Communication, Renegotiation and Coordination with Private Values
aa r X i v : . [ ec on . T H ] M a y Renegotiation and Coordination with Private Values *Yuval Heller † and Christoph Kuzmics ‡ May 13, 2020
Abstract
We define and characterize the set of renegotiation-proof equilibria of coordination games withpre-play communication in which players have private preferences over the feasible coordinatedoutcomes. Renegotiation-proof equilibria provide a narrow selection from the large set of qual-itatively diverse Bayesian Nash equilibria in such games. They are such that players never mis-coordinate, play their jointly preferred outcome whenever there is one, and communicate onlythe ordinal part of their preferences. Moreover, they are robust to changes in players’ beliefs,interim Pareto efficient, and evolutionarily stable.
Keywords : coordination games, renegotiation-proofness, equilibrium entrants, secret handshake,incomplete information, evolutionary robustness.
JEL codes : C72, C73, D82
We define and characterize renegotiation-proof equilibria for a class of coordination games withpre-play cheap-talk communication in which agents have private information about what action theywould prefer to coordinate on. An equilibrium is renegotiation-proof if, after the communication andgiven the information that this reveals, players never have a joint deviation into a Pareto-improvingequilibrium (where the deviation may rely on additional communication). We are interested in twotypical kinds of situations for which renegotiation proofness is the appropriate solution concept,albeit for different reasons in the two situations. * We have benefited greatly from discussions with Srinivas Arigapudi, Tilman Borgers, Michael Greinecker, JonathanNewton, Bill Sandholm, and Joel Sobel. We would like to express our gratitude to participants of various conferenceand seminar audiences for many useful comments: LEG2018 & LEG2019 conferences (in Lund and Bar-Ilan univer-sity, respectively), Bielefeld Game Theory 2018 workshop, Israeli Game Theory 2018 conference in IDC, and seminaraudiences at Caltech, Tel Aviv University, University of Cyprus, Haifa University, and UC San Diego. Yuval Heller isgrateful to the European Research Council for its financial support (Starting Grant † Department of Economics, Bar Ilan University, Israel. Email: [email protected]. ‡ Department of Economics, University of Graz, Austria. Email: [email protected]. Players also typically coordinate effectively in our second example: Casual observation suggeststhat pedestrians typically are able to avoid bumping into each other, even though there is no uni-form social norm such as “always stay on the right” as there is for cars (Young, 1998). Moreover,pedestrians often use brief nonverbal communication to signal their preferred direction (e.g., a slightmovement to the left or right, a tilt of the head, a glance in a certain direction), and the (coordinated)direction in which they pass each other depends on this communication. We show that renegotiation-proof equilibria have a very specific structure that is consistent with In fact, the result of this paper that binary communication is all that is needed for successful coordination providesanother argument against allowing even a brief form of explicit communication between oligopolistic competitors. The example is motivated by Goffman (1971, Chapter 1, p. 6): “Take, for example, techniques that pedestriansemploy in order to avoid bumping into one another. [...] There are an appreciable number of such devices; they areconstantly in use and they cast a pattern on street behavior. Street traffic would be a shambles without them.”
Relationship to the literature
Game theorists have long recognized that coordination is an im-portant aspect of successful economic and social interaction, that it requires an explanation evenin complete-information coordination games, and that it does not occur in all circumstances. Onepossible explanation for some, fairly simple,examples of coordination is the concept of a focal point,due to Schelling (1960), which is, loosely speaking, a strategy profile that jumps out at players asclearly the right way to play a game. Perhaps one of the situations in which we most plausibly expect coordination is when peopleplay the same coordination game many times with different people and there is some evolutionary(or learning) process. This approach is already present in the “mass action” interpretation of equi-librium given by Nash (1950), and then taken up more formally in Maynard Smith and Price (1973)who define the notion of evolutionary stability. It is well known that all pure equilibria in coordi-nation games are evolutionarily stable (whereas mixed equilibria are not stable). This literature thussupports the view that while play in the long run will be coordinated, it is not necessarily efficientlycoordinated. Another explanation for coordination is that it is achieved through communication, even if it is Evidence for miscoordination in lab experiments is reported by, e.g., Van Huyck, Battalio, and Beil (1990),Mehta, Starmer, and Sugden (1994), and Blume and Gneezy (2010). According to Farrell and Klemperer (2007, Sec-tion 3.4) miscoordination also occurs regularly in real-life economic interactions. Different attempts at formalizing focal points are given in, e.g., Sugden (1995) and Alós-Ferrer and Kuzmics (2013). See, e.g., Weibull (1995) and Sandholm (2010) for a textbook treatment of the large literature on evolutionary gametheory following these. Kandori, Mailath, and Rob (1993) and Young (1993) show that in the very long run and under persistent low-probability errors an evolutionary (learning) process leads to the risk-dominant, not necessarily Pareto-dominant, purestrategy equilibrium in two-by-two coordination games.
Baseline model and robustness checks
While we take into account incomplete information inthe coordination problem, we try to keep the baseline model tractable by simplifying other aspectsof the problem. We restrict attention in the baseline model to (incomplete information) two-playertwo-action (coded as left and right) coordination problems, in which every type has a coordinationconcern. Different types only differ in how much they prefer coordination on one action over theother one. Players can communicate only once (by sending a message from a large finite set) beforethe game is played.Section 8 provides a series of extensions and robustness checks of our baseline model. First, weshow that all our results hold for any length of pre-play communication. Next, we extend our keyresults to general (possibly asymmetric) coordination games, which might involve more than twoplayers and more than two actions. Finally, we study a variant of our baseline model in which afew types have dominant actions. We show that in this setup, there is a unique renegotiation-proofequilibrium strategy among the strategies satisfying our three key properties (the identity of thisunique strategy depends on the distribution of types with dominant actions).4 tructure
Section 2 presents our baseline model. Section 3 defines Bayesian Nash equilibria andthe three key properties that renegotiation-proof equilibria have. Section 4 defines the concept ofrenegotiation-proofness appropriately adapted to our incomplete-information strategic setting. Sec-tion 5 presents the main result and a sketch of its proof. Section 6 discusses the efficiency propertiesof renegotiation-proof equilibria. Section 7 discusses additional notions of evolutionary stability be-yond the secret-handshake stability implied by renegotiation-proofness. Section 8 provides a seriesof robustness checks and extensions. Section 9 concludes. The formal proofs are presented in theappendices. Appendices C–E are presented in the online supplementary material.
We consider a setup in which two agents with private idiosyncratic preferences play a two-actioncoordination game that is preceded by pre-play cheap talk. Extensions are presented in Section 8.
Players and types
There are two players in ex-ante symmetric positions. Players can choose oneof two actions, L and R . Each player has a privately known “value” or “type.” The two players’values are independently drawn from a common atomless distribution with a continuous cumulativedistribution function F with a full support on the unit interval U = [ , ] and with density f (i.e., f ( u ) > u ∈ U ). Payoff matrix
For any realized pair of types, u and v , the players play a coordination game givenby the following payoff matrix, where the first entry is the payoff of the player of type u (choosingrow) and the second entry is the payoff of the player of type v (choosing column).Table 1: Payoff Matrix of the Coordination GameType vL R Type u L u , 1- v
0, 0 R
0, 0 u , v We call this game the coordination game without communication and denote it by Γ . Remark . Our results are similar without assuming full support. The only difference is that withoutfull support renegotiation-proofness implies binary communication only of messages that are usedwith positive probability, while with full support it implies binary communication also of unusedmessages, which yields the exact “iff” characterization of the renegotiation-proofness in Theorem 1.5 re-play communication
After learning their type, but before playing this coordination game, thetwo players each simultaneously send a publicly observable message from a finite set of messages M (satisfying 2 ≤ | M | < ∞ ), where ∆ ( M ) is the set of all probability distributions over messages in M . We assume that messages are costless. We call the game, so amended, the coordination gamewith communication and denote it by h Γ , M i . Strategies
A player’s (ex-ante) strategy in the coordination game with communication is then apair σ = ( µ , ξ ) , where µ : U → ∆ ( M ) is a (Lebesgue measurable) message function that describeswhich (possibly random) message is sent for each possible realization of the agent’s type, and ξ : M × M → U is an action function that describes the maximal type (cutoff type) that chooses L as afunction of the observed message profile; that is, when an agent who follows strategy ( µ , ξ ) observesa message profile ( m , m ′ ) (message m sent by the agent, and message m ′ sent by the opponent),then the agent plays L if her type u is at most ξ ( m , m ′ ) (i.e., if u ≤ ξ ( m , m ′ ) ), and she plays R if u > ξ ( m , m ′ ) . (The choice that the threshold type plays L does not affect our analysis, given theassumption of F being atomless.)Let Σ be the set of all strategies in the game h Γ , M i . Remark . In principle, we should allow more general action functions ξ : U × M × M → △{ L , R } ,which specify the probability that an agent chooses L as a function of the observed message profileand the agent’s type. It is simple to see, however, and proven in Lemma 1 in Appendix A.1, thatany “generalized” strategy is dominated by a strategy that uses a cutoff action function in the secondstage. The intuition, is that following the observation of any pair of messages, lower types alwaysgain more (less) than higher types from choosing L ( R ). We thus simplify our notation by consideringonly cutoff action functions of the form ξ : M × M → U .Let µ u ( m ) denote the probability, given message function µ , that a player sends message m if sheis of type u . Let ¯ µ ( m ) = IE u [ µ u ( m )] be the mean probability that a player of a random type sendsmessage m (where the expectation is taken with respect to F ). Let supp ( ¯ µ ) = { m ∈ M | ¯ µ ( m ) > } denote the support of ¯ µ . With a slight abuse of notation we write ξ ( m , m ′ ) = L when all types (whosend message m with positive probability) play L (i.e., when ξ ( m , m ′ ) ≥ sup ( u ∈ U | µ u ( m ) > ) ),and we write ξ ( m , m ′ ) = R when all types play R (i.e., when ξ ( m , m ′ ) ≤ inf ( u ∈ U | µ u ( m ) > ) ). We here define the standard notion of (Bayesian Nash) equilibrium strategies, present properties thatrenegotiation-proof equilibria turn out to have, and present examples of equilibria in the coordinationgame with communication with and without these properties.6iven a strategy profile ( σ , σ ′ ) and a type profile u , v ∈ U , let π u , v ( σ , σ ′ ) denote the interim (pre-communication) expected payoff of a player of type u who follows strategy σ and faces an opponentof type v who follows strategy σ ′ . Formally, for σ = ( µ , ξ ) and σ ′ = ( µ ′ , ξ ′ ) , π u , v (cid:0) σ , σ ′ (cid:1) = ∑ m ∈ M ∑ m ′ ∈ M µ u ( m ) µ v (cid:0) m ′ (cid:1) (cid:0) ( − u ) { u ≤ ξ ( m , m ′ ) } { v ≤ ξ ′ ( m ′ , m ) } + u { u > ξ ( m , m ′ ) } { v > ξ ′ ( m ′ , m ) } (cid:1) , where 1 { x } is the indicator function equal to 1 if statement x is true and zero otherwise. Let π u (cid:0) σ , σ ′ (cid:1) = IE v (cid:2) π u , v (cid:0) σ , σ ′ (cid:1)(cid:3) ≡ Z v = π u , v (cid:0) σ , σ ′ (cid:1) f ( v ) dv denote the expected interim payoff of a player of type u who follows strategy σ and faces an opponentwith a random type who follows strategy σ ′ . Finally, let, π (cid:0) σ , σ ′ (cid:1) = IE u (cid:2) π u (cid:0) σ , σ ′ (cid:1)(cid:3) ≡ Z u = π u (cid:0) σ , σ ′ (cid:1) f ( u ) du denote the ex-ante expected payoff of an agent who uses strategy σ against strategy σ ′ .A strategy σ is a (symmetric Bayesian Nash) equilibrium strategy if π u ( σ , σ ) ≥ π u ( σ ′ , σ ) foreach u ∈ U and each strategy σ ′ ∈ Σ . Let E ⊆ Σ denote the set of all equilibrium strategies of h Γ , M i . Three key properties
We call a strategy σ = ( µ , ξ ) ∈ Σ mutual-preference consistent if whenever u , v < / then ξ ( m , m ′ ) = ξ ( m ′ , m ) = L for all m ∈ supp ( µ u ) and all m ′ ∈ supp ( µ v ) and if whenever u , v > / then ξ ( m , m ′ ) = ξ ( m ′ , m ) = R for all m ∈ supp ( µ u ) and all m ′ ∈ supp ( µ v ) . That is, playerswith the same ordinal preference coordinate on their mutually preferred outcome.We call a strategy coordinated if ξ ( m , m ′ ) = ξ ( m ′ , m ) ∈ { L , R } for any pair of messages m , m ′ ∈ supp ( ¯ µ ) . A coordinated strategy never leads to miscoordination after any (used) message pair.For any message m ∈ M , define the expected probability of a player’s opponent playing L con-ditional on the player sending message m ∈ M and the opponent following strategy σ = ( µ , ξ ) ∈ Σ ,as β σ ( m ) = Z u = ∑ m ′ ∈ supp ( µ u ) µ u ( m ′ ) { u ≤ ξ ( m ′ , m ) } f ( u ) du . We say that strategy σ has binary communication if there are two numbers 0 ≤ β σ ≤ β σ ≤ m ∈ M we have β σ ( m ) ∈ [ β σ , β σ ] , for all messages m ∈ M such that there isa type u < / with µ u ( m ) > β σ ( m ) = β σ , and for all messages m ∈ M such that there isa type u > / with µ u ( m ) > β σ ( m ) = β σ . That is, binary communication implies thatplayers (essentially) use just two kinds of messages: any message sent by types u < . L , and any message sent by7ypes u > . R . Note that, as defined here, a strategy with no communication also has binary communication (inwhich the player’s message does not affect the probability of the partner playing L ).In Appendix B we show that no single one of these three properties is implied by the other two.Clearly, a strategy that has binary communication and is coordinated must be an equilibrium. InAppendix B we also show that no combination of any two of these three properties implies that astrategy is an equilibrium.Consider a strategy that is coordinated and mutual-preference consistent and has binary commu-nication. The first two properties determine the behavior of agents with the same ordinal preferences(i.e., when both types are below ⁄ , or both above ⁄ ). The property of binary communication, then,implies that the probability with which the players coordinate on L , conditional on having differentordinal preferences (i.e., conditional on one player having type u < / and the other player havingtype v > / ), is independent on the message sent by the player. We denote this probability by α σ ,and refer to it as the left tendency of the strategy. We can express β and β σ as follows: β σ = F ( / ) α σ and β σ = F ( / ) + ( − F ( / )) α σ . The first equality ( β σ = F ( / ) α σ ) is implied by the fact that when any type u > / sends a messageexpressing her preference for coordination on R , the players coordinate on L only if the opponent’spreferred outcome is L (which happens with a probability of F ( / ) ), and they then coordinate on L with a probability of α σ . The second equality ( β σ = F ( / ) + ( − F ( / )) α σ ) is implied by thefact that when any type u < / sends a message expressing her preference for coordination on L ,the players coordinate on L with probability one if the opponent’s preferred outcome is L , and theycoordinate on L with a probability of α σ if the opponent’s preferred action is R . Remark . The set of strategies with the above three properties (coordination, mutual-preferenceconsistency, and binary communication) is essentially one-dimensional because the left tendency α σ ∈ [ , ] of such a strategy σ describes all payoff-relevant aspects. Two strategies with the sameleft tendency can only differ in the way in which the players implement the joint lottery when theyhave different preferred outcomes, but these implementation differences are nonessential, as theprobability of the joint lottery inducing the players to coordinate on L remains the same. Examples of equilibria satisfying all properties
The following strategies, denoted by σ L , σ R , and σ C , are prime examples (that play a special role in later sections) of strategies that are all mutual-preference consistent, coordinated, and have binary communication.The strategies σ L and σ R are given by the pairs ( µ ∗ , ξ L ) and ( µ ∗ , ξ R ) , respectively. The messagefunction µ ∗ has the property that there are messages m L , m R ∈ M such that message m L indicates a8reference for L and m R a preference for R , and the action functions ξ L and ξ R are defined as follows: µ ∗ ( u ) = m L u ≤ m R u > . ξ L (cid:0) m , m ′ (cid:1) = R m = m ′ = m R L otherwise , ξ R (cid:0) m , m ′ (cid:1) = L m = m ′ = m L R otherwise . This means that the “fallback norm” of σ L (which is applied when the agents have differentpreferred outcomes) is to coordinate on L , while that of σ R is to coordinate on R . In other words theleft tendency of σ L is one and the left tendency of σ R is zero.Strategy σ C = ( µ C , ξ C ) has the “fallback norm” of using a joint lottery to choose the coordinatedoutcome. Each agent simultaneously sends a random bit and the coordinated outcome depends onwhether the random bits are equal or not.Assume that | M | ≥
4. We denote four distinct messages by m L , , m L , , m R , , m R , ∈ M , where weinterpret the first subscript ( R or L ) as the agent’s preferred direction, and the second subscript (0 or1) as a random binary number chosen with probability / each by the agent. Formally, the messagefunction µ C is defined as follows: µ C ( u ) = m L , ⊕ m L , u ≤ m R , ⊕ m R , u > , where α m ⊕ ( − α ) m ′ is a lottery with a probability of α on message m and 1 − α on message m ′ .In the second stage, if both agents share the same preferred outcome they play it. Otherwise,they coordinate on L if their random numbers differ, and coordinate on R otherwise. Formally: ξ C (cid:0) m , m ′ (cid:1) = R ( m , m ′ ) ∈ { ( m R , , m R , ) , ( m R , , m R , ) , ( m R , , m L , ) , ( m R , , m L , )( m R , , m R , ) , ( m R , , m R , ) , ( m L , , m R , ) , ( m L , , m R , ) } L otherwise . Note that of all the strategies that satisfy the three properties, strategies σ L and σ R are the simplest in terms of the number of “bits” needed to implement the message function. Strategy σ C is in acertain sense the fairest : conditional on a coordination conflict, i.e., conditional on one agent havinga type between 0 and / and the other agent having a type between / and 1, both agents expect thesame payoff. By contrast, strategy σ L favors types below / , and strategy σ R favors types above / . Examples of equilibria not satisfying some of the properties
The coordination game with com-munication h Γ , M i admits many more equilibria that satisfy only some or even none of the threeproperties defined above. It has, for instance, two simple babbling equilibria, in which agents ig-nore the communication and apply a uniform norm of always playing L (or R ). These equilibria are9oordinated and trivially have binary communication, but are not mutual-preference consistent.Depending on the distribution of types, the game can also have additional inefficient babblingequilibria, in which agents sometimes miscoordinate. Specifically, if there exists a type x ∈ ( , ) sat-isfying x = F ( x ) , then there is a babbling equilibrium in which agents ignore messages and choose L if and only if their type is below x . Such a babbling equilibrium (trivially) has binary communicationbut does not satisfy the other two properties defined above. Note that, as by assumption F ( ) = F ( ) =
1, all babbling equilibria can be identified with an x ∈ [ , ] that satisfies F ( x ) = x . Remark . When there is no pre-play communi-cation (i.e., | M | = x (with x = F ( x ) ) is less than one. In particular, if the distribu-tion of types satisfies f ( ) , f ( ) >
1, then there exists x ∈ ( , ) satisfying x = F ( x ) and f ( x ) < f ( ) , f ( ) > u ≤ / or u > / ). One simpleexample of such an equilibrium for a specific distribution F is Example 1 in Section 6. It is notcompletely straightforward to construct such examples for all possible distributions F if we consideronly a single round of communication as we do in the main body of this paper. In Section 8, however,we show that our main results continue to hold if we allow multiple rounds of communication. Inthe case of multiple rounds of communication it is relatively straightforward to construct examplesof equilibria that do not have binary communication and that, therefore, reveal some cardinal contentof the players’ preferences. For simplicity, assume that the distribution F is symmetric around / .That is f ( x ) = f ( − x ) for all x ∈ [ , ] or, equivalently, F ( x ) = − F ( − x ) for all x ∈ [ , ] . Inparticular, we have that F ( / ) = / .The equilibrium is such that there is an x satisfying 0 < x < / such that in the first round ofcommunication players indicate whether their preferences are “extreme” (i.e., u ≤ x or u > − x )or “moderately left” ( x < u ≤ / ) or “moderately right” ( / < u ≤ − x ). In the second roundindividuals only reveal real additional information if in the first round one sent the extreme messageand the other a moderate message, in which case the extreme type now reveals which side sheprefers ( u ≤ / or u > / ). In this case, joint play is dictated by the extreme type’s preferences.If both players sent the extreme message in the first round, then there is no more communication10nd both extreme types follow their inclination (play L if u ≤ / and R otherwise). This leads tomiscoordination with a conditional probability of a half.Any two moderate types essentially play the coordinated strategy σ C ; that is if they have the samepreferred outcome, then they play it, and, otherwise, they use communication to induce a joint fairlottery over both playing L and both playing R . In Appendix C we write this strategy down formallyand show that for any symmetric distribution F there is an x ∈ ( , / ) such that this strategy is indeeda Nash equilibrium of the coordination game with two rounds of communication. For any given strategy in Σ employed by both players in the game h Γ , M i , communication and knowl-edge of this strategy lead to updated and possibly, different and asymmetric information about thetwo agents’ types. Suppose that the updated distributions of types are given by some distributionfunctions G and H . The two agents then face a (possibly asymmetric) game of coordination withoutcommunication, which we shall denote by Γ ( G , H ) . Note that the original game (without communi-cation) Γ is then given by Γ ( F , F ) .Let f m be the type density conditional on the agent following a given strategy in the game h Γ , M i and sending a message m ∈ supp ( ¯ µ ) , i.e., f m ( u ) = f ( u ) µ u ( m ) ¯ µ ( m ) , and let F m be the cumulative distribution function associated with density f m .We allow players to renegotiate after communication. Renegotiating players can use any newfinite message set, ˜ M . Given a strategy of the game h Γ , M i employed by both players, we denote theinduced renegotiation game after a positive probability message pair m , m ′ ∈ M by h Γ ( F m , F m ′ ) , ˜ M i .For a pair of strategies σ , σ ′ in renegotiation game h Γ ( G , H ) , ˜ M i , define the post-communication expected payoffs for a type u agent by π Hu (cid:0) σ , σ ′ (cid:1) = IE v ∼ H (cid:2) π u , v (cid:0) σ , σ ′ (cid:1)(cid:3) ≡ Z v = π u , v (cid:0) σ , σ ′ (cid:1) h ( v ) dv . Define E ( G , H ) as the set of all (possibly asymmetric) equilibrium profiles of the coordinationgame with communication h Γ ( G , H ) , ˜ M i for some finite message set ˜ M . Furthermore, let S ( G ) denote the set of all symmetric equilibrium strategies of the coordination game with communication The density f m depends on the given strategy in the game h Γ , M i . For aesthetic reasons we refrain from giving thisstrategy a name and from indicating this obvious dependence in our notation. All of our main results remain the same if one limits renegotiating players to using the original set of messages M ,as long as the set M includes at least four messages (i.e., | M | ≥ Γ ( G , G ) , ˜ M i for some finite message set ˜ M . With a slight abuse of notation for any strategy σ ofthe game h Γ , M i , we denote by σ its babbling prescription after message pair m , m ′ ∈ M , i.e., inthe game h Γ ( F m , F m ′ ) , ˜ M i by σ ; that is, we denote by σ the strategy in h Γ ( F m , F m ′ ) , ˜ M i in which theplayer chooses her message uniformly regardless of her type, and then plays according to what theoriginal strategy σ induced her to play in h Γ , M i after observing the message profile ( m , m ′ ) .The motivation for renegotiation-proofness is this: If there is an observed pair of messages afterwhich the original equilibrium induces the agents to play a low payoff strategy profile payoff, then,arguably, the agents can use an additional round of communication to renegotiate the existing “bad”equilibrium of the current induced game, and to coordinate their play on a Pareto-improving equi-librium (which weakly improves the payoff of all possible types of both players). Our refinement ofrenegotiation-proofness requires that no such Pareto-improving equilibria exist in any induced gamewith additional communication. Formally, Definition 1.
