Extensive Games with Imperfect Information

Extensive Games with Imperfect Information Jeff Rowley Abstract This document will draw heavily on definitions presented in Osborne; An Introduction to Game Theory and, for later definitions, Jehle & Reny; Advanced Microeconomic Theory. This is done mainly for accuracy of definition rather than clarity of presentation. Examples will reference Jehle & Reny by a if they are taken from that textbook, but if no reference is made, assume that the example is from Osborne. The document will first define an extensive game with imperfect information. A more intuitive explanation of what each feature included in the definition means will then be given, and an example included. The reader will then be given formal definitions for Nash equilibrium and subgame perfect equilibrium. An example will be used to demonstrate why subgame perfect equilibrium loses much of its bite when an extensive game is modified such that a player has imperfect information, and why it is necessary to introduce a new equilibrium concept. Up to this point, most of the definitions introduced will involve, or could be extended to involve, mixed strategies. For games with imperfect information it is useful to introduce a new strategy profile: a behavioural strategy. It will be stated that, where every player exhibits perfect recall, the two concepts are a dual approach to the same problem but that behavioural strategies are relatively easier to work with. Sequential rationality and consistency will be defined and imposed as necessary conditions for the existence of a weak sequential equilibrium. The final part of the document will work through some illustrative examples. 1 Extensive Games A significant problem in economics is that of imperfect information, and of particular note, asymmetric information. In the language of games, information is imperfect if some player i is unable to distinguish between two histories h,h of the game. This definition does not exclude a player from possessing imperfect information over two states ω,ω of the world; rather, consider ω,ω to be two 1

1.1 Terminal Histories 1 EXTENSIVE GAMES histories of the game which follow a move by nature 1. Now, if h,h cannot be distinguished by i, then they lie in the same information partition I i 2. A consequence of this is that the actions availableto i after the histories h,h must be the same. Formally, if h,h I i then A i (h) = A i (h ) = A i (I i ) 3. Extensive Games with Imperfect Information. An extensive game with imperfect information is formally defined as consisting of A set of players I = {1,...,H}. A set of sequences (terminal histories) having the property that no sequence is a proper subhistory of any other sequence. A function (the player function) that assigns either a player or nature to every sequence that is a proper subhistory of some terminal history. Standard convention dictates that the player function is denoted P( ). A function that assigns to each history h such that P(h) = Υ a probability distribution over the actions available after that history, where each such distribution is independent of every other such distribution. Less formally, there exists a state space Ω with an accompanying probability distribution. For each player, a partition (the player s information partition) of the set of histories assigned to that player by the player function such that for every history h in any given member of the partition, the set A i (h) of actions available is the same. For each player, preferences over the set of lotteries over terminal histories. The payoff function is Bernoulli, u i : ωk ΩS(ω k ) R. For completeness, an alternative description of the game replaces the notion of information partitions with a signal function τ i : Ω I i and any signals t i (ω) which i receives. The analogous interpretation arises when ω,ω I i if and only if t i (ω) = t i (ω ). Replacing ω with h and Ω with H allows for the possibility of this system being used more generally to describe extensive games with imperfect information. The notion of information sets may be easier to understand and formulate. This definition is complete but demanding. To benefit the understanding of the reader it will be necessary to decompose this definition into each of its required elements and express them in less formal language. Presentation of an extensive game with imperfect information will be illustrative of how the formal description is applied in practice. 1.1 Terminal Histories A terminal history is simply a possible outcome of the game. It is a description of what actions are played at each node of the whole game Γ( ) such that a terminal node is reached, including actions made by nature. For this reason, a terminal history is often referred to as a sequence. 1 I shall denote nature as Υ. This is my own convention and not standard notation. 2 The word partition is used rather than set to highlight the possibility of player i being unable to distinguish between multiple sets of histories which are partitions of the information set 3 Otherwise player i could distinguish between h,h as the set of actions available to her would be different which clearly violates the assumption of h,h I i. 2 Jeff Rowley, 2012

