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Monotone Comparative Statics without Lattices

Yeon-Koo Che, Jinwoo Kim, Fuhito Kojima

TL;DR

The theory is applied to establish existence and monotone comparative statics of Nash equilibria in games with strategic complementarities and of stable many-to-one matchings in two-sided matching problems, allowing for general preferences that accommodate indifferences and incomplete preferences.

Abstract

The theory of Monotone Comparative Statics (MCS) has traditionally required a lattice structure, excluding certain multidimensional environments such as mixed-strategy games where this property fails. We show that this structure is not essential. We introduce a weaker notion, the pseudo-lattice property, and preserve the theory's core results by generalizing the MCS theorems for individual choice and Tarski's fixed-point theorem. Our framework expands comparative statics to pseudo quasi-supermodular games. Crucially, it enables the first MCS analysis of mixed-strategy Nash equilibria and trembling-hand perfect equilibria.

Monotone Comparative Statics without Lattices

TL;DR

The theory is applied to establish existence and monotone comparative statics of Nash equilibria in games with strategic complementarities and of stable many-to-one matchings in two-sided matching problems, allowing for general preferences that accommodate indifferences and incomplete preferences.

Abstract

The theory of Monotone Comparative Statics (MCS) has traditionally required a lattice structure, excluding certain multidimensional environments such as mixed-strategy games where this property fails. We show that this structure is not essential. We introduce a weaker notion, the pseudo-lattice property, and preserve the theory's core results by generalizing the MCS theorems for individual choice and Tarski's fixed-point theorem. Our framework expands comparative statics to pseudo quasi-supermodular games. Crucially, it enables the first MCS analysis of mixed-strategy Nash equilibria and trembling-hand perfect equilibria.

Paper Structure

This paper contains 27 sections, 31 theorems, 74 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Related Literature.
  3. Preliminaries
  4. The structural properties of domain.
  5. Set orders
  6. Individual Choices
  7. MCS characterization of individual choices
  8. Directional weakening.
  9. The structure of the maximizer set
  10. Decision under uncertainty
  11. Fixed Points and Nash Equilibria
  12. Existence and comparative statics of fixed points
  13. Directional weakening.
  14. Pseudo quasi-supermodular games
  15. Application to generalized Bertrand games.
  16. ...and 12 more sections

Key Result

Theorem 1

A chain complete set $X$ is a complete pseudo lattice if and only if it admits both the largest and smallest elements.

Figures (2)

  • Figure 1: A complete pseudo lattice that is not a lattice.
  • Figure 2: Failure of the lattice property

Theorems & Definitions (75)

  • Example 1
  • Example 2: Stochastic dominance order
  • Example 3: Mean-preserving spread/convex order
  • Theorem 1
  • Lemma 1
  • Corollary 1
  • Theorem 2
  • proof
  • Theorem 3: Characterization
  • Proposition 1: Monotonicity
  • ...and 65 more