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lecture8 notes - 6.254 Game Theory with Engineering...

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6.254 : Game Theory with Engineering Applications Lecture 8: Supermodular and Potential Games Asu Ozdaglar MIT March 2, 2010 1
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Game Theory: Lecture 8 Introduction Outline Review of Supermodular Games Potential Games Reading: Fudenberg and Tirole, Section 12.3. Monderer and Shapley, “Potential Games,” Games and Economic Behavior, vol. 14, pp. 124-143, 1996. 2
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Game Theory: Lecture 8 Supermodular Games Supermodular Games Supermodular games are those characterized by strategic complementarities Informally, this means that the marginal utility of increasing a player’s strategy raises with increases in the other players’ strategies . Implication best response of a player is a nondecreasing function of other players’ strategies Why interesting? They arise in many models. Existence of a pure strategy equilibrium without requiring the quasi-concavity of the payoff functions. Many solution concepts yield the same predictions. The equilibrium set has a smallest and a largest element. They have nice sensitivity (or comparative statics) properties and behave well under a variety of distributed dynamic rules. Much of the theory is due to [Topkis 79, 98], [Milgrom and Roberts 90], [Milgrom and Shannon 94], and [Vives 90, 01]. 3
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Game Theory: Lecture 8 Supermodular Games Increasing Differences Key property: Increasing differences . Definition Let X R and T be some partially ordered set. A function f : X × T R has increasing differences in ( x , t ) if for all x x and t t, we have f ( x , t ) f ( x , t ) f ( x , t ) f ( x , t ) . Intuitively : incremental gain to choosing a higher x (i.e., x rather than x ) is greater when t is higher, i.e., f ( x , t ) f ( x , t ) is nondecreasing in t . You can check that the property of increasing differences is symmetric : an equivalent statement is that if t t , then f ( x , t ) f ( x , t ) is nondecreasing in x . The previous definition gives an abstract characterization. The following result makes checking increasing differences easy in many cases. 4
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Game Theory: Lecture 8 Supermodular Games Increasing Differences Lemma Let X R and T R k for some k, a partially ordered set with the usual vector order. Let f : X × T R be a twice continuously differentiable function. Then, the following statements are equivalent: The function f has increasing differences in ( x , t ) . For all t t and all x X , we have f ( x , t ) f ( x , t ) . x x For all x X, t T, and all i = 1, . . . , k, we have 2 f ( x , t ) 0. x t i 5
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Game Theory: Lecture 8 Supermodular Games Monotonicity of Optimal Solutions Key theorem about monotonicity of optimal solutions: Theorem (Topkis) Let X R be a compact set and T be some partially ordered set. Assume that the function f : X × T R is continuous [or upper semicontinuous] in x for all t T and has increasing differences in ( x , t ) .
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