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Unformatted text preview: Nau: Game Theory 1 Updated 9/8/11 CMSC 498T, Game Theory 2. Analyzing NormalForm Games Dana Nau University of Maryland Nau: Game Theory 2 Updated 9/8/11 Some Comments about NormalForm Games The normalform game representation is very restricted No such thing as a conditional strategy (e.g., cross the bay if the temperature is above 70) No temperature or anything else to observe Only two kinds of strategies: Pure strategy : just a single action Mixed strategy : probability distribution over pure strategies i.e., choose an action at random from the probability distribution Much more complicated games can be mapped into normalform games Each pure strategy is a description of what youll do in every situation you might ever encounter in the game Later on, in I'll show you some examples But not until Chapter 4 C D C 3, 3 0, 5 D 5, 0 1, 1 Nau: Game Theory 3 Updated 9/8/11 How to reason about games? In singleagent decision theory, look at an optimal strategy Maximize the agents expected payoff in its environment With multiple agents, the best strategy depends on others choices Deal with this by identifying certain subsets of outcomes called solution concepts This chapter discusses two solution concepts: Pareto optimality Nash equilibrium Chapter 3 will discuss several others Nau: Game Theory 4 Updated 9/8/11 Pareto Optimality A strategy profile s Pareto dominates a strategy profile s if no agent gets a worse payoff with s than with s , i.e., u i ( s ) u i ( s ) for all i , at least one agent gets a better payoff with s than with s , i.e., u i ( s ) > u i ( s ) for at least one i A strategy profile s is Pareto optimal (or Pareto efficient ) if theres no strategy profile s ' that Pareto dominates s Every game has at least one Pareto optimal profile Always at least one Pareto optimal profile in which the strategies are pure Nau: Game Theory 5 Updated 9/8/11 C D C 3, 3 0, 5 D 5, 0 1, 1 Examples The Prisoners Dilemma ( D,C ) is Pareto optimal: no profile gives player 1 a higher payoff ( D,C ) is Pareto optimal: no profile gives player 2 a higher payoff ( C,C ) is Pareto optimal: no profile gives both players a higher payoff ( D,D ) isnt Pareto optimal: ( C,C ) Pareto dominates it Which Side of the Road (Left,Left) and (Right,Right) are Pareto optimal In commonpayoff games, all Pareto optimal strategy profiles have the same payoffs If (Left,Left) had payoffs (2,2), then (Right,Right) wouldnt be Pareto optimal Left Right Left 1, 1 0, 0 Right 0, 0 1, 1 Nau: Game Theory 6 Updated 9/8/11 Best Response Suppose agent i knows how the others are going to play Then i has an ordinary optimization problem: maximize expected utility Well use s i to mean a strategy profile for all of the agents except i s i = ( s 1 , , s i 1 , s i +1 , , s n ) Let s i be any strategy for agent i . Then ....
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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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