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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
quasiconcavity 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 nontopological 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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 Spring '10
 AsuOzdaglar

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