Communication in Bayesian games:
Overview of work on implementing mediators
in game theory
Françoise Forges
June 2010
This is a slightly enriched version of the slides of my talk at the workshop “Decentralized Mechanism Design, Distributed Computing and Cryptogra-phy” organized by I. Abraham, D. Gerardi and J. Halpern, Princeton, June 3-4, 2010. The main reference is Forges (2009).
Plan of the talk
1. Basic de…nitions: Bayesian game, cheap talk, mediator, communica-tion equilibrium outcome (CEO), implementacommunica-tion of CEOs without a mediator
2. From in…nitely repeated games to cheap talk 3. Cheap talk in sender-receiver games
4. From mental poker to cheap talk
5. Some representative implementation results 6. Topics for future research
1
Basic De…nitions
A Bayesian game is described by the parameters N; (Ti)
i2N; p; (Ai)i2N; (ui)i2N,
where
- N is the set of players - Ti
is the (…nite) set of types of player i, i 2 N - pis a probability distribution over T =Qj2NTj
- Ai is the (…nite) set of actions of player i; let A denote the action space:
A =Qj2NAj.
- ui : T A ! R is the (von Neumann-Morgenstern) utility function of player i
is played as follows:
A move of nature chooses t = (tj)
j2N according to p
Player i is only informed of his own type ti The players choose simultaneously an action.
In order to give a de…nition of “cheap talk” in the Bayesian game , we …rst focus on interim, bounded cheap talk extensions of , which take place for a …nite number of stages S, after the move of nature:
At the beginning of stage s = 1; 2; :::; S, every player i selects a single, non-anonymous, message (from a …nite set), which is delivered to a subset of players at the end of stage s; messages from di¤erent players are sent simultaneously. Players perfectly recall all their past messages.
At stage S + 1, the players simultaneously choose an action as in and are rewarded as in , independently of the exchanged messages.
A cheap talk extension of generates a new (“extended”) game, to which the standard implicit common knowledge assumption applies.
Remark: the expression “cheap talk” (or “plain conversation”, or “direct communication”, etc.), has been used, more or less formally, to cover vari-ous ways in which players can exchange messages. Here, a player can only
send one message at every stage, but this message can be received by sev-eral players at the same time and yet not be public (“partial broadcast”). Alternatively, one could for instance just assume that there is a secure com-munication channel between any pair of players. In the description above, messages are delivered without delay.
Let us now introduce communication with a “mediator” in the Bayesian game (see Forges (1986) and Myerson (1986)). We start with the simplest conceivable de…nition. For every N tuple of types t = (tj)j2N, let q(:jt) be a probability distribution over the action space A and let q = (q(:jt))t2T.
The interpretation is that, after the move of nature in , a mediator invites every player i, i 2 N, to report a type ri, then selects an N -tuple of
actions a according to q(:jr), r = (ri)
i2N, and privately recommends ai to
player i. This de…nes a “mediated game”.
A system of type dependent probability distributions q is a communica-tion equilibrium outcome (CEO) of if in the mediated game, none of the players can gain by unilaterally lying on his type or deviating from the rec-ommended action (i.e., to be “honest and obedient”is a Nash equilibrium in the mediated game):
X t i2T i p(t ijti)X a2A q(ajt)ui(t; a) X t i2T i p(t ijti)X a2A q(ajri; t i)ui(t; i(ai); a i) 8i 2 N; 8ti; ri 2 Ti;8 i : Ai ! Ai
This apparently restrictive de…nition is justi…ed by the canonical rep-resentation of communication equilibria (which generalizes the “reve-lation principle”in mechanism design, see, e.g., Myerson (1982)). In order to state it, let us consider an arbitrary ex ante and interim cheap talk extension of . By “ex ante”, we mean that the players can also exchange messages before the move of nature in . Let us add a mediator to the cheap talk extension, namely, a set of inputs Ii
s and a set of outputs Osi for every player
i at every stage s, together with transition probabilities to choose N tuples of outputs at stage s as a function of all past inputs and outputs. A general mediated extension of is de…ned as follows: at every stage, before exchang-ing cheap messages, the players simultaneously send an input to the mediator
and then receive a private output from the mediator. We can now state a basic lemma:
Canonical representation of communication equilibria: The set of all Nash equilibrium outcomes of all general mediated extensions of a Bayesian game coincides with the set of CEOs of :
The communication equilibrium extends the concept of correlated equi-librium, which was originally de…ned by Aumann (1974, 1987) for games with complete information, in which j Ti j= 1, i 2 I. The basic lemma applies. More precisely, let a correlated equilibrium outcome be de…ned in this case as a probability distribution over A. The set of all Nash equilib-rium outcomes of all general mediated extensions of a game G with complete information coincides with the set of correlated equilibrium outcomes of G.
