Game Theory. Wolfgang Frimmel. Perfect Bayesian Equilibrium

Transcription

1 Game Theory Wolfgang Frimmel Perfect Bayesian Equilibrium / 22

2 Bayesian Nash equilibrium and dynamic games L M R 3 2 L R L R 2 2 L R L 2,, M,2, R,3,3 2 NE and 2 SPNE (only subgame!) 2 / 22

3 Non-credible threats (R, R ) depends on a non-credible threat - If player 2 is able to move, then playing L dominates R Player should not be induced to play R by 2 s threat to play R We need a concept that rules out non-credible threats and promises - BNE is not sufficient Combining SPNE with BNE, i.e. BNE in every subgame? In principle yes, but usually useless! Why? Continuation game A continuation game is a game that could begin at any information set rather than only at singleton information set We require a BNE in every continuation game 3 / 22

4 Requirement - Belief At each information set, the player with the move must have a belief about which node in the information set has been reached by the play of the game. For non-singleton information sets, a belief is a probability distribution over the nodes in the information set For singleton information sets, players belief assigns probability one on the single decision node. L R M 3 [p] [-p] 2 L R L R / 22

5 Requirement 2 - Sequential Rationality Given the beliefs, the players strategies must be sequentially rational. At each information set, the action taken by the player with the move must be optimal given the beliefs at that information set... given the other players will play according to that strategy in the continuation game Given player 2 s belief of p and p, the expected payoff from playing L is p + ( p) 2 = 2 p The expected playoff from playing R is p + ( p) = p Since 2 p > p for any value of p, sequential rationality prevents player 2 to choose R Given player 2 plays L, the only best-response of player is L 5 / 22

6 On and off the equilibrium path Requirement and 2 suffice to eliminate the non-credible equilibrium (R, R ) they do not imply that beliefs are reasonable and consistent with equilibrium strategies Definition For a given equilibrium in a given game, an information set is on the equilibrium path if it will be reached with positive probability if the game is played according to the equilibrium strategies For a given equilibrium in a given game, an information set is off the equilibrium path if it is certain not to be reached if the game is played according to the equilibrium strategies 6 / 22

7 Requirement 3 and 4 - consistent beliefs At information sets on the equilibrium path, beliefs are determined by Bayes Rule and the players equilibrium strategies: b(n I ) = Pr(n s) Pr(n s) n I where Pr(n s) is the probability that we reach node n according to strategy profile s In the SPNE (L, L ) of our example, player 2 s consistent belief must be p =. Why? For our example: Apply Bayes rule: Pr(n b(n I ) = (L,L ) Pr(n (L,L )+Pr(n 2 (L,L )) = + = At information sets off the equilibrium path, beliefs are determined by Bayes Rule and the players equilibrium strategies where possible. 7 / 22

8 Perfect Bayesian Equilibrium Definition A perfect Bayesian equilibrium consists not only of a strategy profile but also a belief profile A pair (s, b) of a strategy profile s and belief profile b is a perfect Bayesian equilibrium if (s, b) is sequentially rational and b is consistent with s (s, b) satisfy requirements -4 PBE is a stronger concept than SPNE - eliminates implausible SPNE Optimal action depends on beliefs in that information set and beliefs are determined by actions higher up in the game tree the Generalized backward induction procedure would not work (except if the uninformed player moves before the informed player) 8 / 22

9 How to find a PBE? No standard algorithm Eliminate all dominated strategies for each player List all strategies of the first-mover and then go through each of these strategies For every strategy of the first-mover what does this imply for the beliefs of the uninformed player? Once beliefs are updated, what do they imply for the uninformed player s strategy? Given the updated belief and optimal strategy of the uninformed player, what does this imply for the strategy of the first player? First player has incentive to deviate NO equilibrium First player has no incentive to deviate Perfect Bayesian equilibrium 9 / 22

10 Example Consider the following extensive form game: 4 6 L M R 2 l r l r Find all pure strategy NE! Find all SPNE! Find all PBE! / 22

11 Signaling game Signal jamming: the informed player incurs a cost to prevent revealing his private information Signaling: the informed player is willing to incur a cost to credibly convey his private information Signaling games and signal jamming games have the same structure and involve two players: a Sender (S) a Receiver (R) Applications Limit pricing (signal jamming game) Job market signaling (Spence, 973) Monetary policy (Vickers, 986) Corporate investment (Myers and Majluf, 984) / 22

12 Signaling game (cont.) Timing of the game Nature draws a type t i for player (Sender) from a set of feasible types T = t,..., t I according to a probability distribution p(t i ) with I i= p(t i ) = 2 The sender observes t i and chooses a message m j from a set of feasible actions M S = m,..., m J 3 The receiver observes m j but not the type t i of the sender and chooses an action a k from a set of feasible actions A R = a,..., a K 4 Payoffs are given by u S (t i, m j, a k ) and u R (t i, m j, a k ) 2 / 22

