observes, the concept of menu cannot be extended to multiprincipal
multiagent games. If a principal offers a subset of all possible outcomes to each agent,
they may respond with different outcomes to the same principal, who would not then
know which outcome to implement. There is no simple methodology for the characterization of equilibria in multiprincipal multiagent games.
Epstein and Peters (1999) propose a set of mechanisms for characterizing all equilibria of every multiprincipal multiagent game. They expand the concept of information
by arguing that information should include information about the mechanism offered
by other principals. Hence, in a two-principals framework, a principal should ask the
agent not only his type but also the mechanisms offered by the other principal. This true
for the second principal as well. There is then a difficulty in defining a direct mechanism. Epstein and Peters (1999) have shown that this problem can be overcome and that
mechanisms exist that permit the transfer of the right information from the agents to the
principals.
Yamashita (2010) proposes a set of mechanisms able to sustain a large set of equilibria in games with at least three agents. Our results show that Yamashita’s mechanisms cannot be duplicated by direct mechanisms. Following the same line, Peters and
Troncoso Valverde (2011) show that any incentive compatible and individually rational
allocation can be implemented by mechanisms generalizing Yamashita’s ones. To do
that, they consider mechanisms in which players do not send only a message about their
types but also a correlating message (which is a real number in the set [0, 1]).3 Thus
their mechanisms cannot be seen as “direct”. Our conclusions go in the same direction:
In setting with many principals and many agents, asking for the type is not enough, no
matter how the type is defined. Hence there is no way to get rid of Peters and Troncoso Valverde’s correlating messages or similar correlating devices.
In the present paper it will be argued that one cannot restrict attention to incentive
compatible equilibria even if the concept of information is widened. In a multiprincipal
framework, incentive compatibility leads to restriction in the agents’ behaviors: If every
3 Moreover,
their mechanisms are quite complex since they allow for two communication stages.
3
agent reveals his information (his type and the mechanisms offered, defined appropriately) then by definition he plays a pure strategy. It will be argued that some outcomes
sustained by equilibria in which agents play mixed strategies cannot be sustained by
equilibria in which agents play pure strategies.
2
Multiprincipal Multiagent games
Here, a generalized version of Martimort and Stole (2002) is presented in which several
agents are considered. Each agent knows his type, but other agents and principals have
only uncertain knowledge of this information.
We consider a situation in which a number of principals (indexed by j ∈ N =
{1, . . . , n}) are contracting with several agents (indexed by i ∈ M = {1, . . . , m}). Each
principal j has to take a decision y j ∈ Y j , where Y j is a compact metric space.
Each agent i knows the realization of a random variable ωi . This random variable
is defined over the set Ωi . To keep the analysis simple, all sets will be taken as finite.
Define the set Ω = ∏ Ωi . The principals know only the probability distribution f over
i∈M
set Ω by which the random variable (ωi )i∈N is drawn. The probability distribution f is
common knowledge. The corresponding cumulative distribution function is denoted as
F.
Each principal j can communicate and contract with every agent i. Each agent
i sends privately a message cij in the set Cij to each principal j. Hence each principal
(
)
receives an array of messages c j = c1j , c2j , . . . , cmj , based on which he makes a decision.
The sets Cij are taken to be compact and metric (for all i and j), but they can be arbitrarily
large; in particular, we may have Ci > Ωi .
j
Formally, the set of feasible contracts A j for principal j is taken to be the set of
measurable mappings:
( )
γ j : ∏ Cij → ∆ Y j ,
(1)
i∈M
We denote these mappings as “contract mechanisms;” they describe the strategies available to principal i. Principal i is able to commit based on these mechanisms. A mech4
anism4 can be seen as a rule in operation. There are two standard, but restrictive, assumptions. First, a principal is not able to write a contract contingent on the contracts
offered by other principals. In that sense, contracts are incomplete. Second, contracts
are public: Principals announce simultaneously their mechanisms to all agents. We denote by Γi the set of all available mechanisms to principal i, and by Γ the collection of
those sets Γ = ∏ Γi .
