Dynamic Set Values for Nonzero-Sum Games with Multiple Equilibriums

Zachary Feinstein (), Birgit Rudloff () and Jianfeng Zhang ()
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Zachary Feinstein: Stevens Institute of Technology, School of Business, Hoboken, New Jersey 07030
Birgit Rudloff: Vienna University of Economics and Business, Institute for Statistics and Mathematics, 1020 Vienna, Austria
Jianfeng Zhang: Department of Mathematics, University of Southern California, Los Angeles, California 90089

Mathematics of Operations Research, 2022, vol. 47, issue 1, 616-642

Abstract: Nonzero sum games typically have multiple Nash equilibriums (or no equilibrium), and unlike the zero-sum case, they may have different values at different equilibriums. Instead of focusing on the existence of individual equilibriums, we study the set of values over all equilibriums, which we call the set value of the game. The set value is unique by nature and always exists (with possible value ∅ ). Similar to the standard value function in control literature, it enjoys many nice properties, such as regularity, stability, and more importantly, the dynamic programming principle. There are two main features in order to obtain the dynamic programming principle: (i) we must use closed-loop controls (instead of open-loop controls); and (ii) we must allow for path dependent controls, even if the problem is in a state-dependent (Markovian) setting. We shall consider both discrete and continuous time models with finite time horizon. For the latter, we will also provide a duality approach through certain standard PDE (or path-dependent PDE), which is quite efficient for numerically computing the set value of the game.

Keywords: Primary: 91A25; secondary: 91A15; 91A06; 49L20; nonzero sum game; Nash equilibrium; set value; dynamic programming principle; closed-loop controls; path dependent PDE (search for similar items in EconPapers)
Date: 2022
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Persistent link: https://EconPapers.repec.org/RePEc:inm:ormoor:v:47:y:2022:i:1:p:616-642

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