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Unformatted text preview: ete lattice and f : S → S an increasing function. Then, the set of fixed points of f , denoted by E , is nonempty and (E , ≥) is a complete lattice. f(s) f(s) s s 19 Game Theory: Lecture 7 Supermodular Games Supermodularity of a Function Definition Let (X , ≥) be a lattice. A function f : X → R is supermodular on S if for all x , y ∈ X f (x ) + f (y ) ≤ f (x ∧ y ) + f (x ∨ y ). Note that supermodularity is automatically satisfied if X is single dimensional. 20 Game Theory: Lecture 7 Supermodular Games Monotonicity of Optimal Solutions From now on, we will assume that X ⊆ R. The following analysis and theory extends to the case where X is a lattice. We first study the monotonicity properties of optimal solutions of parametric optimization problems. Consider a problem x (t ) = arg max f (x , t ), x ∈X where f : X × T → R, X ⊆ R, and T is some partially ordered set. We will mostly focus on T ⊆ RK with the usual vector order, i.e., for some x , y ∈ T , x ≥ y if and only if xi ≥ yi for all i = 1, . . . , k . We are interested in conditions under which we can establish that x (t ) is a nondecreasing function of t . 21 Game Theory: Lecture 7 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. 22 Game Theory: Lecture 7 Supermodular Games Increasing Differences Lemma Let X ⊂ R and T ⊂ Rk 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 ∂ ti 23 Game Theory: Lecture 7 Supermodular Games Example I – Network effects (positive externalities) A set I of users can use one of two products X and Y (e.g., Blu-ray and HD DVD). Bi (J , k ) denotes payoff to i when a subset J of users use k and i ∈ J . There exists a p ositive externality if Bi (J , k ) ≤ Bi (J � , k ), when J ⊂ J � , i.e., player i better off if more users use the same technology as him. This leads to a strategic form game with actions Si =...
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This document was uploaded on 03/19/2014 for the course EECS 6.254 at MIT.

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