A_Course_in_Game_Theory_-_Martin_J._Osborne 36

A_Course_in_Game_Theory_-_Martin_J._Osborne 36 - = 0. We...

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Page 21 theorem to prove that there is an action such that is a Nash equilibrium of the game. (Such an equilibrium is called a symmetric equilibrium .) Give an example of a finite symmetric game that has only asymmetric equilibria. 2.5 Strictly Competitive Games We can say little about the set of Nash equilibria of an arbitrary strategic game; only in limited classes of games can we say something about the qualitative character of the equilibria. One such class of games is that in which there are two players, whose preferences are diametrically opposed. We assume for convenience in this section that the names of the players are ''1" and "2" (i.e. N = {1,2}). •Definition 21.1 A strategic game is strictly competitive if for any and we have if and only if . A strictly competitive game is sometimes called zerosum because if player 1's preference relation is represented by the payoff function u 1 then player 2's preference relation is represented by u 2 with u 1 + u 2
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Unformatted text preview: = 0. We say that player i maxminimizes if he chooses an action that is best for him on the assumption that whatever he does, player j will choose her action to hurt him as much as possible. We now show that for a strictly competitive game that possesses a Nash equilibrium, a pair of actions is a Nash equilibrium if and only if the action of each player is a maxminimizer. This result is striking because it provides a link between individual decision-making and the reasoning behind the notion of Nash equilibrium. In establishing the result we also prove the strong result that for strictly competitive games that possess Nash equilibria all equilibria yield the same payoffs. This property of Nash equilibria is rarely satisfied in games that are not strictly competitive. •Definition 21.2 Let be a strictly competitive strategic game. The action is a maxminimizer for player 1 if Similarly, the action is a maxminimizer for player 2 if...
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