1.2 Player Function 1 EXTENSIVE GAMES 1.2 Player Function Consider some history h h t where h t is any terminal history of Γ( ). Then the player function assigns either nature Υ or a player i I to move at the node Γ(h). That is, the player function prescribes the order of play. Formal notation of the player function P(h) = i when player i is required to move (to be replaced by Υ when nature is required to move). 1.3 States Wherever nature is required to move, it assigns some probability distribution p(ω h ) over its availableactions at that history of the game. States are randomly selected according to this distribution. This simplification is sufficient to grasp an understanding the concept of how states are selected with any elaboration merely confusing. 1.4 Information Partition The defining characteristic of games of imperfect information is that a player doesn t know exactly where they are in the game. That is, they do not know which sequence of actions has been played to date. The history h consisting of a sequence of actions a 1,a 2 might not be differentiable from the history h = (ã 1,a 2 ). In contrast, the history h = (a 1,ã 2 ) might be differentiable from h and h. For this example, h,h I while h J; h,h are a partitioned set of the entire information set while h is a singleton contained within the entire information set. 1.5 Preferences Each player has preferences over each outcome of the game. There will exist a utility function which maps these preferences to R such that there is ordinal comparability between any two outcomes. The condition that preferences are Bernoulli is simply a technical assumption on the form of the utility function over different states. Example 315.1 A card game in which nature deals a high or a low card to a gambler who then decides whether to raise (R) or see (S). Her opponent then decides whether to pass (P) or meet (M). Players I = {1,2}. Terminal Histories (H,S), (H,R,P), and (H,R,M); (L,S), (L,R,P), and (L,R,M). Player Function P( ) = Υ, P(H) = P(L) = 1, and P(R) = 2. States Ω = (H,L); p (H) = 1 2 and p (L) = 1 2. Information Partitions Player 1 observes nature s move: she can distinguish 3 Jeff Rowley, 2012

2 EQUILIBRIUM STRATEGIES between the state H and the state L. Player 2 cannot distinguish between the history h H = (H,R) and the history h L = (L,R) as h R,h L belong to the same information partition. Preferences Preferences are according to the following table. P M R,R 1, 1 0,0 R,S 0,0 1 2, 1 2 S,R 1, 1 1 2, 1 2 S,S 0,0 0,0 2 Equilibrium Strategies Let the set of all actions available to some player i following the history h be denoted by A i (h). Recall, that for any histories h,h belonging to the same information partition I i, it must be the case that A i (h) = A i (h ) = A i (I i ). A consequence is that, upon reaching the information set I i, player i can only specify a single action (or mixture of actions) to be played, regardless of whether the history is h,h. This last fact is hopefully intuitive to the reader and yet places some caveats on what constitutes a strategy. Strategy in an Extensive Game. A (pure) strategy of player i in an extensive game is a function that assigns to each of i s information set I i an action in A i (I i ). That is, a strategy for an extensive game is a prescription of what actions player i will take wherever she thinks she is in the game every time she is required to move, regardless of whether that information set is reached or not. Convention is to write a pure strategy as s and a mixed strategy as α. Nash Equilibrium of an Extensive Game. The mixed strategy profile α in an extensive game is a (mixed strategy) Nash equilibrium if, for each player i and every mixed strategy α i of player i, player i s expected payoff to α is at least as large as her expected payoff to (α i,α i ) according to a payoff function whose expected value represents player i s preferences over lotteries. u i (α ) u i (α i,α i ) Generally, to solve for the Nash equilibria of an extensive game, one can reduce the game to normal form and solve for the Nash strategies. However, Nash equilibria need not be credible. Example 317.1 Imagine an entry game in which a challenger can choose to enter a market or to stay out of the market (Out). Prior to entering, the challenger can either make preparations (Ready) or rush in (U nready). The incumbent then has the choice of whether to acquiesce (A) or to fight (F). Fighting is costly for both firms, and preparing is costly for an entering firm. For now, assume that there 4 Jeff Rowley, 2012

