1.1 Normal Form Game

1.1 Normal Form Game - AMS 335/ECO 355 Game Theory Chapter...

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AMS 335/ECO 355 Game Theory Fall 2010 Chapter 1 Games of Complete Information Zhen Xu Page 1 of 5 Static Game : each player simultaneously chooses a strategy 1 Complete information : each player’s payoff function is common knowledge among all the players. Unlike the perfect information , players do not necessarily have the knowledge of the actions inside the game. 1.1 Normal Form games and Nash Equilibrium Definition An n-player normal form game specifies the players’ strategy spaces ሺS ,ڮ,S and their payoff functions U ,ڮ,U . We denote this game by Gൌ ሼS ; U . The n-player normal-form game specifies: 1) The players are numbered from 1,2, ,n and an arbitrary player i 2) The strategy set available to player i : S i . Let s i denote an arbitrary element of this set, i.e. s אS 3) The payoff received by player i is the result of each combination of strategies, i.e. u ൌU ሺs ,s ,ڮ,s Table 1 a two-player normal form game Player 2 s 21 s 22 s 2q Player 1 s 11 U 1 (s 11, s 21 ), U 2 (s 11, s 21 )U 1 (s 11, s 22 ), U 2 (s 11, s 22 ) U 1 (s 11, s 2q ), U 2 (s 11, s 2q ) s 12 U 1 (s 12, s 21 ), U 2 (s 12, s 21 1 (s 12, s 22 ), U 2 (s 12, s 22 ) U 1 (s 12, s 2q ), U 2 (s 12, s 2q ) s 1p U 1 (s 1p, s 21 ), U 2 (s 1p, s 21 1 (s 1p, s 22 ), U 2 (s 1p, s 22 ) U 1 (s 1p, s 2q ), U 2 (s 1p, s 2q ) S 1 =( s 11 , s 12 , , s 1p ), S 2 =( s 21 , s 22 , , s 1q ) Take s 1i S 2 and s 2j S 2 , then u 1 =U 1 (s 1i , s 2j ) and u 2 =U 2 (s 1i , s 2j ). For a two-player normal form game, the above table is common knowledge to the two players in this game. Definition In the normal form game G ൌ ሼS , let s and s be feasible strategies for player i (i.e. s and s ). Strategy s is strictly dominated by strategy s if for each feasible combination of the other players’ strategies, the payoff of player i from playing s is less than the payoff from playing s : U ሺs ୧ିଵ ୧ାଵ ሻ൏U ሺs ୧ିଵ ୧ାଵ for each ሺs ୧ିଵ ୧ାଵ that can be constructed from the other players’ strategy spaces S ୧ିଵ ,S ୧ାଵ The above equation can be simplified into U ሺs ି୧ ሺs ି୧ The subscript –i is known as the players other than i 1 Do not restrict to simultaneous move. The players know each others’ chosen strategies at the same time. They do
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This note was uploaded on 04/27/2011 for the course ECO 355 taught by Professor Xu during the Fall '10 term at SUNY Stony Brook.

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1.1 Normal Form Game - AMS 335/ECO 355 Game Theory Chapter...

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