Game Theory 14.122: Handout #L Finding Pbe in Signaling Games
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, we 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 incentive 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.
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
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, we 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 incentive 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.
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