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Unformatted text preview: Nau: Game Theory 1 Updated 9/19/11 CMSC 498T, Game Theory 3. More about NormalForm Games Dana Nau University of Maryland Nau: Game Theory 2 Updated 9/19/11 Outline Chapter 2 discussed two solution concepts: Ø Pareto optimality and Nash equilibrium Chapter 3 discusses several more: Ø Maxmin and Minmax Ø Dominant strategies Ø Rationalizability Ø Correlated equilibrium Ø Tremblinghand perfect equilibrium Ø εNash equilibrium Ø Rationalizability Ø Evolutionarily stable strategies Nau: Game Theory 3 Updated 9/19/11 WorstCase Expected Utility For agent i , the worstcase expected utility of a strategy s i is the minimum over all possible combinations of strategies for the other agents: Example: Battle of the Sexes Ø Wife’s strategy s w = {( p , Opera), (1 – p , Football)} Ø Husband’s strategy s h = {( q , Opera), (1 – q , Football)} Ø u w ( p,q ) = 2 pq + (1 – p )(1 – q ) = 3 pq – p – q + 1 Ø For any fixed p , u w ( p,q ) is linear in q • e.g., if p = ½ , then u w ( ½ , q ) = ½ q + ½ Ø 0 ≤ q ≤ 1, so the min must be at q = 0 or q = 1 • e.g., min q ( ½ q + ½ ) is at q = 0 Ø min q u w ( p,q ) = min ( u w ( p, 0), u w ( p, 1)) = min (1 – p , 2 p ) Husband Wife Opera Football Opera 2, 1 0, 0 Football 0, 0 1, 2 min s ! i u i s i , s ! i ( ) Why did I write u w ( p , q ) instead of u w ( s w , s h ) ? Nau: Game Theory 4 Updated 9/19/11 Maxmin Strategies A maxmin strategy for agent i Ø A strategy s 1 that makes i ’s worstcase expected utility as high as possible: Ø This isn’t necessarily unique Ø Often it is mixed Agent i ’s maxmin value , or security level , is the maxmin strategy’s worst case expected utility: max s i min s " i u i s i , s " i ( ) argmax s i min s ! i u i s i , s ! i ( ) Also called maximin Nau: Game Theory 5 Updated 9/19/11 Example Wife’s and husband’s strategies Ø s w = {( p , Opera), (1 – p , Football)} Ø s h = {( q , Opera), (1 – q , Football)} Recall that wife’s worstcase expected utility is min q u w ( p,q ) = min (1 – p , 2 p ) Ø Find p that maximizes it Max is at 1 – p = 2 p , i.e., p = 1/3 Ø Wife’s maxmin value is 1 – p = 2/3 Ø Wife’s maxmin strategy is {(1/3, Opera), (2/3, Football)} Similarly, Ø Husband’s maxmin value is 2/3 Ø Husband’s maxmin strategy is {(2/3, Opera), (1/3, Football)} p min q u w ( p,q ) 2 p 1 – p Husband Wife Opera Football Opera 2, 1 0, 0 Football 0, 0 1, 2 Nau: Game Theory 6 Updated 9/19/11 Question Why might an agent i want to use a maxmin strategy? Nau: Game Theory 7 Updated 9/19/11 Answers Why might an agent i want to use a maxmin strategy? Ø Useful if i is cautious (wants to maximize his/her worstcase utility) and doesn’t have any information about the other agents • whether they are rational • what their payoffs are • whether they draw their action choices from known distributions Ø Useful if i has reason to believe that the other agents’ objective is to...
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This note was uploaded on 01/13/2012 for the course CMSC 498T taught by Professor Staff during the Fall '11 term at Maryland.
 Fall '11
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