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Unformatted text preview: nnies are the same s side up then player 1 n gets a dollar fro player 2. om hen pen ands, the pe ennies show opposite sides up th player 2 w hen If, wh they op their ha gets a dollar fro player 1. om Let's represent this game in both the extensive f s form and th normal fo he orm... Rem member that we said th this gam has no p t hat me pure strateg Nash equ gy uilibrium. However, if thes players mix strategies (and the game is f se finite) there will exist a e mixe strategy Nash equilibrium. ed MIXE STRATEGY NASH EQUILIBRIA ED H If a p player is going to mix o a set of pure strategies then a of the stra on all ategies in the s must ha the sam payoff. H set ave me Hence, the player's pr robability dis stribution over these strat tegies is "ar rbitrary". Of co ourse, the other player's payoffs do depend upon the p o probability d distribution and h hence, a player's mixe strategy is complet ed tely determi ined by the other playe ers. Let's clarify how this works by solving the mixed strategy N s w s g d Nash equilib brium for the e matc ching pennies game an then the (slightly) m nd e more compl Battle o the Sexes lex of s game e. 309 ote 1 ng that Player 2 mixes as 1 (H, ) r s Deno Player 1's payoffs from playin H given t and a also denote Player 1's payoffs fro playing T given tha Player 2 mixes as e s om at 1 (T ). T, Now, let's consi ider Player 1's payoff from playin H... ng 1 (H, ) = (1) + (-1) (1 ) ) 1 (H, ) = 1 + 1 (H ) = 2 1 H, and w what about Player 1's payoff from playing T t m T... 1 (T, ) = (- + (1) (1 ) -1) ) 1 (T, = - + 1 ) 1 (T ) = 1 2 T, 2 Now, for Player 2 to make Player 1 in r ndifferent be etween play ying H or T they must ose Player 1's p payoffs the s same for H and T (i.e. 1 (H, ) = choo the that makes P 1 (T )). T, So... ... 1 (H, ) = 1 (T, ) , 2 1 = 1 2 4 = 2 = 310 Denote Player 2's payoffs from playing H given that Player 1 mixes as 2 (, H) and also denote Player 2's payoffs from playing T given that Player 1 mixes as 2 (, T). Now, let's consider Player 2's payoff from playing H... 2 (, H) = (-1) + (1) (1 )...
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This note was uploaded on 05/25/2010 for the course ECON 301 taught by Professor Sning during the Spring '10 term at University of Warsaw.
- Spring '10