Elements of Lattice Games
Partial order and lattice formalize the idea of direction of movement. Correspondence and fixed point provide the mathematical underpinning for best choices and equilibrium. A lattice game is a strategic game in which players have lattice action spaces. F
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Elements of Lattice Games
Abstract Partial order and lattice formalize the idea of direction of movement. Correspondence and fixed point provide the mathematical underpinning for best choices and equilibrium. A lattice game is a strategic game in which players have lattice action spaces. Foundations of monotone games are set in this framework. Concepts from the theory of normal form games provide the tools to analyze and predict outcomes in lattice games. The theory of lattice games develops these ideas and a diversity of examples highlights their applications. Keywords Partial order · Lattice · Nash equilibrium · Dominance solvable · Globally stable · Lattice game
Chapter 1 points out elements of the mathematical structure useful to study monotone interdependent interactions and equilibrium outcomes emerging from these interactions. This chapter presents some of the underlying mathematical framework useful throughout the book. Important components of this framework include the notions of partial order and lattice. These are used to formulate several ideas helpful for the analysis, including interval in a lattice, complete lattice, lattice set order, correspondence, and the set of fixed points of a correspondence.
© The Author(s) 2021 T. Sabarwal, Monotone Games, https://doi.org/10.1007/978-3-030-45513-2_2
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T. SABARWAL
In addition to this mathematical structure, this chapter develops the framework of lattice games. The class of lattice games is a general construct that naturally subsumes games with strategic complements, games with strategic substitutes, and combinations of the two. It provides a foundation for the study of these three classes that comprise monotone games. It includes additional cases as well. Several results for monotone games can be seen more generally at this level.
2.1
Lattices
A general tool to formalize the idea of direction is partial order on a set. A partially ordered set in which every pair of elements has a (smallest) larger element and a (largest) smaller element is a lattice. Standard material on lattices may be found in Birkhoff (1995) and in Topkis (1998). Results that are unavailable or hard to find elsewhere in the form used here are proved. 2.1.1
Partially Ordered Set
A binary relation on a set X is reflexive, if for every x ∈ X , x x, it is antisymmetric, if for every x, y ∈ X , x y and y x ⇒ x = y, and it is transitive, if for every x, y, z ∈ X , x y and y z ⇒ x z. A partial order on a set X is a binary relation that is reflexive, antisymmetric, and transitive. A partially ordered set , or poset , is a set X along with a partial order on the set, denoted (X, ). For a poset (X, ) and subset A of X , the relative partial order on A is defined as follows. For every x, x ∈ A, x A x ⇔ x x . It follows immediately from the definition that (A, A ) is a poset in the relative partial order. For posets (X, X ) and (Y, Y ), the Cartesian product X × Y is a poset under the product partial order given by (x, y) (x , y ) ⇔ x X x a
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