i∈M
Finally, we consider here “stochastic mechanisms,” rather than “deterministic” mechanisms which are mappings from ∏ Cij to Y j .
i∈M
Importantly, if attention is restricted to deterministic mechanisms even in singleprincipal models, the restriction to incentive compatible mechanisms entails loss of
generality. To generalize, therefore, all minimal assumptions needed in single principal
models are retained.5
We solve the game backwards. At the last stage, the agents have observed the mechanisms chosen by the principals. Each agent has to send a message to each principal, so
that a strategy for an agent at this stage is:
λic : Ωi × ∏ A j →
j∈N
∏ Cij .
(2)
j∈N
We denote by Lci the collection of such pure strategies.
The payoffs of principal j ∈ {1, . . . , n} are represented by the von Neumann-Morgenstern
utility functions
v j : ∏ Y j × ∏ Ωi → [0, 1] .
(3)
j∈N
i∈M
For the agents, payoffs are represented by the von Neumann-Morgenstern utility functions:
ui : ∏ Y j × ∏ Ωi → [0, 1] .
(4)
j∈N
i∈M
4 Our model follows the line initiated by Peters (2001) and Martimort and Stole (2002): Principals
cannot write contracts contingent on contracts offered by the other principals. See Peters and Troncoso Valverde (2011) for an alternative approach.
5 See Strausz (2003) and Attar, Campioni, Piaser, and Rajan (2010) for a discussion of the role of
stochastic mechanisms in multiagent models.
5
We denote GΓ the multiple-principal multiple-agent game relative to the set of mechanisms Γ.
(
)
We call an “outcome” the lottery over ∆
∏ Yj
induced by the strategies played
j∈M
by the players. If these strategies constitute an equilibrium, the resulting outcome will
be called an “equilibrium outcome”. We call the expected utilities associated with an
outcome an “output” of the game. In the same way, if an output is associated with an
“equilibrium outcome”, it is called an “equilibrium output”.
Three more concepts are needed: generalized direct mechanisms, incentive compatibility, and an extended version of the Revelation Principle.
First, we define a "generalized" direct mechanism game as a game GΓ in which
every communication set Cij is the set T i × Ωi . Then we can define a "generalized"
direct mechanism. We denote by GD such a game.
( )
d j : T M × Ω → ∆ Yj ,
(5)
We denote by D j the set of all generalized direct mechanism, and the collection of those
sets is D, i.e; D = ∏ j∈N D j . The set T is a “language” describing the mechanisms, i.e. T
is such that a one-to-one map ψ exists from T to D. We do not discuss the construction
of these mechanisms. Epstein and Peters (1999) provide a complete analysis of the
existence of such complex mechanisms.6
Second, in the game GD , a generalized direct mechanism d j is incentive compatible
given the collection of mechanisms d− j for principal j if:
• An equilibrium of the continuation game exists in which for all θ in Θ, each agent
( )
i sends the message t, θi to the principal j.
• If such equilibria exist, we assume that agents coordinate on it.
6 Epstein and Peters (1999) allow agents to take action,
in which respect they are more general than the
present work. Since the present result is a negative one, this loss of generality is problematic. Moreover,
Epstein and Peters (1999) restrict attention to symmetric games to keep their notation simple, although it
has no effect on their results.
6
A mechanism is incentive compatible given a set of mechanisms offered by the other
principals. If this set of mechanisms change, the considered mechanism does not need
to be incentive compatible anymore. Of course, the agents must reveal their true type θi
but also the other relevant information t ∈ T , which is the same for all agents.
Third, in this context a Revelation Principle would be written as:
Revelation Principle Any equilibrium of the GΓ multiprincipal multiagent game GΓ
can be reproduced in the GD game in which principals at equilibrium offer incentive
compatible mechanisms.
But this statement is wrong, as shown in the next section.
Before the presentation of our counter-example, let us remark that if every principal
offers an incentive compatible mechanisms then agents reveal their information and
hence play pure strategies.
