Implementing Mediators with Asynchronous Cheap Talk
aa r X i v : . [ c s . D C ] J un Implementing Mediators with Asynchronous Cheap Talk
Ittai AbrahamVMWARE [email protected]
Danny Dolev ∗ School of Computer Science and EngineeringThe Hebrew University of JerusalemJerusalem, Israel [email protected]
Ivan Geffner † Cornell University [email protected]
Joseph Y. Halpern ‡ Cornell University [email protected]
June 5, 2018
Abstract
A mediator can help non-cooperative agents obtain an equilibrium that may oth-erwise not be possible. We study the ability of players to obtain the same equilibriumwithout a mediator, using only cheap talk , that is, nonbinding pre-play communication.Previous work has considered this problem in a synchronous setting. Here we considerthe effect of asynchrony on the problem, and provide upper bounds for implementingmediators. Considering asynchronous environments introduces new subtleties, includ-ing exactly what solution concept is most appropriate and determining what move isplayed if the cheap talk goes on forever. Different results are obtained depending onwhether the move after such “infinite play” is under the control of the players or partof the description of the game.
Having a trusted mediator often makes solving a problem much easier. For example, aproblem such as Byzantine agreement becomes trivial with a mediator: agents can just sendtheir initial input to the mediator, and the mediator sends the majority value back to allthe agents, which they then output. Not surprisingly, the question of whether a problem ∗ Danny Dolev is Incumbent of the Berthold Badler Chair in Computer Science. † Supported in part by NSF grants IIS-1703846. ‡ Supported in part by NSF grants IIS-1703846 and IIS-0911036, and ARO grant W911NF-17-1-0592, anda grant from Open Philanthropy. n a multiagent system that can be solved with a trusted mediator can be solved by justthe agents in the system, without the mediator, has attracted a great deal of attentionin both computer science (particularly in the cryptography community) and game theory.In cryptography, the focus has been on secure multiparty computation [11, 14]. Here it isassumed that each agent i has some private information x i . Fix functions f , . . . , f n . Thegoal is to have agent i learn f i ( x , . . . , x n ) without learning anything about x j for j = i beyond what is revealed by the value of f i ( x , . . . , x n ). With a trusted mediator, this istrivial: each agent i just gives the mediator its private value x i ; the mediator then sendseach agent i the value f i ( x , . . . , x n ). Work on multiparty computation provides conditionsunder which this can be done in a synchronous system [6, 11, 13, 14] and in an asynchronoussystem [5, 7]. In game theory, the focus has been on whether an equilibrium in a gamewith a mediator can be implemented using what is called cheap talk —that is, just by playerscommunicating among themselves.In the computer science literature, the interest has been in performing multiparty com-putation in the presence of possibly malicious adversaries, who do everything they can tosubvert the computation. On the other hand, in the game theory literature, the assumptionis that players have preferences and seek to maximize their utility; thus, they will subvertthe computation iff it is in their best interests to do so. In [1, 2] (denoted ADGH and ADH,respectively, in the rest of the paper), it was argued that it is important to consider devia-tions by both rational players, who have preferences and try to maximize them, and playersthat we can view as malicious, although it is perhaps better to think of them as rationalplayers whose utilities are not known by the mechanism designer (or other players). ADGHand ADH considered equilibria that are ( k, t ) -robust ; roughly speaking, this means that theequilibrium tolerates deviations by up to k rational players, whose utilities are presumedknown, and up to t players with unknown utilities. Tight bounds were proved on the abilityto implement a ( k, t )-robust equilibrium in the game with a mediator using cheap talk insynchronous systems. These bounds depend on, among other things, (a) the relationshipbetween k , t and n , the total number of players in the system; (b) whether players know theexact utilities of the rational players; and (c) whether the game has a punishment strategy ,where an m -punishment strategy is a strategy profile that, if used by all but at most m play-ers, guarantees that every player gets a worse outcome than they do with the equilibriumstrategy. The following is a high-level overview of results proved in the synchronous settingthat will be of most relevance here. For these results, we assume that the communicationwith the mediator is bounded, it lasts for at most N rounds, and that the mediator can berepresented by an arithmetic circuit of depth c .R1. If n > k + 3 t , then a mediator can be implemented using cheap talk; no punishmentstrategy is required, no knowledge of other agents’ utilities is required, and the cheap-talk protocol has bounded running time O ( nN c ), independent of the utilities.R2. If n > k +3 t , then a mediator can be implemented using cheap talk if there is a ( k + t )-punishment strategy and the utilities of the rational players are known; the cheap-talkprotocol has expected running time O ( nN c ). (In R2, unlike R1, the cheap-talk gamemay be unbounded, although it has finite expected running time.)2n ADH, lower bounds are presented that match the upper bounds above. Thus, forexample, it is shown that n > k + 3 t is necessary in R1; if n ≤ k + 3 t , then we cannotimplement a mediator in general if we do not have a punishment strategy or if the utilitiesare unknown. The proofs of the upper bounds make heavy use of the fact that the settingis synchronous. Here we consider the impact of asynchrony on these results. Once weintroduce asynchrony, we must revisit the question of what it even means to implement anequilibrium using cheap talk. Notions like (Bayesian) Nash equilibrium implicitly assumethat all uncertainty is described probabilistically. Having a probability is necessary to talkabout an agent’s expected utility, given that a certain strategy profile is played. If we werewilling to put a distribution on how long messages take to arrive and on when agents arescheduled to move, then we could apply notions like Nash equilibrium without difficulty.However, it is notoriously difficult to quantify this uncertainty. The typical approach usedto analyze algorithms in the presence of uncertainty that is not quantified probabilistically isto assume that all the non-probabilistic uncertainty is resolved by the environment accordingto some strategy. Thus, the environment uses some strategy to decide when each agent willbe allowed to play and how long each message takes to be delivered. The algorithm isthen proved correct no matter what strategy the environment is following in some class ofstrategies. For example, we might restrict the environment’s strategy to being fair , so thatevery agent eventually gets a chance to move. (See [12] for a discussion of this approach andfurther references.)We follow this approach in the context of games. Note that once we fix the environment’sstrategy, we have an ordinary game, where uncertainty is quantified by probability. Inthis setting, we can consider what is called ex post equilibrium . A strategy is an ex postequilibrium if it is an equilibrium no matter what strategy the environment uses. Ex postequilibrium is a strong notion, but, as we show by example, it can often be attained withthe help of a mediator. We believe that it is the closest analogue to Nash equilibrium in anasynchronous setting.Another issue that plays a major role in an asynchronous setting is what happens if thestrategies of players result in some players being livelocked , talking indefinitely without mak-ing a move in the underlying game, or in some players being deadlocked , waiting indefinitelywithout moving in the underlying game. We consider two approaches for dealing with thisproblem. One way to decide what action to assign to a player that fails to make a decisionin the cheap-talk phase is the default-move approach . In this approach, as part of the de-scription of the game, there is a default move for each player which is imposed if that playerfails to explicitly make a move in the cheap-talk phase. Aumann and Hart [4] considereda different approach, which we henceforth call the AH approach, where a player’s strategyin the underlying game is a function of the (possibly infinite) history of the player in thecheap-talk phase. We can think of this almost as a player writing a will, describing what hewould like to do (as a function of the history) if the game ends before he has had a chanceto move.We believe that both the AH approach and the default-move approach are reasonable indifferent contexts. The AH approach makes sense if the agent can leave instructions that3ill be carried out by an “executor” if the cheap-talk game deadlocks. But if we considera game-theoretic variant of Byzantine agreement, it seems more reasonable to say that if amalicious agent can prevent an agent from making a move in finite time, the agent shouldnot get a chance to make a move after the cheap-talk phase has ended.Our results show that, in the worst case, the cost of asynchrony is an extra k + t in thebounds on n , but we can sometimes save k or even k + t if there is a punishment strategyor if we are willing to tolerate an ǫ “error”. For example, with both the AH approach andthe default-move approach, if the utilities are not known, we can implement a mediatorusing asynchronous cheap talk if n > k + 4 t . Thus, compared to R1, we need an extra k + t . However, if we are willing to accept a small probability of error, so that rather thanimplementing the mediator we get only an ǫ -implementation, and are also willing to accept ǫ -(k,t)-robustness (which, roughly speaking, means that players get within ǫ of the best theycould get), then we can do this if n > k + 3 t , again, using both the AH approach and thedefault-move approach.Just as in the synchronous case, we can do better if we assume that there is a punishmentstrategy and utilities are known (as in R2). Specifically, with the AH approach, we canimplement a mediator if n > k + 4 t (compared to n > k + 3 t in the synchronous case, andcan ǫ -implement a mediator if n > k + 3 t . We use the punishment to deal with deadlock. Ifa good player is waiting for a message that never arrives, then the waiting player instructshis executor to carry out a punishment in his will. Having a punishment does not seem tohelp in the default-move approach unless the default move is a punishment; if it is, then wecan get the same results as with the AH approach.If there is a punishment strategy, these results significantly improve those of Even, Gol-dreich, and Lempel [9]. They provide a protocol with similar properties, but the expectednumber of messages sent is O (1 /ǫ ); with a punishment strategy, a bounded number of mes-sages is sent, with the bound being independent of ǫ . Asynchronous games, mediator games, and cheap talk:
We are interested in imple-menting mediators. Formally, this means we need to consider three games: an underlyinggame
