www.elsevier.com / locate / econbase
Imperfect commitment and the revelation principle: the
multi-agent case
*
Helmut Bester , Roland Strausz
Free University Berlin, Department of Economics, Boltzmannstr. 20, D-14195 Berlin, Germany
Received 6 March 2000; accepted 8 May 2000
Abstract
We consider mechanism design problems with n agents when the mechanism designer cannot fully commit to an allocation function. With a single agent (n51) optimal mechanisms can always be represented by direct mechanisms, under which each agent’s message set is the set of his possible types [Bester, H., Strausz, R., 2000. Contracting with imperfect commitment and the revelation principle: the single agent case. Free University of Berlin, mimeo]. We show that this result does not hold if n$2. That is, in mechanism design problems with multiple agents the use of direct mechanisms may be suboptimal. 2000 Elsevier Science S.A. All rights reserved.
Keywords: Revelation principle; Mechanism design; Limited commitment; Asymmetric information
JEL classification: D82; C72
1. Introduction
The revelation principle (see Gibbard, 1973; Green and Laffont, 1977; Dasgupta et al., 1979; Myerson, 1979) offers a convenient tool to analyze contracting problems under adverse selection. It states that the principal may without loss of generality use a communication game with a direct mechanism in such a way that each agent truthfully reports his type. This allows formulating the principal’s problem as a straightforward optimization problem in which incentive compatibility constraints reflect the restrictions imposed by asymmetric information. Unfortunately, the revelation principle requires that the principal can commit himself to an allocation function, which maps the agents’ messages into final allocations. Thus it is not applicable to environments with imperfect commitment, such as sequential contracting, renegotiation or incomplete contracts.
*Corresponding author. Tel.: 149-30-838-55257; fax: 149-30-838-54142.
E-mail addresses: hbester@wiwiss.fu-berlin.de (H. Bester)., strausz@wiwiss.fu-berlin.de (R. Strausz).
In Bester and Strausz (2000), however, we provide a modified version of the revelation principle for environments with imperfect commitment. The starting point of this modified version is that the contracting parties care about implementable allocations and the communication game only insofar as their payoffs are concerned. For contracting problems between a principal and a single agent, we are indeed able to prove that the payoffs on the Pareto frontier of an arbitrary mechanism may also be obtained by a direct mechanism (see Theorem 1 in Section 2.2 below). Further, under this mechanism it is an optimal strategy for the agent to reveal his type truthfully and he will use this strategy with positive probability. In the same way as the standard revelation principle, this result allows us to formulate the contracting problem as a straightforward optimization problem.
In this paper we present an example to show that our result in Bester and Strausz (2000) cannot be extended to the case where the principal deals with more than one agent. We first consider a communication game in which the dimensionality of the agents’ message sets is larger than the number of possible types. We then show that replacing this communication game by a direct mechanism necessarily makes one of the contracting parties worse off. As a result, using a direct mechanism may not be optimal for the principal: either his payoff is lower than under some alternative mechanism or it may happen that one of the agents’ individual rationality constraints is no longer satisfied.
2. Mechanism design
Consider a contracting problem between a principal and n agents: The principal has no private information. Each agent i51, . . . , n, however, is privately informed about his type ti[T . Fori simplicity, we take T to be finite. The agents’ types ti 5(t , . . . , t , . . . , t )1 i n [T5T1? ? ? 3Ti? ? ? 3
T are drawn from some objective distribution p(t). After agent i observes his type t , his conditionaln i probability about the other agents’ types is p(t2iut ). The principal’s problem consists of selecting ani
allocation y[Y. This allocation together with the agents’ types determines the players’ von
Neumann–Morgenstern utilities: We denote the principal’s payoff by V( y, t) and the agents’ payoffs by U ( y, t), ii 51, . . . , n.
A mechanism or contract specifies for each agent i a message space M and a game in which thei
agents choose their messages. To avoid measure theoretical problems, we take M to be finite andi
denote by } the set of probability distributions over M . After agent i has learned his type t , he
i i i
chooses a message; his reporting strategy m(t ): T →} selects the message m with probability
i i i i
mi(mut ). A mechanism is called a direct mechanism if Mi i5T for all i. With a direct mechanism thei
message space is the agents’ type space and in the message game each agent simply announces some type.
Since the agents’ types are private information, the principal’s choice of y can depend on t only through the agents’ messages m5(m , . . . , m , . . . , m )1 i n [M5M1? ? ? 3Mi? ? ? 3M . We denoten
such an allocation function by y: M→Y.
2.1. Full commitment
about his type t and chooses a reporting strategy. For a giveni hM,yj, the agents are thus engaged in a static Bayesian game in which they simultaneously select messages. These messages determine the allocation y according to the allocation function y. For such a setting, the revelation principle states that the principal can content himself with a direct mechanism under which all agents announce their types truthfully.