2 EQUILIBRIUM STRATEGIES is perfect information. Players I = {Challenger, Incumbent}. Terminal Histories (Ready,A), (Ready,F), (Unready,A), (Unready,F), (Out,A), and (Out,F). Player Function P( ) = Challenger, and P(Ready) = P(Unready) = Incumbent. Preferences Preferences are according to the following table. A F Ready 3, 3 1, 1 Unready 4,3 0,2 Out 2,4 2,4 There are two Nash equilibrium strategies for this game: (U nready, A) and (Out,F). Clearly, the latter is not credible in so much as, if the challenger enters, the optimal strategy of the incumbent is to acquiesce. For games with perfect information, the notion of subgame perfect equilibrium was introduced. Subgame Perfect Nash Equilibrium. The strategy profile s in an extensive game with perfect information is a subgame perfect Nash equilibrium if, for every player i, every history h after which it is player i s turn to move (P(h) = i), and for every strategy s i of player i, the terminal history O h (s ) generated by s after the history h is at least as good according to player i s preferences as the terminal history O h (s i,s i ) generated by the strategy profile (s i,s i ) in which player i chooses s i while every other player j chooses s j. Equivalently, for every player i and every history h after which it is player i s turn to move, u i (O h (s ) u i (O h (s i,s i)) for every strategy s i of player i, where u i is the payoff function that represents player i s preferences and O h (s) is the terminal history consisting of h followed by the sequence of actions generated by s after h. This is a rather involved definition. A concise summary of subgame perfect equilibrium is that the strategy profile s be a Nash equilibrium, and also that the sequence of actions s (h) following the history h be a Nash equilibrium of the proper subgame Γ(h). A necessary condition for subgame perfect equilibrium is that the candidate equilibrium is a Nash equilibrium. This is a useful condition since the set of subgame equilibria must be a subset of the Nash equilibria which have already been found. For Example 317.1, only the Nash equilibrium (U nready, A) is subgame perfect. Unfortunately, when a game is modified such that there is imperfect information, subgame perfect equilibrium can lose much of its bite. The reason is that Γ(h) is a proper subgame of the whole game Γ( ) if and only if the information partition to which h belongs is a singleton set. Formally, I(x) = {x} 5 Jeff Rowley, 2012

3 BEHAVIOURAL STRATEGIES for some node x. Where a history h belongs to a nonsingleton information partition, that history does not constitute the initial node of a proper subgame. In some cases the number of subgames is restricted such that only the whole game remains as an improper subgame of itself. In this extreme case, subgame perfect equilibrium converges to Nash equilibrium. Example 317.1 (Again) Let the game be modified such that the incumbent cannot distinguish between a prepared challenger and an unprepared challenger. States There is a single state which occurs with certainty. Information Partition The challenger has a single information set and is perfectly informed. The incumbent cannot distinguish between the history Ready and the history U nready as these histories belong to the same information partition. Both Nash equilibrium strategies for the game, (Unready,A) and (Out,F), are also subgame perfect equilibrium strategies as the only subgame of Γ( ) is the game itself. 3 Behavioural Strategies It is here that the notion of beliefs is introduced. For games with perfect information, such a concept was implicit in so much as a player s belief about her opponent s strategy was simply equal to the strategy itself. When dealing with extensive games with imperfect information a player s beliefs need not be determined solely by her opponent s strategy. For example, in a variant of the entry game considered in Example 317.1 where the incumbent optimally prefers to acquiesce to a prepared challenger but fight an unprepared one, the Nash equilibrium of the game is (Out,F). However, for F to be the optimal action for the incumbent when the information set containing the histories Ready, U nready is reached, her belief must be that an entering challenger is more likely to be unprepared. As the optimal strategy for the challenger is Out, the challenger has no basis on which to form this belief. The optimal strategy of the challenger is now explicitly dependent upon her beliefs. Belief System. A belief system in an extensive game is a function that assigns to each information set a probability distribution over the histories in that information set. With this definition to hand, it is now that a distinction can be drawn between mixed strategies α and behavioural strategies β. The principal difference between the two strategy profiles is that mixed strategies are defined ex-ante whilst behavioural strategies are determined interim. This is a slightly incorrect statement but one that effectively captures the distinction between the concepts. A more rigorous description is as follows. Let S i be the set of all pure strategies available to player i. A mixed strategy profile assigns a probability distribution overall s i S i where α i (s i ) is the probability that any one strategy s i is played. 6 Jeff Rowley, 2012