Γ, an extension Γ d of Γ with a mediator, and an extension Γ CT of Γ with (asyn-chronous) cheap-talk. We assume that Γ is a normal-form Bayesian game : each player hasa type t taken from some type space T i , such that there is a commonly known distributionon T ⊆ T × · · · × T n , the set of types; each player i chooses an action a ∈ A i , the set ofactions of agent i ; player i ’s utility u i is determined by the type profile of the players andthe actions they take. A strategy for player i in the Bayesian game is just a function T i to A i , which tells player i what to do, given his type. If A = A × · · · × A n , then a strategyprofile ~σ = ( σ , . . . , σ n ) can be viewed as a function ~σ : T → ∆( A ) (where, as usual, ∆( X )denotes the set of probability distributions on X ).The basic notions of a game with a mediator, a cheap-talk game, and implementation arestandard in the game-theory literature. However, since we consider them in an asynchronous4etting, we must modify the definitions somewhat.We first define asynchronous games . In an asynchronous game, we assume that playersalternate making moves with the environment—first the environment moves, then a playermoves, then the environment moves, and so on. The environment’s move consists of choosinga player i to move next and a set of messages in transit to i that will be delivered just before i moves (so that i ’s move can depend on the messages i receives). The environment is subjectto two constraints: all messages sent must eventually be delivered and, for all times m andplayers i , if i is still playing the game at time m , then there must be some time m ′ ≥ m that i is chosen to move. We can describe an asynchronous game by a game tree. Associatedwith each non-leaf node or history is either a player—the player whose move it is at thatnode—or the environment (which can make a randomized move). The nodes where a player i moves are further partitioned into information sets ; intuitively, these are nodes that player i cannot tell apart. We assume that the environment has complete information, so that theenvironment’s information sets just consist of the singletons. A strategy for player i is a(possibly randomized) function from i ’s information sets to actions; we can similarly define astrategy for the environment. We can essentially view the environment strategy as defininga scheduler (and thus we sometimes refer to an environment strategy as a scheduler).For our results, we start with an n -player Bayesian game Γ in normal form (called the underlying game ), with { , . . . , n } being the set of players, and then consider two types ofgames that extend Γ. A game Γ ′ extends Γ if the players have initial types from the sametype space as Γ, with the same distribution over types; moreover, in each path of the gametree for Γ ′ , the players send and receive messages, and perform at most one action from Γ.In a history where each player makes a move from Γ, each player gets the same utility asin Γ (where the utility is a function of the moves made and the types). That leaves openthe question of what happens in a complete history of Γ ′ where some players do not makea move in Γ. As we suggested in the introduction, we consider two approaches to dealingwith this. In the first approach, we assume that the description of Γ ′ includes a function M i for each player i that maps player i ’s type to a move in Γ. In an infinite history h where i has type t and does not make a move in Γ, i is viewed as having made move M i ( t ). Wecan then define each player’s utility in h as above. This is the default-move approach . Inthe AH approach, we extend the notion of strategy so that i ’s strategy in Γ ′ also describeswhat move i makes in the underlying game Γ in any infinite history h where i has not madea move in Γ. In the AH approach, i ’s move in h is under i ’s control; in the default-moveapproach, it is not.Given an underlying Bayesian game Γ (which we assume is synchronous—the playersmove simultaneously), we will be interested in two types of extensions. A mediator game ex-tending Γ is an asynchronous game where players can send messages to and receive messagesfrom a mediator (who can be viewed as a trusted third party) as well as making a move inΓ; “good” or “honest” players do not send messages to each other, but “bad” players (i.e.,one of the k rational deviating players or one of the t “malicious” players with unknownutilities) may send messages to each other as well as to the mediator. We assume that thespace of possible messages that can be sent in a mediator game is fixed and finite.5n an asynchronous cheap-talk game extending Γ, there is no mediator. Players sendmessages to each other via asynchronous channels, as well as making a move in Γ. We assumethat each pair of agents communicates over an asynchronous secure private channel (the factthat the channels are secure means that an adversary cannot eavesdrop on conversationsbetween the players). We assume that each player can identify the sender of each message.Finally, we assume that in both the mediator game and the cheap-talk game, when a playeris first scheduled, it gets a signal that the game has started (either an external signal fromthe environment, or a game-related message from another player or the mediator). Implementation:
In the synchronous setting, a strategy profile ~σ ′ in a cheap-talk Γ CT extending an underlying game Γ implements a strategy ~σ in a mediator game Γ d extendingΓ if ~σ and ~σ ′ correspond to the same strategy in Γ; that is, they induce the same functionfrom T to ∆( A ). The notion of implementation is more complicated in an asynchronoussetting, because the probability on action profiles also depends on the environment strategy.Because Γ CT and Γ d are quite different games, the environment’s strategies in Γ CT are quitedifferent from those in Γ d . So we now say that ~σ ′ implements ~σ if the set of distributionson actions profiles in Γ induced by ~σ and all possible choices of environment strategy is thesame as that induced by ~σ ′ and all possible choices of environment strategy. More precisely,let S Γ ′ ,e and S Γ ′′ ,e denote the the set of environment strategies in Γ ′ and Γ ′′ , respectively. Astrategy σ e ∈ S Γ ′ ,e and a strategy profile ~σ for the players in Γ ′ together induce a function( ~σ, σ e ) from T to ∆( A ). A strategy profile ~σ ′ in Γ ′ implements a strategy profile ~σ ′′ in Γ ′′ if { ( ~σ ′ , σ ′ e ) : σ ′ e ∈ S Γ ′ ,e } = { ( ~σ ′′ , σ ′′ e ) : σ ′′ e ∈ S Γ ′′ ,e } . Since the outcome that arises if the playersuse a particular strategy may depend on what the environment does, this says that the setof outcomes that can result if the players use ~σ ′ is the same as the set of outcomes that canresult if the players use ~σ ′′ .For some of our results, we cannot get an exact implementation; there may be someerror. Given two discrete distributions π and π ′ on some space S , the distance between π and π ′ , denoted dist ( π, π ′ ), is at most ǫ if P s ∈ S | π ( s ) − π ′ ( s ) | ≤ ǫ . As we observed earlier,in the mediator game and the cheap-talk game, Recall that a strategy profile ~σ for theplayers and a strategy σ e for the environment together induce a mapping from type profilesto ∆( A ). We lift the notion of distance to such function by defining dist (( ~σ, σ e )) , ( ~σ ′ , σ ′ e )) =max ~x ∈T dist (( ~σ, σ e )( ~x ) , ( ~σ ′ , σ ′ e )( ~x )). Say that ~σ ′ ǫ - implements ~σ ′′ if • for all σ ′ e ∈ S Γ ′ ,e there exists σ ′′ e ∈ S Γ ′′ ,e such that dist (( ~σ ′ , σ ′ e ) , ( ~σ ′′ , σ ′′ e )) ≤ ǫ ; and • for all σ ′′ e ∈ S Γ ′′ ,e there exists σ ′ e ∈ S Γ ′ ,e such that dist (( ~σ ′′ , σ ′′ e ) , ( ~σ ′ , σ ′ e )) ≤ ǫ .Note that ~σ ′ implements ~σ ′′ iff ~σ ′ ~σ ′′ .The notion of implementation is quite strong. For example, if ~σ ′ involves fewer roundsof communication than ~σ ′′ , there may be far fewer distinct schedulers in the game involving ~σ ′ than in the game involving ~σ ′′ . Thus, we may not be able to recover the effect of allpossible schedulers. (Indeed, for some of our results the implementation needs to be quitelong precisely in order to capture all possible schedulers.) This suggests the following notion:a strategy profile ~σ ′ in Γ ′ weakly implements a strategy profile ~σ ′′ in Γ ′′ if { ( ~σ ′ , σ ′ e ) : σ ′ e ∈S Γ ′ ,e } ⊆ { ( ~σ ′′ , σ ′′ e ) : σ ′′ e ∈ S Γ ′′ ,e } . Thus, if ~σ ′ weakly implements ~σ ′′ , then every outcome of ~σ ′
6s one that could also have arisen with ~σ ′′ , but the converse may not be true. Specifically,there may be some behaviors of the environment with ~σ ′′ that cannot be simulated by ~σ ′ .As we shall see, this may actually be a feature: we can sometimes simulate the effect of only“good” schedulers. In any case, note that in the synchronous setting, implementation andweak implementation coincide. We can also define a notion of weak ǫ -implementation inthe obvious way; we leave the details to the reader. Termination:
We will be interested in asynchronous games where, almost surely, thehonest players stop sending messages and make a move in the underlying game. In themediator games that we consider, this happens after only a bounded number of messageshave been sent. But even with this bound, there may not be a point in a history whenplayers know that they can stop sending messages; although a player i may have moved inthe underlying game, i may still need to keep checking for incoming message, and may needto respond to them, to ensure that other players can make the appropriate move.For some of our results, we must assume that, in the mediator game, there comes apoint when all honest players know that they have terminated the protocol; they will notget further messages from the mediator and can stop sending messages to the mediator, andshould make a move in the underlying game if they have not done so yet. For simplicity, forthese results, we restrict the honest players and the mediator to using strategy profiles thathave the following canonical form : Using a canonical strategy, player i sends a message tothe mediator in response to a message from the mediator that does not include “STOP” if ithas not halted, and these are the only messages that i sends, in addition to an initial messageto the mediator. If player i gets a message from the mediator that includes “STOP”, then i makes a move in the underlying game and halts. We assume that, as long as the honestplayers and mediator follow their part of the canonical strategy profile, there is a constant r such that, no matter what strategy the rational and malicious players and the environmentuse, the mediator sends each player i at most r messages in each history, and the final messageincludes “STOP”. We conjecture that the assumption that players and mediator are usinga strategy in canonical form in the mediator game is without loss of generality; that is, a( k, t )-robust strategy profile in a mediator game Γ d can be implemented by a ( k, t )-robuststrategy profile in Γ d that is in canonical form. However, we have not proved this conjectureyet. In this section, we review the solution concepts introduced in ADGH and extend them toasynchronous settings.Note that in an asynchronous game Γ, the utility of a player i can depend not onlyon the strategies of the agents, but on what the environment does. Since we consider anunderlying game, a mediator game, and a cheap-talk game, it is useful to include explicitlyin the utility function which game is being considered. Thus, we write u i (Γ , ~σ, σ d , σ e , ~x ) todenote the expected utility of player i in game Γ when players play strategy profile ~σ , themediator plays σ d , the environment plays σ e , and the type profile is ~x . We typically say7input profile” rather than “type profile”, since in our setting, the type of player i is just i ’s initial input. Note that if Γ is the underlying game, the σ e component is unnecessary,since the underlying game is assumed to be synchronous. We occasionally omit the mediatorstrategy σ d when it is clear from context.Given a type space T , a set K of players, and ~x ∈ T , let T ( ~x K ) = { ~x ′ : ~x ′ K = ~x K } . If Γis a Bayesian game over type space T , ~σ is a strategy profile in Γ, and Pr is the probabilityon the type space T , let u i (Γ , ~σ, σ e , ~x K ) = X ~x ′ ∈T ( ~x K ) Pr( ~x ′ | T ( ~x K )) u i (Γ , ~σ, σ e , ~x ′ ) . Thus, u i (Γ , ~σ, σ e , ~x K ) is i ’s expected payoff if everyone uses strategy ~σ and type profiles arein T ( ~x K ). k -resilient equilibrium: In a standard game, a strategy profile is a Nash equilibriumif no player can gain any advantage by using a different strategy, given that all the otherplayers do not change their strategies. The notion of k -resilient equilibrium extends Nashequilibrium to allow for coalitions. Definition 3.1. ~σ is a k -resilient equilibrium (resp., strongly k -resilient equilibrium ) in anasynchronous game Γ if, for all subsets K of players with ≤ | K | ≤ k , all strategy profiles ~τ K for the players in K , all type profiles ~x ∈ T , and all strategies σ e of the environment, u i (Γ , ( ~σ − K , ~τ K ) , σ e , ~x K ) ≤ u i (Γ , ~σ, σ e , ~x K ) for some (resp., all) i ∈ K . Thus, ~σ is k -resilient if, no matter what the environment does, no subset K of at most k players can all do better by deviating, even if they share their type information (so thatif the true type is ~x , the players in K know ~x K ). It is strongly k -resilient if not even one ofthe players in K can do better if all the players in K deviate.For some of our results we will be interested in equilibria that are “almost” k -resilient, inthe sense that no player in a coalition can do more than ǫ better if the coalition its strategy,for some small ǫ . Definition 3.2.