We say that an equilibrium strategy σ ∈ E is post-communication equilibrium Pareto-dominated if either there is a message m ∈ supp ( ¯ µ ) and an equilibrium ˜ σ ∈ S ( F m ) such that π F m u ( σ , σ ) ≤ π F m u ( ˜ σ , ˜ σ ) for all u ∈ supp ( F m ) with strict inequality for some u ∈ supp ( F m ) , or there is apair of messages m = m ′ ∈ supp ( ¯ µ ) and an equilibrium profile ˜ σ ∈ E ( F m , F m ′ ) such that π F m ′ u ( σ , σ ) ≤ π F m ′ u ( ˜ σ ) and π F m v ( σ , σ ) ≤ π F m v ( ˜ σ ) for all u ∈ supp ( F m ) and all v ∈ supp ( F m ′ ) with strict inequalityfor some u ∈ supp ( F m ) or some v ∈ supp ( F m ′ ) . Definition 2.
An equilibrium strategy σ ∈ E is renegotiation-proof if it is not post-communicationequilibrium Pareto-dominated.We have chosen to define a mild notion of renegotiation-proofness because it already suffices forthe strict characterization given in Theorem 1. Our refinement is mild in the following ways: weallow players to renegotiate only after they observe their realized messages, and when players playa symmetric induced game, we allow them only to implement an alternative symmetric equilibrium.All of our results remain the same with a stronger refinement that allows players to: renegotiatealso before communicating (see Remark 6), and play asymmetric equilibria when renegotiating insymmetric induced games. With all this in place we can state our main result.
Theorem 1.
A strategy of the game with communication h Γ , M i is a renegotiation-proof equilibriumstrategy if and only if it is mutual-preference consistent, coordinated, and has binary communication. ketch of proof. The “if” part, i.e., that any strategy satisfying the three key properties must berenegotiation-proof, is fairly straightforward. We here provide a sketch of the proof of the “onlyif” part. The proof in Appendix A.2 is split into three lemmas, each showing that one of the threeproperties must hold.Lemma 2 proves that a renegotiation-proof equilibrium strategy must be coordinated: if play afterany message pair is not coordinated then it is Pareto inferior (given the information about players’types implicit in this message pair) to either σ L , σ R , or σ C . To see this, suppose first that bothplayers use thresholds below / . Then this strategy is Pareto-dominated by σ R as types above / gain because σ R induces their first-best outcome, and types below / gain because σ R yields a highercoordination probability and a higher probability of the opponent playing this type’s preferred action L . Analogously, an equilibrium in which both players use thresholds above / is Pareto-dominatedby σ L . Suppose, finally, that player one uses threshold x < / , while player two uses threshold x ′ > / . Observe that x < / (resp., x ′ > / ) can be an equilibrium threshold only if player two(resp., player one) plays L with an average probability of less (resp., more) than / . This, impliesthat players in these equilibria coordinate with a probability of at most / , and one can show thatsuch a low coordination probability implies that these equilibria are Pareto-dominated by σ C .Next, we show in Lemma 3 that a renegotiation-proof equilibrium strategy must have binarycommunication. The reason for this is that if a strategy is coordinated, then different messages canonly lead to different ex-ante probabilities of coordination on L (and R ). Thus, any type who favors L , i.e., any type u < / , will choose a message to maximize this probability, while any type u > / will choose a message to minimize this probability. Thus, essentially only two kinds of messagesare used in a coordinated equilibrium strategy.Finally, we show in Lemma 4 that a renegotiation-proof equilibrium strategy must be mutual-preference consistent. Given that it is coordinated, we know that any message pair will lead to eithercoordination on L or on R . If it is not mutual-preference consistent then, without loss of generality,there are two types u , u ′ < / that, with positive probability, send a message pair ( m , m ′ ) that leadsthem to coordinate on R . But then all types who send this message pair would be weakly better off(and some strictly better off) if instead of coordinating on R they use strategy σ R , which would allowthem to coordinate on L if and only if both types are below / . Remark . Note that the set of renegotiation-proof equilibria is completely independent of the distri-bution of private preferences. This implies that any renegotiation-proof equilibrium strategy remains In our proof we actually prove a slightly stronger result. Any equilibrium that is not renegotiation-proof is in factsuch that the equilibrium strategy is Pareto-dominated by a renegotiation-proof equilibrium strategy after some (positiveprobability) message pair, while no renegotiation-proof equilibrium strategy is Pareto-dominated by any equilibriumstrategy after any message pair. See the related discussion of internal and external consistency requirements for adefinition of renegotiation-proof strategies in repeated games in, e.g., Ray (1994).
In this section we investigate the efficiency properties of renegotiation-proof equilibria. We firstargue that the first-best outcome cannot be achieved by any equilibrium of any coordination gamewith communication. We then provide an example of an equilibrium with high ex-ante payoffs thatis not, however, renegotiation-proof. We then show that all renegotiation-proof equilibria, whilenot necessarily ex-ante payoff optimal, are at least interim (pre-communication) Pareto efficient.Finally, we show that at least one of the two “extreme” renegotiation-proof equilibria, σ L and σ R ,provides the highest ex-ante payoff of all the coordinated equilibria, and that any equilibrium withoutcommunication is Pareto-dominated by either one of these extreme renegotiation-proof equilibria orby the action-symmetric renegotiation-proof equilibrium σ C . First best
The first-best ex-ante payoff can be induced only by a strategy that is coordinated andsuch that the coordinated outcome depends heavily on the cardinal preferences of the two agents.Specifically, the first-best strategy is one that induces coordination on L whenever u + v ≤ R otherwise. The first-best ex-ante payoff can thus be induced only by a strategy inwhich each agent reveals her type, and the two agents then choose the favorite outcome, L or R , ofthe more extreme type (i.e., the type that is farther away from / ). Note that this strategy is not anequilibrium: each player has an incentive to present a more extreme type than her real type (e.g., alltypes u > / would claim to have type 1). High payoff of non-coordinated equilibria
Equilibria with miscoordination (which cannot berenegotiation-proof due to Theorem 1) may induce agents to credibly reveal some cardinal infor-mation about their type. This can happen if there is a message that induces a higher probabilityof coordinating on the agent’s preferred outcome but also a higher probability of miscoordinationcompared with some other available message. Such a message can then be chosen by extreme typeswith u far from / , while moderate types with u closer to / choose the other message. Such equi-libria with miscoordination may induce a higher ex-ante payoff, if the benefit from signaling theextremeness of the type outweighs the loss due to miscoordination. Consider the following example. Example 1.
For simplicity we let the distribution of types F be discrete with four atoms / + ε , / − ε , / + ε , / − ε , with a probability of / for each atom and ε > The One can easily adapt the example to an atomless distribution of types, in which each atom is replaced with acontinuum of nearby types. L , always coordinating on R , bothwith an ex-ante payoff of / , and playing L if and only if the type is less than / with an ex-antepayoff of / < / for all ε sufficiently small. Theorem 1 (together with the symmetry of thedistribution F ) implies that with communication, any renegotiation-proof equilibrium strategy (inparticular σ L or σ R ) induces the same expected ex-ante payoff of / > / for all ε sufficiently small.This game also has a (non-renegotiation-proof) equilibrium strategy with miscoordination thatyields a higher ex-ante payoff than the renegotiation-proof payoff of / , provided that the messageset M has sufficiently many elements. To simplify the presentation we here allow the players touse public correlation devices to determine their joint play after sending messages, which can beapproximately implemented by a sufficiently large message set (à la Aumann and Maschler, 1968;see the similar construction in Case (II) of the proof of Proposition 8 in Appendix E.7.3). Let m L , m l , m r , m R ∈ M and consider strategy σ = ( µ , ξ ) as follows. Let µ ( / + ε ) = m L , µ ( / − ε ) = m l , µ ( / + ε ) = m r , and µ ( / − ε ) = m R , and let ξ ( m a , m b ) = L if a , b ∈ { L , l } , ξ ( m a , m b ) = R if a , b ∈ { r , R } , ξ ( m L , m r ) = L , ξ ( m l , m R ) = R , ξ ( m l , m r ) be a joint lottery to coordinate on L or R with probability / each, and, finally, let ξ ( m L , m R ) be a joint lottery to coordinate on L or R with probability / each, and to play the inefficient mixed equilibrium (in which each type playsher preferred outcome with probability / − ε ) with probability / . It is straightforward to verifythat for, say ε = / , this strategy is indeed an equilibrium strategy with an ex-ante payoff ofaround 0 . / of all the renegotiation-proof equilibria.This equilibrium strategy is not coordinated (nor does it satisfy the other two properties of mutual-preference consistency and binary communication) and hence, by Theorem 1, it is not renegotiation-proof. Interim (pre-communication) Pareto optimality
An (ex-ante symmetric) social choice function is a function φ : [ , ] → ∆ (cid:16) { L , R } (cid:17) assigning to each pair of types a possibly correlated actionprofile with the condition that φ u , v ( a , b ) = φ v , u ( b , a ) for any a , b ∈ { L , R } , where φ u , v ≡ φ ( u , v ) .We interpret φ u , v as the correlated action profile played by the two players when a player of type u interacts with a player of type v . Let Φ be the set of all such functions.Any strategy of any coordination game with communication (with any finite message set) inducesa social choice function in Φ , but not all social choice functions in Φ can be generated by a strategyof a given coordination game with communication. One can interpret Φ as the set of outcomes thatcan be implemented by a social planner who perfectly observes the types of both players and, canforce the players to play arbitrarily. We restrict attention to symmetric social choice functions here for two reasons. First, it makes the paper conceptuallyconsistent, given that the subject of the paper, coordination games with communication, is a class of (ex-ante) symmetricgames. Second, it makes it unnecessary for us to here introduce player subscripts that we do not need anywhere else inthe paper. Proposition 1 below, however, also holds even if we allow asymmetric social choice functions. u ∈ [ , ] , let π u ( φ ) denote the expected payoff of a player of type u under socialchoice function φ , i.e., π u ( φ ) = IE v [( − u ) φ u , v ( L , L ) + u φ u , v ( R , R )] . A strategy is interim (pre-communication) Pareto-dominated if there is a social choice function thatis weakly better for all types, and strictly better for some types.
Definition 3.
A strategy σ ∈ Σ is interim (pre-communication) Pareto-dominated by a social choicefunction φ ∈ Φ if π u ( σ , σ ) ≤ π u ( φ ) for each type u ∈ [ , ] , with the inequality being strict for apositive measure set of types. A strategy σ ∈ Σ is interim (pre-communication) Pareto optimal if itis not interim (pre-communication) Pareto-dominated by any φ ∈ Φ . Proposition 1.
Every renegotiation-proof strategy of a coordination game with communication isinterim (pre-communication) Pareto optimal.Sketch of proof; see Appendix A.3 for the formal proof.
Recall that by Theorem 1 and Remark 3,any renegotiation-proof equilibrium strategy σ is characterized by its left tendency α σ . In orderfor a social choice function φ to improve the payoff of any type u < / (resp., u > / ) relative tothe payoff induced by σ , it must be that φ induces any type u < / (resp., u > / ) to coordinate on L with probability larger (resp., smaller) than α σ . This, in turn, implies that the probability of twoplayers coordinating on L , conditional on the players having different preferred outcomes, must belarger (resp., smaller) than α σ . However, these two requirements contradict each other.Earlier in this section we have given an example of an equilibrium strategy that provides a higherex-ante payoff than any renegotiation-proof equilibrium. This strategy involved a certain degree ofmiscoordination. In the following proposition we show that any equilibrium without miscoordina-tion, i.e., any coordinated equilibrium, must provide an ex-ante expected payoff that is less than orequal to the maximal ex-ante payoff of the two “extreme” renegotiation-proof strategies σ L and σ R . Proposition 2.
Let σ ∈ E be a coordinated equilibrium strategy. Then π ( σ , σ ) ≤ max { π ( σ L , σ L ) , π ( σ R , σ R ) } . Sketch of proof; see Appendix A.3 for the formal proof.
Let α σ be the probability of two players whoeach follow σ to coordinate on L , conditional on the players having different preferred outcomes. Itis easy to see that σ is dominated by the renegotiation-proof equilibrium strategy with the same lefttendency α σ , and that the payoff of the latter strategy is a convex combination of the payoffs of σ L and σ R , which implies that π ( σ , σ ) ≤ max { π ( σ L , σ L ) , π ( σ R , σ R ) } .16 emark . [All-stage renegotiation-proofness à la Benoit and Krishna, 1993] One could refine thenotion of renegotiation-proofness to allow agents to renegotiate to a Pareto-improving equilibriumalso in earlier stages: in the interim stage before observing the realized messages induced by theoriginal equilibrium, and in the ex-ante stage before each agent observes her own type. This morerestrictive definition of renegotiation-proofness à la Benoit and Krishna (1993) enables us to view astrategy satisfying our definition of renegotiation-proofness as ex-post renegotiation-proof strategy ;or as interim renegotiation-proof strategy if, in addition, it is not Pareto-dominated (in the originalgame after each agent observes her own type, yet before observing the realized message profile)by any ex-post renegotiation-proof strategy; or as all-stage renegotiation-proof strategy if there isno other interim renegotiation-proof strategy that induces a higher ex-ante expected payoff to bothplayers (before each player knows her own type). Clearly, any ex-post renegotiation-proof strat-egy that satisfies interim Pareto optimality is an interim renegotiation-proof strategy. Proposition 2implies that either σ L or σ R provides the maximal ex-ante payoff of all the interim renegotiation-proof strategies, and hence either σ L or σ R is an all-stage renegotiation-proof strategy. Moreover, if π ( σ R , σ R ) = π ( σ L , σ L ) , then one can show that either σ L or σ R is the unique coordinated strategythat maximizes the ex-ante payoff, and, thus, the unique all-stage renegotiation-proof strategy.Next, we show that σ L or σ R provides a strictly higher ex-ante expected payoff than any equi-librium of the game without communication (and therefore than any babbling equilibrium of thegame with communication). Recall from Remark 4 and the text preceding it that in the coordinationgame without communication any equilibrium is characterized by a cutoff value x ∈ [ , ] such that x = F ( x ) with the interpretation that types u ≤ x play L and types u > x play R .Let π u ( x , x ′ ) denote the payoff of an agent with type u who follows a strategy with cutoff x andfaces a partner of unknown type who follows a strategy with cutoff x ′ , which is given by π u (cid:0) x , x ′ (cid:1) = { u ≤ x } F (cid:0) x ′ (cid:1) ( − u ) + { u > x } (cid:0) − F (cid:0) x ′ (cid:1)(cid:1) u , and let π ( x , x ′ ) = IE u [ π u ( x , x ′ )] be the ex-ante expected payoff of an agent who follows strategy x andfaces a partner who follows x ′ . Our final result shows that any (possibly asymmetric) equilibrium inthe game without communication is Pareto-dominated by either σ L , σ R , or σ C . Corollary 1.
