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# lecture2 - CMSC 498T Game Theory 2 Analyzing Normal-Form...

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Nau: Game Theory 1 Updated 9/8/11 CMSC 498T, Game Theory 2. Analyzing Normal-Form Games Dana Nau University of Maryland

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Nau: Game Theory 2 Updated 9/8/11 Some Comments about Normal-Form Games The normal-form game representation is very restricted Øඏ No such thing as a conditional strategy (e.g., cross the bay if the temperature is above 70) Øඏ No temperature or anything else to observe Only two kinds of strategies: Øඏ Pure strategy : just a single action Øඏ 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 normal-form games Øඏ Each pure strategy is a description of what you’ll do in every situation you might ever encounter in the game Later on, in I'll show you some examples Øඏ 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 single-agent decision theory, look at an optimal strategy Øඏ Maximize the agent’s 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: Øඏ Pareto optimality Øඏ Nash equilibrium Chapter 3 will discuss several others

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Nau: Game Theory 4 Updated 9/8/11 Pareto Optimality A strategy profile s Pareto dominates a strategy profile s ʹஒ if Øඏ no agent gets a worse payoff with s than with s ʹஒ , i.e., u i ( s ) u i ( s ʹஒ ) for all i , Øඏ at least one agent gets a better payoff with s than with s ʹஒ , i.e., u i ( s ) > u i ( s ʹஒ ) for at least one i A strategy profile s is Pareto optimal (or Pareto efficient ) if there’s no strategy profile s ' that Pareto dominates s Øඏ Every game has at least one Pareto optimal profile Øඏ 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 Prisoner’s 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 ) isn’t Pareto optimal: ( C,C ) Pareto dominates it Which Side of the Road (Left,Left) and (Right,Right) are Pareto optimal In common-payoff games, all Pareto optimal strategy profiles have the same payoffs Øඏ If (Left,Left) had payoffs (2,2), then (Right,Right) wouldn’t be Pareto optimal Left Right Left 1, 1 0, 0 Right 0, 0 1, 1

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Nau: Game Theory 6 Updated 9/8/11 Best Response Suppose agent i knows how the others are going to play Øඏ Then i has an ordinary optimization problem: maximize expected utility We’ll 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 ( s i , s i ) = ( s 1 , …, s i 1 , s i , s i +1 , …, s n ) s i is a
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lecture2 - CMSC 498T Game Theory 2 Analyzing Normal-Form...

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