For ǫ > , ~σ is an ǫ - k -resilient equilibrium (resp., strongly ǫ - k -resilient equi-librium ) if, for all subsets K of players, all strategy profiles ~τ K for the players in K , all typeprofiles ~x ∈ T , and all strategies σ e of the environment, we have u i (Γ , ( ~σ − K , ~τ K ) , σ e , ~x K )
0. We have used this slightly nonstandard definition to make thestatements of our theorems cleaner.
Robustness:
A standard assumption in game theory is that utilities are (commonly)known; when we are given a game we are also given each player’s utility. When players As usual, the strategy profile ( ~σ − K , ~τ K ) is the one where each player i ∈ K plays τ i and each player i / ∈ K plays σ i . Definition 3.3.
A strategy profile ~σ is t -immune in a game Γ if, for all subsets T of playerswith | T | ≤ t , all strategy profiles ~τ , all i / ∈ T , all type profiles ~x ∈ T , and all strategies σ e ofthe environment, we have u i (Γ , ( ~σ − T , ~τ T ) , σ e , ~x T ) ≥ u i (Γ , ~σ, σ e , ~x T ) . Intuitively, ~σ is t -immune if there is nothing that player in a set T of size at most t cando to give the players not in T a worse payoff, even if the players in T share their typeinformation.The notion of t -immunity and k -resilience address different concerns. For t -immunity,we consider the payoffs of the players K . It is natural to combine both notions. Given astrategy profile ~τ , let Γ T~τ be the game which is identical to Γ except that the players in T are fixed to playing strategy ~τ T . Definition 3.4. ~σ is a (strongly) ( k, t )-robust equilibrium in a game Γ if ~σ is t -immune and,for all subsets T of players with | T | ≤ t and all strategy profiles ~τ , ( ~σ − T , ~τ T ) is a (strongly) k -resilient equilibrium of Γ T~τ . We can define “approximate” notions of t -immunity and ( k, t )-robustness analogous toDefinition 3.2: Definition 3.5.
For ǫ > , a strategy profile ~σ is ǫ - t -immune in Γ if, for all subsets T of players with | T | ≤ t , all strategy profiles ~τ , all i / ∈ T , all type profiles ~x ∈ T , and allstrategies σ e of the environment, we have u i (Γ , ( ~σ − T , ~τ T ) , σ e , ~x T ) > u i (Γ , ~σ, σ e , ~x T ) − ǫ . Definition 3.6.
For ǫ ≥ , ~σ is a (strongly) ǫ -( k, t )-robust equilibrium in Γ if ~σ is ǫ - t -immune and, for all subsets T of players with | T | ≤ t and strategy profiles ~τ T , ( ~σ − T , ~τ T ) is a(strongly) ǫ - k -resilient equilibrium of Γ T~τ . In this section, we state our results formally. Just as with R1 and R2, for these resultswe assume that the communication in the mediator game is bounded. But since “rounds”is not meaningful in asynchronous systems, we express the bounds in terms of number of9essages. Specifically, we assume that at most N messages are sent in all histories of themediator game, and that the mediator can be represented by an arithmetic circuit with atmost c gates.We begin with a result that is an analogue of R1 in the asynchronous setting. We saythat a game Γ ′ is a utility variant of a game Γ if Γ ′ and Γ have the same game tree, but theutilities of the players may be different in Γ and Γ ′ . We use the notation Γ( ~u ) if we wantto emphasize that ~u is the utility function in game Γ. We then take Γ( ~u ′ ) to be the utilityvariant of Γ with utility function ~u ′ .One more technical comment before stating the theorems: in the mediator game we canalso view the mediator as a player (albeit one without a utility function) that is following astrategy. Thus, when we talk about a strategy profile that is a ( k, t )-robust equilibrium inthe mediator game, we must give the mediator’s strategy as well as the players’ strategies.We sometimes write ~σ + σ d if we want to distinguish the players’ strategy profile ~σ from themediator’s strategy σ d . We occasionally abuse notation and drop the σ d if it is clear fromcontext, and just talk about ~σ being a ( k, t )-robust equilibrium. Theorem 4.1. If Γ is a normal-form Bayesian game with n players, ~σ + σ d is a strategyprofile for the players and the mediator in an asynchronous mediator game Γ d that extends Γ , and n > k + 4 t , then with both the default-move approach and the AH approach, thereexists a strategy profile ~σ CT that implements ~σ + σ d in the asynchronous cheap-talk game Γ CT such that for all utility variants Γ d ( ~u ′ ) of Γ d , if ~σ + σ d is a (strongly) ( k, t ) -robust equilibriumin Γ d ( ~u ′ ) , then ~σ CT is a (strongly) ( k, t ) -robust equilibrium in Γ CT ( ~u ′ ) , and the number ofmessages sent in a history of ~σ CT is O ( nN c ) , independent of ~u ′ . The proof of Theorem 4.1 uses ideas from the multiparty computation protocol of Ben-Or,Canetti, and Goldreich [5] (BCG from now on). Our construction actually needs strongerproperties than these provided by BCG; we show that we can get protocols with thesestronger properties in a companion paper [10]; see Section 5 for further discussion.We can obtain better bounds if we are willing to accept ǫ -equilibrium, using ideas due toBen-Or, Kelmer, and Rabin [7]. Theorem 4.2. If Γ is a normal-form Bayesian game with n players, ~σ + σ d is a strategyprofile for the players and mediator in an asynchronous mediator game Γ d that extends Γ , M > , and n > k + 3 t , then with both the default-move approach and the AH approach, forall ǫ > , there exists a strategy profile ~σ CT in the asynchronous cheap-talk game Γ CT that ǫ -implements ~σ such that for all utility variants Γ d ( ~u ′ ) of Γ d bounded by M/ (i.e., where therange of u ′ i is contained in [ − M/ , M/ ), if ~σ + σ d is a (strongly) ( k, t ) -robust equilibriumin Γ d ( ~u ′ ) , then ~σ CT is a (strongly) ǫ - ( k, t ) -robust equilibrium in Γ CT ( ~u ′ ) , and the number ofmessages sent in a history of ~σ CT is O ( nN c ) , independent of ~u ′ . If we have a punishment strategy and utilities are known, we can do better with the AHapproach. To make this precise, we need the definition of an m -punishment strategy [1](which generalizes the notion of punishment strategy defined by Ben Porath [8]). Beforedefining this carefully, note that in an asynchronous setting (i.e., in the mediator game and10he cheap-talk game, but not in the underlying game), the utility of players depends on theenvironment’s strategy as well as the players’ strategy profile and the players’ type profile. Definition 4.3. If Γ ′ is an extension of an underlying game Γ , a strategy profile ~ρ in Γ isa k -punishment strategy with respect to a strategy profile ~σ ′ in Γ ′ if for all subsets K ofplayers with ≤ | K | ≤ k , all strategy profiles ~σ in Γ , all strategies σ e for the environment,all type profiles ~x ∈ T , and all players i ∈ K , we have u i (Γ ′ , ~σ ′ , σ e , ~x K ) > u i (Γ , ( ~σ K , ~ρ − K ) , ~x K ) . Thus, if ~ρ is a k -punishment strategy with respect to ~σ ′ , if all but k players play theirpart of ~ρ in the underlying tame, then all of the remaining players will be worse off thanthey would be in Γ ′ if everyone had played ~σ ′ , no matter what they do in the underlyinggame. Theorem 4.4. If Γ is a normal-form Bayesian game with n players, ~σ + σ d is a strategyprofile in canonical form for the players and mediator in an asynchronous mediator game Γ d that extends Γ , there is a ( k + t ) -punishment strategy with respect to ~σ + σ d , and n > k + 4 t ,then with the AH approach, there exists a strategy profile ~σ CT that implements ~σ + σ d in theasynchronous cheap-talk game Γ CT , and if ~σ + σ d is a (strongly) ( k, t ) -robust equilibrium in Γ d , then ~σ CT is a (strongly) ( k, t ) -robust equilibrium in Γ CT . If there exists a strong ( k + t ) -punishment strategy or we require only that ~σ CT is a weak implementation, then the numberof messages in a history of ~σ CT is O ( nc ) (and ~σ CT continues to be a (strongly) ( k, t ) -robustequilibrium in Γ CT if ~σ is a (strongly) ( k, t ) -robust equilibrium in Γ d ). Note that in Theorem 4.4, the running time of the algorithm is significantly affected bywhether we want ~σ CT to implement ~σ or whether a weak implementation suffices.If we assume both that there is a (2 k + 2 t )-punishment strategy and that utilities areknown, we can get an analogue to R2, but with an ǫ error. Theorem 4.5. If Γ is a normal-form Bayesian game with n players, ~σ + σ d is a strategyprofile in canonical form for the players and mediator in an asynchronous mediator game Γ d that extends Γ , there is a (2 k +2 t ) -punishment strategy with respect to ~σ + σ d , and n > k +3 t ,then with the AH approach, for all ǫ > there is a strategy ~σ CT that ǫ -implements ~σ in theasynchronous cheap-talk game Γ CT