4 WEAK SEQUENTIAL EQUILIBRIUM It must be that α i (s i ) [0,1] for all s i, and s i S i α i (s i ) = 1. In contrast, a behavioural strategy specifies a probability distribution over the actions available to some player at one of their information sets. Formally, β i (a i,i i ) [0,1] for all a i A i (I i ), and a i A i(i i) β i(a i,i i ) = 1. Behavioural Strategy in an Extensive Game. A behavioural strategy of player i in an extensive game is a function that assigns to each of i s information sets I i a probability distribution over the actions A(I i ), with the property that each probability distribution is independent of every other distribution. In the above definition, if the reader is more comfortable with the terminology information partition then they may replace information set with information partition. Technically, set is the more correct terminology to use; a partitioned set is a collection of proper subsets (or partitions) that forms the original set. In most cases, mixed strategy profiles and behavioural strategy profiles amount to a primal and dual approach to the same problem; that is, they are equivalent. The necessary condition for this to be true is that all players exhibit perfect recall. Perfect Recall. An extensive form game has perfect recall if whenever two nodes x and y = (x,a,a 1,...,a k ) belong to a single player, then every node in the same information set as y is of the form w = (z,a,a 1,...,a k ) for some node z in the same information set as x. The reader will notice that y and w contain the common member a. Perfect recall demands that each player always remembers what he knew in the past about the history of play. In particular, any two information sets which a player s information set do not allow her to distinguish between can differ only in the actions taken by other players 4. In other words, perfect recall fails when the player takes some action a at x and some action a at z and yet y,w lie in the same information set. 4 Weak Sequential Equilibrium The dependency between a player s strategy and her beliefs means that listing the sequence of actions that she will choose at any point in the game is no longer sufficient to completely describe her play; a complete description of her strategy will necessarily include her beliefs. Assessment. An assessment in an extensive game is a pair consisting of a profile of behavioural strategies and a belief system. Prior to defining the conditions of sequential rationality and consistency under which an assessment is an equilibrium, it will be a useful exercise for the reader to think of an appropriate equilibrium concept. For extensive games with perfect information, subgame perfect equilibrium imposed additional caveats on candidate equilibria by requiring that strategy profiles were Nash equilibria for every subgame of the game. The problem when subgame perfection was applied to games with imperfect information was that 4 Jehle & Reny. 7 Jeff Rowley, 2012

4.1 Sequential Rationality 4 WEAK SEQUENTIAL EQUILIBRIUM 1 L M R [µ] 2 [1 µ] 0,5 l m r l m r 4,0 1,1 0,4 0,4 1,1 4,0 Figure 1: The game presented in example 7.27 of Jehle & Reny. There is one Nash equilibrium (L, m). This is also the unique subgame perfect equilibrium. However, this equilibrium does not satisfy sequential rationality. nodes belonging to nonsingleton information sets could not be considered subgames. To summarise, subgame perfection places no restriction on the strategy following nonsingleton information sets as these cannot be considered subgames. A natural extension would therefore be to impose optimal behaviour following every information set. To incorporate the importance of beliefs in determining behavioural strategies, some notion of optimality given beliefs must also be incorporated. 4.1 Sequential Rationality Sequential Rationality. Each player s strategy is optimal in the part of the game that follows each of her information sets, given the strategy profile and her belief about the history in the information set that has occurred. Precisely, for each player i and each information set I i of player i, player i s expected payoff to the probability distribution O Ii (β,µ) over terminal histories generated by her belief µ i at I i and the behaviour prescribed subsequently by the strategy profile β is at least as large as her expected payoff to the probability distribution O Ii ((γ i,β i ),µ) generated by her belief µ i at I i and the behaviour prescribed subsequently by the strategy profile (γ i,β 1 ), for each of her behavioural strategies γ i. Simply put, sequential rationality imposes that for any arbitrary belief system µ, each player is maximising their expected payoff given the other players strategies, after every information set. Example 7.27 The game tree is presented in Figure 1. There is one Nash equilibrium (L,m). This is also the unique subgame perfect equilibrium. However, this equilibrium does not satisfy sequential rationality. Suppose that player 2 s information set is reached with positive probability. Sequential rationality states that the action m should be optimal given her belief about how the information set was reached 5. Clearly, regardless of which value is assigned to µ, such that µ [0,1], either 5 The belief µ corresponds to the probability with which player 2 thinks the history is M. 8 Jeff Rowley, 2012

