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Denition given a set s and a binary relation the pair

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Unformatted text preview: unction 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]. 15 Game Theory: Lecture 7 Supermodular Games Lattices and Tarski’s Theorem The machinery needed to study supermodular games is lattice theory and monotonicity results in lattice programming. Methods used are non-topological and they exploit order properties We first briefly summarize some preliminaries related to lattices. Definition Given a set S and a binary relation ≥, the pair (S , ≥) is a partially ordered set if ≥ is reflexive (x ≥ x for all x ∈ S ), transitive (x ≥ y and y ≥ z implies that x ≥ z ), and antisymmetric (x ≥ y and y ≥ x implies that x = y ). A partially ordered set (S , ≥) is (completely) ordered if for x ∈ S and y ∈ S , either x ≥ y or y ≥ x . 16 Game Theory: Lecture 7 Supermodular Games Lattices Definition A lattice is a partially ordered set (S , ≥) s.t. any two elements x , y have a least upper bound (supremum), supS (x , y ) = inf {z ∈ S |z ≥ x , z ≥ y }, and a greatest lower bound (infimum), inf S (x , y ) = sup{z ∈ S |z ≤ x , z ≤ y } in the set. Supremum of {x , y } is denoted by x ∨ y and is called the join of x and y . Infimum of {x , y } is denoted by x ∧ y and is called the meet of x and y . Examples: Any interval of the real line with the usual order is a lattice, since any two points have a supremum and infimum in the interval. However, the set S ⊂ R2 , S = {(1, 0), (0, 1)}, is not a lattice with the vector ordering (the usual componentwise ordering: x ≤ y if and only if xi ≤ yi for any i ), since (1, 0) and (0, 1) have no joint upper bound in S . S � = {(0, 0), (0, 1), (1, 0), (1, 1)} is a lattice with the vector ordering. Similarly, the simplex in Rn (again with the usual vector ordering) {x ∈ Rn | ∑i xi = 1, xi ≥ 0} is not a lattice, while the box {x ∈ Rn | 0 ≤ x1 ≤ 1} is. 17 Game Theory: Lecture 7 Supermodular Games Lattices Definition A lattice (S , ≥) is complete if every nonempty subset of S has a supremum and an infimum in S . Any compact interval of the real line with the usual order is a complete lattice, while the open interval (a, b ) is a lattice but is not complete [indeed the supremum of (a, b ) does not belong to (a, b )]. 18 Game Theory: Lecture 7 Supermodular Games Tarski’s Fixed Point Theorem We state the lattice theoretical fixed point theorem due to Tarski. Let (S , ≥) be a partially ordered set. A function f from S to S is increasing if for all x , y ∈ S , x ≥ y implies f (x ) ≥ f (y ). Theorem (Tarski) Let (S , ≥) be a compl...
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