such that if ~σ + σ d is a (strongly) ( k, t ) -robust equilibriumin Γ d , then ~σ CT is a (strongly) ǫ - ( k, t ) -robust equilibrium in Γ CT , and the number of messagessent in a history of ~σ CT is O ( n N c ) . If there exists a strong ( k, t ) -punishment strategy or werequire only that ~σ CT is a weak implementation, then the number of messages in a historyof ~σ CT is O ( nc ) (and ~σ CT continues to be a (strongly) ( k, t ) -robust equilibrium in Γ CT if ~σ is a (strongly) ( k, t ) -robust equilibrium in Γ d ). We prove these results using ideas in the spirit of ADGH, but much more care must betaken to deal with asynchrony. Among other things, we need stronger security guaranteesthan are traditionally provided for multiparty communication; see Section 5 for details. Weprovide proofs of all the results in Section 6.11 t -bisimulation and t -emulation To construct the cheap-talk protocol for Theorems 4.1 and 4.2, we use ideas from a companionpaper [10], where we provide constructions that extend the security guarantees given by themulti-party computation protocols for the synchronous case by Ben-Or, Goldwasser, andWigderson [6] (BGW from now on) and the asynchronous case by Ben-Or, Canetti, andGoldreich [5] (BCG from now on). We briefly review the main details here.BGW/BCG show that if a function f of n inputs provided by n players can be computedusing a mediator, then it can be computed by the players without the mediator and withoutrevealing any information beyond the function value, even when some of the players aremalicious. BGW deals with the synchronous case and tolerates up to n/ n/
4. The notion of notrevealing any information is made precise by defining a set of ideal distributions over possiblevalues of the function, and ensuring that the real distribution is identically distributed toone of those (see BGW and BCG for formal definitions and details).We can view a mediator game as computing an action profile in the underlying game; theideal distributions are the possible distributions over action profiles when the honest playersplay their component of the ( k, t )-robust equilibrium strategy profile in the mediator game.BCG’s protocol then essentially gives us a strategy in the cheap-talk game. However, theBCG protocol is not sufficient for our purposes for two reasons: it does not guarantee thatthe real protocol is an implementation of the ideal protocol in the sense of the definition inSection 2 (although it does suffice for weak implementation), nor does it guarantee that theprotocol is a ( k, t )-robust equilibrium. To prove these stronger results, we show that ~σ CT can be constructed so as to satisfy some additional security properties, which we now define. Definition 5.1 ( t -bisimulation) . Take an adversary A to be a pair ( ~τ T , σ e ) consisting of astrategy for the malicious players and an environment strategy. Let O ( ~π + π d , A, ~x ) be thedistribution over outputs when running strategy ~π with adversary A = ( τ T , σ e ) . Protocol ~π ′ t -bisimulates ~π + π d if, for all T with | T | ≤ t and inputs ~x : • for all adversaries A = ( ~τ T , σ e ) , there exists an adversary A ′ = ( ~τ ′ T , σ ′ e ) such that O ( ~π + π d , A, ~x ) and O ( ~π ′ , A ′ , ~x ) are identically distributed; • for all adversaries A ′ = ( ~τ ′ T , σ ′ e ) , there exists an adversary A = ( ~τ T , σ e ) such that O ( ~π + π d , A, ~x ) and O ( ~π ′ , A ′ , ~x ) are identically distributed. For one direction of the simulation, we need an even tighter correspondence betweendeviations in the cheap-talk game and deviations in the mediator game. This is made precisein the following definition.
Definition 5.2 ( t -emulation) . The protocol ~π ′ t -emulates ~π if, for every scheduler σ ′ e , thereexists a function H from strategies to strategies such that H ( π ′ i ) = π i for all players i and, forall sets T of players with | T | ≤ t and all adversaries A ′ = ( ~τ ′ T , σ ′ e ) , there exists an adversary A = ( ~τ T , σ e ) such that, for all input profiles ~x , O ( ~π, A, ~x ) and O ( π ′ , A ′ , ~x ) are identicallydistributed, where ~τ T = H ( ~τ ′− T ) (and we take H ( τ ′ , . . . , τ ′ m ) = ( H ( τ ) , . . . , H ( τ ′ m )) ). t -bisimulation and t -emulation. As thename suggests, with t -bisimulation, we require simulation in both directions (for every ~τ and σ e there is a ~τ ′ and σ ′ e , and vice versa); for emulation, we have only one direction. Onthe other hand, with t -emulation, the strategy ~τ ′ i depends only on ~τ i and σ e , whereas with t -bisimulation, ~τ ′ i can also depend on σ ′ e and all the strategies in ~τ .Note that 0-bisimulation is equivalent to implementation, while 0-emulation is equivalentto weak implementation. Implementation and weak implementation consider only whathappens when there is no malicious behavior; bisimulation and emulation generalize thesenotions by taking malicious behavior into account. For some of our results (specifically,Theorems 4.1 and 4.2), we use these notions, and show that they are achievable under theconditions of these theorems. In fact, although we don’t need it for our proof, we can showthat we can get t -bisimulation and t -emulation under the conditions of Theorems 4.4 and 4.5as well. We briefly comment on how this can be done when we present the proof.We have required schedulers to deliver each message eventually. Because we assumethat protocols in the mediator game are bounded, all protocols in the mediator game mustterminate. This means that we can’t hope to emulate a protocol in the cheap-talk gamethat deadlocks. (We assume that if the protocol deadlocks, it has a special output that wedenote ⊥ . Given our constraints, we can never get an output of ⊥ in the mediator game.) Todeal with this situation, we relax this requirement on schedulers somewhat, but only in themediator game. We take a relaxed scheduler to be one that may never deliver some messages.However, we require that if the mediator sends several messages at the same step, then arelaxed scheduler either delivers all of them or none of them. (There is no requirementon messages sent by the players, since they send messages only to the mediator, and wecan assume without loss of generality that they send only one message at each step.) Wecan define relaxed t -emulation just as we defined t -emulation, except that we now allow thescheduler σ e in the definition to be a relaxed scheduler. Finally, we define ( t, t ′ ) -emulation just as we defined relaxed t -emulation except that σ e must be non-relaxed if | T | ≤ t ′ .We need a further property to deal with protocols that involve punishment strategies.For a punishment strategy to be effective, all the honest players have to play it. In ourprotocols, the punishment strategy is played when there is a deadlock (so some players neverterminate); that is, the punishment strategy is in the honest players’ “wills”. Thus, wewant it to be the case that either none of the honest players terminate (in which case thepunishment strategy will be effective) or all of them terminate; we do not want it to be thecase that only some of the honest players terminate. Definition 5.3 ( t -cotermination) . A protocol ~π t -coterminates if, for all schedulers σ e , allsubsets T of at most t players, and all strategy profiles ~τ T for the players in T , in all historiesof the protocol ( ~π − T , ~τ T , σ e ) , either all the players not in T terminate or none of them do. For some of our results, we need “approximate” versions of t -bisimulation, t -emulation, re-laxed t -emulation, and t -cotermination that allow an ǫ probability of error. For t -bisimulation, t -emulation, ( t, t ′ )-emulation, and relaxed t -emulation, this means that the distance betweenthe distribution over outputs in the cheap-talk game and the distribution in the media-tor game is at most ǫ (where the notion of distance is that used in the definition of ǫ -13mplementation in Section 2) while for t -cotermination it means that the property holdswith probability 1 − ǫ . We call these properties ( ǫ, t )-bisimulation, ( ǫ, t )-emulation, relaxed( ǫ, t )-emulation and ( ǫ, t )-cotermination.In [10], the following results are proved: Theorem 5.4.
Given a mediator game Γ d extending Γ and a strategy profile ~σ + σ d , thereexists a strategy profile ~σ CT for Γ CT such that ~σ CT t -coterminates, t -emulates (resp., relaxed ( t, t ′ ) -emulates), and t -bisimulates ~σ + σ d if t < n/ , t < n/ (resp., t < n/ and t ′ < n/ ),and t < n/ respectively, and the expected number of messages in histories of ~σ CT is O ( nN c ) . Theorem 5.5.