More generally, correlated equilibria also account for ex ante cheap talk in the Bayesian game (which can be represented in strategic form, with strategy sets ATi
i , i 2 N); typically, a mediator recommends then a complete
strategic plan (in ATi
i ) to every player i 2 N before the beginning of the
game (i.e., even before the move of nature). In particular, we can study correlated equilibria in an interim cheap talk extension of the Bayesian game . However some game theorists (e.g., Myerson (2004)) argue that a Bayesian game has no ex ante stage.
We can now formulate some natural questions:
Implementation problem: Can we achieve all CEOs of as (possibly “re…ned”, e.g., “sequential”) Nash (or correlated) equilibrium outcomes of a cheap talk extension of ?
Variant: Can we achieve a speci…c class of CEOs of as (possibly “re…ned”) Nash (or correlated) equilibrium outcomes of a cheap talk extension of ? Characterization problem: Can we characterize all (possibly “re…ned”) Nash (or correlated) equilibrium outcomes of a speci…c cheap talk extension of ?
2
From in…nitely repeated games to cheap
talk
In the late sixties, Aumann and Maschler introduced the model of in…nitely repeated games with incomplete information in order to study unmediated communication in strategic games. This seminal work …rst appeared as a (rather con…dential) report to the U.S. Arms Control and Disarmament Agency (see Aumann and Maschler (1995)).
Extending Aumann and Maschler’s model, Aumann, Maschler and Stearns (1968) consider an in…nitely repeated two-person game 1 in which player 1 learns his type t1
2 T1
at stage 0, jT2
j = 1 and both players simultane-ously choose actions at every stage 1, 2,...The players observe each other’s action but not their stage utilities. In the game 1, information transmis-sion is cheap as far as utilities are not discounted : the players can use as many stages as they wish for communication purposes. Simultaneous moves allow for jointly controlled lotteries, a device which will turn out to be cru-cial later on to implement mediators by cheap talk. A major insight in Aumann, Maschler and Stearns (1968) is that “several stages of information transmission make a di¤erence”in the sense that the set of Nash equilibrium outcomes that the players can achieve strictly increases with the number of communication stages. But the game 1 is a complex one, since in addition to cheap talk, the game allows for long term cooperation, as in the so-called “Folk theorem”.
S. Hart (1985), together with Aumann and Hart (1986), provides a full characterization of all Nash equilibrium outcomes of 1. In particular, building on a geometric example of Aumann and Hart (1986), Forges (1984) demonstrates that more equilibrium outcomes can be achieved with an “un-bounded” number of stages of information transmission.
The previous results gave rise to “remakes”, namely, they were reformu-lated in terms of Nash equilibrium outcomes of long cheap talk extensions of a (one-shot) Bayesian game . Forges (1990a) considers a job market example with the same basic features as in Forges (1984). Aumann and Hart (2003) characterize the Nash equilibrium outcomes of in…nite cheap talk extensions of any Bayesian game in which jNj = 2, jT2
j = 1.
Coming back to the in…nitely repeated game, Forges (1985) proposes a characterization of all correlated equilibria of 1 under the further assump-tion that player 1’s acassump-tions have no impact on utilities, i.e., that ui(t1; a1; a2)
ui(t1; a2), i = 1; 2.
1then amounts to a “sender-receiver game”. As a direct
by-product of Forges (1985), one gets the following result on the implemen-tation of a mediator by cheap talk:
Proposition 1: Let be a two-person Bayesian game with jT2
j = jA1
j = 1. Every communication equilibrium outcome (CEO) of can be achieved as a correlated equilibrium of an interim cheap talk extension of , in which the informed player sends a single message from a …nite set to the uninformed player.
Forges (1988a) extends the result to the case of multiple senders; Forges (1988b) characterizes the correlated equilibria in all games 1in which jNj = 2 and jT2j = 1, i.e., allowing player 1 to make utility relevant decisions.
