lecture7a - CMSC 498T, Game Theory 7a....

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Nau: Game Theory 1 Updated 11/9/11 CMSC 498T, Game Theory 7a. Incomplete-Information Games Dana Nau University of Maryland
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Nau: Game Theory 2 Updated 11/9/11 Introduction All the kinds of games we’ve looked at so far have assumed that everything relevant about the game being played is common knowledge to all the players: Ø the number of players Ø the actions available to each Ø the payoff vector associated with each action vector True even for imperfect-information games Ø The actual moves aren’t common knowledge, but the game is We’ll now consider games of incomplete ( not imperfect ) information Ø Players are uncertain about the game being played
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Nau: Game Theory 3 Updated 11/9/11 Example Consider the payoff matrix shown here Ø ε is a small positive constant; Agent 1 knows its value Agent 1 doesn’t know the values of a, b, c, d Ø Thus the matrix represents a set of games Ø Agent 1 doesn’t know which of these games is the one being played Agent 1 wants a strategy that makes sense despite this lack of knowledge If Agent 1 thinks Agent 2 is malicious, then Agent 1 might want to play a maxmin, or “safety level,” strategy minimum payoff of T is 1– ε minimum payoff of B is 1 Ø So agent 1’s maxmin strategy is B
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Nau: Game Theory 4 Updated 11/9/11 Regret (Section 3.2) Suppose Agent 1 doesn’t think Agent 2 is malicious Agent 1 might reason as follows: Ø If Agent 2 plays R , then 1’s strategy changes 1’s payoff by only a small amount Payoff is 1 or 1– ε ; Agent 1’s difference is only ε Ø If Agent 2 plays L , then 1’s strategy changes 1’s payoff by a much bigger amount Either 100 or 2, difference is 98 Ø If Agent 1 chooses T , this will minimize 1’s worst-case regret Maximum difference between the payoff of the chosen action and the payoff of the other action
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Nau: Game Theory 5 Updated 11/9/11 Minimax Regret Suppose i plays action a i and the other agents play action profile a –i Ø i ’s regret : amount i lost by playing a i instead of i ’s best response to a i i doesn’t know what a –i will be, but can consider the worst case: Ø maximum regret for a i , maximized over every possible a –i Minimax regret action : an action with the smallest maximum regret Can extend to a solution concept Ø All agents play minimax regret actions regret( a i , a ! i ) = max " a i # A i u i " a i , a ! i ( ) $ % & ' ( ) ! u i a i , a ! i ( ) max a ! i " A ! i regret( a i , a ! i ) = max a ! i " A ! i max # a i " A i u i # a i , a ! i ( ) $ % & ' ( ) ! u i a i , a ! i ( ) * + , - . / argmin a i ! A i max a " i ! A " i regret( a i ) = argmin a i ! A i max a " i ! A " i max # a i ! A i u i # a i , a " i ( ) $ % & ' ( ) " u i a i , a " i ( ) * + , - . /
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