Monotone Comparative Statics without Lattices

TL;DR

This paper extends Monotone Comparative Statics (MCS) theory beyond lattice structures by introducing the pseudo-lattice property, enabling analysis of complex environments like mixed-strategy games and equilibria.

Contribution

It introduces the pseudo-lattice property, generalizing MCS theory to broader settings including mixed-strategy and trembling-hand equilibria.

Findings

Generalized MCS theorems for individual choice and fixed points

Expanded MCS applicability to pseudo quasi-supermodular games

First MCS analysis of mixed-strategy Nash and trembling-hand perfect equilibria

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.