3
Cheap talk in sender-receiver games
Crawford and Sobel (1982) (and, independently, Green and Stokey (1980/ 2007)) considered a speci…c, economically relevant, class of sender-receiver games, in which T1 = A2 = [0; 1]. They characterized, as “partitional
equi-librium outcomes”, all Nash equiequi-librium outcomes of the one-shot interim cheap talk extension of , in which the informed player sends a single mes-sage to the uninformed player. The main mesmes-sage at the time was that “Cheap talk matters! ”, which came as a surprise to economists who, in the light of e.g. Spence (1973), considered that signalling costs were crucial to fruitful information transmission.
Krishna and Morgan (2004) show that “several stages of information transmission make a di¤erence”in Crawford and Sobel’s model by exhibiting a three-stage cheap talk equilibrium which Pareto-improves on single stage ones.
Goltsman, Hörner, Pavlov and Squintani (2009) provide necessary and su¢ cient conditions for cheap talk implementation of e¢ cient CEOs.
Blume (2010) establishes that e¢ cient CEOs can be implemented as cor-related equilibria of the one-shot cheap talk game (relying on a construction which di¤ers from Forges (1985)).
4
From mental poker to cheap talk
In 1982-1983, Imre Bárány was visiting CORE (Belgium) and presented his paper “Mental poker”with Z. Füredi. J.-F. Mertens suggested that the tech-niques used in that paper could be useful to implement correlated equilibria by “plain conversation” between su¢ ciently many players.
Proposition 2(Bárány (CORE DP 1987, 1992)): Let G be a strategic form game with n 4 players. Every rational (i.e., in Q) correlated equilibrium outcome of G can be achieved as a Nash equilibrium of a cheap talk+PVR (ex ante) extension of G with …nitely many stages.
In Bárány’s original statement, cheap talk is enriched by possible public veri…cation of the record (PVR): at every stage, every player can ask for the revelation of all exchanged messages. But this assumption is in fact not necessary, thanks to “logical signatures”(Ben-Or, Goldwasser and Widgerson (1988), Vida (2007)).
Observe that the implementation is in Nash, not sequential, equilibrium; indeed, in Bárány’s construction, players need not optimize out of the equi-librium path.
5
Some (further) representative
implementa-tion results
The following result is in the same vein as proposition 1:
Proposition 3(Forges (1990b)): Let be a Bayesian game with n 4 play-ers. Every communication equilibrium outcome (CEO) of can be achieved as a (“sequential”) correlated equilibrium of an interim two stage cheap talk extension of .
Several de…nitions of “sequential”correlated equilibrium are conceivable; in the previous statement, we just mean that the construction does not in-volve any irrational punishments. The same remark already applied to propo-sition 1.
By combining the variant of proposition 2 (without PVR), the previous proposition 3 and Gerardi (2000), we get the following
Corollary: For n 4, every rational CEO can be implemented as a Nash equilibrium with interim cheap talk in …nitely many stages.
A direct application of proposition 2 and proposition 3, as in Forges (1990b), leads the players to perform the cheap talk phase achieving cor-relation at the ex ante stage. Gerardi (2000) modi…es the construction to guarantee that cheap talk only takes place at the interim stage. The corollary is consistent with recent results, like Abraham, Dolev, Gonen and Halpern (2006) and Abraham, Dolev and Halpern (2008), which will be presented in more details later in the workshop.
The corollary inherits the main drawback of proposition 2, namely that the implementation of the correlation phase makes use of possibly non-rational punishments. The problem disappears once there are at least 5 players:
Proposition 4 (Gerardi (2004)): Let be a Bayesian game with n 5 players. Every rational (i.e., in Q), full support, CEO of can be achieved as a sequential Nash equilibrium of an interim cheap talk extension of with …nitely many stages.
The solution to the implementation problem in this proposition satis…es three essential requirements: (essentially) all CEO’s are implemented, cheap talk takes place exclusively at the interim stage and the implementation is achieved with an appropriate re…nement of Nash equilibrium (see Gerardi and Myerson (2007) for a detailed analysis of sequential CEOs, in particular for the role of the full support assumption).
The assumption of 5 players in proposition 4 is of course a strong one. Possibility results are also available for n 4players:
Proposition 5 (Ben Porath (2003+corrigendum 2006)): Let be a Bayesian game with n 4 players. Let q be a rational (i.e., in Q), full support, CEO of which interim strictly dominates a Nash equilibrium of :
8i 2 N; 8ti 2 Ti : Ui(qj ti) > Ui( j ti)
q can be achieved as a sequential Nash equilibrium of an interim cheap talk extension of with …nitely many stages.