Given a mediator game Γ d extending Γ , a strategy profile ~σ + σ d in Γ d , anda real number ǫ ∈ (0 , , there exists a strategy profile ~σ CT in Γ CT such that ~σ CT ( ǫ, t ) -coterminates, ( ǫ, t ) -emulates (resp., ( ǫ, t, t ′ ) -emulates), and ( ǫ, t ) -bisimulates ~σ + σ d if t Clearly, if (1) holds for all σ e , σ ′ e , ~x , and ~x ′ , then ~σ is t -immune. For the converse,suppose by way of contradiction that ~σ is t -immunte but for some T , ~τ , σ e , σ ′ e , and i / ∈ T ,(1) does not hold. Consider a scheduler σ ′′ e that acts just like σ e , except that if some player i sends a message to itself it acts like σ ′ e . Then players in T can effectively decrease i ’scontradicting that payoff with scheduler σ ′′ e by sending a message to themselves and playingas if they had input ~x ′ T ; that is, there is a strategy ~τ ′ T such that u i (Γ d , ( ~σ − T , ~τ ′ T ) , σ ′′ e , ~x T ) = u i (Γ d , ( ~σ − T , ~τ T ) , σ ′ e , ~x ′ T ) < u i (Γ d , ~σ, σ e , ~x T ) = u i (Γ d , ~σ, σ ′′ e , ~x T ) , contradicting the assumption that ~σ is t -immune. The details for the proof are leftA similar argument shows that ( k, t )-robust strategy profiles satisfy a correspondinglystronger condition, made precise in the following proposition: Proposition 6.2. A strategy profile ~σ is ( k, t ) -robust (resp., strongly ( k, t ) -robust) if andonly if it is t -immune and for all disjoint sets K and T with ≤ | K | ≤ k and | T | ≤ t , allstrategy profiles ~τ K , ~τ T , and ~τ ′ T for the players in K and T , respectively, all environmentstrategies σ e and σ ′ e , and all input profiles ~x and ~x ′ , we have that u i (Γ d , ( ~σ − ( K ∪ T ) , ~τ K , ~τ ′ T ) , σ ′ e , ~x ′ ( K ∪ T ) ) ≤ u i (Γ d , ( ~σ − T , ~τ T ) , σ e , ~x T ) (2) for some i ∈ K (resp., for all i ∈ K ).Proof. Again, it is clear that if (2) holds for all K and T with 1 ≤ | K | ≤ k and | T | ≤ t , all ~τ K , ~τ T , ~τ ′ T , ~x , and ~x ′ , and some (resp., all) i ∈ K , then ~σ is (strongly) ( k, t )-robust.For the converse, assume by way of contradiction that ~σ is ( k, t )-robust, but for somedisjoint sets K and T with 1 ≤ | K | ≤ k and | T | ≤ t , ~τ K , ~τ T , ~τ ′ T , ~x , and ~x ′ , and all i ∈ K , (2)does not hold. Again, we use the fact that the rational players can effectively communicate15ith malicious players and with the scheduler. Consider a scheduler σ ′′ e that acts like σ e unless some player sends a message to itself, in which case it acts like σ ′ e and a strategyprofile ~τ ′′ T in which each i ∈ T acts as if it was using strategy ( τ T ) i , except that it switchesto ( τ ′ T ) i and to pretend have input x ′ i if it receives a message from a rational player askingit to do so. Then, given input profile ~x , strategy profile ~τ ′′ T for T , and scheduler σ ′′ e , player i can gain by sending a message to itself and sending a message to players in T asking themto follow ~τ ′ T and pretends to have input ~x ′ T , and by making players in K play ~τ K as if theyhad input ~x ′ K , rather than playing ~σ . This contradicts the assumption that ~σ is ( k, t )-robust.The argument for strong ( k, t )-robustness.Another property interesting in its own right that follows from this argument is that( k, t )-robust strategies must be scheduler-proof : the expected payoff for all players is thesame regardless of the scheduler: Corollary 6.3. If ~σ is ( k, t ) -robust for some k ≥ , then for all sets T with | T | ≤ t , strategyprofiles ~τ T for the players in T , environment strategies σ e and σ ′ e , input profiles ~x , and players i / ∈ T , we have u i (Γ , ( ~σ − T , ~τ T ) , σ e , ~x T ) = u i (Γ , ( ~σ − T , ~τ T ) , σ ′ e , ~x T ) . We have analogous strengthenings of ǫ - t -immunity and ǫ -( k, t )-robustness.The proofs are essentially identical to that of Proposition 6.1, so we omit them here. Proposition 6.4. If ǫ > and ~σ is ǫ - t -immune in game Γ , then for all sets T of playerswith | T | ≤ t , strategy profiles τ T for the players in T , environment strategies σ e and σ ′ e , inputprofiles ~x and ~x ′ , and players i / ∈ T , we have that u i (Γ , ( ~σ − T , ~τ T ) , σ ′ e , ~x ′ T ) > u i (Γ , ~σ, σ e , ~x T ) − ǫ. Proposition 6.5. A strategy profile ~σ is ǫ - ( k, t ) -robust (resp., strongly ǫ - ( k, t ) -robust) ingame Γ if and only if it is ǫ - t -immune and, for all disjoint sets K and T of players with ≤ | K | ≤ k and | T | ≤ t , all strategy profiles ~τ K and ~τ T for players in K and T , respectively,all environment strategies σ e and σ ′ e , and all input profiles ~x and ~x ′ , we have that u i (Γ , ( ~σ − ( K ∪ T ) , ~τ K , ~τ T ) , σ ′ e , ~x ′ ( T ∪ K ) ) < u i (Γ , ( ~σ − T , ~τ T ) , σ e , ~x T ) + ǫ for some i ∈ K (resp., for all i ∈ K ). It will be useful for our later results that we can actually improve on the bound of ǫ inPropositions 6.4 and 6.5. Proposition 6.6. If ~σ is an ǫ - t -immune strategy in a finite game Γ , then there exists ǫ with < ǫ < ǫ such that for all sets of players T with | T | ≤ t , strategy profiles ~τ T for theplayers in T , environment strategies σ e and σ ′ e , input profiles ~x and ~x ′ , and players i / ∈ T ,we have that u i (Γ , ( ~σ − T , ~τ T ) , σ ′ e , ~x ′ T ) > u i (Γ , ~σ, σ e , ~x T ) − ǫ . roof. Since, by Proposition 6.4, for each choice of ~τ T , σ e , and σ ′ e , we have u i (Γ , ~σ, σ e , ~x T ) − u i (Γ , ( ~σ − T , ~τ T ) , σ ′ e , ~x ′ T ) < ǫ, and the space of player strategy profiles, environment strategies, and input value profilesis compact, if we take the sup of the left-hand side over all choices of strategy profiles ~τ T ,environment strategies σ e and σ ′ e , and input profiles ~x and ~x ′ , it takes on some maximumvalue ǫ < ǫ . We can then take ǫ = ( ǫ + ǫ ) / ǫ -( k, t )-robustness. The proof is analo-gous to that of Proposition 6.6. Proposition 6.7. If Γ is a finite game and ~σ is a (strongly) ǫ - ( k, t ) -robust strategy in Γ d ,then there exists ǫ with < ǫ < ǫ such that for all disjoint sets K and T of players with ≤ | K | ≤ k and | T | ≤ t , all strategy profiles ~τ K and ~τ T for players in K and T , respectively,all environment strategies σ e and σ ′ e , and all input profiles ~x and ~x ′ , we have that u i (Γ , ( ~σ − ( K ∪ T ) , ~τ K , ~τ T ) , σ e , ~x ′ ( K ∪ T ) ) < u i (Γ , ( ~σ − T , ~τ T ) , σ ′ e , ~x T ) + ǫ for some i ∈ K (resp., all i ∈ K ). By Theorem 5.4, if n > k +4 t , there exists a strategy ~σ CT that ( k + t )-bisimulates and ( k + t )-emulates ~σ + σ d , in which the expected number of messages is O ( nN c ). It is immediate fromthe definition of ( k + t )-bisimulation that ~σ CT implements ~σ + σ d . Since the probability ofdeadlock is 0, the action that players play in case of deadlock are irrelevant, so this approachworks equally well for the AH approach and the default-move approach. It remains to showthat, for each utility variant Γ d ( ~u ′ ) of Γ d , if ~σ + σ d is a (strongly) ( k, t )-robust equilibriumin Γ d ( ~u ′ ), then ~σ CT is a (strongly) ( k, t )-robust equilibrium in Γ CT ( ~u ′ ). We start by showingthat ~σ CT is t -resilient in Γ CT ( ~u ′ ).Given T with | T | ≤ t , ~τ T , and σ e , by Theorem 5.4, there exists a function H fromstrategies to strategies and σ ′ e such that, for all input profiles ~x , we have u ′ i (Γ CT ( ~u ′ ) , (( ~σ CT ) − T , τ T ) , σ e , ~x ) = u ′ i (Γ d ( ~u ′ ) , ( ~σ − T , H ( ~τ T )) , σ ′ e , ~x )for all players i . There also exists a scheduler of the form σ ′′ e such that u ′ i (Γ CT ( ~u ′ ) , ~σ CT , σ ′ e , ~x ) = u ′ i (Γ d ( ~u ′ ) , ~σ, σ ′′ e , ~x ) . Since ~σ is t -immune, for all i / ∈ T we have u ′ i (Γ CT ( ~u ′ ) , (( ~σ CT ) − T , ~τ T ) , σ e , ~x T )= u ′ i (Γ d ( ~u ′ ) , ( ~σ − T , H ( ~τ T )) , σ ′ e , ~x T ) ≥ u ′ i (Γ d ( ~u ′ ) , ~σ, σ ′′ e , ~x T ) [by Lemma 6.1]= u ′ i (Γ CT ( ~u ′ ) , ~σ CT , σ ′ e , ~x T ) . ~σ CT is t -immune.To show (strong) ( k, t )-robustness, taking ~τ T , σ e , and σ ′ e as above, suppose that K is aset of players disjoint from T such that | K | ≤ k , and the players in K play ~τ K . Then, byTheorem 5.4, there exists σ ∗ e such that u ′ i (Γ CT ( ~u ′ ) , (( ~σ CT ) − ( K ∪ T ) , ~τ K , ~τ T ) , σ e , ~x ) = u ′ i (Γ d ( ~u ′ ) , ( ~σ − ( K ∪ T ) , H ( ~τ K ) , H ( ~τ T )) , σ ∗ e , ~x ( K ∪ T ) )for all players i . By Corollary 6.2, if ~σ + σ d is ( k, t )-robust (resp., strongly ( k, t )-robust) inΓ d ( ~u ′ ), then u ′ i (Γ d ( ~u ′ ) , ( ~σ − ( K ∪ T ) , H ( ~τ K ) , H ( ~τ T )) , σ ∗ e , ~x ( K ∪ T ) ) ≤ u ′ i (Γ d ( ~u ′ ) , ( ~σ − T , H ( ~τ T )) , σ ′ e , ~x T )for some (resp., all) i ∈ K . For those i ∈ K for which this inequality holds, we have u ′ i (Γ CT ( ~u ′ ) , (( ~σ CT ) − ( K ∪ T ) , τ K , τ T ) , σ e , ~x ( K ∪ T ) )= u ′ i (Γ d ( ~u ′ ) , ( ~σ − ( K ∪ T ) , H ( ~τ K ) , H ( ~τ T )) , σ ∗ e , ~x ( K ∪ T ) )= u ′ i (Γ d ( ~u ′ ) , ( ~σ − T , H ( ~τ T )) , σ ′ e , ~x T )= u ′ i (Γ CT ( ~u ′ ) , (( ~σ CT ) − T , σ e , ~x T ) . It follows that ~σ CT is (strongly) ( k, t )-robust in Γ CT ( ~u ′ ). The argument is essentially the same as that used for Theorem 4.1, except that we nowuse Theorem 5.5 instead of Theorem 5.4. By Theorem 5.5, for all ǫ ′ ∈ (0 , ~σ CT that ǫ -( k + t )-bisimulates ~σ , ǫ -( k + t )-emulates ~σ , and uses O ( nN c ) messagesin expectation. It follows that ~σ CT ǫ ′ -implements ~σ and has at most a probability ǫ ′ ofdeadlock. We show next that we can make ǫ ′ sufficiently small so that it becomes irrelevantif we work with the AH approach or the default-move approach.We can now prove ǫ -( k, t )-robustness. Suppose that ~σ + σ d is a (strongly) ǫ -( k, t )-robustequilibrium in the utility variant Γ d ( ~u ′ ) of Γ d . We show that ~σ CT is ǫ - t -immune in Γ CT ( ~u ′ ).Since ~σ CT ǫ ′ -( k + t ) emulates ~σ + σ d , given a set T of players with | T | ≤ t , a strategy profile ~τ T for the players in T , and an environment strategy σ ′ e in the cheap-talk game, by Theorem 5.5,there exists a function H as in the definition of t -emulation. Moreover, for all inputs ~x , wecan associate histories in the mediator game and histories in the cheap-talk game in sucha way that the set of histories where the outcomes differ has probability at most ǫ ′ . Sinceall utilities are in the range [ − M/ , M/ M . Thus, there exists an environmentstrategy σ ′ e such that for all input profiles ~x , we have u ′ i (Γ CT ( ~u ′ ) , (( ~σ CT ) − T , H ( ~τ T )) , σ ′ e , ~x ) > u ′ i (Γ d ( ~u ′ ) , ( ~σ − T , ~τ T ) , σ e , ~x ) − ǫ ′ M for all i T . Theorem 5.5 also guarantees that there exists an environment strategy σ ′′ e suchthat u ′ i (Γ CT ( ~u ′ ) , ~σ CT , σ ′ e , ~x ) < u ′ i (Γ d ( ~u ′ ) , ~σ, σ ′′ e , ~x ) + ǫ ′ M. ~σ is ǫ - t immune in Γ d ( ~u ′ ), by Proposition 6.6, there exists a value ǫ with 0 < ǫ < ǫ such that u ′ i (Γ CT ( ~u ′ ) , (( ~σ CT ) − T , ~τ ′ T ) , σ ′ e , ~x T ) > u ′ i (Γ d ( ~u ′ ) , ( ~σ − T , H ( ~τ T )) , σ e , ~x T ) − ǫ ′ M> u ′ i (Γ d ( ~u ′ ) , ~σ, σ ′′ e , ~x T ) − ǫ − ǫ ′ M> u ′ i (Γ CT ( ~u ′ ) , ~σ CT , σ ′ e , ~x T ) − ǫ − ǫ ′ M. If we take ǫ ′ = ( ǫ − ǫ ) / M , this shows that ~σ CT is ( ǫ, t )-immune with both the AH approachand the default-move approach.To show (strong) ǫ -( k, t )-robustness, keeping T , ~τ T , H , σ e , and σ ′ e as above, for all sets K of players disjoint from T with 1 ≤ | K | < k and strategy profiles ~τ K , there exists anenvironment strategy σ ∗ e and a value ǫ with 0 < ǫ < ǫ such that for all input profiles ~x , u ′ i (Γ CT ( ~u ′ ) , (( ~σ CT ) − ( K ∪ T ) ) , ~τ K , ~τ T ) , σ ′ e , ~x ( K ∪ T ) ) < u ′ i (Γ d ( ~u ′ ) , ( ~σ − ( K ∪ T ) , H ( ~τ K ) , H ( ~τ T )) , σ e , ~x ( K ∪ T ) ) + ǫ ′ M< u ′ i (Γ d ( ~u ′ ) , ( ~σ − T , H ( ~τ T )) , σ ′′ e , ~x T ) + ǫ + ǫ ′ M [by Proposition 6.7] < u ′ i (Γ CT ( ~u ′ ) , ( ~σ CT ) − T , ~τ ) , σ e , ~x T ) + ǫ + 2 ǫ ′ M [by Theorem 5.5].for some (for all) i ∈ K . This shows that if we take ǫ ′ := ( ǫ − ǫ ) / M , then ~σ CT is ǫ -( k, t )-robust. Note that this argument works for both the AH approach and the default-moveapproach since it does not depend on the actions played in deadlock. The proof of Theorem 4.4 is similar in spirit to that of Theorem 4.1. The main problem wehave to deal with is that of ensuring that rational players participate. To force participation,we have the honest players put the punishment strategy in their “wills”, so that if ~σ CT endsin deadlock, the rational players will be punished. Unfortunately, a naive implementation ofthis approach does not work, as the following example shows.Consider an underlying game Γ for n > k players where the set of actions is A := { , , ⊥} . If at least k + 1 players play ⊥ , all players get a payoff of 1.1; if k or fewer playersplay ⊥ and all players play either 0 or ⊥ , then all players get a payoff of 1; if k or fewerplayers play ⊥ and all players play either ⊥ or 1, then all players get a payoff of 2; otherwise,all players get 0. Let Γ d be an extension of Γ with a mediator. Suppose that the mediator d uses the following strategy: First, d chooses a value b ∈ { , } with equal probability. Then d chooses a ∈ { , } with equal probability and sends the message a + bi (mod 2) to player i (note that the same a is used in all these messages). Finally, d sends the message “output b ; STOP” to all players (so the strategy is in canonical form).Let σ i be the strategy where player i ignores the message a + bi and plays b after receivingthe message “output b ”. It is easy to check that ~σ is a k -resilient equilibrium in the mediatorgame, and gives players an expected payoff of 1.5. Moreover, playing ⊥ is a k -punishmentstrategy with respect to ~σ , since if all but k players play ⊥ , then everyone gets a payoff of1.1 (since at least k + 1 players play ⊥ ), which is less than 1.5.The naive approach to implementing the mediator does not work for this game, at leastwith the punishment strategy ⊥ . For example, suppose that after receiving the messages19 + bi (mod 2), the rational players communicate with each other. Moreover, suppose thatthe set K of rational players includes i and j such that i − j is odd. Then the rationalplayers can compute b . If b = 0, they actually prefer their payoff with the punishmentstrategy to their payoff with ~σ CT . Thus, they will stop sending messages. The simulationwill not terminate, so the punishment strategy in the players’ wills will be applied, makingthe rational players better off. Thus, we cannot simulate the mediator with this approach.Of course, there are punishment strategies in this game that would lead to cooperation (e.g.,randomizing between 0 and 1). Nevertheless, this example shows that using an arbitrarypunishment strategy may not suffice to force the rational players to cooperate.The problem here is that the mediator tells each player i what a + bi is. We do not wantthe mediator to send such unnecessary information. But what counts as unnecessary? As wenow show, for each strategy profile ~σ + σ d of a mediator game we can construct a strategy ~σ m + σ md that implements ~σ + σ d and leaks no information. More precisely, there existsa function f from strategy profiles to strategy profiles such that, for all strategy profiles ~σ + σ d , f ( ~σ + σ d ) implements ~σ + σ d and essentially all the mediator sends each player whenplaying f ( ~σ + σ d ) is the action to play in the underlying game. (If we require only weakimplementation, then this is exactly the case; for implementation, the messages can alsoinclude a round number.) Moreover, if ~σ + σ d is ( k, t )-robust (resp., strongly ( k, t )-robust, ǫ -(k,t)-robust, strongly ǫ -(k,t)-robust), then is f ( ~σ + σ d ). The construction of f proceeds asfollows:Let S det Γ d ,e / ∼ denote the set of ∼ -equivalence classes. Thus, S det Γ d ,e / ∼ is essentially the setof deterministic environment strategies that result in different outcomes in the underlyinggame when the players and the mediator use ~σ + σ d in Γ d .The intuition underlying f ( σ d ) is that the mediator chooses an equivalence class in S det Γ d ,e / ∼ , chooses an environment strategy σ e in the equivalence class, and simulates theoutcome of ( ~σ + σ d , σ e ). In order to get an implementation, we must ensure that it is possi-ble for f ( σ d ) to choose all possible equivalences classes in S det Γ d ,e / ∼ . To do this, let R be theleast integer such that ( Rn )! ≥ |S det Γ d ,e / ∼| . We show below that we can take R = 2 rn log( n ) .The mediator σ d sends R messages to each player. As we shall see, this suffices for themediator to choose all possible equivalence classes in S detΓ d ,e / ∼ .The strategy f ( σ i ) is straightforward: player i initially sends the mediator the message( i, , x i ), where x i is i ’s input. Then for 1 ≤ r < R , after receiving a message with content r from the mediator, player i sends the mediator ( i, r, x i ). When i receives a message of theform (STOP, a i ) from the mediator, i plays a i and halts.The mediator f ( σ d ) proceeds as follows: it sends each player i R − r th message just says “ r ”. It then waits until there are at least n − k − t players from whichit has received a valid and complete set of messages, where a set of messages from a player i is valid and complete if for all r with 0 ≤ r ≤ R − 1, the mediator has received exactlyone message of the form ( i, r, x ), where x is an input value that i could have, and all the x values are the same in these R messages. The next time that the mediator is scheduled, itsends a STOP message to each player with an action to perform. We next explain how themediator calculates which actions the players perform.20et P be the set of players from whom the mediator has received