Let ( x , x ′ ) be a (possibly asymmetric) equilibrium in the coordination game withoutcommunication. Then π u ( x , x ′ ) ≤ π u ( σ L , σ L ) for all types u ∈ U , or π u ( x , x ′ ) ≤ π u ( σ R , σ R ) for alltypes u ∈ U , or π u ( x , x ′ ) ≤ π u ( σ C , σ C ) for all types u ∈ U . Moreover, all the inequalities are strictfor almost all types.
Corollary 1 is immediately implied by Lemma 2 in Appendix A.2, and the sketch of proof of thelemma is presented as part of the sketch of the proof of Theorem 1.17
Evolutionary Stability
A common interpretation of a Nash equilibrium is a convention that is reached as a result of a pro-cess of social learning when similar games are repeatedly played within a large population. Thisinterpretation seems very apt, for instance, if we think of our motivating example of how pedestriansavoid bumping into each other. Specifically, consider a population in which a pair of agents from alarge population are occasionally randomly matched and play the coordination game with commu-nication h Γ , M i . The agents can observe past behavior of other agents who played similar games inthe past. It seems plausible that the aggregate behavior of the population would gradually convergeinto a self-enforcing convention, which is a symmetric Nash equilibrium of h Γ , M i (see the “massaction” by Nash (1950); and see Weibull (1995) and Sandholm (2010) for a textbook introduction).We have argued that renegotiation-proofness is a necessary condition of evolutionary stabilityby capturing the idea of stability with respect to secret-handshake mutations as in Robson (1990).In this section we report results from Appendix D in which we investigate the evolutionary stabilityproperties of both σ L and σ R (the results can be extended to all renegotiation-proof strategies).In Appendix D.1 we show that strategies σ L and σ R are neutrally stable strategies (NSS) in thesense of Maynard Smith and Price (1973), and evolutionarily stable strategies (ESS) if | M | =
2. Thisimplies that σ L and σ R are robust to the presence of a small proportion of experimenting agents whobehave differently than the rest of the population.We are not quite satisfied with this result for three reasons. First, neutral stability is not thestrongest form of evolutionary stability, although in games with cheap talk it is typically the strongestform of stability one can expect owing to the freedom that unused messages provide mutants; see,e.g., Banerjee and Weibull (2000). Second, our game, owing to the incomplete information mod-eled here as a continuum variable, has a continuum of strategies, especially in the action phase aftermessages are observed. But with a continuum of strategies the notion of even an ESS is not suffi-cient to imply local convergence to the equilibrium from nearby states. The reason for this, see e.g.,Oechssler and Riedel (2002), is that ESS for continuum models considers only the possibly largestrategy deviation of a small proportion of individuals and not the small strategy deviation of possi-bly a large proportion of individuals. Lastly, our cheap-talk game is a two-stage game and, hence,an extensive-form game. It is well known that extensive-form games do not admit ESSs of the entiregame (unless they are strict equilibria; see Selten, 1980), and hence it seems reasonable to explorethe stability of equilibrium behavior in each stage separately.To address these issues we investigate two additional evolutionary stability properties. We inves- We here do not consider set-valued concepts of evolutionarily stability such as evolutionary stable sets (Swinkels,1992 and the related analysis in, e.g., Balkenborg and Schlag, 2001, 2007), nor the perturbation-based concept of limit-ESS (Selten, 1983; Heller, 2014), which lies in between ESS and NSS. Evolutionwill not necessarily place huge restrictions on play in unreached induced games (see, e.g., Nachbar,1990; Gale, Binmore, and Samuelson, 1995), but should do so for induced games reached with pos-itive probability. To address this, we study the evolutionary stability properties of strategies σ L and σ R at the message level (taking as given the action functions in the second stage), and at the actionlevel (after messages that are observed with positive probability). In Appendix D.2 we show that,when the action is fixed to be either ξ L or ξ R , the message function µ ∗ used by σ L and σ R is weaklydominant. This is a substantially stronger property than the message function being an NSS. In Appendix D.3 we investigate the evolutionary stability properties of the action choice inducedby σ L and σ R after the players observe the message pair. As choosing an action as a function of aplayer’s type is equivalent to choosing a cutoff from a continuum (the unit interval), we employ astability concept designed for such cases. The issue is further complicated by the fact that, owingto the asymmetry after unequal messages, we need to employ a multidimensional stability con-cept. The literature provides one in the form of a neighborhood invader strategy developed for thedouble-population case by Cressman (2010), building on earlier work by Eshel and Motro (1981)and Apaloo (1997), among others. We show that the action choice induced by σ L and σ R indeedconstitutes a neighborhood invader strategy after each pair of possible messages. In this section we present informally various extensions and robustness checks of our main results.We postpone the detailed formal analysis to Appendix E.
Long communication
In Appendix E.1 we show that all of our results hold in a setup in whichthe pre-play communication phase includes multiple rounds. Players observe messages after eachround and can condition their message choice and their final action choice on the history of observedmessage pairs. Renegotiation takes place once, just before the final action choices are made.
Multidimensional sets of types
In Appendix E.2 we study general symmetric two-action two-player coordination games, where miscoordination may result in different payoffs to the L and R Our renegotiation-proofness concept does not impose any restrictions on unreached induced games other than thatthe strategy in the whole game must be an equilibrium strategy: we do not require play in unreached induced games tobe a Nash equilibrium nor do we require renegotiation-proofness in unreached induced games. Therefore, we here donot demand evolutionary stability in unreached induced games. For dynamic evolutionary processes weakly dominated strategies are not always eliminated. See, e.g., Weibull(1995), Hart (2002), Kuzmics (2004), Kuzmics (2011), Laraki and Mertikopoulos (2013), Bernergård and Mohlin(2019) for a discussion of this issue. Note also that Kohlberg and Mertens (1986) made it a desideratum that a con-cept of strategic stability should not include weakly dominated strategies.
More than two players
Suppose that there are n ≥ u is equal to u if all players play R , it is equal to 1 − u if all players play L , and it is equal to zero if notall the players play the same action. In Appendix E.3 we show that our main result (Theorem 1) aswell as appropriate versions of the efficiency results hold in this setup. Asymmetric coordination games
Appendix E.4 shows that all our results hold in asymmetriccoordination games in which the distributions of the types of the two players’ positions may differ.
Coordination games with more than 2 actions
In Appendix E.5 we analyze coordination gameswith more than two actions. In this setup we are able to prove somewhat weaker variants of ourmain result. First, we show that σ C remains a renegotiation-proof equilibrium strategy in this moregeneral setup (by contrast, strategies σ L and σ R might not be equilibria in this setup). While we donot have a full characterization of the set of renegotiation-proof equilibrium strategies, we are ableto show that renegotiation-proof equilibrium strategies must satisfy mutual-preference consistencyand coordination whenever both players have sent the same message. Extreme types with dominant actions
Appendix E.6 extends our analysis to a setup in whichsome types find that one of the actions is a dominant action for them. We show that in the presenceof these extreme types there exists an essentially unique renegotiation-proof equilibrium strategythat satisfies the three key properties, where the left tendency of this strategy is equal to the share ofextreme types whose dominant action is L . In this setup moderate “leftists,” i.e., types u ∈ ( , / ) ,gain if there are more extreme “leftists” than extreme “rightists,” in the sense that the above essen-tially unique renegotiation-proof strategy induces a higher probability of coordination on action L when two agents with different preferred outcomes meet.20 Discussion
Renegotiation-proofness concepts have been developed in the context of infinitely and finitely re-peated games with complete information in, e.g., Farrell and Maskin (1989), Van Damme (1989),Bernheim and Ray (1989), Evans and Maskin (1989), and in, e.g., Benoit and Krishna (1993) andWen (1996), respectively. There is a sizable literature on the renegotiation-proofness of contractsin the presence of asymmetric information going back to Hart and Tirole (1988) and Dewatripont(1989) and including recent contributions by Maestri (2017) and Strulovici (2017).A close concept to our notion of renegotiation proofness is that of posterior efficiency of Forges(1994) (building on Holmström and Myerson, 1983). Forges (1994) argues that the final outcomeof a mechanism (or here strategy profile) will not necessarily fully reveal all initially privately heldinformation. Posterior efficiency then only demands that the outcome be efficient given the informa-tion that the people can infer from the outcome of the mechanism alone. Similarly we here demandthat the strategy profile prescribe an action profile after messages are sent that is efficient given theinformation revealed by the messages sent. Thus, our players have more information than just theprescribed action profile as they in fact observe also the messages that typically provide additionalinformation; see also Kawakami (2016, page 897). Our definition of renegotiation-proofness nec-essarily also differs from Forges’s posterior efficiency in that in our strategic setting we impose theadditional (sequential rationality) requirement that agents play an equilibrium action profile giventhe information they have.Starting with the secret handshake argument provided in Robson (1990) (see also the earlierrelated notion of “green beard effect” in Hamilton, 1964; Dawkins, 1976), there is a sizable literatureon the evolutionary analysis of costless pre-play communication before players engage in a completeinformation coordination game. This includes, e.g., Sobel (1993), Blume, Kim, and Sobel (1993),Wärneryd (1993), Kim and Sobel (1995), Bhaskar (1998), and Hurkens and Schlag (2003). Suppose that a complete information coordination game has two Pareto-rankable equilibria.Then the Pareto-inferior equilibrium is not evolutionarily stable as it can be invaded by mutantswho use a previously unused message as a secret handshake: if their opponent does not use thesame handshake they simply play the Pareto-inferior equilibrium (as do all incumbents), but if theiropponent also uses the secret handshake both sides play the Pareto-superior equilibrium.Suppose that the game has an equilibrium that is not Pareto-dominated by another equilibriumbut is Pareto-dominated by some non-equilibrium strategy profile. Then the same argument wouldsuggest that not only the Pareto-dominated equilibrium is unstable, but also the mutant strategy pro-file – by virtue of not being an equilibrium – is itself unstable. To sidestep this issue, one can appeal In the same spirit, but using different equilibrium refinements, is the literature that models repeated pre-play com-munication as structured negotiation that influences which equilibrium to play in a complete-information game (see, e.g.,Farrell and Maskin, 1989; Rabin, 1994; and Safronov and Strulovici, 2019).
21o the notion of “robustness to equilibrium entrants” introduced by Swinkels (1992), that considersonly those mutants that play an equilibrium strategy. In the prisoner’s dilemma, for example, mutualdefection is the unique strategy profile that is robust with respect to equilibrium entrants. Our notionof renegotiation-proofness has a similar flavor as we require that a renegotiation-proof equilibriumnot to be dominated by another equilibrium. Given the incomplete information in our model webelieve that the most appropriate place to apply this idea is at the stage after messages are sent, i.e.,at the posterior stage in the language of Forges (1994), as explained above. Another related literature deals with stable equilibria in coordination games with private values,but without pre-play communication. Sandholm (2007) (extending earlier results of Fudenberg and Kreps,1993; Ellison and Fudenberg, 2000) shows that mixed Nash equilibria of the game with complete in-formation can be purified in the sense of Harsanyi (1973) in an evolutionarily stable way (see also Re-mark 4). Finally, two related papers analyze stag-hunt games with private values. Baliga and Sjöström(2004) show that introducing pre-play communication induces a new equilibrium in which thePareto-dominant action profile is played with high probability. Jelnov, Tauman, and Zhao (2018)show that in some cases a small probability of another interaction can substantially affect the set ofequilibrium outcomes in stag-hunt games with private values.
A Proofs of Main Results
A.1 Undominated Action Strategies
In this subsection we show that our restriction to threshold action functions is without loss of gener-ality, in the sense that each generalized action function is dominated by a threshold strategy.Let Γ ( F , G ) be a coordination game without communication (possibly played after a pair of mes-sages is observed in the original game h Γ , M i ). A generalized strategy in this game is a measurablefunction η : U → ∆ ( { L , R } ) that describes a mixed action as a function of the player’s type. A gen-eralized strategy in Γ ( F , G ) corresponds to a generalized action function ξ : U × M × M → △{ L , R } (see Remark 2), given a specific pair of observed messages ( m , m ′ ) , i.e., η ( u ) ≡ ξ ( u , m , m ′ ) .A pair of generalized strategies η , ˜ η are almost surely realization equivalent (abbr., equivalent ),which we denote by η ≈ ˜ η , if they induce the same behavior with probability one, i.e., ifIE u ∼ F [ η ( u ) = ˜ η ( u )] ≡ Z u ∈ U f ( u ) { η ( u ) = ˜ η ( u ) } du = . In a recent paper, Newton (2017) provides an evolutionary foundation for players developing the ability to renego-tiate into a Pareto-better outcome (“collaboration” in the terminology of Newton). See also Neary and Newton (2017) who study coordination games without communication played on a graph, andprovide sufficient conditions for heterogeneous equilibria with miscoordination to be stable.
22t is immediate that two equivalent generalized strategies always induce the same (ex-ante) payoff,i.e., that π ( η , η ′ ) = π ( ˜ η , η ′ ) for each generalized strategy η ′ .A generalized strategy is a cutoff strategy if there exists a type x ∈ [ , ] such that η ( u ) = L for each u < x and η ( u ) = R for each u > x . A generalized strategy η is strictly dominated bygeneralized strategy ˜ η if π ( η , η ′ ) < π ( ˜ η , η ′ ) for any generalized strategy η ′ of the opponent.The following result shows that any generalized strategy is either equivalent to a cutoff strategy,or it is strictly dominated by a cutoff strategy. Lemma 1.
Let η be a generalized strategy. Then there exists a cutoff strategy ˜ η , such that either η is equivalent to ˜ η , or η is strictly dominated by ˜ η .Proof. If IE u ∼ F [ η u ( L )] = u ∼ F [ η u ( L )] = η is equivalent to the cutoff strategy ofalways playing L (resp., R ). Thus, suppose that IE u ∼ F [ η u ( L )] ∈ ( , ) . Let x ∈ ( , ) be such that F ( x ) = IE u ∼ F [ η u ( L )] = R u η u ( L ) f ( u ) du . Let ˜ η then be the cutoff strategy with cutoff x , i.e.,˜ η u ( L ) = u ≤ x u > x . Assume that η and ˜ η are not equivalent, i.e., η ˜ η . Let η ′ be an arbitrary generalized strategy ofthe opponent. By construction, strategies η and ˜ η induce the same average probability of choosing L .Strategies ˜ η and η differ in that ˜ η induces lower types to choose L with higher probability, and highertypes to choose L with lower probability, i.e., η u ( L ) ≤ ˜ η u ( L ) for any type u ≤ x and η u ( L ) ≥ ˜ η u ( L ) for any type u > x . Since η ˜ η and IE u ∼ F [ η u ( L )] ∈ ( , ) , it follows that the inequalities are strictfor a positive measure of types, i.e.,0 < Z u < x f ( u ) { η ( u ) < ˜ η ( u ) } du and 0 < Z u > x f ( u ) { η ( u ) > ˜ η ( u ) } du . The fact that lower types always gain more (less) from choosing L ( R ) relative to higher types,with a strict inequality unless the opponent always plays R ( L ), implies that π ( η , η ′ ) < π ( ˜ η , η ′ ) . A.2 Proof of Theorem 1
We first prove the “if” part of the theorem. Suppose that σ = ( µ , ξ ) ∈ Σ is mutual-preferenceconsistent, coordinated, and has binary communication.As σ is mutual-preference consistent it must satisfy supp ( F m ) ⊆ [ , / ] or supp ( F m ) ⊆ [ / , ] for any message m ∈ supp ( ¯ µ ) . Consider any message pair m , m ′ ∈ supp ( ¯ µ ) . There are three casesto consider. Suppose first that both supp ( F m ) , supp ( F m ′ ) ⊆ [ , / ] . Then as σ is mutual-preferenceconsistent we have that ξ ( m , m ′ ) = ξ ( m ′ , m ) = L . Thus ξ describes best-reply behavior after thismessage pair. Moreover this behavior is the best possible outcome for any type in [ , / ] and thus for23ny type in supp ( F m ) and supp ( F m ′ ) . The second case of supp ( F m ) , supp ( F m ′ ) ⊆ [ / , ] is analogous.Suppose, finally, that, w.l.o.g., supp ( F m ) ⊆ [ , / ] and supp ( F m ′ ) ⊆ [ / , ] . As σ is coordinated wehave that ξ ( m , m ′ ) = ξ ( m ′ , m ) = L or ξ ( m , m ′ ) = ξ ( m ′ , m ) = R . Action function ξ , therefore, againdescribes best-reply behavior. Moreover, one player always obtains her most preferred outcome. Inorder for a new strategy profile to improve the opponent’s outcome, this new profile must require theformer player to deviate from her most preferred outcome. Thus, there is no message set ˜ M such thatan equilibrium σ ′ in the game h Γ ( F m , F m ′ ) , ˜ M i Pareto dominates σ after this message pair.All this shows that action function ξ is a best response to µ and to itself given µ and that,moreover, it cannot be post-communication equilibrium Pareto-dominated. It remains to show thatthe message function µ is optimal when the opponent chooses σ = ( µ , ξ ) .Consider type u ∈ [ , / ] and consider this type’s choice of message. As σ has binary com-munication and is coordinated, different messages m ∈ M can only trigger different probabilities ofcoordinating on L with a highest likelihood of such coordination for any message m ∈ supp ( µ u ) .Therefore, type u is indifferent between any message m ∈ supp ( µ u ) and weakly prefers sending anymessage m ∈ supp ( µ u ) to sending any message m ′ supp ( µ u ) . An analogous statement holds fortypes u ∈ [ / , ] . This concludes the proof of the “if” part of the theorem.We prove the “only if” part in three lemmas, one for each of the three properties. Lemma 2.
