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Unformatted text preview: aying poor str rategies is s to less t set than in the mix of pure strategie ( > 0). e es, nition: Defin A perfect equilib brium is the limit point of an per e rfect equilib brium as (no 0. uniqu ueness of equilibrium is assured) e ). his ee... So how does th work for an example? Let's se 317 So, if Player 2 mixes again then 2 (, B) < 2 (, B) for all > 0. f m nst at will r y g n This means tha Player 2 w set their probability of playing B to less than when st 1 they mix agains . i.e. (1 ) < Now, consider Player 1's p P payoffs from mixing ag m gainst . 1 (t, ) = (100) + (50 (1 ) 0) 1 (t, = 150 50 ) what about Player 1's payoff from playing b... t m and w (b, 00) 00) ) 1 ( ) = (10 + (10 (1 ) 1 ( ) = 100 (b, 0 f m nst So, if Player 1 mixes again then 1 (t, ) < 1 (b, ) for all < 1. This means tha Player 1 w set their probability of playing t to less th when at will r y g han they mix agains . i.e. < st In the limit, as 0, the pu strategy T for playe 2 emerge (i.e. 1) and the e ure y er es pure strategy b for player 1 emerges (i.e. (1 ) )1). This gives us th Perfect he Nash Equilibrium at (b, T). h Hope efully, we notice that th success he sive elimina ation of wea akly domina ated strate egies would have resu d ulted in the same equilibrium with hout all of th effort. his This is only coin ncidental since the exa ample is de esigned to b simple a be and illustrative. 318 RE S Y QUILIBRIA A MOR MIXED STRATEGY NASH EQ Rem member, the were no pure strate Nash E ere o egy Equilibria for the "Rock Paper, k, Sciss sors" game e. We p proceed jus like we did with a 2X matrix. F st X2 First we find Player 1's payoffs d s from playing each of their o options aga ainst the d distribution. 1 (R, ) = (0) 1 + (1) (2) + (1) (1 1  2) 1 (R, ) =  2 + (1  1  2) 1 (R, ) = 1  1  22 (1) 1 (P, ) = (1) 1 + (0) (2) + (1) (1 1  2) 1 (P, ) = 1  1 + 1 + 2 1 (P, ) = 21 + 2  1 (2) 1 (S, ) = (1) 1 + (1) (2) + (0) (1 1  2) 1 (S, ) =  1 + 2 1 (P, ) = 2 1 (3) column play yer's choice of 's tha will make the row e at Now we can solve for the c er nt ons s h e playe indifferen among...
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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
 sning
 Economics

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