4.2 Consistency 4 WEAK SEQUENTIAL EQUILIBRIUM (1) 1 (0) 2 (1 ( 2 3 ) 3 ) [µ] 3 [1 µ] Figure 2: The game presented in example 7.29 of Jehle & Reny. Given the behavioural strategy of player 1, player 3 s information set is never reached. Nonetheless, consistency still places some restrictions on her beliefs. the action l or the action r is optimal. Hence, the subgame perfect equilibrium is not sequentially rational. 4.2 Consistency Beliefs are obviously pivotal to the optimality of behavioural strategies; if beliefs are altered slightly, it might be that a Nash profile that was previously sequentially rational is no longer. Yet no restriction has been placed upon how beliefs are formed, or their accuracy. Consistency. For every information set I i reached with positive probability given the strategy profile β, the probability assigned by the belief system to each history h in I i is given by Pr(h ) = Pr(h β) h I i Pr(h β) (Bayes rule). Consistency imposes quite strong restrictions on beliefs for those information sets that are reached with positive probability. To see this, the reader should look at Bayes rule and think what it means. The numerator is the probability of some history h being reached given the behavioural strategy profile β. What should immediately be apparent is that consistency imposes that beliefs be realistic in the sense that they reflect play. The denominator is the probability that the information set that contains h is reached given the behavioural strategy profile β. 4.2.1 Trembling Hand The question then is how beliefs should be derived such that they satisfy consistency when an information set is not reached given the behavioural strategy profile β. It is here that the idea of a trembling hand is introduced. Suppose that, in Figure 2, the behavioural strategy of player 1 was instead to play left with probability 1 ǫ and right with probability ǫ. Then, the information set would be reached with positive probability. µ,1 µ are then 9 Jeff Rowley, 2012

4.2 Consistency 4 WEAK SEQUENTIAL EQUILIBRIUM 1 (0) (0) (1) [µ] [1 µ] 2 Figure 3: A game in which Bayes rule does not uniquely define the beliefs µ,1 µ. A trembling hand can be used but does not place strict constraints on the formation of the beliefs. uniquely defined by Bayes rule. µ = 1 µ = ǫ 1 3 ǫ 1 3 +ǫ 2 3 ǫ 2 3 ǫ 1 3 +ǫ 2 3 Bayes rule thus uniquely defines µ = 1 3,1 µ = 2 3. Using the idea of a trembling hand which assigns positive probability to every information set being reached is useful in determining consistent beliefs if an information set is never reached under an arbitrary behavioural strategy profile β. This is not to say that the notion of a trembling hand can be used in every circumstance; in many games, the trembling hand demonstrates that some arbitrary beliefs can be assigned to some histories contained in an information set that is never reached. Consider the game in Figure 3. If probabilities ǫ 1,ǫ 2 are assigned to left and middle, then the beliefs µ,1 µ depend on the relative magnitudes of ǫ 1,ǫ 2. To proceed further, the exact constraints placed upon the beliefs by consistency must be formalised. Consistent Assessments. An assessment (β, µ) for a finite extensive form game Γ is consistent if there is a sequence of completely mixed behavioural strategies, β n, converging to β, such that the associated sequence of Bayes rule induced systems of beliefs, µ n, converges to µ. According to this definition, whenever a trembling hand is applied to some behavioural strategy profile β and the trembles ǫ 0 such that β n β, then the beliefs µ n derived using Bayes rule for the behavioural strategy profile β n will be consistent for µ. To illustrate, consider again Figure 3. Suppose that the ǫ 1,ǫ 2 are the probabilities assigned by player 1 to the actions left and middle as part of some behavioural strategy profile β n. µ = ǫ 1 ǫ 1 +ǫ 2 1 µ = ǫ 2 ǫ 1 +ǫ 2 10 Jeff Rowley, 2012

4.3 Weak Sequential Equilibrium 5 INTUITIVE CRITERION In this case, any arbitrary beliefs can be assigned to the events left and middle. To see this, suppose that ǫ 1 = ǫ 2. Then as ǫ 1,ǫ 2 0, µ n,1 µ n 1 2, 1 2. Alternatively, if ǫ 1 = ǫ 2 2 then µ n,1 µ n 0,1. The reader is invited to try other combinations of ǫ 1,ǫ 2 such that any belief system can be generated. To summarise, consistency requires that beliefs be derived using Bayes rule wherever possible. Wherever an information set is not reached, Bayes rule is undefined. A trembling hand can be applied to a behavioural strategy profile β such that Bayes rule is defined as the trembles ǫ 0. In some cases, the trembling hand does indeed define consistent beliefs but in others merely illustrates the point that beliefs can be arbitrarily assigned whenever an information set is not reached according to the behavioural strategy profile β. 4.3 Weak Sequential Equilibrium As sequential rationality and consistency have been defined, it is now possible to define a weak sequential equilibrium of an extensive game with imperfect information. Weak Sequential Equilibrium. An assessment (β, µ) (consisting of a behavioural strategy profile β and a belief system µ) is a weak sequential equilibrium if it satisfies sequential rationality and weak consistency of beliefs with strategies. 5 Intuitive Criterion Consider the Beer-Quiche game set out in Figure 4. This type of game is a signalling game as the receiver forms her beliefs over which type of the sender she based upon what type of breakfast she has eaten (the message she receives). An assessment for the receiver will thus consist of a strategy following the signal Beer and a strategy following the signal Quiche, and a belief system over the probability that the sender is Strong given the signal she receives. This is consistent with the definition previously given that a player must specify a strategy at every information set, and the beliefs she forms over histories in each information set. There are two broad types of equilibria that could exist for this game: pooling equilibria in which both types of sender consume the same breakfast, and separating equilibria in which each type of sender consumes a different breakfast. In the latter, the receiver has the advantage of knowing exactly which type of her opponent she faces based on the breakfast they consume. It is claimed that there exist no separating equilibria for this game. The receiver would always choose to Fight a weak opponent and to Acquiesce to a strong opponent. For any separating equilibria, the W eak type would always prefer to consume whatever breakfast the Strong type is consuming as she would derive a strictly higher payoff, regardless of the suggested separating equilibrium. Let p > 0.5 so that the probability of asender being strong is greater than the probability of an opponent being weak. There are two candidate pooling equilibria. The first is where both types of sender drink Beer and the second is where both types of sender eat Quiche. It should be obvious that Bayes 11 Jeff Rowley, 2012