Every renegotiation-proof equilibrium strategy σ = ( µ , ξ ) is coordinated.Proof. We need to show that for any message pair m , m ′ ∈ supp ( ¯ µ ) ,either ξ ( m , m ′ ) ≥ sup { u | µ u ( m ) > } or ξ ( m , m ′ ) ≤ inf { u | µ u ( m ) > } . Let m , m ′ ∈ supp ( ¯ µ ) and assume to the contrary thatinf { u | µ u ( m ) > } < ξ ( m , m ′ ) < sup { u | µ u ( m ) > } . As σ is an equilibrium, we have inf { u | µ u ( m ′ ) > } < ξ ( m ′ , m ) < sup { u | µ u ( m ′ ) > } becauseotherwise the sender of m ′ would play L with probability one or R with probability one, in whichcase the the best reply of the sender of message m would be to play L (or R ) regardless of her type.Let x = ξ ( m , m ′ ) and x ′ = ξ ( m ′ , m ) . We now show that the equilibrium ( x , x ′ ) of the game withoutcoordination Γ ( F m , F m ′ ) is post-communication Pareto-dominated by either σ L , σ R , or σ C .There are three cases to be considered. Case 1: Suppose that x , x ′ ≤ / . We now show that in thiscase the equilibrium ( x , x ′ ) is Pareto-dominated by σ R . Consider the player who sent message m .Case 1a: Consider a type u ≤ x . Then we have ( − u ) F m ′ ( ) + u (cid:0) − F m ′ ( ) (cid:1) ≥ ( − u ) F m ′ ( x ′ ) , u agent’s payoff under strategy profile σ R and the right-hand side thepayoff under strategy profile ( x , x ′ ) . The inequality follows from the fact that u ( − F m ′ ( / )) ≥
0, and F m ′ ( / ) ≥ F m ′ ( x ′ ) follows from the fact that F m ′ is nondecreasing (as it is a cumulative distributionfunction). This inequality is strict for all u except for u = x ′ = / .Case 1b: Now consider a type u with x < u ≤ / . Then we have ( − u ) F m ′ ( ) + u (cid:0) − F m ′ ( ) (cid:1) > u ( − F m ′ ( x ′ )) , where the left-hand side is a type u agent’s payoff under strategy profile σ R and the right-hand sideis the payoff under strategy profile ( x , x ′ ) . The inequality follows from the fact that by u ≤ / wehave that 1 − u ≥ u , and therefore ( − u ) F m ′ ( / ) + u ( − F m ′ ( / )) ≥ u .Case 1c: Finally, consider a type u > / . Then we have u > u ( − F m ′ ( x ′ )) , where the left-handside is a type u agent’s payoff under strategy profile σ R and the right-hand side is the payoff understrategy profile ( x , x ′ ) .The analysis for the player who sent message m ′ is analogous.Case 2: Suppose that x , x ′ ≥ / . The analysis is analogous to Case 1 if we replace σ R with σ L .Case 3: Suppose, w.l.o.g. for the remaining cases, that x ≤ / ≤ x ′ . The equilibrium ( x , x ′ ) inthis case is Pareto-dominated by σ C . To see this, consider the player who sent message m .Case 3a: Consider a type u ≤ x . Then we have ( − u ) (cid:2) F m ′ ( ) + (cid:0) − F m ′ ( ) (cid:1)(cid:3) + u (cid:0) − F m ′ ( ) (cid:1) > ( − u ) F m ′ ( x ′ ) , where the left-hand side is a type u agent’s payoff under strategy profile σ C and the right-handside the payoff under strategy profile ( x , x ′ ) . The inequality follows from the fact that we have F m ′ ( x ′ ) = x ≤ / due to ( x , x ′ ) being an equilibrium.Case 3b: Now consider a type u with x < u ≤ / . Then we have ( − u ) (cid:2) F m ′ ( ) + (cid:0) − F m ′ ( ) (cid:1)(cid:3) + u (cid:0) − F m ′ ( ) (cid:1) > u ( − F m ′ ( x ′ )) , where the left-hand side is a type u agent’s payoff under strategy profile σ C and the right-hand sidethe payoff under strategy profile ( x , x ′ ) . The inequality follows from the fact that by u ≤ / we have1 − u ≥ u and thus ( − u ) (cid:2) F m ′ ( ) + (cid:0) − F m ′ ( ) (cid:1)(cid:3) + u (cid:0) − F m ′ ( ) (cid:1) ≥ u . Case 3c: Finally, consider a type u > / . Then we have u (cid:2)(cid:0) − F m ′ ( ) (cid:1) + F m ′ ( ) (cid:3) + ( − u ) F m ′ ( ) > u ( − F m ′ ( x ′ )) , where the left-hand side is a type u agent’s payoff under strategy profile σ C and the right-hand side isthe payoff under strategy profile ( x , x ′ ) . The inequality follows from the fact that we have F m ′ ( / ) > F m ′ ( / ) ≤ F m ′ ( x ′ ) .The analysis for the player who sent message m ′ is analogous. Lemma 3.
Every renegotiation-proof equilibrium strategy σ has binary communication.Proof. Let σ be a renegotiation-proof equilibrium strategy. Recall that β σ ( m ) = Z u = ∑ m ′ ∈ M µ u ( m ′ ) { u ≤ ξ ( m , m ′ ) } f ( u ) du . As σ is coordinated by Lemma 2, the payoff to a type u from sending message m ∈ supp ( µ ) is ( − u ) β σ ( m ) + u ( − β σ ( m )) . For a type u < / the problem of choosing a message to maximize her payoffs is thus equivalentto choosing a message that maximizes β σ ( m ) . We thus must have that there is a β σ ∈ [ , ] such thatfor all u < / and all m ∈ supp ( µ u ) , we have β σ ( m ) = β σ . Analogously, we must have a β σ ∈ [ , ] such that for all u > / and all m ∈ supp ( µ u ) , we have α σ ( m ) = β σ . Clearly also β σ ≤ β σ . To extendthe argument to unused messages m supp ( µ ) we rely on the full support assumption. Assume tothe contrary that there is a message m supp ( µ ) with β σ ( m ) > β σ (resp., β σ ( m ) < β σ ). Thenany sufficiently high (resp., low) type u would strictly earn by deviating to sending message m andplaying L (resp., R ), which contradicts the supposition that σ is an equilibrium strategy. Lemma 4.
Every renegotiation-proof equilibrium strategy σ is mutual-preference consistent.Proof. By Lemma 2 a renegotiation-proof equilibrium strategy σ = ( µ , ξ ) is coordinated. Supposethat it is not mutual-preference consistent. Then there is either a message pair ( m , m ′ ) such that thereare types u , v < / with m ∈ supp ( µ u ) and m ′ ∈ supp ( µ v ) such that play after ( m , m ′ ) is coordinated on R , or a message pair ( m , m ′ ) such that there are types u , v > / with m ∈ supp ( µ u ) and m ′ ∈ supp ( µ v ) such that play after ( m , m ′ ) is coordinated on L . In the former (resp., latter) case strategy σ ispost-communication Pareto dominated by strategy σ R (resp., σ L ) in the game h Γ ( F m , F m ′ ) , { m L , m R }i because strategy σ R (resp., σ L ) does not affect the payoff of all types u ≥ / (resp., u ≤ / ), and itstrictly improves the payoff to all types u < / (resp., u > / ). A.3 Proofs of Section 6 (On Efficiency)
Proof of Proposition 1.
By Theorem 1 and Remark 3 a renegotiation-proof strategy σ ’s equilibriumpayoff is determined by its left tendency α ≡ α σ ∈ [ , ] . This equilibrium payoff is given by π u ( σ , σ ) = ( − u ) (cid:2) F ( ) + α (cid:0) − F ( ) (cid:1)(cid:3) + u ( − α ) (cid:2) − F ( ) (cid:3) , u ∈ ( , / ] , and it is given by π u ( σ , σ ) = ( − u ) α F ( ) + u (cid:2)(cid:0) − F ( ) (cid:1) + F ( )( − α ) (cid:3) . for each type u ∈ ( / , ] . The payoff to a type u from a given social choice function φ is given by π u ( φ ) = ( − u ) IE v φ u , v ( L , L ) + u IE v φ u , v ( R , R ) . Now suppose that φ interim (pre-communication) Pareto dominates σ . Then π u ( φ ) ≥ π u ( σ , σ ) for all u ∈ [ , ] with a strict inequality for a positive measure of u . As π u ( σ , σ ) is a convex combi-nation of two payoffs, this implies that:IE v φ u , v ( L , L ) ≥ F ( ) + α (cid:0) − F ( ) (cid:1) for any u ≤ / , and (1)IE v φ u , v ( R , R ) ≥ (cid:0) − F ( ) (cid:1) + F ( )( − α ) for any u > / , (2)with at least one of the inequalities holding strictly for a positive measure of types. We can writeIE v φ u , v ( L , L ) = F ( ) IE { v ≤ / } φ u , v ( L , L ) + (cid:0) − F ( ) (cid:1) IE { v > / } φ u , v ( L , L ) , where, for instance, IE { v > / } denotes the expectation conditional on v > / . Substituting this lastequality in Eq. (1) yields the following inequality F ( ) IE { v ≤ / } φ u , v ( L , L ) + (cid:0) − F ( ) (cid:1) IE { v > / } φ u , v ( L , L ) ≥ F ( ) + α (cid:0) − F ( ) (cid:1) for any u ≤ / . The fact that IE { v ≤ / } φ u , v ( L , L ) ≤ { v > / } φ u , v ( L , L ) ≥ α for any u ≤ / . An analogous argument (applied to Eq. (2)) implies that IE { v < / } φ u , v ( R , R ) ≥ − α , for any u > / , with at least one of these inequalities holding strictly for a positive measure of types.This implies thatIE { u < / } IE { v > / } φ u , v ( L , L ) ≥ α and IE { u > / } IE { v < / } φ u , v ( R , R ) ≥ − α , with at least one of the two inequalities holding strictly. By the symmetry of φ we have φ u , v ( R , R ) = φ v , u ( R , R ) and thus IE { u < / } IE { v > / } φ u , v ( L , L ) + IE { u < / } IE { v > / } φ u , v ( R , R ) > , which contradicts φ u , v being a social choice function.The proof of Proposition 2 uses the following lemma (which is of independent interest).27 emma 5. Let σ ∈ E be a coordinated equilibrium strategy. Then there is a renegotiation-proofstrategy σ ′ such that either σ and σ ′ are interim (pre-communication) payoff equivalent or σ ′ in-terim (pre-communication) Pareto dominates σ .Proof. Let σ = ( µ , ξ ) ∈ E be coordinated. For each message m ∈ M , let p m ∈ [ , ] be the probabilitythat the players coordinate on L , conditional on the agent sending message m : p m = ∑ m ′ ∈ M µ (cid:0) ¯ m ′ (cid:1) { ξ ( m , m ′ )= L } . As σ is coordinated, it follows that 1 − p m is the probability that the players coordinate on R , condi-tional on the agent sending message m .Let ¯ p = max m ∈ M p m be the maximal probability, and let p = min m ∈ M p m be the minimal prob-ability. By definition, p ≤ ¯ p . As σ is an equilibrium strategy, p < ¯ p implies that all types u < / send a message inducing probability ¯ p and all types u > / send a message inducing probability p .Therefore, the expected payoff of a type u ≤ / is given by π u ( σ , σ ) = ¯ p ( − u ) + ( − ¯ p ) u , and theexpected payoff of any type u > / is equal to π u ( σ , σ ) = p ( − u ) + (cid:0) − p (cid:1) u . This is also true if p = ¯ p . Note that for types u < / , the expected payoff strictly increases in ¯ p and for types u > / the type’s expected payoff strictly decreases in p .We consider three cases. Suppose first that p ≤ ¯ p ≤ F ( / ) . Then let σ ′ = σ R . This strategy isalso coordinated and its induced payoffs can be written in the same form as those for strategy σ with p ′ = p ′ = F ( / ) . Thus, we get that π u ( σ ′ , σ ′ ) ≥ π u ( σ , σ ) for every u ∈ [ , ] . This impliesthat σ is either interim (pre-communication) payoff equivalent to or Pareto-dominated by σ ′ = σ R .The second case where F ( / ) ≤ p ≤ ¯ p is analogous to the first one, with σ ′ = σ L .In the final case p < F ( / ) < ¯ p . Let α ∈ [ , ] be such that F ( / ) + ( − F ( / )) α = ¯ p and let σ ′ be a renegotiation-proof strategy with left tendency α . Then p ≥ α F ( / ) and by construction σ is either interim (pre-communication) payoff equivalent to or Pareto dominated by σ ′ . Proof of Proposition 2.
By Lemma 5 we have that every coordinated equilibrium strategy σ is in-terim (pre-communication) Pareto-dominated by some renegotiation-proof strategy with some lefttendency α ∈ [ , ] denoted by σ α . We thus have that π ( σ , σ ) ≤ π ( σ α , σ α ) .The ex-ante expected payoff of to a u type under strategy σ α is given by π u ( σ α , σ α ) = ( − u ) (cid:2) F ( ) + α (cid:0) − F ( ) (cid:1)(cid:3) + u ( − α ) (cid:0) − F ( ) (cid:1) for u ≤ / and π u ( σ α , σ α ) = ( − u ) α F ( ) + u (cid:2) − F ( ) + ( − α ) F ( ) (cid:3) for u > / . It is straightforward to verify that π u ( σ α , σ α ) = απ u ( σ , σ ) + ( − α ) π u ( σ , σ ) . for every u .As σ = σ L and σ = σ R and as for all u ∈ [ , ] π u ( σ α , σ α ) is the same convex combination of28 u ( σ L , σ L ) and π u ( σ R , σ R ) , we have π ( σ α , σ α ) = απ ( σ , σ ) + ( − α ) π ( σ , σ ) , which impliesthat π ( σ , σ ) ≤ π ( σ α , σ α ) ≤ max { π ( σ L , σ L ) , π ( σ R , σ R ) } . B More on Properties of Strategies
In this appendix we demonstrate that no single one of the three properties (mutual-preference con-sistency, coordination, and binary communication) is implied by the other two. Clearly a strategythat has binary communication and is coordinated must be an equilibrium. No other combinationof two of the three properties implies that a strategy is an equilibrium. Finally, we also define whatit means for a strategy to be ordinal preference-revealing and show that this is implied by it beingmutual-preference consistent.Consider the following strategy σ = ( µ , ξ ) in the game with communication with a message set M that contains at least three elements. Let m L , m L , m R ∈ M , let µ ( u ) = m L if u ≤ m L if < u ≤ m R if u > , and let ξ be such that ξ ( m iL , m jL ) = L for all i , j ∈ { , } , ξ ( m R , m R ) = R , ξ ( m L , m R ) = ξ ( m R , m L ) = R ,and ξ ( m L , m R ) = ξ ( m R , m L ) = L . This strategy is mutual-preference consistent and coordinated butdoes not have binary communication. It is not an equilibrium as types u ≤ / would strictly preferto send message m L .Consider the following strategy σ = ( µ , ξ ) in the game with communication with a message set M that contains at least two elements. Let m L , m R ∈ M , let µ ( u ) = ( m L if u ≤ m R if u > , and let ξ be such that ξ ( m L , m L ) = L , ξ ( m R , m R ) = R , ξ ( m L , m R ) = / , and ξ ( m R , m L ) = / . Thisstrategy is mutual-preference consistent, has binary communication, but is not coordinated. Foralmost all type distributions F this is not an equilibrium: it is only an equilibrium if F satisfies ( F ( / ) − F ( / )) / ( − F ( / )) = / and F ( / ) / F ( / ) = / .Finally, for a strategy that has binary communication and is coordinated but not mutual-preferenceconsistent, consider the equilibrium strategy that always leads to coordination on action L for anypair of messages.Note also that an equilibrium does not necessarily satisfy any of the three properties. The interiorcutoff babbling equilibria mentioned in Section 3 are not coordinated and not mutual-preference29onsistent. The equilibrium of Example 1 does not have binary communication.Call a strategy σ = ( µ , ξ ) ∈ Σ ordinal preference-revealing if there exist two nonempty, disjoint,and exhaustive subsets of supp ( ¯ µ ) denoted by M L and M R (i.e., supp ( ¯ µ ) = ˙ M L S M R ) such that if u < / , then µ u ( m ) = m ∈ M R , and if u > / , then µ u ( m ) = m ∈ M L . With anordinal preference-revealing strategy a player indicates her ordinal preferences. A strategy σ that ismutual-preference consistent is also ordinal preference-revealing (but not vice versa). Suppose not.Then there is a message m and two types u < / and v > / such that µ u ( m ) , µ v ( m ) >
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Non-binary Communication Equilibrium (Online Appendix)
We here formally present the example, which is discussed informally at the end of Section 3, of anequilibrium in which agents reveal some information about the cardinality of their preferences.Suppose that | M | ≥ e , m L , m R ∈ M and let σ = ( µ , µ , ξ ) , with µ : U → ∆ ( M ) , µ : M × M × U → ∆ ( M ) , and ξ : ( M × M ) → U be asfollows. For the first round of messages there is an x ∈ [ , ] such that µ ( u ) = e if u ≤ x or u > − xm L if x < u ≤ m R if < u ≤ − x . The second round of messages depends on the outcome of the first round and is best described in thefollowing table. e m L m R e e µ ∗ µ ∗ m L m L m L µ C m R m R µ C m R Each entry in this table describes the message function that a player follows if her first-stage message isthe one indicated on the left and her opponent’s first-stage message is the one indicated at the top. Themessage function µ ∗ (after for instance a message pair of ( e , m L ) ) is just as in the definition of σ L and σ R (in Section 3). The message function µ C is as in the definition of σ C with an appropriate relabelingof four messages in M . The action function is also best given in table form as a function of the result ofthe first round of communication (or the second round when so indicated). e m L m R e ( L if u ≤ R if u > ( L if u ≤ R if u > ( L if u ≤ R if u > m L ( L if µ = ( m L , m L ) R otherwise L ξ C m R ( R if µ = ( m R , m R ) L otherwise ξ C R Action function ξ C is as defined for σ C applied to the second round of communication only.We can complete the description of this strategy by requiring that all other messages in M be treatedexactly the same as one of the messages e , m L , m R . Proposition 3.
Let F be a nondegenerate symmetric distribution around / , i.e., F ( x ) = − F ( − x ) forall x ∈ [ , ] . Then there is an x ∈ ( , / ) such that the above-defined strategy of the coordination game, ith two rounds of communication and with | M | ≥ , is a Nash equilibrium.Proof. Consider the given strategy for an arbitrary x ∈ ( , / ) . First note that whenever messages leadthe players to coordinate their action then clearly both players are best replying to each other with theiractions. This is so in all cases except when both players send message e in the first round. In this caseplayers choose L if their type u ≤ / and R otherwise. Each player, in this case, faces an opponentthat either has u ≤ x or u > − x . In the first (second) case the opponent plays L ( R ). Given that F ( x ) = − F ( − x ) both cases are equally likely. Given this, players’ actions are indeed best replies.Thus, all action choices are best responses to the given strategy. We now turn to message choices.Consider the second round. After moderate messages in the first round messages in the second roundeither do not affect play at all (after ( m L , m L ) and ( m R , m R ) ) or do so as in strategy σ C . In either caseplayers are indifferent between all messages. After message pairs ( m L , e ) and ( m R , e ) the sender of themoderate message has a strict incentive to send the same message again, while the sender of the extrememessage has a strict incentive to send m L if her type u < / or to send m R if her type u > / as thisinduces coordination on her preferred outcome. After both players send message e , play will not dependon messages in the second round either and so both players will be indifferent between all messages.Thus, the behavior in the second round of communication is a best response to the given strategy.Finally, we need to consider the incentives to send messages in the first round. It is obvious thatany type u < / prefers sending message m L to sending message m R and vice versa for types u > / .The only remaining thing to show is that types u ≤ x and u > − x and only these weakly prefer tosend message e in the first round. Given the symmetry it is without loss of generality to consider a type u ≤ / . Given the strategy, sending message e yields to this type a payoff of F ( x )( − u ) + (cid:0) F ( ) − F ( x ) (cid:1) ( − u ) + (cid:0) F ( − x ) − F ( ) (cid:1) ( − u ) + ( − F ( − x )) , where F ( x ) is the probability that her opponent is an extreme left type, ( F ( / ) − F ( x )) is the probabilitythat the opponent is a moderate left type, ( F ( − x ) − F ( / )) is the probability that the opponent is amoderate right type, in all of which cases both players eventually play L , and where ( − F ( − x )) isthe probability that her opponent is an extreme right type, in which case the two players miscoordinate.Sending message m L yields a payoff of F ( x )( − u ) + (cid:0) F ( ) − F ( x ) (cid:1) ( − u ) + (cid:0) F ( − x ) − F ( ) (cid:1) + ( − F ( − x )) u . A type u ≤ / therefore weakly prefers sending message e to sending message m L if and only if D x ( u ) ≡ (cid:0) F ( − x ) − F ( ) (cid:1) ( − u ) − (cid:0) F ( − x ) − F ( ) (cid:1) − ( − F ( − x )) u ≥ . Using the symmetry of F we can rewrite D ( x ) as D x ( u ) = ( / − F ( x )) ( / − u ) − F ( x ) u . Note that D x ( u ) is linear and downward sloping in u if x ∈ ( , / ) . In an equilibrium we then must2ave that D x ( x ) =
0. This implies D x ( x ) = (cid:0) − F ( x ) (cid:1) (cid:0) − x (cid:1) − F ( x ) x = , or, equivalently, D x ( x ) = − F ( x ) − x = . As D ( ) = / > D / ( / ) = − / <
0, and D x ( x ) is a continuous function in x ,there is an x ∈ ( , / ) such that D x ( x ) =
0. For this x the given strategy is thus an equilibrium. D Evolutionary Stability Analysis (Online Appendix)
In this appendix we analyze the stability of strategies σ L and σ R (the results can be extended to otherrenegotiation-proof equilibria, but we omit the details here for brevity). D.1 Evolutionary/Neutral Stability
We say that two strategies are almost surely realization equivalent (abbr., equivalent) if they induce thesame behavior in almost all types (regardless of the opponent’s behavior).