13 Signaling game (cont.) Pooling equilibrium In a signaling game, a pooling equilibrium is a perfect Bayesian equilibrium in which all types of the Sender play the same action In a pooling equilibrium, the receiver does not learn anything from sender s actions on the equilibrium path, i.e. the beliefs at the information set on the path are just the prior beliefs Separating equilibrium In a signaling game, a separating equilibrium is a perfect Bayesian equilibrium in which every type of the Sender play a different action If a type t plays message m in a separating equilibrium, then by consistency the receiver assigns probability to t when he observes m. Therefore, after the sender sends the message, the receiver learns the type (hence, putting probability on the correct type). 3 / 22

14 Signaling game (cont.) a Sender m t m 2 a a 2 p a 2 Receiver Nature Receiver a -p a a 2 m t 2 m 2 a 2 We could easily represent the game also in a more standard extensive form. How would the game tree look like? 4 / 22

15 Signaling game (cont.) Entry deterrence Firm (the incumbent) could either have high costs (c h ) or low costs (c l ) Firm 2 (the entrant) cannot observe incumbent s costs Firm first decides to charge a high (p h ) or a low price (p l ) (message) Firm 2 observes the price and then decides whether to enter (E) the market or not (N ) Entrant s profits are dependent on incumbent s costs Assume that cost types are equally likely Incumbent has an incentive to send the information that he has low cost Independent of the real cost type, a way to signal low cost is charging a low price 5 / 22

16 Signaling game (cont.) 5 2 E p p h Incumbent c h p l q E N Entrant x=.5 Nature N Entrant E -x=.5 E N -p p h c l p l -q N 6 / 22

17 Signaling game (cont.) How many strategies does the incumbent have? There are four possible equilibrium candidates: Separating equilibrium: s (s (c h ), s (c l )) = (p h, p l ) 2 Separating equilibrium: s (s (c h ), s (c l )) = (p l, p h ) 3 Pooling equilibrium: s (s (c h ), s (c l )) = (p h, p h ) 4 Pooling equilibrium: s (s (c h ), s (c l )) = (p l, p l ) How to find the PBE? check for each of the four possible candidates separately! 7 / 22

18 Signaling game (cont.) Separating equilibrium s (s (c h ), s (c l )) = (p h, p l ) What does this strategy of the incumbent imply for the entrant s belief? p = and q = Given the updated beliefs, what does this imply for the optimal strategy of the entrant? s 2 (p h, p l ) = (E, N ) Does the incumbent have an incentive to deviate, given the strategy of the entrant? Yes, the high-cost type has an incentive to deviate to low price (payoff 55 > 5) No perfect Bayesian equilibrium 8 / 22

19 Signaling game (cont.) Separating equilibrium s (s (c h ), s (c l )) = (p l, p h ) What does this strategy of the incumbent imply for the entrant s belief? p = and q = Given the updated beliefs, what does this imply for the optimal strategy of the entrant? s 2 (p h, p l ) = (N, E) Does the incumbent have an incentive to deviate, given the strategy of the entrant? Yes, both types have an incentive to deviate (payoff 75 > 7 and 75 > 35, respectively) No perfect Bayesian equilibrium 9 / 22

20 Signaling game (cont.) Pooling equilibrium s (s (c h ), s (c l )) = (p h, p h ) What does this strategy of the incumbent imply for the entrant s belief? p = 2 and q = Given the updated beliefs, what does this imply for the optimal strategy of the entrant? U 2 (E) = = 2.5 < = U 2(N ) s 2 (p h, p l ) = (N, ) Does the incumbent have an incentive to deviate? Yes, the low cost type has an incentive to deviate to low price (payoff 75 > 7 if entrant would play E and > 7 if entrant would play N in case of p l ) No perfect Bayesian equilibrium 2 / 22

21 Signaling game (cont.) Pooling equilibrium s (s (c h ), s (c l )) = (p l, p l ) What does this strategy of the incumbent imply for the entrant s belief? p = and q = 2 Given the updated beliefs, what does this imply for the optimal strategy of the entrant? U 2 (E) = = 2.5 < = U 2(N ) s 2 (p h, p l ) = (, N ) Does the incumbent have an incentive to deviate? No, if entrant plays E s 2 (p h, p l ) = (E, N ) iff 2p + ( p)( 25), hence iff p 5 9 Perfect Bayesian equilibrium at (E, N ) with beliefs p 5 9 and q = 2 2 / 22

22 Signaling game Solve for all Perfect Bayesian Equilibria of the following signaling game: 3 l p L Player t R q l r Player 2 x=.5 Nature r Player l -x=.5 l 2 r -p L t 2 R -q r 2 22 / 22