Revelation principle. Suppose thathM,yjsupports a Bayesian equilibrium in which the principal has
*
the ex ante expected payoff V * and agent i with type t has the interim expected payoff U (t ). Theni i i ˜ ˆ
there exists a direct mechanism hT, yj which supports a Bayesian equilibrium with payoffs V and
˜ ˜
*
˜U (t ) such that V5V * and U (t )5U (t ) for all i and t . Moreover, in the Bayesian equilibrium
i i i i i i i
under the direct mechanism each agent i reports his type t truthfully, i.e.i mi(tiut )i 51 for all i and
ti[T .i
The revelation principle implies that for studying the set of implementable allocations it suffices to consider only direct mechanisms for which truthtelling is incentive compatible. This insight has turned the revelation principle into a standard tool, as it reduces the study of optimal mechanisms to a straightforward maximization problem in which the principal’s lack of information is reflected by incentive compatibility constraints.
2.2. Imperfect commitment
The standard revelation principle requires the principal to commit himself to an allocation function
y before the agents select their reporting strategies. This requirement may be unrealistic for a large
class of contractual relationships. If the principal cannot commit himself, he chooses an allocation
after observing the agents’ messages. In this situation, allocations are implemented by the Perfect
Bayesian equilibrium of a two-stage game: First, the agents simultaneously choose their messages at
the interim stage. Second, the principal updates his beliefs about the agents types and selects y[Y.
We denote the principal’s conditional belief about the agents’ types as p(tum). Under imperfect
commitment, the allocation functiony is determined by the principal’s equilibrium strategy and has to
satisfy
y(m)[argmax
O
p(tum)V( y, t) (1)y t[T
for all m[M. Moreover, in a Perfect Bayesian equilibrium the principal’s belief p(tum) has to be
consistent with Bayesian updating for all messages m that occur in equilibrium with positive probability.
Because of restriction (1) the proof of the standard revelation principle fails. Yet, in Bester and Strausz (2000) we obtain the following modified version of the revelation principle for the single agent case:
Theorem 1. Let n51. Suppose that M supports a Perfect Bayesian equilibrium in which the
principal has the ex ante expected payoff V * and the agent with type t has the interim expected1
˜
*
payoff U (t ). Then the direct mechanism T supports a Perfect Bayesian equilibrium with payoffs V1 1 1
˜ ˜
*
˜and U (t ) such that V$V * and U (t )$U (t ) for all t [T . Moreover, in the Perfect Bayesian
equilibrium under the direct mechanism the agent reports his type truthfully with positive probability,
i.e. m1(t1ut )1 .0 for all t1[T .1
That is, a direct mechanism can at least match the equilibrium payoffs of any other mechanism. Therefore, if one is interested in optimal mechanisms or attainable payoffs, it suffices to consider only direct mechanisms. Moreover, in Bester and Strausz (2000) we show that one may use this result to formulate the problem of finding an optimal mechanism as a straightforward maximization problem.
3. A counterexample
In this Section we will show that Theorem 1 of Section 2.2 does not extend to contracting problems with more than one agent. We consider an example where the principal faces two agents. Agent 1 is privately informed about his type; he is equally likely to be either of type t or of type t . There is noa b uncertainty about agent 2. In this case, a mechanism merely specifies a message space M for agent 1.
After receiving some message m[M from agent 1, the principal chooses y[Y5[0, 2].
The principal’s payoff V( y, t ) depends on agent 1’s type t1 1[T15ht , ta bj and the allocation y according to
2 2
V( y, t )a 5 2y , V( y, t )b 5 2(22y) . (2)
The payoff of type t of agent 1 is U ( y, t ), where1 1 1
2 2
U ( y, t )1 a 5 2(0.52y) , U ( y, t )1 b 5 2(1.52y) . (3)
Agent 2’s payoff is independent of agent 1’s type and is given by
2
U ( y)2 5 210(12y) . (4)
Note that, if after receiving a message m the principal believes that agent 1 is of type ta with
probability p(taum)512p(tbum), condition (1) implies that the allocation function satisfies
y(m)52 1
f
2p(taum) .g
(5)Moreover, the principal’s conditional belief is consistent with Bayes’ rule if
m1(mut )a ]]]]]]
p(taum)5 (6)
m1(mut )a 1m1(mut )b
for all m[M such that m1(mut )a 1m1(mut )b .0.
Our argument proceeds as follows: We first construct a Perfect Bayesian equilibrium for a message
space M5hm , m , mA B Cj with three messages. Then we show that one of the players must become
worse off in comparison with this equilibrium when M is replaced by the direct mechanism T15ht ,a
tbj. Our first observation shows that with three messages for agent 1 there is a partial pooling
equilibrium: Whereas the messages m and m reveal the agent’s type, message m is uninformativeA C B
Observation 1. The message space M5hm , m , mA B Cj supports a Perfect Bayesian equilibrium such
Proof. Let the principal’s beliefs and the allocation function satisfy
p(taum )A 51, p(taum )B 50.5, p(taum )C 50;y(m )A 50,y(m )B 51,y(m )C 52. (7)
It is easily verified that p(taum) and y(m) satisfy conditions (5) and (6). Given the allocation function, the agent’s reporting strategy is optimal because
We now characterize the Perfect Bayesian equilibria that can be supported by a direct mechanism
T15ht , ta bj. The following observation shows that T supports a Perfect Bayesian equilibrium in1 which agent 1 reveals his type with probability one.