Definition 4.
A condition holds for almost all types if the set of types that satisfy the condition ˜ U ⊆ U has mass one (as measured by the distribution f ), i.e., R u ∈ ˜ U f ( u ) du = . Definition 5.
Strategies σ = ( µ , ξ ) and ˜ σ = (cid:16) ˜ µ , ˜ ξ (cid:17) are almost surely realization equivalent (abbr., equivalent ) if for almost all types u ∈ [ , ] : µ u ( m ) = ˜ µ u ( m ) for every message m ∈ M , and F m ( ξ ( m , m ′ )) = F m (cid:16) ˜ ξ ( m , m ′ ) (cid:17) for all messages m , m ′ ∈ supp ( ¯ µ ) .If σ and ˜ σ are equivalent strategies we denote this by σ ≈ ˜ σ . It is immediate that equivalent strategiesalways obtain the same ex-ante expected payoff.An equilibrium strategy σ is neutrally (evolutionarily) stable if it achieves a weakly (strictly) higherex-ante expected payoff against any (non-equivalent) best-reply strategy, relative to the payoff that thebest-reply strategy achieves against itself. Definition 6 (adaptation of Maynard Smith and Price, 1973) . Equilibrium strategy σ ∈ E is neutrallystable if π ( ˜ σ , σ ) = π ( σ , σ ) ⇒ π ( σ , ˜ σ ) ≥ π ( ˜ σ , ˜ σ ) for any nonequivalent strategy ˜ σ σ . It is evolu-tionarily stable if this last weak inequality is replaced by a strict one.The refinement of neutral stability is arguably a necessary requirement for an equilibrium to be astable convention in a population (see, e.g., Banerjee and Weibull, 2000). If σ is an equilibrium strategythat is not neutrally stable, then a few experimenting agents who play a best-reply strategy σ ′ can invadethe population. These experimenting agents would fare the same against the incumbents, whereas theywould outperform the incumbents when being matched with other experimenting agents. This impliesthat, on average, these experimenting agents would be more successful than the incumbents, and theirfrequency in the population would increase in any payoff-monotone learning dynamics. This, in turn,implies that the population will move away from σ .Our first result shows that both σ L and σ R are neutrally stable, and, moreover, they are evolutionarilystable if there are two feasible messages. 3 roposition 4. Strategies σ L and σ R are neutrally stable strategies of the coordination game with com-munication h Γ , M i . Moreover, if | M | = , then σ L and σ R are evolutionarily stable strategies.Proof. We here prove this result for σ L . The proof for σ R proceeds analogously and is omitted. In orderto prove this result we first characterize all strategies σ that are best responses to σ L and thus satisfy π ( σ , σ L ) = π ( σ L , σ L ) . Consider the message and action choice of a type u when her opponent usesstrategy σ L . If our type u chooses any message other than m R , her opponent, sending message m L or m R , plays action L in either case. Our type u could then choose action L (as prescribed by σ L ), whichprovides a payoff of 1 − u , or action R , which provides a payoff of zero. Thus all types u < L in this case. Also note that sending any message other than m L leads to abest possible payoff of 1 − u .If our type u chooses to send message m R then there are two cases. First, suppose that her opponentsends message m L , in which case her opponent chooses action L . Our type u could then choose action L (as prescribed by σ L ), which provides a payoff of 1 − u , or action R which provides a payoff of zero. Thusall types u < L in this case. Second, suppose that her opponentsends message m R , in which case her opponent chooses action R . Our type u could then choose action R (as prescribed by σ L ), which provides a payoff of u , or action L which provides a payoff of zero. Thusall types u > R in this case. Note that sending message m R thusprovides a best possible payoff of F ( / )( − u ) + ( − F ( / )) u .For type u it is then a strict best response to send message m R if F ( / )( − u ) + ( − F ( / )) u > − u ,which is the case if and only if u > / (as F ( / ) ∈ ( , ) by assumption). For the case of | M | = σ L is equivalent to σ L as only three possible types have an alternative bestreply: types u = u = / , and u = F ). Any strategy that differs from σ L for a positive measure of types yields a strictly inferior payoffagainst σ L than σ L does. This proves that σ L is evolutionarily stable in the case of | M | = σ that satisfy π ( σ , σ L ) = π ( σ L , σ L ) .Suppose from now on that | M | >
2. Our type u then has a choice of messages m = m R when u < / .All of these messages can at best lead to a payoff of u (from playing L , L ) and, therefore, all of them areequally good when playing against σ L . As her opponent never chooses any message other than m L or m R (each has probability zero under σ L ) our type u < / when best responding can play anything afterany message pair ( m , m ′ ) when both m , m ′
6∈ { m L , m R } . Let σ be a strategy that satisfies all the previousrestrictions, where all types u play a (in most cases unique and strict) best response against σ L . Thenwe have that π u ( σ , σ ) = π u ( σ L , σ ) for all u ≥ / (as the behavior under σ for types u ≥ / (except forpossibly types u = / and u =
1) is identical to that under σ L ), π u ( σ , σ ) ≤ − u for all u < / (sincethis type can achieve at best 1 − u ), and π u ( σ L , σ ) = − u (since σ similarly to σ L prescribes playing L in this case). We thus have for any such σ by construction π ( σ , σ L ) = π ( σ L , σ L ) . We also have that π ( σ L , σ ) ≥ π ( σ , σ L ) for any such σ . Finally any best-reply strategy to strategy σ L must be equivalent tosome such strategy σ and thus σ L is neutrally stable.4 .2 Message Function is Dominant Next we show that the first-stage behavior induced by strategy σ L (resp., σ R ), namely, the messagefunction µ ∗ , is weakly dominant (and strictly dominant when | M | = ξ L (resp., ξ R ). This suggests that thebehavior in the first stage that is induced by σ L (resp., by σ R ) is robust to any perturbation that keeps thebehavior in the second stage unchanged. Specifically, it implies that even if the message function usedby the population is perturbed in an arbitrary (and possibly significant) way, then the original function µ ∗ yields a weakly higher payoff than any other message function, which suggests that the behavior inthe first stage would converge back to play µ ∗ under any payoff-monotone learning dynamics.Proposition 5 shows that message function µ ∗ yields a weakly higher payoff relative to any othermessage function when the action function is given by ξ L or ξ R . Moreover, the inequality is strictwhenever the alternative message function is essentially different from µ ∗ in the sense of inducing lowtypes to play m R or inducing high types to play m = m R . Proposition 5.
Let µ ′ , µ ′′ be arbitrary message functions. Then for, ξ ∈ { ξ L , ξ R } and for any type u = / , π u (cid:0) ( µ ∗ , ξ ) , (cid:0) µ ′ , ξ (cid:1)(cid:1) ≥ π u (cid:0)(cid:0) µ ′′ , ξ (cid:1) , (cid:0) µ ′ , ξ (cid:1)(cid:1) . This inequality is strict for ξ = ξ L if µ ′ u ( m R ) > for a positive measure of types u and, either µ ′′ u ( m R ) > for a positive measure of types u < / , or µ ′′ u ( m R ) < for a positive measure of types u > / . Thisinequality is strict for ξ = ξ R if µ ′ u ( m L ) > for a positive measure of types u and, either µ ′′ u ( m L ) > fora positive measure of types u > / , or µ ′′ u ( m L ) < for a positive measure of types u < / .Proof. Consider the case of ξ = ξ L (the other case is proven analogously). Let γ denote the probabilitythat a player following strategy ( µ ′ , ξ L ) sends message m R . Then sending any message other than m R when the partner sends ( µ ′ , ξ L ) yields a payoff of 1 − u , and sending message m R yields a payoff of γ u + ( − γ )( − u ) . Thus any type u > / weakly prefers sending message m R to sending any othermessage (and strictly prefers this if γ > u < / weakly prefers sending any messageother than m R to sending message m R (and strictly prefers this if γ < µ ′ of the opponent, µ ∗ optimizes the message choice for every type u universally. D.3 Action Function is a Neighborhood Invader Strategy
In the induced second-stage game Γ ( F m , F m ′ ) (the game played after players observe a pair of messages ( m , m ′ ) ), players choose a cutoff to determine whether to play action L (if their type is below or equal tothat cutoff) or action R (otherwise). Thus, players essentially choose a number (their cutoff) from the unitinterval. Note also that this induced game is asymmetric whenever the message profile is asymmetric,i.e., when m = m ′ . As argued by Eshel and Motro (1981) and Eshel (1983), when the set of strategies5s a continuum, a stable convention should be robust to perturbations that slightly change the strategyplayed by all agents in the population. Cressman (2010) formalizes this requirement using the notion ofneighborhood invader strategy (adapting the related notion of Apaloo, 1997). In what follows we showthat the action function induced by σ L and σ R is a neighborhood invader strategy in any induced game Γ ( F m , F m ′ ) on the path of play.Fix a message function µ and a pair of messages m , m ∈ supp ( ¯ µ ) . We identify a strategy in theinduced game Γ ( F m , F m ) with thresholds x i , which is interpreted as the maximal type for which player i ∈ { , } plays L . We say that strategy x i of player i is equivalent to x ′ i (denoted by x i ≈ x ′ i ) in the inducedgame Γ ( F m , F m ) , if F m i ( x i ) = F m i ( x ′ i ) , which implies that both thresholds induce the same behavior withprobability one. Let π m , m ( x , x ) denote the expected payoff of an agent with a random type sampledfrom f m who uses threshold x when facing a partner with a random unknown type sampled from f m who uses threshold x .A strategy profile ( x , x ) is a strict equilibrium if any best reply to x j is equivalent to x i , i.e., π m , m ( x ′ , x ) ≥ π m , m ( x , x ) ⇒ x ′ ≈ x , and π m , m ( x ′ , x ) ≥ π m , m ( x , x ) ⇒ x ′ ≈ x .We say that the strict equilibrium ( x , x ) is a neighborhood invader strategy in the induced game Γ ( F m , F m ) if the population converges to ( x , x ) from any nonequivalent nearby strategy profile ( x ′ , x ′ ) in two steps: (1) strategy x i yields a strictly higher payoff against x j relative to the payoff of x ′ i against x j (which implies convergence from (cid:16) x ′ i , x ′ j (cid:17) to (cid:16) x i , x ′ j (cid:17) ), and (2) due to ( x , x ) being a strict equilibrium,strategy x j yields a strictly higher payoff against x i relative to the payoff of x ′ j against x i (which impliesthe convergence from (cid:16) x i , x ′ j (cid:17) to (cid:0) x i , x j (cid:1) ). Definition 7 (Adaptation of Cressman, 2010, Def. 5) . Fix a message function µ and a pair of mes-sages m , m ∈ supp ( ¯ µ ) . A strict Nash equilibrium ( x , x ) is a neighborhood invader strategy profile in Γ ( F m , F m ) if there exists ε >
0, such that for each ( x ′ , x ′ ) satisfying x ′ x , x ′ x , | x ′ − x | < ε and | x ′ − x | < ε , then either π m , m ( x , x ′ ) > π m , m ( x ′ , x ′ ) or π m , m ( x , x ′ ) > π m , m ( x ′ , x ′ ) .Proposition 6 shows that the profile of action functions induced by σ L (or, similarly, by σ R ) is aneighborhood invader strategy in any induced game. Proposition 6.
Let m , m ∈ supp ( ¯ µ ∗ ) . Then both strategy profiles ( ξ L ( m , m ) , ξ L ( m , m )) and ( ξ R ( m , m ) , ξ R ( m , m )) are strict equilibria and neighborhood invader strategy profiles in Γ F m , F m .Proof. We present the proof for ( ξ L ( m , m ) , ξ L ( m , m )) (the proof for ξ R is analogous). Observe that m , m ∈ supp ( ¯ µ ∗ ) implies one of three cases: m = m = m L , m = m = m R , or m = m R , m = m L .We analyze each case as follows.Suppose first that m = m = m L . This implies that ξ L ( m , m ) = ξ L ( m , m ) = F m ( / ) = F m ( / ) =
1. Let ¯ x < / be sufficiently close to / such that F m ( ¯ x ) , F m ( ¯ x ) > / . Observe that π m , m ( , x ) > π m , m ( y , x ) for any x > ¯ x and any y
1. This proves that ( ξ L ( m , m ) , ξ L ( m , m )) is a strict equilibrium and a neighborhood invader strategy profile. Now suppose that m = m = m R .6his implies that ξ L ( m , m ) = ξ L ( m , m ) = F m ( / ) = F m ( / ) =
0. Let ¯ x > / be sufficientlyclose to / such that F m ( ¯ x ) , F m ( ¯ x ) < / . Observe that π m , m ( , x ) > π m , m ( y , x ) for any x < ¯ x andany y
0. This proves that ( ξ L ( m , m ) , ξ L ( m , m )) is a strict equilibrium and neighborhood invader.Suppose finally that m = m R , m = m L . This implies that ξ L ( m , m ) = ξ L ( m , m ) = F m ( / ) =
0, and F m ( / ) =
1. Observe that π m , m ( , ) > π m , m ( x , ) for any x π m , m ( , ) > π m , m ( x , ) for any x
1, which implies that ( ξ L ( m , m ) , ξ L ( m , m )) is a strict equilibrium. Let ¯ x > / be suffi-ciently close to / such that F m ( ¯ x ) < / . Observe that π m , m ( , x ) > π m , m ( y , x ) for any x < ¯ x and any y
1. This proves that ( ξ L ( m , m ) , ξ L ( m , m )) is a neighborhood invader strategy profile. D.4 Remark on Evolutionary Robustness
Oechssler and Riedel (2002) present a strong notion of stability, called evolutionary robustness, thatrefines both evolutionary stability and the neighborhood invader strategy. An evolutionary robust strategy σ ∗ is required to be robust against small perturbation in the strategy played by the population, which maycomprise both (1) a few experimenting agents who follow arbitrary strategies, and (2) many agents whofollow strategies that are only slightly different than σ ∗ . Specifically, if σ is a distribution of strategiesthat is sufficiently close to σ ∗ (in the L norm induced by the weak topology), evolutionary robustness àla Oechssler and Riedel requires that π ( σ ∗ , σ ) > ( σ , σ ) .One can show that σ L and σ R do not satisfy this condition (and, we conjecture, that no strategy cansatisfy this condition in our setup). However, we conjecture that one can show that σ L and σ R satisfy asomewhat weaker notion of evolutionary robustness: for each strategy distribution σ sufficiently close to σ L ( σ R ), there exists a finite sequence of strategy distributions σ , σ , . . . , σ k , such that π ( σ , σ ) ≥ ( σ , σ ) , π ( σ , σ ) ≥ ( σ , σ ) , . . . , π ( σ k , σ k − ) ≥ ( σ k − , σ k − ) , and π ( σ L , σ ) ≥ ( σ , σ ) (resp., π ( σ R , σ ) ≥ ( σ , σ ) ), with strict inequalities if | M | = σ is not realization equivalent to σ L ( σ R ). E Analysis of the Extensions of Section 8 (Online Appendix)
E.1 Multiple Rounds of Communication
Consider a variant of the coordination game with communication in which players have T ≥ M . Playersobserve messages after each round and can, thus, condition their message choice and then their final ac-tion choice on the history of observed message pairs up to the point in time where they take their messageor action decision. Renegotiation then possibly takes place once at the end of this communication phasebut before the final action choices are made. Let M = S T − t = ( M × M ) t , where ( M × M ) = /0.A (pure) message protocol is a function m : M → M that describes the message sent by an agent asa deterministic function of the message profiles observed in the previous rounds of communication.7et M be the set of all message protocols. A strategy σ = ( µ , ξ ) is a pair where µ : U → ∆ ( M ) denotes the message function , prescribing a (possibly random) message protocol for each type, and ξ : ( M × M ) T → U denotes the action function by means of describing the cutoff (the highest possiblevalue of u ) for the two players to choose action L after observing the final message history. Renegotiationis modeled, as in the main text, as a possibility for the two players to play an equilibrium of a new gamewith another round of communication after all messages are sent, possibly using a different message set.Next, we adapt the notion of binary communication to fit multiple rounds of communication. Forany message protocol m ∈ M , let β σ ( m ) denote the expected probability of a player’s opponent playing L conditional on the player following message protocol m ∈ M and the opponent following strategy σ = ( µ , ξ ) ∈ Σ . We say that strategy σ has binary communication if there are two numbers 0 ≤ β σ ≤ β σ ≤ m ∈ M we have β σ ( m ) ∈ [ β σ , β σ ] , for all message protocols m ∈ M such that there is a type u < / with µ u ( m ) > β σ ( m ) = β σ , and for all messageprotocols m ∈ M such that there is a type u > / with µ u ( m ) > β σ ( m ) = β σ . That is, binarycommunication implies that players use just two kinds of message protocols: any message protocolused by types u < / induces the consequence of maximizing the probability of the opponent to play L , and any message protocol used by types u > / induces the opposite consequence of maximizing theprobability of the opponent to play R .Theorem 1, together with Propositions 1 and 2, holds in this setting with minor adaptations to theproof (omitted for brevity). Thus, regardless of the length of the pre-play communication, agents canreveal only their preferred outcome (but not the strength of their preference), and, regardless of havingaccess to additional rounds of communication, they cannot improve the ex-ante expected payoff relativeto the payoff induced by a single round of communication with a binary message. E.2 Multidimensional Sets of Types
In our model we made the simplifying assumption that miscoordination provides the same payoff (nor-malized to zero) to both players. This is not completely innocuous. In this section we explore whichresults are still true in this more general setting. Consider the following multidimensional set of types.Let ˆ U , a subset of IR , be the set of payoff matrices of binary coordination games, with u ab being thepayoff if a player chooses action a ∈ { L , R } while her opponent chooses action b ∈ { L , R } :ˆ U = { ( u LL , u LR , u RL , u RR ) | u LL > u RL and u RR > u LR } . Thus, all types strictly prefer coordination on the same action as the partner to miscoordination. Notethat any affine transformation of all payoffs neither changes the player’s incentives nor changes howshe compares any two outcome distributions ∈ ∆ ( { L , R } ) . We can thus subtract min { u RL , u LR } from allpayoffs and then divide all payoffs by some number such that the sum of the diagonal entries is equalto one. This leaves two parameters to describe a payoff vector in ˆ U . This means that for our purposes8he set ˆ U is two-dimensional. Let ˆ Γ = ˆ Γ ( G ) denote the coordination game with the two-dimensionaltype space ˆ U , endowed with an atomless CDF G over ˆ U with a density g . Similarly, let h ˆ Γ , M i be thecorresponding game with communication.Given a type u = ( u LL , u LR , u RL , u RR ) , let ϕ u ∈ [ , ] denote type u ’s indifference threshold , which isthe probability of the opponent playing L that induces an agent of type u to be indifferent: ϕ u = u RR − u LR u LL − u RL + u RR − u LR . Observe that an agent with indifference threshold ϕ u , where ϕ u is a number always between 0 and 1,prefers to play L ( R ) if her partner plays L with probability larger (smaller) than ϕ u . In other words, fora given probability of her partner playing L , a type u prefers to play L if and only if ϕ u is less than thatprobability. Thus, the indifference threshold ϕ u replaces what we denoted by u in the main model. Inparticular, in this setting we can also restrict attention to cutoff action functions. These are now appliedto ϕ u instead of to u . Thus, under a strategy σ = ( µ , ξ ) a player plays action L after observing a messagepair ( m , m ′ ) if and only if ϕ u ≤ ξ ( m , m ′ ) .Recall, that action L is risk-dominant (Harsanyi and Selten, 1988) if it is a best reply against theopponent randomizing equally over the two actions, i.e., if ϕ u ≤ / ⇔ u LL − u LR ≥ u RR − u RL . The crucial assumption that we implicitly made in our (one-dimensional) main model is that for anytype of player the action that she prefers to coordinate on is also risk-dominant for her.
