Lecture8-new

Lecture8-new - 6.254 Game Theory with Engineering...

Info iconThis preview shows pages 1–7. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 6.254 : Game Theory with Engineering Applications Lecture 8: Supermodular and Potential Games Asu Ozdaglar MIT March 2, 2010 1 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 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 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 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 ) ∂ x ≥ ∂ f ( x , t ) ∂ x . For all x ∈ X, t ∈ T, and all i = 1, . . . , k, we have ∂ 2 f ( x , t ) ∂ x ∂ t i ≥ 0. 5 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....
View Full Document

This note was uploaded on 05/08/2010 for the course CS 6.254 taught by Professor Asuozdaglar during the Spring '10 term at MIT.

Page1 / 25

Lecture8-new - 6.254 Game Theory with Engineering...

This preview shows document pages 1 - 7. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online