a valid and completeof messages when it is next scheduled. There are two cases. If P consists of all players,this means that the mediator has received Rn messages. There are ( Rn )! orders that thesemessages could have come in. Moreover, for each possible order of messages, there is adeterministic scheduler σ ′ e that delivers the messages in just this order. By choice of R ,( Rn )! ≥ |S det Γ d ,e / ∼ ′ | , so there is a surjective mapping H P from each message order to ascheduler σ e in the game Γ d . The mediator then simulates a computation of ( ~σ, σ d ) withthe scheduler σ e corresponding to the message order it actually received (generating therandomness for all the players and for σ d ) using the input ~x that it received from the players,and sends each player i the action that results from this simulation as its R th message.Now suppose P , the set of players from whom the mediator has received a valid andcomplete of messages when it is next scheduled, does not consist of all players. Let σ ′ P be ascheduler in the game Γ ′ d that resulted in this message order. Consider a fixed scheduler σ P in the game Γ d where the messages of all players not in P are delayed until after the mediatorhas sent all STOP messages. (There must be such a scheduler, since | P | ≥ n − k − t ). Let ~x P be the profile of inputs that the mediator has received from the players in P , and extendit arbitrarily to an input profile ~x . The mediator f ( σ d ) simulates a computation of ( ~σ, σ d )with the scheduler σ P (again, generating the randomness for all the players and for σ d ) usingsome profile ~x that extends the profile ~x P it received from the players in P (it doesn’t matterwhich vector ~x is used, since the mediator σ d does not receive messages from players not in P ), and sends each player i the action that results from this simulation as its R th message.If we require only weak implementation, we can make do with far fewer messages. Eachplayer i just sends the mediator f ( σ d ) an initial message of the form ( i, x i ). The mediatorwaits until it has these initial messages from n − k − t players. If the mediator has receivedmessages from the players in P when it is next scheduled, it simulates the mediator σ P onsome input ~x that extends ~P .We now show that this construction has the required properties. Lemma 6.8. Given a finite mediator game Γ d that extends an underlying game Γ , a canon-ical strategy profile ~σ for the players, and a strategy σ d for the mediator in Γ d , then f ( ~σ + σ d ) implements (resp., weakly implements) ( ~σ, σ d ) , and if ~σ + σ d is ( k, t ) -robust (resp., strongly ( k, t ) -robust, ǫ -(k,t)-robust, strongly ǫ -(k,t)-robust), then so is f ( ~σ + σ d ) . Moreover, thenumber of messages sent in each history of f ( ~σ + σ d ) is O ( N log( N )) (resp., O ( n ) ).Proof. Implementation (resp. weakly implementation) follows from the construction. Sup-pose that ~σ + σ d is ǫ -( k, t )-robust (resp., strongly ǫ -( k, t )-robust, ( k, t )-robust, strongly ( k, t )-robust). We must show that f ( ~σ + σ d ) has the corresponding property. The proofs areessentially the same in all cases; moreover, the same argument works for both constructions(i.e., both the one the gives implementation and the one that gives weak implementation).For definiteness, we deal with ǫ -( k, t )-robustness here.We start by showing ǫ - t -immunity. If f ( ~σ + σ d ) is not ǫ - t -immune, then there must bescheduler σ ′ e , a set T of players with | T | ≤ t , a strategy ~τ T for the players in T , and an inputprofile ~x T that shows this. That is, for some i / ∈ T , we have u i (Γ ′ d , ( f ( ~σ T ) , τ T ) , σ ′ e , ~x T ) ≤ u i (Γ ′ d , f ( ~σ ) , σ ′ e , ~x T ) − ǫ. 21e can assume without loss of generality that σ ′ e is deterministic. Moreover, we can furtherassume without loss of generality that, when playing with ~τ , the malicious players deviateonly by sending an input other than their actual input. All other deviations correspond toplaying with a different (deterministic) scheduler. Specifically, since the mediator ignores allmessages that are not sent by a player in the set P , the outcome is equivalent to playingwith a scheduler that delays these messages until after the mediator sends its R th message.And if a deviating player j does not send a message that it should send or sends a messagelate, we can just consider the scheduler that sends messages in the order that f ( σ d ) actuallyreceived them. It now follows that there exists a scheduler σ ′′ e and an input profile ~x ′ T suchthat u i (Γ ′ d , ( f ( ~σ ) T , τ T ) , σ ′ e , ~x T ) = u i (Γ ′ d , f ( ~σ ) , σ ′′ e , ~x ′ T ) . Also, by construction, there exists a scheduler σ e such that u i (Γ ′ d , f ( ~σ ) , σ ′′ e , ~x ′ T ) = u i (Γ d , ~σ, σ e , ~x ′ T )for some scheduler σ e . Similarly, there exists a scheduler σ ∗ e such that u i (Γ ′ d , f ( ~σ ) , σ ′ e , ~x ′ T ) = u i (Γ d , ~σ, σ ∗ e , ~x ′ T ) . Thus, we have u i (Γ d , ~σ, σ e , ~x T ) ≤ u i (Γ d , ~σ, σ ∗ e , ~x ′ T ) − ǫ. By Proposition 6.4, this contradicts the assumption that ~σ is ǫ - t -immune.Now let K be any subset disjoint from T such that 1 ≤ | K | ≤ k . Then, using ananalogous argument, there exists a scheduler σ ′′′ e and an input profile ~x ′′ T such that u i (Γ ′ d , f ( ~σ ) − ( T ∪ K ) , σ ′ e , ~x ( T ∪ K ) ) = u i (Γ ′ d , f ( ~σ ) , σ ′′′ e , ~x ′′ ( T ∪ K ) ) , for all players i . There also exists a scheduler σ ∗∗ e and an input profile ~x ′′′ T such that u i (Γ d , f ( ~σ ) − T , ~τ T , σ ′ e , ~x ( T ∪ K ) ) = u i (Γ d , f ( ~σ ) , σ ∗∗ e , ~x ′′′ ( T ∪ K ) )for all players i . Thus, u i (Γ ′ d , f ( ~σ ) − ( T ∪ K ) , σ ′ e , ~x ( T ∪ K ) )= u i (Γ d , f ( ~σ ) , σ ′′′ e , ~x ′′ ( T ∪ K ) ) < u i (Γ d , f ( ~σ ) , σ ∗∗ e , ~x ′′′ ( T ∪ K ) ) + ǫ [by Proposition 6.5]= u i (Γ d , f ( ~σ ) − T , ~τ T , σ ′ e , ~x ( T ∪ K ) )which shows that f ( ~σ + σ d ) is ǫ -( k, t )-robust.It remains to compute the bound on the number of messages sent. Clearly, if all we needis a weak implementation, then n messages suffice. In the case of implementation, at most2 Rn messages are sent (at most R by each player, and at most Rn by the mediator), where R is the least integer such that ( Rn )! ≥ |S det Γ d ,e / ∼| . Thus, we must compute an estimate forthe number of equivalence of deterministic schedulers.22 deterministic scheduler is a function from message patterns to a choice of messageto be delivered, where a message pattern describes which messages have been sent so far,which were delivered, and the order that the messages were sent and delivered, but not thecontents of the message. For example, taking ( s, i, j, k ) (resp., ( d, i, j, k )) to denote the k thmessage sent by player i to player j (resp., that the k th message sent by player i to j isreceived by j ), and taking the mediator to be player 0, then a typical message pattern mightbe ( s, , , s, , , s, , , d, , , 2) is the message pattern where the mediatorfirst send a message to player 3, then player 1 sends a message t the mediator, then themediator sends a second message to player 3, and then then mediator’s second message toplayer 3 is delivered. Given this message pattern, the scheduler can choose to deliver themediator’s first message to player 3 or the message from player 1 to the mediator. Since themediator sends at most r messages to each player, and each player sends at most r messagesto the mediator, message patterns have length at most 4 rn . The messages sent by a playerto the mediator are numbered consecutively, as are the messages sent by the mediator toeach player. A straightforward computation then shows that there are at most (4 rn )! / ( r !) n message patterns of length 4 rn . It is easy to see that there are fewer message patternsof length k for k < rn , so there are clearly at most (4 rn )(4 rn )! / ( r !) n message patternsof length at most 4 rn . A message pattern can have at most 2 rn undelivered messages, sothere are at most (2 rn ) (4 rn )(4 rn )! / ( r !) n equivalence classes of schedulers. A straightforwardapplication of Stirling’s approximation formula ( n ! ∼ ( n/e ) n √ πn ) shows that R = (4 rn ) rn suffices for our purposes. Since rn is a bound on the number of messages sent, we have N = rn , completing the proof.Thus, without loss of generality, we can assume that the players and mediator use sucha strategy profile. We call f ( ~σ + σ d ) the minimally informative strategy correspondingto ~σ + σ d . More generally, we say that ~σ m + σ md is a minimally informative strategy if ~σ m + σ md = f ( ~σ + σ d ) for some strategy profile ~σ + σ d .Since we consider only mediator games in canonical form, this guarantees terminationfor all honest players regardless of the strategy of rational and malicious