Definition 8.
An atomless distribution over the payoff space U with density function g : U → IR satisfies unambiguous coordination preferences if for any u ∈ U with g ( u ) > u LL ≥ u RR ⇔ ϕ u ≤ / .Under a distribution over types with unambiguous coordination preferences, every type in its supportprefers coordinating on action L iff that type finds action L risk dominant. Under the assumption that thedistribution satisfies unambiguous coordination preferences, Thm. 1 goes through unchanged if we set F ( ϕ ) = Z { u ∈ U : ϕ u ≤ ϕ } g ( u ) du to be the implied distribution over the players’ indifference threshold induced by density g . As in thebaseline model, we assume that F ( ϕ ) has full support on the interval [ , ] . Theorem 2 (Theorem 1 adapted to a multidimensional set of types) . A strategy σ of a game h ˆ Γ , M i thatsatisfies unambiguous coordination preferences is a renegotiation-proof equilibrium strategy if and onlyif it is mutual-preference consistent, coordinated, and has binary communication. The proof is presented in Appendix E.7.1. The intuition is the same as in Theorem 1. The adaptationof Lemma 2, to the current setup relies on having unambiguous coordination preferences.While we cannot say that the restriction of unambiguous coordination preferences is necessary forthe result, we present an example that suggests that if this restriction is not satisfied, then equilibria with9iscoordination may be renegotiation-proof.
Example 2.
There are four possible preference types as follows: u L L RL 2 ⁄ R 0 1 u L L RL 2 -8R 0 1 u R L RL 1 0R ⁄ u R L RL 1 0R -8 2Define the distribution of types F such that P ( u L ) = P ( u R ) = / and P ( u L ) = P ( u R ) = / .Let M = { m L , m R } and let σ = ( µ , ξ ) be such that µ ( u L ) = µ ( u L ) = m L and µ ( u R ) = µ ( u R ) = m R (making σ mutual-preference consistent), and ξ ( m L , m L ) = L , ξ ( m R , m R ) = R , ξ ( u L , m L , m R ) = ξ ( u R , m R , m L ) = L , and ξ ( u L , m L , m R ) = R as well as ξ ( u R , m R , m L ) = R .It is straightforward to verify that σ is an equilibrium. Note that it is mutual-preference consistentand has binary communication, but it is not coordinated. We now show that σ is not Pareto-dominated byany coordinated equilibrium strategy in any induced post-communication game. To see this note that thenon-coordinated equilibrium following ( m L , m R ) is not Pareto-dominated by any coordinated equilibrium σ α with left tendency of α (with additional communication): under σ α the expected payoff of an agent,conditional on observing message pair ( m L , m R ) , is given by ( / ) · + ( / ) · ( / ) = + / for types u L and u R , and equal to / for types u L and u R . In order to induce a payoff of at least 1 + / to type u L with any coordinated equilibrium strategy σ α , it must be that α ≥ / , while in order to induce a payoffof at least 1 + / to type u L , it must be that α ≤ / . Thus, there is no α that satisfies both requirements.Note, however, that any strategy that is coordinated and mutual-preference consistent and has bi-nary communication is renegotiation-proof also in the general setting, and that only the other directionmay fail without the assumption of unambiguous coordination preferences. There may be additionalrenegotiation-proof equilibria in the general setting. One can show that any such renegotiation-proofequilibrium strategy must satisfy mutual-preference consistency, but need not satisfy the other two prop-erties (namely, coordination and binary communication). E.3 More Than Two Players
Consider a variant of the coordination game in which there are n ≥ { L , R } for everyplayer and the payoff to player i is equal to u i if every player chooses action R , equal to 1 − u i if everyplayer chooses L , and equal to zero otherwise. The payoff to type u i is independent and identically drawnfrom some given distribution F with support in the unit interval [ , ] . Before players choose actions, This distribution is discrete, but could easily be modified to a nearby atomless distribution without changing the result. The expected payoffs are: · + · (cid:0) · + · (cid:1) = + for types u L and u R , and · + · · = + for types u L and u R . This implies that no type wants to misreport her preferred outcome in round one. In particular, a misreportingtype u L will get a payoff of · + · · = < + . M and observe all these messages. Let h Γ n , M i denote this n -player coordination game with pre-play communication.In this setting the appropriate version of Theorem 1 still holds. Theorem 3 (Theorem 1 adapted to more than two players) . A strategy σ of the n-player coordinationgame h Γ n , M i is a renegotiation-proof equilibrium strategy if and only if it is mutual-preference consis-tent, coordinated, and has binary communication.Sketch of proof; for the formal proof see Appendix E.7.2. The proof of the “only if” direction has to beadapted (the proof of the “if” direction remains, essentially, the same). In this setting it is not generallytrue that any play that involves miscoordination is post-communication Pareto-dominated by σ L , σ R ,or σ C . The proof instead first establishes that miscoordination after all players send the same message mustbe Pareto-dominated by either σ L or σ R (Lemma 7). This is then used to show that a renegotiation-proof equilibrium strategy must be mutual-preference consistent (Lemma 8). Then one can show that arenegotiation-proof strategy must be coordinated and must have binary communication (Lemma 9).Prop. 1 and Cor. 1 also hold in the multi-player setting: renegotiation-proof equilibrium strategiesare interim (pre-communication) Pareto-undominated and Pareto-improving relative to all symmetricequilibria of the game without communication. By contrast, Prop. 2 does not extend to this setting: withthree players, for instance, for some distributions of values F , the strategy that determines the fallbackoption by majority vote (in the case of messages that indicate different preferred actions) is an ex-antepayoff improvement over a simple fallback norm of choosing, say, action L in every case of disagreement. E.4 Asymmetric Coordination Games
Adapted Model
Consider a setup similar to our baseline model except that the distributions of the typesof the two players’ positions differ: the type of player 1 is distributed according to F and the type ofplayer 2 is distributed according to F . As in the baseline model, both distributions are assumed to beatomless with full support in [ , ] . Let h Γ ( F , F ) , M i denote the asymmetric coordination game withcommunication (to ease notation, we assume that both players have the same set of messages at theirdisposal). Let Σ i denote the set of all strategies of player i ∈ { , } . We let i denote the index of oneplayer and j denote the index of the opponent. Remark . The game h Γ ( F , F ) , M i in which both players have the same distribution of types correspondsto a setup, in which the payoff-irrelevant position of player 1 or player 2 is identifiable, and the playerscan condition their play on their positions.Given a strategy profile ( σ , σ ) , let π iu ( σ , σ ) denote the (interim) payoff of type u of player i ∈{ , } , and let π i ( σ , σ ) = IE u ∼ F i (cid:2) π iu ( σ , σ ) (cid:3) denote the ex-ante payoff of player i ∈ { , } . A strategy11rofile ( σ , σ ) is an equilibrium if π u ( σ , σ ) ≥ π u ( σ ′ , σ ) for each strategy σ ′ ∈ Σ and for each type u of player 1, and π u ( σ , σ ) ≥ π u ( σ , σ ′ ) for each strategy σ ′ ∈ Σ and for each type u of player 2. Adapted Properties
We adapt the three key properties of Section 3 as follows. Let µ iu ( m i ) denotethe probability, given message function µ i , that player i sends message m i if she is of type u i . Let µ i ( m i ) = IE u ∼ F i (cid:2) µ iu ( m i ) (cid:3) be the average (ex-ante) probability of player i sending message m i . A strategyprofile ( σ , σ ) is mutual-preference consistent if whenever u , u < / then ξ ( m , m ) = ξ ( m , m ) = L for all m ∈ supp (cid:0) µ u (cid:1) and m ∈ supp (cid:0) µ u (cid:1) , and whenever u , u > / then ξ ( m , m ) = ξ ( m , m ) = R for all m ∈ supp (cid:0) µ u (cid:1) and m ∈ supp (cid:0) µ u (cid:1) .A strategy profile ( σ , σ ) is coordinated if ξ ( m , m ) = ξ ( m , m ) ∈ { L , R } for each pair of mes-sages m ∈ supp (cid:0) µ (cid:1) and m ∈ supp (cid:0) µ (cid:1) .For any strategy profile σ = (cid:0) ( µ , ξ ) , ( µ , ξ ) (cid:1) ∈ Σ × Σ and any message m j ∈ M , define β σ i ( m j ) = E u ∼ F i " ∑ m i ∈ M µ iu ( m i ) { u ≤ ξ i ( m i , m j ) } as the expected probability of player i playing L conditional on player j sending message m j ∈ M . Wesay that strategy profile σ = ( σ , σ ) has (essentially) binary communication if there are two pairs ofnumbers 0 ≤ β σ ≤ β σ ≤ ≤ β σ ≤ β σ ≤ m ∈ M and each player i ∈ { , } we have β σ i ( m ) ∈ [ β σ i , β σ i ] ; for all messages m ∈ M such that there is a type u < / with µ ju ( m ) > β σ i ( m ) = β σ i ; and for all messages m ∈ M such that there is a type u > / with µ u ( m ) > β σ i ( m ) = β σ i .Consider a strategy profile σ = ( σ , σ ) that is coordinated and mutual-preference consistent and hasbinary communication. Then there are α σ , α σ ∈ [ , ] such that, for each i ∈ { , } , β σ i = (cid:0) − F j ( ) (cid:1) α σ i and β σ i = F j ( ) + (cid:0) − F j ( ) (cid:1) α σ i , where α σ i is the probability of coordination on L conditional on player i having type u i < / and player j having type u i > / . We refer to α σ = (cid:0) α σ , α σ (cid:1) as the left-tendency profile of a strategy profile σ that is coordinated and mutual-preference consistent and has binary communication. It is simpleto see that the set of strategies satisfying the above three properties (coordination, mutual-preferenceconsistency, and binary communication) is essentially two-dimensional because the left-tendency profile α σ = (cid:0) α σ , α σ (cid:1) of such a strategy profile σ describes all payoff-relevant aspects. Two such strategyprofiles σ and σ ′ with the same left-tendency profile (i.e., with α σ = α σ ′ ) can only differ in the wayin which the players implement the joint lottery when they have different preferred outcomes, but theseimplementation differences are not payoff-relevant, as the probability of the joint lottery inducing theplayers to play L remains the same. Adaptation of Renegotiation-proofness
Given a strategy profile of the game h Γ , M i we denote theinduced “renegotiation” game after a positive probability message pair m , m ∈ M is sent by h Γ ( F m , F m ) , ˜ M i .12or a strategy profile σ ′ of such a renegotiation game h Γ ( G , G ) , ˜ M i , define the post-communication ex-pected payoffs for a player i of type u by π i , G u (cid:0) σ ′ (cid:1) = IE v ∼ G (cid:2) π iu , v (cid:0) σ ′ (cid:1)(cid:3) ≡ Z v = π iu , v (cid:0) σ ′ (cid:1) g ( v ) dv . Define E ( G , G ) as the set of all (possibly asymmetric) equilibrium profiles of the coordination gamewith communication h Γ ( G , G ) , ˜ M i for some finite message set ˜ M .We say that a strategy profile σ is post-communication equilibrium Pareto-dominated if there is apair of messages m ∈ supp ( µ ) and m ∈ supp ( µ ) and an equilibrium profile ˜ σ ∈ E ( F m , F m ) suchthat π i , F mi u ( σ ) ≤ π i , F mi u ( ˜ σ ) for each player i ∈ { , } and all u ∈ supp ( F m i ) with strict inequality for some u ∈ supp ( F m i ) of some player i ∈ { , } . A strategy profile σ is renegotiation-proof if it is not post-communication equilibrium Pareto-dominated. Adapted Results
Our main result remains the same in the setup of asymmetric coordination games.The proof, which is analogous to the proof of Theorem 1, is omitted for brevity.
Theorem 4.
A strategy profile σ of h Γ ( F , F ) , M i is a renegotiation-proof equilibrium strategy if andonly if it is mutual-preference consistent, coordinated, and has binary communication. Prop. 1–2 and Cor. 1 can be adapted to the current setup analogously (proofs are omitted for brevity).
E.5 Coordination Games with More Than 2 Actions
Next we extend our main model to games with more than two actions. We consider a coordination gamewith two players in which the two players first send one message from a finite message set M and then,after observing the message pair, choose one action from the ordered set A = ( a , . . . , a k ) with 2 < k < ∞ .A player’s type is now a vector u = ( u , . . . , u k ) ∈ [ , ] k , where we interpret the i -th component u i as the payoff of the agent if both players choose action a i . If the players choose different actions(miscoordinate), then they both get a payoff of zero. We assume that the distribution of types F is acontinuous (atomless) distribution with full support in [ , ] k . For each action a i , let p i be the probabilitythat the preferred action of a random type is a i (i.e., the probability that u i = max ( { u , . . . , u k } ) ). Let h Γ A , M i = h Γ A ( F ) , M i be the coordination game with set of actions A and pre-play communication.A player’s (ex-ante) strategy is a pair σ = ( µ , ξ ) , where µ : U → ∆ ( M ) is a message function thatdescribes which message is sent for each possible realization of the player’s type, and ξ : M × M × U → ∆ ( A ) is an action function that describes the distribution of actions chosen as a measurable function ofthe player’s type and the observed message profile. That is, when a player of type u who follows strategy ( µ , ξ ) observes a message profile ( m , m ′ ) , then this player plays action a i with probability ξ u ( m , m ′ ) ( a i ) .As in the main model, this game has many equilibria. For every action there is a babbling equilibriumin which players of all types after observing any message pair play this action. For every pair of actions13 i , a j there are also equilibria in which players send only one of two messages, one message indicating apreference for a i and another for not a i with play coordinated on a i if both players send the appropriatemessage and play coordinated on a j otherwise. None of these equilibria are renegotiation-proof as theyare not mutual-preference consistent and mutual-preference consistency is a necessary condition for astrategy to be renegotiation-proof also in the present context, as we shall see below.It is more difficult to find equilibria that are mutual-preference consistent, that is equilibria in whicheach player indicates her most preferred action out of all k actions and play is coordinated on that actionif both players indicate a preference for it. Simple adaptations of σ L and σ R are not equilibria in thepresent context. To see this, consider a strategy in which there is a “fallback” action, say action a , inwhich players indicate their most preferred action (the action with the highest u i ), and in which play iscoordinated on either action a i if both players indicate a preference for it, or coordinated on action a otherwise. Suppose that the distribution of types is such that there are two actions a i and a j (unequalto each other and unequal to a ) with p j > p i . But then there is a player type u = ( u , u , . . . , u k ) with u i = max { u , . . . , u k } , u j very close to u i , and u < u j , who would prefer to indicate a preference foraction a j . Indicating a preference for action a i , under the given strategy, provides her with a payoffof p i u i + ( − p i ) u . Indicating a preference for a j yields a payoff of p j u j + ( − p j ) u . But thenfor a suitably chosen vector u = ( u , u , . . . , u k ) the latter expression is greater than the former, whichcontradicts the supposition that the given strategy is an equilibrium.Next we show that a simple adaptation of σ C remains a renegotiation-proof equilibrium strategyalso in this setup. Let m , m , . . . , m k , m , m , . . . , m k ∈ M be 2 k distinct messages, where the index i ofmessage m bi is interpreted as denoting that the agent’s preferred outcome is the i -th outcome, and theindex b ∈ { , } is interpreted as a random binary number. Let σ C = ( µ C , ξ C ) be extended to the currentsetup as follows. Define µ C ( u ) = / m i ⊕ / m i , where i = argmin j (cid:8) u j | u j = max { u , .., u k } (cid:9) . Thus,the message function µ C induces each agent to reveal her preferred outcome, and to uniformly choose abinary number (either, zero or one). In the second stage, if both agents share the same preferred outcomethey play it. Otherwise, they coordinate on the preferred action with the smaller index if both agents havechosen the same random number, and they coordinate on the preferred outcome with the larger index ifboth agents have chosen different random numbers, i.e., ξ C (cid:16) m bi , m cj (cid:17) = a i ( i ≤ j and b = c ) OR ( i ≥ j and b = c ) a j otherwise . We then have the following proposition.
Proposition 7.
Strategy σ C is a renegotiation-proof equilibrium strategy in the game h Γ A , M i .Proof. Observe that an agent who sends message m bi obtains an expected payoff of / u i + / ∑ kj = p j u j when facing a partner who follows strategy σ C . As the second term in this sum is the same for all14essages, an agent of type u sends this message only if u i = max { u , . . . , u k } , as required. The remainingarguments as to why the second-stage behavior is a best reply and why σ C is renegotiation-proof areanalogous to the proof of the “if” part of Theorem 1 and are omitted for brevity.A strategy σ = ( µ , ξ ) is same-message coordinated if for all messages m ∈ supp ( ¯ µ ) there is anaction a i such that for all u with µ u ( m ) > ξ ( u , m , m ) = a i . In what follows we show that anecessary condition for a strategy to be a renegotiation-proof equilibrium strategy is that this strategy issame-message coordinated and mutual-preference consistent. Proposition 8.