In other words on the equilibrium path, agents’ behavior is deterministic. If the
principals each play a pure strategy, they all agents send their type and the same message
t ∈ T . Our argument is based on this observation.
3
The Impossibility of a Revelation Principle
There are three principals (called Principal 1, Principal 2 and Principal 3) and two agents
(called Agent 1 and Agent 2). There is no incomplete information, so that
∀i ∈ M, Ωi = 1.
(6)
{
}
Principal 1 chooses an allocation in the set ∆ (Y1 ), where Y1 = y11 , y21 . Similarly, Princi{
}
pal 2 chooses an allocation in the set ∆ (Y2 ), where Y2 = y12 , y22 , and Principal 3 chooses
}
{
an allocation in the set ∆ (Y3 ) where Y3 = y13 , y23 .
Agents have no private information, and do not take any action. In a game without
communication, they would not play any role.
The payoffs are given by the following matrices; if Principal 3 implements y13 , the
payoffs are:
7
y12
y22
y11
(1, 1, 2, 1, 2)
(−50, −50, −50, −50, −50)
2
y1 (−50, −50, −50, −50, −50)
(1, 1, 0, 1, 0)
and if Principal 3 implements y23
y12
y22
y11
(1, 1, 0, 1, 0)
(−50, −50, −50, −50, −50)
y21 (−50, −50, −50, −50, −50)
(1, 1, 2, 1, 2)
The five elements in each cell respectively denote the utilities of Principal 1, Principal 2,
Principal 3, Agent 1 and Agent 2.
Consider now the following indirect mechanism. Principal 1, Principal 2 and Principal 3 are allowed to communicate with the agents, the message spaces being the same
∀i, j Cij = {m1 , m2 , m3 , m4 }. Suppose that Principal 1 (resp. Principal 2) chooses the
following mechanisms (denoted γ1 , resp γ2 ): If he receives message m1 or m4 from
Agent 1, he implements the decisions y11 (resp. y12 ). If from the same agent he receives
the message m2 or m3 , he implements the allocation y21 (resp. y22 ). Principal 1’s mechanism (resp. Principal 2’s mechanism) is independent of Agent 2’s message. Principal 1
and Principal 2 choose their allocation according to the message sent by Agent 1 only.
Principal 3 chooses the allocation y13 whatever messages he receives. We denote this
mechanism by γ3 .
Agent 1 sends message m1 to both Principal 1 and Principal 2 with probability 1/2,
and message m2 to both Principal 1 and Principal 2 with the same probability. He sends
message m3 to principal 3 with probability 1. He maintains this behavior no matter what
Principal 3’s strategy is. For our argument, it is not necessary to describe Agent 2’s
strategy. We shall suppose that at equilibrium he sends message m3 to all principals.
Principal 1 and Principal 2 have no profitable deviation: They get their maximal
payoff. Principal 3 also has no deviation. To improve his utility, Principal 3 would like
to take a decision correlated with Agent 1’s messages to the other principals. If Agent 1
sends message m1 , he would like to implement y13 and y23 in the other case. Since Agent 1
8
does not send to Principal 3 messages correlated with the message he sent to the other
principal, this is never possible.
We shall argue that the outcome sustained by the former equilibrium cannot be sustained by an equilibrium in which the agents play pure strategies.
First, given the payoffs that we want to implement, the outcome must be stochastic.
Since agents are playing pure strategies, it is necessary to consider an equilibrium in
which Principal 1 and Principal 2 mix mechanisms.
Suppose that Principal 1 mixes equally over two mechanisms, γ′1 and γ′′1 . Principal 2
similarly mixes equally over two mechanisms, γ′2 and γ′′2 .7 Suppose the mechanisms are
such that, if Agent 1 sends message mi to principal j, he implements the decision yij . As
above, the message sent by Agent 2 has no importance.
Let us assume that Agent 1 send the message m1 to both principals if γ′1 and γ′2
are realized or if γ′′1 and γ′′2 are realized. In other cases he sends message m2 to both
principals. He maintains this behavior for every mechanism offered by Principal 3.
