C 07 2.5 7,0 6.6 R252 3.3 5,2 2,2 7,0 2.5 0. 4.4 R66 2.2 4.4 10.3 Multiple retionalizable strategies
0,7 2,5 7,0 6,6 5,2 3,3 5,2 2,2 7,0 2,5 0,7 4,4 6,6 2,2 4,4 10,3 R1 R2 R3 R4 C1 C2 C3 C4 Multiple retionalizable strategies
Nash equilibrium(1) Nash equilibrium is the most common equilibrium concept used A Nash equilibrium is a strategy profile such that every player's strategy is a best response to the strategies of all the other players For any permissible s. we can define the best-response function S =B, s i where si is the best response to s Then a Nash equilibrium strategy profile is an S* where S*=B, (s- *)for all i A Nash equilibrium requires that each player play a best response and that expectations regarding the play of their rivals are correct From the definition of a best response, for the strategy profile s* to be a Nash equilibrium (s S*2Ti (si, S-*)for all s, in S, and for all players i
Nash equilibrium(1) • Nash equilibrium is the most common equilibrium concept used. • A Nash equilibrium is a strategy profile such that every player’s strategy is a best response to the strategies of all the other players. • For any permissible s-i we can define the best-response function si=Bi (s-i ) where si is the best response to s-i . • Then a Nash equilibrium strategy profile is an s* where si *=Bi (s-i *) for all i. • A Nash equilibrium requires that each player play a best response and that expectations regarding the play of their rivals are correct. • From the definition of a best response, for the strategy profile s* to be a Nash equilibrium πi (si *, s-i *)≥πi (si ’, s-i *) for all si ’ in Si and for all players i
Nash equilibrium(2) For each player, given the Nash equilibrium strategies of all her rivals her best choice must be her Nash equilibrium strategy-there must be not any other available strategies: no player has any ex post regret Given the play of other players each player is doing as well as he can and hence no player has a reason to change strategy even if he has the opportunity In a Nash equilibrium, no player can unilaterally deviate and do better. This provides us with a way to find nash equilibria in simple finite games
Nash equilibrium(2) • For each player, given the Nash equilibrium strategies of all her rivals, her best choice must be her Nash equilibrium strategy-there must be not any other available strategies: no player has any ex post regret. • Given the play of other players, each player is doing as well as he can and hence no player has a reason to change strategy even if he has the opportunity. • In a Nash equilibrium, no player can unilaterally deviate and do better. This provides us with a way to find Nash equilibria in simple finite games
Hockey game Ballet Hockey 0.0 game Ballet 0.0 The battle of the sexes
3,1 0,0 0,0 1,3 Hockey game Ballet Hockey game Ballet The battle of the sexes
Discussion and interpretation of nash equilibria(1) Practical limitations There are 2 practical difficulties associated with the use of the concept of nash equilibrium: (a) there may be multiple Nash equilibria and ( b) an equilibrium may not exist When there are multiple Nash equilibrium strategy profiles, each player has a set of Nash equilibrium strategies
Discussion and interpretation of Nash equilibria (1) • Practical limitations • There are 2 practical difficulties associated with the use of the concept of Nash equilibrium: (a) there may be multiple Nash equilibria and (b) an equilibrium may not exist. • When there are multiple Nash equilibrium strategy profiles, each player has a set of Nash equilibrium strategies