If strategy σ of the game h Γ A , M i with action set A is renegotiation-proof, then it issame-message coordinated and mutual-preference consistent.Sketch of proof; for the formal proof see Appendix E.7.3. To show that a renegotiation-proof equilibriumstrategy is same-message coordinated we cannot, in fact, use the proof of the main theorem becauseLemma 2 crucially depends on the game having only two actions. Instead, we suppose to the contrary thatthere is a renegotiation-proof equilibrium strategy in which play is not coordinated after both players havesent the same message m . This strategy thus induces some nondegenerate probability distribution overactions after both players send message m . We then construct a post-communication Pareto-dominatingequilibrium of the induced game that is fully coordinated and has a probability of coordination on everyaction exactly equal to the probability of this action being played under the original strategy conditionalon observing ( m , m ) , which contradicts the supposition. The construction is achieved by players sendingrandom messages in such a way that they are indifferent between all messages and this joint lottery isimplemented. The proof that a renegotiation-proof equilibrium strategy must be mutual-preferenceconsistent is then achieved straightforward (by a simple adaptation of Lemma 8).We are able neither to show nor to provide a counterexample that a renegotiation-proof strategy mustbe coordinated after observing a pair of different messages, and that it must have binary communication.
E.6 Extreme Types with Dominant Actions
In this subsection we show how to extend our analysis to a setup in which some types have an extremepreference for one of the actions such that it becomes a dominant action for them.Let a < b >
1. We extend the set of types to be the interval [ a , b ] . Observe that action L ( R )is a dominant action for any type u < u >
1) as coordinating on R ( L ) yields to such a type a negativepayoff of u < − u < u < u > extreme , and types Using simultaneous communication to implement a jointly controlled lottery was introduced in Aumann and Maschler(1968) (see also Heller, 2010 for a recent implementation, which is robust to joint deviations). The original implementationworks perfectly if the probabilities of different actions are rational numbers. In Appendix E.7.3 we present a more elaborateimplementation that allows to deal also with irrational numbers in the current setup. u ∈ [ , ] ) moderate . We assume that the cumulativedistribution of types F is continuous (atomless) and has full support in the interval [ a , b ] .We further assume that the extreme types are a minority both among the agents who prefer action R and among the agents who prefer L , i.e., F ( ) < / F ( / ) and 1 − F ( ) < / ( − F ( / )) . Next,we adapt the definitions of coordination and binary communication to the current setup. The originaldefinition of coordination is too strong in the current setup, as, clearly, when extreme types with differentpreferred outcomes meet they must miscoordinate. Thus, we present a milder notion. A strategy is weaklycoordinated if whenever two moderate types meet they never miscoordinate. Note that the definition doesnot impose any restriction on what happens when an extreme type meets a moderate type.The original definition of binariness is too weak in the current setup. This is because coordinatedstrategies must allow for some miscoordination between extreme types, which implies that an agentcares not only about the average probability of the opponent playing left (i.e., β σ ( m ) ), but also aboutthe total probability of miscoordination. Thus, we strengthen binariness by requiring that there exist twodistributions of messages, which are used by all types below / and all types above / , respectively.Formally, a strategy σ = ( µ , ξ ) has strongly binary communication if µ ( u ) = µ ( u ′ ) if either u , u ′ ≤ / or u , u ′ > / . It is easy to see that the strategies σ L , σ R , σ C defined in Section 3 all satisfy stronglybinary communication. Moreover, one can show, for any α ∈ [ , ] , that if there exists a strategy σ thatis coordinated, mutual-preference consistent, and has binary communication with left tendency α , thenthere also exists strategy ˜ σ with the same properties that is strongly binary communication.Our next result shows that there exists, essentially, a unique renegotiation-proof equilibrium strategythat is coordinated, mutual-preference consistent, and has strongly binary communication. Proposition 9.
In a coordination game with communication and with dominant action types, a strat-egy σ that is coordinated, mutual-preference consistent, and has strongly binary communication is arenegotiation-proof equilibrium strategy if and only if it has a left tendency of α = F ( ) F ( )+( − F ( )) . The formal proof is presented in Appendix E.7.4. The key intuition is that given the frequency ofdominant action types F ( ) > L -dominant action types) and 1 − F ( ) > R -dominant actiontypes) to make the agent of type u = / indifferent between signaling a lower than half or higher than halftype we must have a strategy that counterbalances these frequencies of dominant action types. To seethis, consider an adaptation of σ L = ( µ ∗ , ξ L ) to this setting by having extreme types follow their dominantaction in the second stage (and moderate types play in the same way as in the baseline model). Note that σ L is no longer an equilibrium with extreme types. Observe that having a moderate type send message m R leads to coordination with probability one (sometimes on R and sometimes on L depending on theopponent’s message), while having a moderate type send message m L leads to coordination (on L only)with probability F ( ) <
1. This implies that agents of type u < / sufficiently close to / strictly prefersending message m R to sending message m L (as the former induces a higher probability of coordinationon the same action as the partner), which contradicts the supposition that σ L is an equilibrium strategy.16ppendix E.7.4 also shows that a left tendency α renegotiation-proof strategy that is coordinated andhas strongly binary communication can be implemented whenever α is a rational number and the set ofmessages M is sufficiently large (and irrational α -s can be approximately implemented by ε -equilibria).Observe that in the symmetric case ( F ( ) = − F ( ) ), σ C is the essentially unique renegotiation-proof strategy with the above two properties. Further observe that in the asymmetric case, the moderatetypes gain if the extreme types with the same preferred outcome are more frequent than the extreme typesof the opposite preferred outcome. Specifically, if there are more extreme “leftists” than extreme “right-ists” (i.e., F ( ) > − F ( ) ), then the essentially unique renegotiation-proof strategy with properties ofcoordination and strongly binary communication induces higher probability to coordinate on action L (rather than on action R ) whenever two moderate agents with different preferred outcomes meet. E.7 Formal Proofs of Extensions
E.7.1 Proof of Theorem 2 (Multidimensional types, Section E.2)
The proof of Thm. 2 mimics the proof of Thm. 1 except that Lemma 2 has to be adapted somewhat asfollows (this is the only place where one uses the assumption of unambiguous coordination preferences).
Lemma 6.
Assume that the atomless distribution of types have unambiguous coordination preferences.Let σ = ( µ , ξ ) be a renegotiation-proof equilibrium strategy. Then it is coordinated.Proof. We need to show that for any message pair m , m ′ ∈ supp ( ¯ µ ) ,either ξ ( m , m ′ ) ≥ sup { ϕ u | µ u ( m ) > } or ξ ( m , m ′ ) ≤ inf { ϕ u | µ u ( m ) > } . Let m , m ′ ∈ supp ( ¯ µ ) and assume to the contrary thatinf { ϕ u | µ u ( m ) > } < ξ ( m , m ′ ) < sup { ϕ u | µ u ( m ) > } . As σ is an equilibrium, we must have inf { ϕ u | µ u ( m ′ ) > } < ξ ( m ′ , m ) < sup { ϕ u | µ u ( m ′ ) > } . (Oth-erwise the m ′ message sender would play L with probability one or R with probability one, in which casethe m message sender’s best response would be to play L (or R ) regardless of her type). Let x = ξ ( m , m ′ ) and x ′ = ξ ( m ′ , m ) . In what follows we will show that the equilibrium ( x , x ′ ) of the game without com-munication Γ ( F m , F m ′ ) is Pareto-dominated by either σ L , σ R , or σ C (all based on ϕ u instead of u ).There are three cases to be considered. Case 1: Suppose that x , x ′ ≤ / . We now show that in thiscase the equilibrium ( x , x ′ ) is Pareto-dominated by σ R . Consider the player who sent message m .Case 1a: Consider a type u with ϕ u ≤ x . Then we have u LL F m ′ ( x ′ ) + (cid:0) − F m ′ ( x ′ ) (cid:1) u LR ≤ u LL F m ′ ( ) + u LR (cid:0) − F m ′ ( ) (cid:1) ≤ u LL F m ′ ( ) + u RR (cid:0) − F m ′ ( ) (cid:1) , u agent’s payoff under strategy profile ( x , x ′ ) and the last expressionis her payoff under strategy profile σ R . The first inequality follows from u LL ≥ u LR and F m ′ ( / ) ≥ F m ′ ( x ′ ) by the fact that F m ′ is nondecreasing (as it is a CDF), and the second inequality follows from u RR ≥ u LR .This inequality is strict when u LL > u LR and F m ′ ( / ) > F m ′ ( x ′ ) or when u RR > u LR .Case 1b: Now consider a type u with x < ϕ u ≤ / . Then we have u RL F m ′ ( x ′ ) + u RR (cid:0) − F m ′ ( x ′ ) (cid:1) ≤ u LL F m ′ ( x ′ ) + u RR (cid:0) − F m ′ ( x ′ ) (cid:1) ≤ u LL F m ′ ( ) + u RR (cid:0) − F m ′ ( ) (cid:1) , where the first expression is the type u agent’s payoff under strategy profile ( x , x ′ ) and the last expressionis her payoff under strategy profile σ R . The first inequality follows from u LL ≥ u RL and the second onefrom F m ′ ( / ) ≥ F m ′ ( x ′ ) and u LL ≥ u RR . Note also that the second inequality follows from the assumptionof unambiguous coordination preferences and ϕ u ≤ / . This inequality is strict when u LL > u RL or when F m ′ ( / ) > F m ′ ( x ′ ) and u LL > u RR .Case 1c: Finally, consider a type u with ϕ u > / . Then we have u RR > u RL F m ′ ( x ′ ) + u RR ( − F m ′ ( x ′ )) ,where the right-hand side is the type u agent’s payoff under ( x , x ′ ) and the left-hand side is her payoff un-der σ R . The inequality follows from the observation that u RR > u RL because u RR > u LL by the assumptionof unambiguous coordination preferences, and u LL ≥ u RL by the fact that it is a coordination game.The analysis for the player who sent message m ′ is analogous.Case 2: Suppose that x , x ′ ≥ / . The analysis is analogous to Case 1 if we replace σ R with σ L .Case 3: Suppose, without loss of generality for the remaining cases, that x ≤ / ≤ x ′ . We show thatthe equilibrium ( x , x ′ ) in this case is Pareto-dominated by σ C . Consider the player who sent message m .Case 3a: Consider a type u such that ϕ u ≤ x . Then we have u LL (cid:2) F m ′ ( ) + (cid:0) − F m ′ ( ) (cid:1)(cid:3) + u RR (cid:0) − F m ′ ( ) (cid:1) > u LL F m ′ ( x ′ ) + u LR ( − F m ′ ( x ′ )) , where the right-hand side is the type u agent’s payoff under strategy profile ( x , x ′ ) and the left-hand sideis her payoff under strategy profile σ C . The inequality follows from the observation that u RR ≥ u LR and F m ′ ( x ′ ) ≤ / by the fact that F m ′ ( x ′ ) = x when ( x , x ′ ) is an equilibrium.Case 3b: Now consider a type u with x < ϕ u ≤ / . Then we have u RL F m ′ ( x ′ ) + u RR (cid:0) − F m ′ ( x ′ ) (cid:1) ≤ u LL F m ′ ( x ′ ) + u RR (cid:0) − F m ′ ( x ′ ) (cid:1) ≤ u LL (cid:2) + F m ′ ( x ′ ) (cid:3) + u RR (cid:0) − F m ′ ( ) (cid:1) , where the first expression is the type u agent’s payoff under strategy profile ( x , x ′ ) and the last expressionis her payoff under strategy profile σ C . The first inequality follows from u LL ≥ u RL and the secondone from u LL ≥ u RR by the assumption of unambiguous coordination preferences given ϕ u ≤ / and F m ′ ( x ′ ) = x by ( x , x ′ ) being an equilibrium and x < / . The inequality is strict if u LL > u RL or u LL > u RR .18ase 3c: Finally, consider a type u with ϕ u > / . Then we have u RL F m ′ ( x ′ ) + u RR (cid:0) − F m ′ ( x ′ ) (cid:1) < u RL F m ′ ( ) + u RR (cid:2)(cid:0) − F m ′ ( ) (cid:1) + F m ′ ( ) (cid:3) ≤ u LL F m ′ ( ) + u RR (cid:2)(cid:0) − F m ′ ( ) (cid:1) + F m ′ ( ) (cid:3) , where the first expression is a u type’s payoff under strategy profile ( x , x ′ ) and the last expression is herpayoff under strategy profile σ C . The first inequality follows from u RR > u LL ≥ u RL by the assumption ofunambiguous coordination preferences and from ( − F m ′ ( / )) ≥ ( − F m ′ ( x ′ )) as F m ′ is nondecreasing.The analysis for the player who sent message m ′ is analogous. E.7.2 Proof of Theorem 3 (Multiple Players, Section E.3)
The “if” part is analogous to the proof of the “if” part of Theorem 1. The proof of the “only if” part doesnot extend directly and has to be adapted as follows. The following lemma states that play is coordinatedwhenever all players send the same message.
Lemma 7.
Let σ = ( µ , ξ ) be a renegotiation-proof equilibrium strategy. Let m ∈ supp ( ¯ µ ) and let m = ( m , . . . , m ) be the vector with n identical entries of m, which represents the case of all n playerssending message m. Then either ξ ( m ) ≥ sup { u | µ u ( m ) > } or ξ ( m ) ≤ inf { u | µ u ( m ) > } .Proof. Let m ∈ supp ( ¯ µ ) and assume to the contrary that inf { u | µ u ( m ) > } < ξ ( m ) < sup { u | µ u ( m ) > } .Let x = ξ ( m ) . We now show that the symmetric equilibrium in which all players use cutoff x after send-ing the identical message m , denoted by x = ( x , . . . , x ) , is equilibrium Pareto-dominated by σ L or σ R .There are two cases to be considered. Case 1: Suppose that x ≤ / . We now show that in this casethe equilibrium x is Pareto-dominated by σ R .Case 1a: Consider a type u ≤ x . Then we have ( − u ) (cid:0) F m ( ) (cid:1) n − + u (cid:16) − (cid:0) F m ( ) (cid:1) n − (cid:17) ≥ ( − u ) ( F m ( x )) n − , where the left-hand side is the type u agent’s payoff under strategy profile σ R and the right-hand side isher payoff under strategy profile x . The inequality follows from u (cid:16) − ( F m ( / )) n − (cid:17) ≥ F m ( / ) ≥ F m ( x ) by the fact that F m is nondecreasing (as it is a cumulative distribution function). Note also that thisinequality is strict for all u except for u = x = / .Case 1b: Now consider a type u with x < u ≤ / . Then we have ( − u ) (cid:0) F m ( ) (cid:1) n − + u (cid:16) − (cid:0) F m ( ) (cid:1) n − (cid:17) ≥ u (cid:16) − ( F m ( x )) n − (cid:17) , where the left-hand side is the type u agent’s payoff under strategy profile σ R and the right-hand side isher payoff under strategy profile x . The inequality follows from the fact that given u ≤ / we have that19 − u ≥ u and therefore ( − u ) (cid:0) F m ( ) (cid:1) n − + u (cid:16) − (cid:0) F m ( ) (cid:1) n − (cid:17) ≥ u . Note that this inequality actually holds strictly for all u .Case 1c: Finally, consider a type u > / . Then we have u > u ( − F m ( x )) n − , where the left-handside is the type u agent’s payoff under strategy profile σ R and the right-hand side is her payoff under x .Case 2: Suppose that x ≥ / . The analysis is analogous to Case 1 if we replace σ R with σ L . Lemma 8.
Every renegotiation-proof equilibrium strategy σ = ( µ , ξ ) is mutual-preference consistent.Proof. The proof of this lemma involves two steps. In the first step we show that a renegotiation-proofequilibrium strategy σ is ordinal preference revealing, i.e., such that for any message m ∈ supp ( ¯ µ ) , F m ( / ) ∈ { , } . We then use this to show that σ is mutual-preference consistent.Assume, first, that σ is a renegotiation-proof but not ordinal preference-revealing equilibrium strat-egy. That is, suppose to the contrary that F m ( / ) ∈ ( , ) . Then there are types u < / as well as types u > / who both send message m with positive probability. By Lemma 7 play after message pair ( m , m ) must be either L or R . If it is L then the equilibrium strategy σ L Pareto-dominates playing L , with a strictpayoff improvement for all types u > / (and unchanged payoffs for all types u ≤ / ). If it is R thenthe equilibrium strategy σ R Pareto-dominates playing R , with a strict payoff improvement for all types u < / (and unchanged payoffs for all types u ≥ / ).Given a renegotiation-proof equilibrium strategy σ = ( µ , ξ ) , we can classify messages in the sup-port of µ into two distinct sets, M L = M L ( σ ) = { m ∈ supp ( µ ) | F m ( / ) = } and M R = M R ( σ ) = { m ∈ supp ( µ ) | F m ( / ) = } , where M L ∩ M R = /0 and M L ∪ M R = supp ( ¯ µ ) .To show that a renegotiation-proof equilibrium strategy σ = ( µ , ξ ) is mutual-preference consistent,then consider any profile of types ( u , u , . . . , u n ) such that u i < / for all i ∈ { , . . . , n } . They must eachsend a message in M L , which we denote by the profile m = ( m , . . . , m n ) . Any play after message profile m that is not coordinated on L is now clearly Pareto-dominated (given that all types ≤ / ) by playing theequilibrium strategy L . The case for a profile of types u i > / for all players is proven analogously.The following lemma shows that, in a renegotiation-proof equilibrium strategy, agents never misco-ordinate after observing any message profile. Lemma 9.
Every renegotiation-proof equilibrium strategy σ is coordinated.Proof. Suppose that σ = ( µ , ξ ) is a renegotiation-proof equilibrium strategy. Given Lemmas 7 and 8 itonly remains to prove that play under σ is coordinated even after mixed messages are sent, i.e., whenthere is at least one player who sends a message in M L and another player who sends a message in M R ,where M L and M R are as defined in the proof of Lemma 8. Suppose that this is the case. Then let I ⊂ { , . . . , n } be the set of all players who send a message m i ∈ M L . Let I c denote its complement. ByLemma 8 all i ∈ I c satisfy m i ∈ M R . Let x i = ξ ( m i , m − i ) be the cutoff used by player i after observing20essage profile ( m , m , . . . , m n ) . Then by Lemma 8 we have x i ≤ / for all i ∈ I and x i ≥ / for all i ∈ I c . For this profile x = ( x , . . . , x n ) to be an equilibrium after the players observe message profile ( m , m , . . . , m n ) , we must have that for each i = , , . . . , n , the probability that player i ’s opponentscoordinate their action on L conditional on them coordinating (on either L or R ) is x i = ∏ j = i F m j ( x j ) ∏ j = i F m j ( x j ) + ∏ j = i (cid:0) − F m j ( x j ) (cid:1) . But then, all types of all players, after observing message profile ( m , m , . . . , m n ) , weakly (and somestrictly) prefer to play σ C , which is a payoff identical in this case to a public fair coin toss to determinewhether coordination should be on L or R . To see this, consider a player i who sent a message in M L (i.e., u i ≤ / , which implies that x i ≤ / ) and consider the following two cases.Case 1: Suppose that u i ≤ x i . Under the given strategy, this type’s payoff is ( − u i ) ∏ j = i F m j ( x j ) with ∏ j = i F m j ( x j ) ≤ / . The equilibrium strategy σ C yields to this type a payoff of / ( − u i ) + / u i , whichexceeds the former payoff, which contradicts the supposition that σ is renegotiation-proof.Case 2: Suppose that x i < u i ≤ / . Then, under the given strategy, this type’s payoff is u i ∏ j = i (cid:0) − F m j ( x j ) (cid:1) .The equilibrium σ C yields to this type a payoff of / ( − u i ) + / u i , which, by virtue of 1 − u i ≥ u i , againexceeds the former payoff, which contradicts the supposition that σ is renegotiation-proof.The analysis for a player who sent a message in M R is proven analogously.To complete the proof of Theorem 1 for the case of many players, we need to prove that anyrenegotiation-proof equilibrium strategy also has binary communication. The proof of this statementis analogous to the proof of Lemma 3 and therefore omitted. E.7.3 Proof of Proposition 8 (Multiple Actions, Section E.2)
For the proof, two lemmas about approximating real numbers by rational numbers will be useful. Thefirst lemma shows that any discrete distribution with at least three elements in its support can be ap-proximated from below by a vector of rational numbers, such that the profile of differences (between theirrational exact probability and its rational approximation from below) is roughly uniform in the sensethat no difference is larger than the half the sum of all the differences.