5 INTUITIVE CRITERION 1,1 F B S Q F 1,0 0,3 N p N 0,2 1,0 F 2 Υ 1 p 2 F 1,1 0,2 N B W Q N 0,3 Figure 4: A Beer-Quiche game in which the receiver prefers to Fight a weak opponent but Acquiesce to a strong one. Each type of her opponent has a favourite breakfast but she would forgo that breakfast if it meant not fighting. The receiver s payoffs are listed first. rule uniquely determines that, for any information set reached with positive probability, the belief of the receiver that the sender is Strong is simply equal to p. It is claimed that there exist two pooling equilibria for this game: ((B,B;N,F),(p,1 p;λ,1 λ)), ((Q,Q;F,N),(λ,1 λ;p,1 p)). Here, the first pair list the strategies of the two types of sender, the second pair list the strategies of the receiver following the signals Beer and Quiche, the third pair list the belief of the receiver that the sender is Strong or Weak given the signal Beer, and the fourth pair list the belief of the receiverthat the sender is Strong or Weak given the signal Quiche. Consistency demands that λ 1 26. It is left up to the reader to show that these assessments do indeed constitute weak sequential equilibria. Careful examination of these pooling equilibria will reveal that one of them is not entirely credible; the pooling equilibrium in which both types of sender eats Quiche demands that µ be such that the receiver thinks it more likely a sender drinking Beer is W eak. A further mechanism with which to impose credibility on equilibria is required. The notion of an intuitive criterion is introduced. Prior to defining the intuitive criterion, some discussion must be entered into to determine what constitutes an incredible equilibrium. Consider again the Quiche pooling equilibrium. What makes this equilibrium incredible? Put simply, given the equilibrium behavioural strategy profile β, the belief that a 6 Note that a trembling hand does nothing to tighten the constraint on these beliefs. 12 Jeff Rowley, 2012

5 INTUITIVE CRITERION W eak type will be more likely to play Beer seems unrealistic. In equilibrium, the Weak type getsapayoffof3. IfaWeak type wereeverto deviate to sending the message Beer, the most she could ever hope to get would be a payoff of 2. The belief system which supports this equilibrium is thus incredible. Any intuitive criterion must incorporate some means of ruling out such equilibria. Equilibrium Dominance. A message m can be eliminated for some sender type θ if U (θ) > maxu(m,a,θ) a where U (θ) is the equilibrium payoff to θ from playing m. This definition formalises the argument presented in the paragraph above. A message m is equilibrium dominated for a sender type θ if the payofffrom sending m always yields less utility than the payoff from sending the equilibrium message m. Applying this to the game in Figure 4: the possible messages are Beer, Quiche, the possible sender types are Strong, W eak, and the possible actions for the receiverare Fight,Acquiesce. Fixing θ = Weak, the messagebeer is equilibrium dominated by the message Quiche as, regardless of whether the receiver chooses to F ight or to Acquiesce, Quiche always yields a higher payoff given that the receiver opts to Acquiesce when she sees the sender consumes Quiche. Intuitive Criterion. No message m which is equilibrium dominated will ever be sent by a type θ. The belief system µ must be consistent with the probability that the message m being sent is zero. 13 Jeff Rowley, 2012