Elsevier

Games and Economic Behavior

Volume 126, March 2021, Pages 453-459
Games and Economic Behavior

On sequences of iterations of increasing and continuous mappings on complete lattices

https://doi.org/10.1016/j.geb.2021.01.011Get rights and content

Abstract

We generalize the famous Tarski result by showing that: if X is a complete lattice, and f:XX is an increasing and continuous mapping, then for all points x0X, the limits of sequences (fn(limsupkfk(x0)))n=1 and (fn(liminfkfk(x0)))n=1 are fixed points of f. These limits are the tight fixed-point bounds between which sufficiently large iterations fk(x0) are located. We provide an application of this result to studying best-response dynamics.

Introduction

The celebrated Tarski (1955) fixed-point theorem1 says that an increasing (or order-preserving) mapping f on a complete lattice has a fixed point. Moreover, the set of fixed points is also a complete lattice. The lowest fixed point is the “limit” of the sequence of iterations of the lowest element of the lattice, and the highest fixed point is the “limit” of the sequence of iterations of the highest element of the lattice. In the general case, these sequences have to be transfinite, but if mapping f is in addition continuous, it suffices to study sequences indexed by natural numbers.

The Tarski theorem has numerous applications in studying discrete dynamic processes. In economics, it has also been used to prove equilibrium existence in supermodular games (see Topkis (1979) and Vives (1990)), and to prove the existence of stable matchings (Adachi (2000) and Fleiner (2003)). Echenique (2007) provides an algorithm for finding all Nash equilibria in the games of strategic complements by referring to arguments analogous to those yielding the Tarski theorem. Nishimura and Ok (2012) provide some other applications to optimization and games.

However, Tarski's theorem is not fully satisfying for the following reason. It says nothing about the sequences of iterations of x, when x is neither the lowest nor the highest element of a lattice. Such sequences often appear in the analysis of dynamic processes, e.g., the best-response dynamics in games. Olszewski (forthcoming) fills the gap by showing that the properly defined limit superior and limit inferior of the sequences of iterations are fixed points of f for all x. These limits are the tight fixed-point bounds between which all sufficiently large transfinite iterations are located. Olszewski's result generalizes Tarski's theorem to all x. However, as the Tarski theorem itself, it requires using transfinite sequences.

In present paper, we show that if mapping f is in addition continuous, finding the tight fixed-point bounds does not require using any transfinite sequences. The fact that continuity lets us achieve fixed points without referring to transfinite iterations is important in applications. Indeed, the methods of achieving Nash equilibria in supermodular games (or the other equilibrium concepts in economics) via Tarski's theorem became popular partly because of its computability.2 However, the limits superior and inferior (denoted by limsupkfk(x) and liminfkfk(x)) of the sequence of iterations (fk(x))k=1 indexed by natural numbers are not necessarily fixed points of f. So, the tight fixed-point bounds for increasing and continuous mappings are obtained by taking the limits of the sequences (fn(limsupkfk(x)))n=1 and (fn(liminfkfk(x)))n=1, which are shown to be fixed points.

This result delivers another method of finding fixed points. As an application, we provide tight bounds in Nash equilibria for the location of players action profiles under best-response dynamics for any given initial action profile x0. We prove this result for a large class of games, which includes what we call continuous supermodular games. Milgrom and Roberts (1990) showed that the lowest and the highest Nash equilibria are bounds for supermodular games, which in addition are uniform for all x0. The bounds in this paper are of course tighter, usually strictly, but unlike the Milgrom and Roberts bounds they depend on x0. In the online appendix, we propose another “application,”3 namely, an improvement to the Echenique (2007) algorithm for finding all Nash equilibria in games of strategic complements. Unlike the Echenique algorithm, my algorithm finds not only Nash equilibria, but also all 2-period cycles in two-player games, that is, it finds all pairs of action profiles (x1,x2)(y1,y2) such that xi is player i's best response to yi, and yi is player i's best response to xi for i=1,2. The drawback of my algorithm is that it applies only to two-player games.

Section snippets

Preliminaries

Throughout the paper (X,) is always a complete lattice. Completeness postulates that for all subsets AX there exist an infimum (the greatest lower bound) and a supremum (the least upper bound). They will be denoted by ⋀A and ⋁A, respectively. A monotonic (increasing or decreasing) sequence (xk)k=1X converges to an element x0X ifx0=k1xk or x0=k1xk,respectively. We then write that limkxk=x0. Of course, all monotonic sequences converge by completeness. The notation xkx0 or xkx0 says

Main result

The following theorem is the main result of this paper.

Theorem 1

Suppose that (X,) is a complete lattice, and f:XX is a continuous and increasing mapping. For any given x0X, let xω+n=fn(limsupkfk(x0)) and x_ω+n=fn(liminfkfk(x0)) for n=1,2,....

Then: (i) sequence (xω+n)n=1 is weakly increasing, and sequence (x_ω+n)n=1 is weakly decreasing; (ii) x=limnxω+n and x_=limnx_ω+n are fixed points of f.

In addition, if y_ and y are fixed points of f for which there exist an increasing sequence (y_k)k=

Adaptive dynamics Nash-equilibrium bounds

The purpose of this section is to provide the tightest possible Nash-equilibrium bounds for the location of players' actions under best-response dynamics. We will provide these bounds for any normal-form game with the following properties:

(A) The set of actions Xi of each player i is a complete lattice.

(B) Each player i has the lowest and the highest best response to each action profile of the opponents.

Property (B) allows for defining the following two best-response mappings: f_i:XiXi

References (10)

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