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Unformatted text preview: Economics 405/505 Introduction to Game Theory Rui Zhao University at Albany, SUNY 1 Strategic Games: Theory 1.1 General Idea A strategic game has three essential components. 1. There is a set of players , 1, ..., i, ..., I. 2. Each player i has a set of strategies S i . 3. Each player i has a payoff function u i : S→ < , where S = S 1 × ··· × S I . So each player i will receive a payoff u i ( s ) if the strategies chosen by all players turn out to be s = ( s 1 ,...,s I ) . Such a list s is also called a strategy profile . Example 1 Prisoners’ Dilemma. Payoffs Confess Not Confess5, 5 0, 10 Not10, 01, 1 Example 2 Stag Hunt A group of hunters wish to catch a stag and share it equally. Each hunter can either hunt a stag or a hare, but not both. A stag can be caught only if all hunters devote to hunting stag: it escapes if one hunter tries to hunt a hare. For each hunter, an even share of the stag is better than a hare. Players: the hunters. Strategies: each player has two { Stag, Hare } . Payoffs: Payoffs Stag Hare Stag 2, 2 0, 1 Hare 1, 0 1, 1 1.2 Dominated Strategies Recall that a strategy profile s = ( s 1 ,...,s i ,...,s I ) ∈ S specifies a particular strategy s i for each player i. Denote by s i = ( s 1 ,...,s i 1 ,s i +1 ,...,s I ) the profile of strategies of all players other than player i . Set S i = S 1 ×··· × S i 1 × S i +1 ×··· S I consists of all such profiles. Thus a strategy profile s can also be written as ( s i ,s i ), indicating player i ’s strategy being s i and others’ being s i . Definition 1 A strategy s i is strictly dominated by another strategy s i if u i ( s i ,s i ) > u i ( s i ,s i ) for all s i ∈ S i . In words, s i is a better strategy than s i for player i , regardless what other players are doing. We also say that strategy s i strictly dominates s i ....
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This note was uploaded on 09/14/2011 for the course ECON 505 taught by Professor Zhao during the Spring '11 term at SUNY Albany.
 Spring '11
 Zhao
 Game Theory

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