players, providedthat the scheduler is standard (i.e., not relaxed). However, once we allow relaxed schedulers,there is a possibility of deadlock. We assume for the purposes of the proof that we use theAH approach in the mediator game, and have the players play the punishment punishmentstrategy in their wills. Since ~σ CT guarantees t -cotermination for t < n/ 3, it follows thatin the cheap-talk game, either all honest players terminate or all honest players play thepunishment strategy. This guarantees that the players get the same payoff in correspondinghistories in the mediator game and the cheap-talk game.The next step in proving Theorem 4.4 is to show that rational players playing with arelaxed scheduler cannot get an expected payoff that is higher than their expected payoffwhen they play such a minimally informative ( k, t )-robust equilibrium strategy with a non-relaxed scheduler. Proposition 6.9. If ǫ > , ~σ + σ d is a minimally informative (strongly) ǫ - ( k, t ) -robustequilibrium in a mediator game Γ d for which a ( k + t ) -punishment strategy exists, σ E is a elaxed scheduler, K and T are disjoint sets of players with ≤ | K | ≤ k and | T | ≤ t , and ~τ ( K ∪ T ) is a strategy profile for the players in K ∪ T , then there exists a value ǫ < ǫ such thatfor all non-relaxed schedulers σ e and all input profiles ~x , we have that u ′ i (Γ d , ( ~σ − ( K ∪ T ) , ~τ ( K ∪ T ) ) , σ E , ~x ( K ∪ T ) ) < u ′ i (Γ d , ( ~σ − T , ~τ T ) , σ e , ~x T ) + ǫ for some (for all) i / ∈ T . To prove Proposition 6.9, we need a preliminary lemma that characterizes deadlocks inmediator games in canonical form with relaxed schedulers. It also shows that the schedulercan detect when such a deadlock happens. Lemma 6.10. A run in a mediator game in canonical form with a relaxed scheduler σ E ends in deadlock iff at some point in the run no player has received a STOP message andthe scheduler does not deliver any of the messages not yet delivered.Proof. Clearly, if no player has received a STOP messages and all the messages not yetdelivered are never delivered, then the run ends in deadlock. To show that all deadlocksmust be of this form, assume that a player receives eventually a STOP message. Then, sincethe mediator sent all STOP messages at the same time, all other players are guaranteedto receive a STOP message as well, given our assumption that a relaxed scheduler deliverseither all or none of the messages sent at the same time. Proof of Proposition 6.9. We can view the adversary’s strategy ( ~τ ( K ∪ T ) , σ E ) as a combinationof (possibly infinitely many) deterministic strategies ( ~τ ′ ( K ∪ T ) , σ ′ E ). Thus, it suffices to showthe desired result for each of such deterministic strategies.Let ( ~τ ′ ( K ∪ T ) , σ ′ E ) be a deterministic strategy for the adversary in the support of ( ~τ ( K ∪ T ) , σ E ).By the properties of our construction, the fact that the adversary is deterministic, and thefact that a relaxed scheduler must either deliver all the STOP messages from the mediatoror deliver none, it follows that, for a given input ~x ( K ∪ T ) , either all runs end in deadlock orall honest players terminate.Suppose that for some deterministic adversary ( ~τ ′ ( K ∪ T ) , σ ′ E ) and input ~x K ∪ T all honestplayers terminate. Consider a non-relaxed scheduler σ ′ e that acts just like σ ′ E , except thatwhenever it detects a deadlock (as characterized by Lemma 6.10, using the fact that themediator’s r th message to each player includes STOP), it instead delivers a message chosenat random. Then, σ ′ E and σ ′ e are indistinguishable when the players in K ∪ T play ~τ ′ ( K ∪ T ) .Therefore, there exists a value ǫ ′ < ǫ such that u ′ i (Γ d , ( ~σ − ( K ∪ T ) , ~τ ′ ( K ∪ T ) ) , σ ′ E , ~x ( K ∪ T ) )= u ′ i (Γ d , ( ~σ − ( K ∪ T ) , ~τ ′ ( K ∪ T ) ) , σ ′ e , ~x ( K ∪ T ) ) < u ′ i (Γ d , ( ~σ − T , ~τ T ) , σ e , ~x T ) + ǫ ′ [by Proposition 6.7]for some (for all) i ∈ T and all non-relaxed schedulers σ e . Suppose instead that all runs withadversary ( ~τ ′ ( K ∪ T ) , σ ′ E ) end in deadlock. Then, there exists a value ǫ ′ < ǫ such that u ′ i (Γ d , ( ~σ − ( K ∪ T ) , ~τ ( K ∪ T ) ) , σ ′ E , ~x ( K ∪ T ) ) < u ′ i (Γ d , ~σ, σ e , ~x ( K ∪ T ) ) [by definition of ( t + k )-punishment strategy] < u ′ i (Γ d , ( ~σ − T , ~τ T ) , σ e , ~x T ) + ǫ ′ [by Proposition 6.6]24sing a compactness argument analogous to that of Proposition 6.6, there exists a value ǫ < ǫ such that ǫ ′ ≤ ǫ for all relaxed adversaries ( ~τ ′ ( K ∪ T ) , σ ′ E ) in the support of ( ~τ ( K ∪ T ) , σ E )and all inputs ~x ( K ∪ T ) . So u ′ i (Γ d , ( ~σ − ( K ∪ T ) , ~τ ( K ∪ T ) ) , σ E , ~x ( K ∪ T ) ) < u ′ i (Γ d , ( ~σ − T , ~τ T ) , σ e , ~x T ) + ǫ for all inputs ~x ( K ∪ T ) , all non-relaxed schedulers σ e , all strategies ~τ ( T ∪ K ) , and for some (forall) i / ∈ T .An analogous argument can be used for the errorless case: Proposition 6.11. If ~σ + σ d is a minimally informative ( k, t ) -robust strategy in a mediatorgame Γ d ( u ′ i ) for which a ( k + t ) -punishment strategy exists, σ E is a relaxed scheduler, T and K are disjoint sets of players with | T | ≤ t and ≤ | K | ≤ k , and ~τ ( K ∪ T ) is a strategy profilefor the players in K ∪ T , then there exists a (non-relaxed) scheduler σ e such that for all inputprofiles ~x and all i / ∈ T , u ′ i (Γ d , ( ~σ − ( K ∪ T ) , ~τ ( K ∪ T ) ) , σ d , σ E , ~x ( K ∪ T ) ) ≤ u ′ i (Γ d , ( ~σ − T , ~τ T ) , σ d , σ e , ~x T ) . We can now prove (strong) ( k, t )-robustness. Let Γ d ( ~u ′ ) be any utility variant of Γ d suchthat ~σ + ~σ d is a ( k, t )-robust equilibrium of Γ d ( ~u ′ ), let σ ′ e be any scheduler in Γ CT ( ~u ), andlet K and T be disjoint subsets of players with 1 ≤ | K | ≤ k and | T | ≤ t , respectively, suchthat 3 k + 4 t < n . Let ~τ K and ~τ T be strategy profiles for players in K and T , respectively. ByTheorem 5.4, there exists a function H and a relaxed scheduler σ E in the cheap-talk gamesuch that u ′ i (Γ CT ( ~u ′ ) , ( ~σ CT ) − ( K ∪ T ) , ~τ K , ~τ T ) , σ ′ e , ~x ( K ∪ T ) ) = u ′ i (Γ d ( ~u ′ ) , ( ~σ − ( K ∪ T ) , H ( ~τ K ) , H ( ~τ T )) , σ d , σ E , ~x ( K ∪ T ) )for some (for all) i / ∈ T and all input profiles ~x . We can assume without loss of generalitythat ( ~σ, σ d ) is minimally informative. Thus, by Proposition 6.11, there exists a non-relaxedscheduler σ e such that u ′ i (Γ d ( ~u ′ ) , ( ~σ − ( K ∪ T ) , H ( ~τ T ) , H ( ~τ K )) , σ d , σ E , ~x ( K ∪ T ) ) ≤ u ′ i (Γ d ( ~u ′ ) , ( ~σ − T , H ( ~τ T )) , σ d , σ e , ~x ( K ∪ T ) ) . Finally, by Theorem 5.4, there exists a non-relaxed scheduler σ ′′ e such that u ′ i (Γ CT ( ~u ′ ) , (( ~σ CT ) − T , ~τ T ) , σ ′ e , ~x T )= u ′ i (Γ d ( ~u ′ ) , ( ~σ − T , H ( ~τ T )) , σ d , σ ′′ e , ~x T ) [by Theorem 5.4]= u ′ i (Γ d ( ~u ′ ) , ( ~σ − T , H ( ~τ T )) , σ d , σ e , ~x T ) [by Corollary 6.3].Therefore, u ′ i (Γ CT ( ~u ′ ) , (( ~σ CT ) − ( K ∪ T ) , ~τ K , ~τ T ) , σ ′ e , ~x ( K ∪ T ) ) ≤ u ′ i (Γ CT ( ~u ′ ) , (( ~σ CT ) − T , ~τ T ) , σ ′ e , ~x T ) , as desired. ⊓⊔ We remark that, with a little more effort, we can show that the minimally informativestrategy f ( ~σ + σ d ) that implements ~σ + σ d is actually a t -bisimulation and a t -emulationof of ~σ + σ d . Moreover, the cheap-talk strategy that implements f ( ~σ + σ d ) preserves theseproperties. Thus, under the conditions of Theorem 4.4, we can get a cheap-talk strategythat t -bisimulates and t -emulates a strategy in the mediator game.25 .5 Proof of Theorem 4.5 To prove Theorem 4.5, we use an analogous strategy to that used to prove Theorem 4.4,using Theorem 5.5 instead of Theorem 5.4. The same argument as that used in the proofTheorem 4.2 shows that for all ǫ ′ ∈ (0 , 1] there exists a protocol ~σ CT that ǫ ′ -implements ~σ + σ d and that ~σ CT is ( ǫ, t )-immune.To prove (strong) ǫ -( k, t )-robustness, fix an adversary A = ( ~τ K , ~τ T , σ e ) for subsets K, T such that 1 ≤ | K | ≤ k , | T | ≤ t and K ∩ T = ∅ . By Theorem 5.5, there exists a function H from strategies to strategies and a relaxed scheduler σ E such that, for all input profiles ~x , u i (Γ CT , (( ~σ CT ) − ( K ∪ T ) , ~τ K , ~τ T ) , σ e , ~x ( K ∪ T ) ) < u i (Γ d , ( ~σ − ( K ∪ T ) , H ( ~τ K ) , H ( ~τ T )) , σ d , σ E , ~x ( K ∪ T ) )+ ǫ ′ M for some (resp. for all) i ∈ K .By Theorem 5.5, there exists a non-relaxed scheduler σ ′ e such that u i (Γ CT , (( ~σ CT ) − T , ~τ T ) , σ e , ~x T ) > u i (Γ d , ( ~σ − T , H ( ~τ T )) , σ d , σ ′ e , ~x T ) − ǫ ′ M. Thus, we have that u i (Γ CT , (( ~σ CT ) − ( K ∪ T ) , ~τ K , ~τ T ) , σ e , ~x ( K ∪ T ) ) < u i (Γ d , ( ~σ − T , H ( ~τ T )) , σ d , σ ′ e , ~x T ) + ǫ + ǫ ′ M [by Proposition 6.9] < u i (Γ CT , (( ~σ CT ) − T , ~τ T ) , σ e , ~x T ) + ǫ + 2 ǫ ′ M. for some ǫ < ǫ . 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