Observation 2. The direct mechanism T15ht , ta bjsupports a Perfect Bayesian equilibrium such that
m1(taut )a 5m1(tbut )b 51. In this equilibrium the principal has the ex ante expected payoff V *50; the
Since each type reveals himself, the beliefs satisfy (6) and by (5) the principal responds optimally. Revealing his type is optimal for agent 1 because
U (1 y(t ), t )a a 5 20.25$ 22.255U (1 y(t ), t )b a (10)
Note that, when ignoring the presence of agent 2, the equilibrium described by Observation 2 can be viewed as an example of our Theorem: The principal’s equilibrium payoff under the direct
mechanism T1 is increased in comparison with the equilibrium under the mechanism M in
We now show that there are pooling equilibria, in which both types of agent 1 adopt the same
Agent 1’s reporting strategy entails no information revelation. Therefore, the principal’s beliefs and
the allocation function satisfy (6) and (5). Since y(t )a 5y(t ), agent 1 is indifferent betweenb
*
*
announcing t and t . He receives the payoff U (t )a b 1 a 5U (t )1 b 5 20.25. Becausey(t )a 5y(t )b 51 and
*
V(1, t )a 5V(1, t )b 5 21, the principal and agent 2 realize the payoffs V *5 21, U2 50. h
Our final observation shows that under a direct mechanism there are no other Perfect Bayesian equilibria beyond those described in our last two observations.
Observation 4. The direct mechanism T15ht , ta bj does not support a Perfect Bayesian equilibrium such that m(t ut )±m (t ut ) and m(t ut )[(0, 1) or m(t ut )[(0, 1).
1 a a 1 a b 1 a a 1 a b
Proof. Consider an equilibrium where one type of agent 1 employs a mixed reporting strategy. We
will show that then the other type must mimic this strategy. Assume for instance that type t ’sa
reporting strategy is given by m1(taut )a 512m1(tbut )a 5a[(0, 1). Note that this requires U (1 y(t ),a
t )a 5U (1 y(t ), t ).b a
First consider the possibility that type t chooses the pure reporting strategyb m1(tbut )b 51. Then Bayesian updating yields p(taut )a 51.0.5.p(taut ). By (5), with these beliefs the principal choosesb
y(t )a 50 and y(t )b .1. However, U (0, t )1 a 5U (1 y(t ), t )a a 5U (1 y(t ), t ) implies that eitherb a y(t )b 50 ory(t )b 51. Thus it cannot happen in equilibrium thatm1(tbut )b 51. The same argument eliminates the possibility that m1(taut )b 51 in equilibrium.
We are thus left with the possibility that also type tb chooses a mixed reporting strategy
m1(taut )b 5b[(0, 1), This strategy is optimal for type t only if U (b 1 y(t ), t )a b 5U (1 y(t ), t ). Thisb b equality in combination with U (1 y(t ), t )a a 5U (1 y(t ), t ) yieldsb a y(t )a 5y(t ). By (5) the principal’sb beliefs therefore must satisfy p(taut )a 5p(taut ). This is consistent with Bayes’ rule only ifb a5b. By
repeating the entire argument for an assumed mixing behavior of tb we may conclude that the
equilibrium described in Observation 3 is the only one in which some type of agent 1 uses a mixed
strategy. h
equilibrium of the game with the direct mechanism either the principal or agent 2 is worse off relative to the Perfect Bayesian equilibrium of the game with three messages. This demonstrates that Theorem 1 of Section 2.2 for the single agent case does not apply in an environment with multiple agents.
4. Conclusion
This paper studies mechanism design problems in environments where the mechanism designer cannot fully commit to the outcome induced by the mechanism. In this context, most of the literature focuses on contracting problems between a principal and a single agent (see e.g. Hart and Tirole, 1988; Laffont and Tirole, 1990) and bases its analysis on the optimality of direct mechanisms. Our example demonstrates that a direct mechanism may no longer be optimal when a principal contracts with multiple agents. It is therefore unclear to what extent the results in the literature apply to situations with more than one agent. More importantly, it remains an open question how one can characterize optimal mechanisms for contracting problems with limited commitment and multiple agents.
References
Bester, H., Strausz, R., 2000. Contracting with imperfect commitment and the revelation principle: the single agent case. Free University of Berlin, mimeo.
Dasgupta, P., Hammond, P., Maskin, E., 1979. The implementation of social choice rules. Review of Economic Studies 46, 185–216.
Gibbard, A., 1973. Manipulation for voting schemes. Econometrica 41, 587–601.
Hart, O., Tirole, J., 1988. Contract renegotiation and coasian dynamics. Review of Economic Studies 55, 509–540. Green, J., Laffont, J.-J., 1977. Characterization of satisfactory mechanisms for the revelation of preferences. Econometrica
45, 427–438.
Laffont, J.-J., Tirole, J., 1990. Adverse selection and renegotiation in procurement. Review of Economic Studies 57, 579–625.