If Principal 3 adopts the former mechanisms, the outcome is preserved. But now he
has no incentive to do so. Since mechanisms are public, Agent 2 knows the realisation
of the mixed strategies. He is able to send a message to Principal 3, who then can implement the right outcome from his point of view. Principal 3 could adopt a mechanism
having the following form: If Agent 2 sends message m1 , Principal 3 implements y13 . If
he receives the message m2 from Agent 2, he implements y23 .
Does there exist a way of punishing Principal 3 if he adopts this mechanism? If
we keep the constraint that agents must play pure strategies, the answer is no. To get
an output equivalent equilibrium, we would like Principal 1 and Principal 2 to mix the
mechanisms γ′1 , γ′2 , γ′′1 and γ′′2 , and Principal 3 plays the mechanism γ3 . As we have
seen, γ3 is not necessarily a best reply for Principal 3. For Principal 3 to adopt such
us define γ′1 as the following mechanism: If Principal 1 receives the message m1 or m3 from
Agents 1, he implements the decisions y11 . If from the same agent he receives the message m2 or m4 , he
implements the allocation y21 . The message sent by Agent 2 has no importance. The mechanism γ′′1 is
similar, except that Principal 1 implements y11 when he receives messages m1 , m3 or m4 . Mechanisms γ′2
and γ′′2 are constructed in the same way, with y12 and y22 respectively replacing y11 and y21 .
7 Let
9
a strategy, we should make any deviation from γ3 unprofitable. Since Agent 2 has the
same preference as Principal 3, the only way to do that is through Agent 1. Let us
assume that if Principal 3 plays the mechanism γ3 , Agent 1 behaves as described above:
• He sends a message according to the realization of the mixed strategies of Principal 1 and Principal 2.
• If Principal 3 plays the mechanism γ′3 then Agent 1 sends message m3 to both
Principal 1 and Principal 2.
Such behavior is optimal for Agent 1 only if he gets a payoff of 1 by sending the message m3 , which he can get by ignoring Principal 3’s deviation. This means that the
mechanisms γ′1 , γ′2 , γ′′1 and γ′′2 are such that Principal 1 and Principal 2 respectively implement y11 and y12 when they receive the message m3 (or equivalently y21 and y22 ). If, then,
Agent 1 adopts such a (pure) strategy, Agent 2 easily adapts his strategy by sending the
appropriate message to Principal 3, here m1 .
It follows that the previous equilibrium outcome cannot be sustained by an equilibrium in which the agents are playing pure strategies. The argument does not depend on
the message sets and hence can be reproduced if we consider the set T . We conclude
that our reference outcome cannot be sustained by incentive compatible mechanisms,
even by enlarging the concept of incentive compatibility.
It is true that direct mechanisms convey less information that indirect mechanisms
if we define direct mechanisms as mechanisms in which agents send their type only.
Such mechanisms are not sufficient because agents have more information than their
type, they observe the all offered mechanisms. That why it is natural to try to generalize revealing mechanisms to obtain "direct" mechanisms in which agents send to
principals all their relevant information. The example shows that one cannot generalized the Revelation Principle following that intuition. Whatever the message sets are,
some equilibrium outcomes can be sustained only by equilibria in which agents mixes
over messages. Hence those outcome cannot be sustained by direct mechanisms even if
we generalize the concept of type: If agents report truthfully their relevant information
to principals, by definition they do not mix over messages.
10
4
Discussion
We have considered an example of an outcome sustained by mixed strategy equilibrium.
Specifically, an agent play a mixed strategy. It is not possible to reproduce an equivalent
equilibrium in which only principals would play mixed strategies, because information
plays a crucial role: If agents mix strategies then no player knows the realization of
these mixed strategies. If principals mix mechanisms, agents observe the realization
of the mixed strategies and can therefore transmit some strategic information to the
principals. Hence, direct mechanisms, even in their extended version, are not as general
as indirect mechanisms.
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