lec2_17.pdf - Nash Equilibrium Review and More Examples...

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Nash Equilibrium: Review and More Examples De°nition: A pure strategy Nash equilibrium (NE) of the game ° I; ( S i ) i 2 I ; ( u i ) i 2 I ± is a pro- °le/vector of actions ( s ° 1 ; s ° 2 ; :::; s ° I ) s.t. for all players i : u i ( s ° i ; s ° ± i ) ° u i ( s 0 i ; s ° ± i ) for all s 0 i 2 S i (*) Other ways to say this: s ° = ( s ° 1 ; :::; s ° I ) is a NE if ± s ° i 2 B i ( s ° ± i ) for all players i (in words, the action that player i is supposed to choose, s ° i ; is a best response to s ° ± i (the opponents±equilibrium actions) ± assuming that everyone else follows s ° ; no one wants to change only his own action ± everyone best-responds to his belief about what the opponents do, and this belief is correct ± no one can °nd any pro°table deviations: nobody can increase his payo/ by changing just his own action, assuming everyone else sticks with the plan 1 Tips ± to °nd all NE in a 2-player game, one way is to look at the matrix: ± °rst, indicate all outcomes where P1 is best-responding with a *: in each column, °nd all rows where P1±s payo/ (the 1st #) is highest ± then, indicate all outcomes where P2 is best-responding with a *: in each row, °nd all columns where P2±s payo/ (2nd # is highest) ± any outcome with a * for both players is a NE, as you have shown that both players are best-responding here ± in general, if asked to show that a vector of actions is a NE, just show that no pro°table deviations are possible: show that no player could gain by changing his action, assuming that everyone else sticks with the plan (need to check this for all of the players) ± if asked to show that a vector of actions is not a NE, need to °nd a pro°table deviation: °nd some player who could gain by changing just his own action (assuming everyone else sticks with the plan) 2
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