A Game. Game Theory Review. What if player 2 can t observe what player 1 chose? (D;d,u) is subgame perfect. Player 1 asks What if player 2 plays u?

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1 A Game Game Theory Review Solving for Nash an Sbgame Perfect Eqilibria in Extensive an Strategic Form Games (-5; 0) (0,-5) (-0,-0) (;,) is sbgame perfect What if player can t observe what player chose? (-5; 0) (-5; 0) (0,-5) (0,-5) (-0,-0) (-0,-0) Strategic form game Player Player asks What if player plays? Player Player 5,5-5,0 Player 5,5-5,0 0,-5-0,-0 0,-5-0,-0

2 Player asks what if player plays? Player Player asks what if player plays? Player Player 5,5-5,0 Player 5,5-5,0 0,-5-0,-0 0,-5-0,-0 Player asks what if player plays? Player There are two Nash eqilibria to the game (;) an (;) Player Player 5,5-5,0 Player 5,5-5,0 0,-5-0,-0 0,-5-0,-0 Now, lets retrn to extensive form; player asks what if player plays? An, what if player plays? (-5; 0) (-5; 0) (0,-5) (0,-5) (-0,-0) (-0,-0)

3 An player asks what if player plays? What if player plays? (-5; 0) (-5; 0) (0,-5) (0,-5) (-0,-0) (-0,-0) (,) is Nash (,) is Nash (-5; 0) Best response for (-5; 0) Best response for (0,-5) (-0,-0) Best response for (0,-5) (-0,-0) Best response for What are the Nash eqilibria if player can observe player s move? (-5; 0) (0,-5) (-0,-0) The key to answering this qestion correctly is realizing that player now has 4 niqe strategies (,), (,), (,), (,), 3

4 Which strategies are player s best responses? (-5; 0) (-5; 0) (0,-5) (0,-5) (-0,-0) (-0,-0) (-5; 0) (0,-5) (-0,-0) (-5; 0) (0,-5) (-0,-0) Which strategies are player s best responses? (-5; 0) (-5; 0) (0,-5) (0,-5) (-0,-0) (-0,-0) (-5; 0) (0,-5) (-0,-0) (-5; 0) (0,-5) (-0,-0) So from player s perspective, (,) is a best response to player playing (,) is a best response to player playing (,) is a best response to player playing (,) is a best response to player playing (Note, (,) is a player s ominant strategy, that is, it is a best response no matter what player oes.) We re only half-way home Becase (,), (,) an (,) constitte all the best responses on the party of player, they might be part of a Nash eqilibrim an any eqilibrim that has player oing something else (,) is efinitely part of a nash eqilibrim. Bt we also nee to consier the sitation from player s perspective. Specifically, what is player s best response to each of the propose eqilibrim strategies for player? What shol player o When plays (,)? (recalling that this was s best response to ()) What shol player o when plays (,)? (recalling that this was s best response to ()) (-5; 0) (0,-5) (-0,-0) (-5; 0) (0,-5) (-0,-0) (;,) is a Nash Eqilibrim 4

5 What shol player o is player is expecte to play (,)? What shol player o is player is expecte to play (,)? (Recalling that this is a best response to player playing ) (Recalling that this is a best response to player playing ) (-5; 0) (0,-5) (-0,-0) (-5; 0) (0,-5) (-0,-0) (;,) is a Nash Eqilibrim What shol player o is player is expecte to play (,)? What shol player o is player is expecte to play (,)? (Recalling that this is a best response to player playing or ) (Recalling that this is a best response to player playing or ) (-5; 0) (0,-5) (-0,-0) (-5; 0) (0,-5) (-0,-0) (;,) is a Nash Eqilibrim (;,), (;,) an (;,) are all Nash eqilbria. Bt which one is or best preiction of behavior? (-5; 0) (0,-5) (-0,-0) (-5; 0) (0,-5) (-0,-0) (-5; 0) (0,-5) (-0,-0) (;,), (;,) an (;,) are all Nash eqilbria. Bt which one is or best preiction of behavior? Player makes an increible threat that actally works against his interest! (-5; 0) (0,-5) (-0,-0) (-5; 0) (0,-5) Bt it epens on an believing an increible threat (-0,-0) Player gets his best otcome if this eqilibria is playe (-5; 0) (0,-5) (-0,-0) 5

6 Of the three Nash eqilibria, only (;,) is sbgame perfect (-5; 0) (0,-5) (-0,-0) If only he ha the chance, player wol be only to happy to play Thogh he wishes he in t have to; player plays in response to Conclsions Not all Nash Eqilibria are sb-game perfect eqilibria Those that are not involve a player that moves later making increible threats or promises. I.e. threats or promises that he or she col not be expecte to carry ot when the time came to implement them. Conclsions A very large part of politics involves. trying to convince other actors that the threats or promises that yo are making are creible or selfenforcing. (Signalling). Trying to etermine if other actor s threats or promises are creible. (Screening) 3. Trying to change the environment so that formerly increible threats or promises are now creible. (Instittional esign or solving Commitment problems. The benefits of commitment In the game where players chose at the same time, there were two Nash eqilibria (;) an (;) an, since there was only one sb-game, they were both sb-game perfect. Player Player 5,5-5,0 0,-5-0,-0 Benefits of commitment, cont Bt when player was able to move first (I.e. was able to creibly commit to a strategy), then the only sbgame perfect strategy was (-5; 0) (0,-5) (-0,-0) What got when she coln t commit What got when she col 6