Lemma 10.
Let p ∈ ∆ ( A ) be a distribution satisfying | supp ( p ) | ≥ . Then there exists a function q : A → IR + such that, for each ≤ i ≤ k, q ( a i ) is a rational number, q ( a i ) ≤ p ( a i ) , andp ( a i ) − q ( a i ) ≤ ∑ ≤ j ≤ k (cid:0) p (cid:0) a j (cid:1) − q (cid:0) a j (cid:1)(cid:1) . Proof.
Let δ < min { p ( a ) / | a ∈ supp ( p ) } . As the rational numbers are dense in the reals, for eachreal number ˆ p > δ , there exists a rational number ˆ q ∈ ( , ˆ p ) such that ˆ p − ˆ q ∈ (( / ) δ , δ ) . Call this ˆ q a rational approximation of ˆ p . For each a ∈ supp p , let q ( a ) be a rational approximation of p ( a ) . For each21 supp ( p ) let q ( a ) = q ( a ) =
0. Then it follows that, for each 1 ≤ i ≤ k , q ( a i ) is a rational number, and q ( a i ) ≤ p ( a i ) . Finally we get, for each 1 ≤ i ≤ k , that p ( a i ) − q ( a i ) ≤ δ ≤ | supp ( p ) | δ ≤ ∑ ≤ j ≤ k (cid:0) p (cid:0) a j (cid:1) − q (cid:0) a j (cid:1)(cid:1) , where the first inequality follows directly from the definition of a rational approximation, the secondone follows from the assumption that | supp ( p ) | ≥
3, and the last one from the assumption that, for each a j ∈ supp ( p ) , p ( a j ) − q ( a j ) > ( / ) δ by the definition of a rational approximation.Note that the closer δ is to zero, the better the rational approximation constructed in this proof. Note,however, that this does not matter in the proof of Proposition 8 below, which simply uses any (possiblyquite rough) rational approximation.The second lemma is utilized in the proof of Proposition 8 for the case where the distribution inquestion has exactly two elements in its support. Lemma 11.
Let p , q ∈ ( , ) . Then there exists a rational number α ∈ ( , ) satisfyingp − q − q < α < pq . Proof.
Note that, as p < ≤ ( p − q ) = p − pq + q < p − pq + q .The inequality 0 < p − pq + q then implies that qp − q < p − pq ⇔ q ( p − q ) < p ( − q ) ⇔ p − q − q < pq . The result then follows from the fact that p − q < − q .We now turn to the proof of Proposition 8. Let σ = ( µ , ξ ) be a renegotiation-proof equilibriumstrategy of h Γ A , M i . We begin by showing that σ is same-message coordinated. Let m ∈ supp ( ¯ µ ) letand p ∈ ∆ ( A ) be the distribution of play under σ conditional on message pair ( m , m ) being observed.Assume to the contrary that p ( a ) < a ∈ A (i.e., that there is miscoordination). Case I:
Assume that | supp ( p ) ≥ | . Let q : A → IR + be a rational approximation of p satisfyingthe requirements of Lemma 10. The fact that all q ( a i ) -s are rational numbers implies that there are l , . . . , l k , n ∈ IN, such that q ( a i ) = l i / n for each i and l + . . . + l k ≤ n . Consider the following equilibriumstrategy ˜ σ = (cid:16) ˜ µ , ˜ ξ (cid:17) of the game induced after players observe message pair ( m , m ) with an additionalcommunication round with the set of messages ˜ M = (cid:8) m B , i , b | ≤ B ≤ n , ≤ i ≤ k , b ∈ { , } (cid:9) . We let1 ≤ B ≤ n denote a random integer used for a joint lottery, 1 ≤ i ≤ k denote the index of the player’spreferred outcome (i.e., u i = max (cid:8) u j | ≤ j ≤ k (cid:9) , and b ∈ { , } denotes a random bit). The messagefunction ˜ µ induces each agent to choose the indexes B and b randomly (uniformly, and independently ofeach other), and to choose i such that u i = max (cid:8) u j | ≤ j ≤ k (cid:9) is her preferred outcome.22he action function ˜ ξ (cid:0) m B , i , b , m B ′ , i ′ , b ′ (cid:1) is defined as follows. Let ˆ B = ( B + B ′ ) mod n be the sum ofthe random B -s sent by the players. Both players play action a j if l + . . . + l j − ≤ ˆ B < l + . . . + l j . If ˆ B ≥ l + . . . + l k , then both players play the action in { a i , a i ′ } with the smaller index if action if b = b ′ and theaction with the larger index if b = b ′ . Strategy ˜ σ induces both players to coordinate on a random actionwith probability ¯ q ≡ q ( a ) + . . . + q ( a k ) (and, conditional on that, the random action is chosen to be a j with probability q (cid:0) a j (cid:1) / ¯ q ), and to coordinate on the preferred action of one of the two players (chosenuniformly at random) with probability 1 − ¯ q , which can be written as 1 − ¯ q = ∑ ≤ j ≤ k (cid:0) p (cid:0) a j (cid:1) − q (cid:0) a j (cid:1)(cid:1) ,by the fact that p is a probability distribution and, thus, ∑ ≤ j ≤ k p (cid:0) a j (cid:1) = σ is an equilibrium is analogous to the proof of Prop. 7 and thus omitted. The expectedpayoff that the original equilibrium σ yields to each type u = ( u , . . . , u k ) ∈ U in the game induced afterobserving ( m , m ) is max j (cid:8) p (cid:0) a j (cid:1) u j (cid:9) . The expected payoff that ˜ σ induces for each type u is at least ∑ j q (cid:0) a j (cid:1) u j + ( − ¯ q ) max j (cid:8) u j (cid:9) ≥ max j { q (cid:0) a j (cid:1) u j } + max j (cid:8) u j (cid:9) ∑ j (cid:0) p (cid:0) a j (cid:1) − q (cid:0) a j (cid:1)(cid:1) . Thus, the difference between the payoff of ˜ σ and the payoff of σ for a type u is at least max j (cid:8) u j (cid:9) ∑ j (cid:0) p (cid:0) a j (cid:1) − q (cid:0) a j (cid:1)(cid:1) − (cid:0) max j (cid:8) p (cid:0) a j (cid:1) u j (cid:9) − max j (cid:8) q (cid:0) a j (cid:1) u j (cid:9)(cid:1) ≥ max j (cid:8) u j (cid:9) ∑ j (cid:0) p (cid:0) a j (cid:1) − q (cid:0) a j (cid:1)(cid:1) − ∑ ≤ j ≤ k (cid:0) p (cid:0) a j (cid:1) − q (cid:0) a j (cid:1)(cid:1) u j ≥ , where the first inequality is due tomax j (cid:8) p (cid:0) a j (cid:1) u j (cid:9) − max j (cid:8) q (cid:0) a j (cid:1) u j (cid:9) = p ( a l ) u l − max j (cid:8) q (cid:0) a j (cid:1) u j (cid:9) ≤ p ( a l ) u l − q ( a l ) u l , where l = argmax j (cid:8) p (cid:0) a j (cid:1) u j (cid:9) ; and the second inequality is due to q being a rational approximation of p as given by Lemma 10.This then implies that σ is post-communication equilibrium Pareto-dominated by ˜ σ , which contra-dicts the supposition that σ is a renegotiation-proof equilibrium. Case II:
We are left with the case of | supp ( p ) | =
2. Let supp ( p ) = (cid:8) a i , a j (cid:9) . Let q ∈ ( , ) be theposterior probability of a player having a type u with u i ≥ u j , conditional on sending m . Let p ≡ p ( a i ) .By Lemma 11, there exists a rational number α ≡ k / n ∈ ( , ) satisfying p − q / − q < α < p / q . Consider thefollowing symmetric equilibrium ˜ σ = (cid:16) ˜ µ , ˜ ξ (cid:17) of the game induced after players observe message pair ( m , m ) with an additional communication round with the set of messages ˜ M = { i , j } × { , . . . , n } .The first component of the message of each player is interpreted as her preferred coordinated outcomeof a i and a j , and the second component is a random number between 1 and n . When following strategy˜ σ the players send message component i if and only if u i ≥ u j , send a random number between 1 and n according to the uniform distribution, play a i after observing (( i , a ) , ( i , b )) for any numbers a and b ,play a j after observing (( j , a ) , ( j , b )) for any numbers a and b , play a i after observing (( i , a ) , ( j , b )) if23 + b < k mod n , and play a j after observing (cid:0) ( a i , a ) , (cid:0) a j , b (cid:1)(cid:1) if a + b ≥ k mod n .Observe that ˜ σ is indeed an equilibrium of the induced game, because following any pair of messagesthe players coordinate for sure, each agent with u i > u j (resp., u j > u i ) strictly prefers to report that herpreferred outcome is a i (resp., a j ) as this induces her to coordinate on a i (resp., a j ) with a high probabilityof q + α ( − q ) (resp., 1 − q + ( − α ) q ) instead of with a low probability of α q (resp., ( − q ) ( − α ) ),and each agent is indifferent between sending any random number, as this has no effect on the probabilityof coordinating on a i (which is equal to α = k / n ), given that her opponent chooses his random numberuniformly as well.Recall that the payoff of each type who follows σ in the game induced after observing ( m , m ) is equalto max (cid:8) u i p , u j ( − p ) (cid:9) . The payoff of each type u with u i ≥ u j in equilibrium ˜ σ is given by ( q + α ( − q )) u i + ( − ( q + α ( − q ))) u j > pu i + ( − p ) u j ≥ max (cid:8) u i p , u j ( − p ) (cid:9) , where the first inequality is implied by u i ≥ u j and p − q − q < α ⇔ p < q + α ( − q ) .The payoff of each type u with u i < u j in equilibrium ˜ σ is given by (( − q ) + ( − α ) q ) u j + ( − (( − q ) + ( − α ) q )) u i > ( − p ) u j + pu i ≥ max (cid:8) ( − p ) u j , pu i (cid:9) , where the first inequality is implied by qp > α ⇔ − α > q − pq ⇔ ( − q ) + ( − α ) q > − p and u i < u j .This implies that all types obtain a strictly larger payoff in ˜ σ (relative to the expected payoff of σ in thegame induced after players observe message pair m , m ), which implies that ˜ σ Pareto-dominates σ , whichcontradicts the supposition that σ is renegotiation-proof.Next, we show that σ = ( µ , ξ ) is mutual-preference consistent. For each i ∈ { , . . . , k } , let U i ⊂ [ , ] k be the set of types such that u i ≥ max j = i u j . Assume, first, that σ is not ordinal preference-revealing.That is, suppose that there is a message m ∈ supp ( ¯ µ ) such that there are action indices i , j with i = j and µ u ( m ) > u ∈ U i and some u ∈ U j . We have shown above that the play after playersobserve ( m , m ) must be coordinated on some action a l ∈ A . Now consider the following strategy withnew message space ˜ M = { m i , m ¬ i } in which players of type u ∈ U i send message m i , while all otherssend m ¬ i and play is a l unless both players send message m i , in which case it is a i . This is an equilibriumstrategy of the induced game after players observe ( m , m ) and it Pareto-dominates σ , a contradiction.This proves that a renegotiation-proof equilibrium strategy σ must be ordinal preference-revealing.Given a renegotiation-proof equilibrium strategy σ = ( µ , ξ ) , we can classify messages in the supportof µ into k distinct sets M i = M i ( σ ) = { m ∈ supp ( ¯ µ ) | u ∈ U i } for each i = , . . . , k , where for each i , j with i = j M i ∩ M j = /0 and S ki = M i = supp ( ¯ µ ) .To show that a renegotiation-proof equilibrium strategy σ is mutual-preference consistent, considera message pair ( m , m ′ ) with m , m ′ ∈ M i for some i = , . . . , k . Since σ is ordinal preference-revealing, theupdated support of types who observe either m or m ′ is then in U i . But then any joint action distribution24hat the two players could play after ( m , m ′ ) is Pareto-dominated by the equilibrium of playing action a i . E.7.4 Proof of Proposition 9 (Extreme Types, Section E.6)
Proof of Proposition 9.
Any strategy σ that is coordinated, mutual-preference consistent, and has stronglybinary communication can be characterized by its left tendency (see Section 3) as follows. Under sucha mutual-preference consistent strategy players indicate whether their type is below or above / . Thismeans that there are two disjoint sets of messages, M L and M R , such that players of type u ≤ / send amessage in M L and players of type u > / send a message in M R . Also whenever two players both sendmessages in M L they then play L and if both send messages in M R they both play R . The left tendency α = α σ then describes how moderate players coordinate if one of them sends a message from M L andthe other sends a message from M R . The left tendency α is then the probability that the two playerscoordinate on L (through random message selection within the respective sets of messages), while 1 − α is then the remaining probability that they coordinate on R .To prove the “only if” part, consider an arbitrary left tendency of α ∈ [ , ] . Then consider a playerof type / who needs to be indifferent between sending a message in M L and sending a message in M R for this strategy to be an equilibrium strategy. If she sends a message in M L she coordinates on L whenever either her opponent sends a message in M L (which happens with probability F ( / ) ), orher moderate opponent sends a message in M R (which happens with probability F ( ) − F ( / ) ) and thejoint lottery yields the outcome L (which happens with probability α ). By contrast, she coordinateson R whenever her opponent sends a message in M R and the joint lottery yields the outcome R (whichhappens with probability 1 − α ). Therefore, her expected payoff from sending a message in M L is givenby F (cid:0) (cid:1) + α (cid:0) F ( ) − F (cid:0) (cid:1)(cid:1) + ( − α ) (cid:0) − F (cid:0) (cid:1)(cid:1) . Similarly, her expected payoff from sending amessage in M R is given by α F (cid:0) (cid:1) + ( − α ) (cid:0) F (cid:0) (cid:1) − F ( ) (cid:1) + (cid:0) − F (cid:0) (cid:1)(cid:1) . Clearly, her expectedpayoff from sending a message in M L is equal to her expected payoff from sending a message in M R iff α = F ( ) / ( F ( ) + ( − F ( ))) , as required. This proves the “only if” direction.To prove the “if” direction, we need to show that a coordinated and mutual-preference consistentstrategy with strongly binary communication and with a left tendency of α = F ( ) / ( F ( ) + ( − F ( ))) is both an equilibrium and a renegotiation-proof strategy. To prove the latter condition the same argu-ments as in the relevant parts of the proof of the “if” direction of Theorem 1 apply directly. It remains toshow that such a strategy is an equilibrium strategy. We have already shown that the message functionis a best reply to itself and the action function. All that remains to prove is that the action function is abest reply to the given strategy. It is easy to see that playing L is the optimal strategy when both playerssend a message in M L and thus are of type u < / . In doing so, they coordinate on their most preferredoutcome with probability one. Similarly, playing R after two messages in M R is clearly optimal. Nowsuppose that one player sends a message in M L and the other player sends a message in M R . There aretwo possibilities. Either they are now supposed to both play L (unless they are an extreme R type) or25hey are now supposed to both play R (unless they are an extreme L type). Consider first the person whosends a message in M L and therefore be of type u < / . Suppose that the two players are expected tocoordinate on R . Since her opponent sent a message in M R , our M L sender expects R with a probabilityof one (as all M R senders are of type u > / , which excludes L -dominant action types). But then our M L sender of any type u > R as well. Now suppose that the two players areexpected to coordinate on L . Then our M L sender expects her opponent to play L with a probability of ( F ( ) − F ( / )) / ( − F ( / )) , which is the conditional probability of an R -type to be moderate, whichby assumption is greater than or equal to / . Playing L in this case is therefore optimal for all M L senders.That the M R sender has the correct incentives in her choice of action after any mixed-message pair (onein M R and one in M L ) is proven analogously and requires the assumption that F ( ) / F ( / ) ≤ / .We now show when one can implement a coordinated, mutual-preference consistent strategy with bi-nary communication with the required left tendency of α = F ( ) / ( F ( ) + ( − F ( ))) . This implemen-tation requires two things. First, α needs to be a rational number, and second, the message space needsto be sufficiently large. Note that in the case of a symmetric distribution F (i.e., F ( ) = − F ( ) )the required left tendency is exactly α = / and the required strategy is σ C , as described in Sec-tion 3. More generally, let α = k / n , and assume that | M | ≥ n . Denote 2 n distinct messages as (cid:8) m L , , . . . , m L , n , m R , , . . . , m R , n (cid:9) ∈ M , where we interpret sending messages m L , i as expressing a prefer-ence for L and sending messages m R , i as expressing a preference for R and choosing at random the num-ber i from the set of numbers { , . . . , n } in the joint lottery described below. We arbitrarily interpret anymessage m ∈ M \ (cid:8) m L , , . . . , m L , n , m R , , . . . , m R , n (cid:9) as equivalent to m L , . Given message m ∈ M , let i ( m ) denote its associated random number, e.g., i (cid:0) m L , j (cid:1) = j . Let M R = (cid:8) m R , , . . . , m R , n (cid:9) and M L = M \ M R .Then σ α = ( µ α , ξ α ) can be defined as follows: µ α ( u ) = n m L , + . . . + n m L , n u ≤ n m R , + . . . + n m R , n u > , ξ α (cid:0) m , m ′ (cid:1) = ( m , m ′ ) ∈ M R × M R ( m , m ′ ) M L × M L and ( i ( m ) + i ( m ′ ) mod n ) > k . Thus, µ α induces each agent to reveal whether her preferred outcome is L or R , and to uniformlychoose a number between 1 and n . In the second stage, if both agents share the same preferred outcomethey play it. Otherwise, moderate types coordinate on L if the sum of their random numbers modulo n isat most k , and coordinate on R otherwise. Extreme types play their strictly dominant action. The method for implementing a binary joint lottery of α and 1 − α is based on Aumann and Maschler (1968). In orderto deal with irrational α -s one needs either to slightly weaken the result to show that there exists a renegotiation-proof ε -equilibrium strategy (in which each type of each player gains at most ε from deviating) for any ε >
0, or to allow an infiniteset of messages or a continuous “sunspot.”0, or to allow an infiniteset of messages or a continuous “sunspot.”