In the previous statement, the restriction on the CEO’s to be implemented is extremely useful, since allows to credibly punish all players, whatever
their types, at the same time (in , i.e., at the prior probability distribution p). The proofs of propositions 3, 4 and 5 all make use of the number of players to rely on majority rule at some point: a receiver expects the same message from three di¤erent senders and decides by majority rule; this makes useless any sender’s unilateral deviation. Ben Porath (2003) states proposition 5 for n = 3, but, as explained in Ben Porath (2006), for n = 3, an additional assumption is required. Before we go on with n = 3, let us mention that for n 4, we can still hope for possibility results that would not put any restriction on the implementable CEO’s:
Conjecture (Vida (2010, personal communication)): for n 4, every CEO can be implemented as a sequential Nash equilibrium with interim cheap talk if messages can be chosen from a continuum.
When the number of players is n = 3, several impossibility results indicate that satisfactory implementation require careful assumptions (see Bárány (1992), Forges (1990b), Ben Porath (2003), Abraham, Dolev, Gonen and Halpern (2006), Abraham, Dolev and Halpern (2008), Vida (2007)). For instance, Ben Porath (2006) shows that proposition 5 holds for n = 3 if it is understood that cheap talk allows for an appropriate form of PVR (every player i can safely deposit at stage t a message to be publicly revealed at stage t + s). Other remedies are: cheap talk using a continuum of messages (Vida (2010, personal communication)), cheap talk with a possibly in…nite number of rounds (see Abraham, Dolev and Halpern (2008) and the talks by Joe Halpern and Ittai Abraham in this workshop).
For n = 2 players, there are even more impossibility results. For instance, with complete information, there is no hope to implement more than lotteries over Nash equilibrium outcomes.
Proposition 6 (Vida (2007)): Let be a Bayesian game with n 2 players; essentially every CEO of can be implemented as a correlated equilibrium with interim, long, a.s. …nite, cheap talk.
Ben Porath (1998) and R.V. Krishna (2007) propose implementation re-sults for n = 2 players by relaxing the rules of cheap talk so as to allow the players to make use of urns or envelopes. Dodis, Halevy and Rabin (2000) and Urbano and Vila (2002, 2004) rely on cryptography.
6
Topics for future research
In…nitely repeated games with incomplete information: as poin-ted out in section 2, this model was conceived to study cheap infor-mation transmission at the same time as cooperation but was mostly studied for two person games with lack of information on one side. An important feature of this framework is that it …xes the rules of the implicit cheap talk allowed to the players (which can be seen as an ad-vantage or a drawback). Another important property of this model is that it implicitly captures the e¤ects of commitment. Just to mention a possible recent interest in the topic: Golosov, Skreta, Tsyvinski and A. Wilson (2009) consider a repeated version of Crawford and Sobel (1882)’s game.
Strong (or coalition-proof ) correlated/communication equilib-ria: in this talk, we focused on communication protocols which resist to every player’s unilateral deviations (captured by Nash equilibrium). Starting with Aumann’s strong Nash equilibrium, game theory has de-veloped tools to deal with deviations by coalitions of players. Some pa-pers (Milgrom and Roberts (1996), Moreno and J. Wooders (1996), I. Ray (1996)) proposed a de…nition of coalition-proof correlated equilib-ria, while Einy and Peleg (1995) made an attempt at de…ning coalition-proof communication equilibria. A main issue is to determine the cheap talk extension in which to apply game-theoretical solution concepts dealing with possible deviations by coalitions: the revelation principle typically does not survive in this environment. More generally, the devi-ations by coalitions are handled with a very speci…c scenario in mind. In computer science, it is customary to design protocols that are robust to deviations by coalitions of a su¢ ciently small size (see Abraham, Dolev, Gonen and Halpern (2006), Abraham, Dolev and Halpern (2008) and the talks by Joe Halpern and Ittai Abraham in this workshop).
Mediation with commitment: in this talk, mediated communica-tion has only been used to correlate strategies and to exchange infor-mation, but it can also serve as a commitment device. Moulin and Vial (1978) is an early paper using this idea. Recent papers on the topic are Monderer and Tennenholtz (2009), Ashlagi, Monderer and Tennenholtz (2009).
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