A lattice-theoretic approach to a class of dynamic games
type: Item
instance of: scholarly article
title: A lattice-theoretic approach to a class of dynamic games
zbMATH Open document ID: 0684.90108
DOI: 10.1016/0898-1221(89)90100-4
author: Yanyan Li
publication date: 1989-01-01T00:00:00Z
review text: A dynamic game is considered in which the dynamics of common-property capital is governed by \(x_{t+1}=f(x_ t-c^ 1_ t-c^ 2_ t)\), where \(c^ i_ t\) is the consumption of player i at stage t. Player i wants to maximize its payoff \[ \sum^{\infty}_{t=0}\delta^ t_ iu_ i(c^ t_ i), \] where \(\delta_ i\) is a discount factor. The functions f, \(u_ 1\) and \(u_ 2\) are given and the most essential assumptions with respect to these functions is that they are isotone (concave and increasing). For this two-person nonzero-sum game the existence of stationary Nash strategies is proved. These strategies, \(c^ i_ t=\gamma_ i(x_ t)\), are lower semi-continuous. The proof itself uses the Schauder- Tychonoff fixed-point theorem. It rests on a number of lemmas in which Tarski's fixed-point theorem and a weak version of Topkis' theorem are used. This latter theorem is cast within lattice-theoretic concept. An extension to n-player games is probably not difficult.
Mathematics Subject Classification ID: 91A20
Mathematics Subject Classification ID: 91A05
zbMATH DE Number: 4121799
zbMATH Keywords: dynamic game
zbMATH Keywords: stationary Nash strategies
zbMATH Keywords: fixed-point theorem
zbMATH Keywords: lattice- theoretic concept
reviewed by: Geert Jan Olsder
MaRDI profile type: Publication
full work available at URL: 0898-1221(89)90100-4
OpenAlex ID: W2048232860
cites work: Non-cooperative games
cites work: Equilibrium Points in Nonzero-Sum n-Person Submodular Games
cites work: Dynamic noncooperative game theory
cites work: Discounted Dynamic Programming
cites work: Minimizing a Submodular Function on a Lattice
cites work: A lattice-theoretical fixpoint theorem and its applications
cites work: Dynamic programming and stochastic control
cites work: Q3968042
published in: Computers & Mathematics with Applications
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description: scientific article; zbMATH DE number 4121799