game theory

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Iterated Elimination of Strictly Dominated Strategies
Last Example
Player 1
Up Down
Left 1,1 2,1
Player 2 Middle Right 0,2 1,-1 1,0 0,0
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Iterated Elimination of Strictly Dominated Strategies
Game Theory and Information Economics
Lecture 1
Ker Zhang
September 2, 2013
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Introduction
Game theory is a study of decision making under a multi-person environment. Applications:
u1 (talk , quiet ) > u1 (quiet , quiet ) and u1 (talk , talk ) > u1 (quiet , talk ) Talk strictly dominates Quiet for player 1. Player 2 knows player 1 is rational and will not play Quiet. By IESDS, player 2 will choose between Quiet and Talk assuming player 1 will choose talk. (Talk,Talk) is the outcome of this game.
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Iterated Elimination of Strictly Dominated Strategies
Another Example
Left 1,5 2,3 Player 2 Middle Right 3,1 6,4 4,4 3,2
Player 1
Up Down
u1 (down, left ) > u1 (up , left ), u1 (down, middle ) > u1 (up , middle ) and u1 (down, right ) < u1 (up , right ) No strategy dominates each other for player 1. However for player 2, u2 (up , left ) > u2 (up , right ) and u1 (down, left ) > u1 (down, right ) ’right’ is dominated by ’left’. ’middle’ is not dominated Player 1 knows player is rationl and will not play ’right’, therefore ’right’ is eliminated. Similarly for Player 2 The outcome is (down,middle).
Note: Strategy vs Action: Strategy is a set of specific action for each state.
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Static Game of Complete Information
Normal form representation
Payoff functions {u1 (s ), . . . , un (s )}
Payoff matrix. Player 2 Quiet Talk -2,-2 -10,-1 -1,-10 -6,-6
Player 1
Quiet Talk
Normal form game is defined as G = {S1 , . . . , Sn ; u1 , . . . , un }
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Oligopolies Auctions Biology Political Computer Science
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Static Game of Complete Information
Normal form representation
Players {1, . . . , n}.
{1, 2}
Strategy space {S1 , . . . , Sn }
Payoffs may depend on the strategies of all the players ui (s1 , . . . , sn ) u1 (talk , talk ) = u2 (talk , talk ) = −6 u1 (talk , quiet ) = −1, u2 (talk , quiet ) = −10
Crit of rationality Often imprecise Dove left right 5,-5 -5,5 -5,5 5,-5
Hawk
left right
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Iterated Elimination of Strictly Dominated Strategies
How it works
In the normal form game G = {S1 , . . . , Sn ; u1 , . . . , un }. si , si are two feasible strategies of player i . si strictly dominates si if ui (s1 , . . . , si −1 , si , si +1 , . . . , sn ) > ui (s1 , . . . , si −1 , si , si +1 , . . . , sn ) for any strategies {s1 , . . . , si −1 , si +1 , . . . , sn } that can be constructed from the strategy space of other players {S1 , . . . , Si −1 , Si +1 , . . . , Sn }. (1)
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Iterated Elimination of Strictly Dominated Strategies
2x2 Example Prisoners’ Dilemma
Player 1
Quiet Talk
Player 2 Quiet Talk -2,-2 -10,-1 -1,-10 -6,-6
si ∈ Si denotes a particular strategy from the set of strategies available to player i . S1 = {Talk , Quiet } S2 = {Talk , Queiet }
Player 1
Quiet Talk
Player 2 Quiet Talk -2,-2 -10,-1 -1,-10 -6,-6