41 Pareto Ranked Coordination G B G 22 00 B 00 11 3 equilibria BB OO and 1 3 2

41 pareto ranked coordination g b g 22 00 b 00 11 3

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Pareto Ranked Coordination G B G 2,2 0,0 B 0,0 1,1 3 equilibria: ( B,B ) , ( O,O ) , and (( 1 3 , 2 3 ) , ( 1 3 , 2 3 )) 42
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A Special Game G B G 1,0 -1,0 B 0,0 0,0 How many equilibria? 43
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A Special Game G B G 1,0 -1,0 B 0,0 0,0 Infinite equilibria: (( a, 1 - a ) , ( 1 2 , 1 2 )) for any a [ 0 , 1 ] 44
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Fact 2 For any game, the set of mixed strategy profiles is convex. For a finite game, the set of mixed strategy profiles is compact. Fact 3 The best response correspondence is convex valued. For a finite game, the best response correspondence is upper hemi-continuous and compact valued. Thus, in a nutshell, allowing for mixed strategies guarantees the existence of Nash equilibria in finite games. 50
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Minmax and Interchangeability Theorem 4 (Minmax Theorem) For every two-person, zero-sum game with finite pure strategies, there exists a value V and a mixed strategy for each player, such that (a) Given player 2’s strategy, the best payoff possible for player 1 is V , and (b) Given player 1’s strategy, the best payoff possible for player 2 is - V . Interchangeability. All equilibrium strategies for either given player in a two-player constant sum game are equivalent and any pair of equilibrium strategies is an equilibrium: For a two player, constant-sum game, if ( s * 1 ,s * 2 ) is an equilibrium and ( s ** 1 ,s ** 2 ) is an equilibrium, then ( s * 1 ,s ** 2 ) and ( s ** 1 ,s * 2 ) are also equilib- ria. 51
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How Many Equilibria? The Oddness Theorem Theorem 5 (Wilson (1971)) Almost all finite games have a finite and odd number of equilibria. (“Almost all” means it is not the case for a set of games possessing Lebesqgue measure zero in R i N Ai .) 52
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The Generic Oddness of Fixed Points 53
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The Generic Oddness of Fixed Points 54
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Equilibrium Refinements Pareto dominance/efficiency Symmetry Consider Battle of the Sexes: symmetric equilibrium is Pareto dominated. 55
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Iterated Dominance p -Beauty Contest Suppose that 3 individuals each simultaneously an- nounce a number x i [ 0 , 100 ] . Player i ’s payoff is u i ( a 1 ,a 2 ,a 3 ) = 1 if a i - p 3 3 i = 1 a i < min j i a j - p 3 3 i = 1 a i 0 if a i - p 3 3 i = 1 a i > min j i a j - p 3 3 i = 1 a i (and the prize is split evenly if more than one individual wins.) 56
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