A lattice-theoretic approach to a class of dynamic games
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.
scientific article; zbMATH DE number 4121799
scientific article