MIT_pbe - Game Theory 14.122: Handout #l Finding PBE in...

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Game Theory 14.122: Handout #l Finding PBE in Signaling Games 1 General Strategy In a 2 x 2 signaling game, there can be any or all of the following Perfect Bayesian Equilibria (PBE): both types of Player 1 may play pure strategies in equilibrium (if they play the same strategy, we say it is a pooling equilibrium; if they differ, say it is a separating equilibrium); one type of Player 1 may play a pure strategy while the other plays a mixed strategy (leading to a semi-separating equilibrium); or both types of Player 1 may play mixed strategies. (We won’t deal with the latter case.) When looking for a PBE. .. 1. Decide whether you’re looking for a separating, pooling, or semi-separating equilibrium. 2. Assign a strategy (a message for each type) to Player 1; make sure it is not strictly dominated. 3. Derive beliefs for Player 2 according to Bayes’ rule at each information set reached with positive probability along the equilibrium path. Set arbitrary beliefs for Player 2 at information sets that are never reached along the equilibrium path. 4. Determine Player 2’s best response. 5. In view of Player 2’s response, check to see whether Player 1 has an in- centive to deviate from the strategy you assigned her in any state of the world (in other words, for all types of Player 1). If she does not, you have found a PBE. If she does, this is not an equilibrium - return to step 2 and assign Player 1 a different strategy. 6. Once you have exhausted all possible strategies within an equilibrium subset, return to step 1 and select a different type of PBE.
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2 Example Nature moves first. With equal probabilities, Nature assigns Player 1 type t E {t1,t2). Player 1 knows her type, and chooses whether to play L or R. Player 2 does not know what player’s 1 type is, but he sees whether she plays L or R. He chooses U or D. Figure 1: Game The extensive form of the game and its payoffs are presented in Figure 1. For example, if Player 1 is of type tr and plays R, and Player 2 plays U, the payoffs are 5 to Player 1 and 4 to Player 2. Note that if Player 1 is of type t2 and plays R, and Player 2 plays U, the payoff to Player 1 is x. Different values of x will be used at different points. Note also that for Player 1 of type tr, R strictly dominates L: playing L, she will always get 2; playing R, she will get at least 3. This means that in any PBE, type tr will always play R. This is intended to simplify the search for equilibria by eliminating half of the possible equilibria of each type (more than half of the semi-separating equilibrium possibilities). In a more general set up, these additional possibilities would need to be explored.
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3 Finding a Separating PBE 3.1 Player 1 Strategy Assume for now that x = 2. In a separating PBE, each Player 1 type chooses a different message, so that the message perfectly identifies the player type. There are only two possibilities for separating strategies for Player 1. One possibility is that type 1 will choose R and type 2 will choose L. The other possibility is that type 1 will choose L and type 2 will choose R. If either type has one message which strictly dominates
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MIT_pbe - Game Theory 14.122: Handout #l Finding PBE in...

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