Portfolio Construction

Robust Game Theory

Topics - Portfolio Construction

${ numberSection } ${ text }
Robust Game Theory

Classical game theory, which is useful in understanding market behavior, relies on what have been described as four “essential elements” — who is playing, how much information they possess, the actions available to them, and the payoffs for each outcome.

In this paper, we propose a model of games in which players have incomplete information, and we offer ideas on how to optimize outcomes. We address incomplete-information games without private information as well as those involving potentially private information.

Our robust optimization model relaxes the assumptions of Harsanyi’s Bayesian games model and gives a notion of equilibrium that subsumes the ex post equilibrium concept. We also prove the existence of equilibria in any such robust finite game, when the payoff uncertainty set is bounded.

For any robust finite game with bounded polyhedral payoff uncertainty set and finite type spaces, we formulate the set of equilibria as the dimension-reducing, component-wise projection of the solution set of a system of multilinear equations and inequalities. We suggest a computational method for approximately solving such systems and give numerical results of the implementation of this method.

Furthermore, we describe a special class of robust finite games, whose equilibria are precisely those of a related complete-information game with the same number of players and the same action spaces. Using illustrative examples of robust games from this special class, we compare properties of robust finite games with those of their Bayesian-game counterparts. Moreover, we prove that symmetric equilibria exist in symmetric, robust finite games with bounded uncertainty sets.

Published in

Mathematical Programming

This document is not intended to, and does not relate specifically to any investment strategy or product that AQR offers. It is being provided merely to provide a framework to assist in the implementation of an investor’s own analysis and an investor’s own view on the topic discussed herein.

This document has been provided to you solely for information purposes and does not constitute an offer or solicitation of an offer or any advice or recommendation to purchase any securities or other financial instruments and may not be construed as such. The factual information set forth herein has been obtained or derived from sources believed by the author and AQR Capital Management, LLC (“AQR”) to be reliable but it is not necessarily all-inclusive and is not guaranteed as to its accuracy and is not to be regarded as a representation or warranty, express or implied, as to the information’s accuracy or completeness, nor should the attached information serve as the basis of any investment decision. This document is not to be reproduced or redistributed to any other person. The information set forth herein has been provided to you as secondary information and should not be the primary source for any investment or allocation decision. Past performance is not a guarantee of future performance. 

This material is not research and should not be treated as research. This paper does not represent valuation judgments with respect to any financial instrument, issuer, security or sector that may be described or referenced herein and does not represent a formal or official view of AQR. The views expressed reflect the current views as of the date hereof and neither the author nor AQR undertakes to advise you of any changes in the views expressed herein. 

The information contained herein is only as current as of the date indicated, and may be superseded by subsequent market events or for other reasons. Charts and graphs provided herein are for illustrative purposes only. The information in this presentation has been developed internally and/or obtained from sources believed to be reliable; however, neither AQR nor the author guarantees the accuracy, adequacy or completeness of such information. Nothing contained herein constitutes investment, legal, tax or other advice nor is it to be relied on in making an investment or other decision. There can be no assurance that an investment strategy will be successful. Historic market trends are not reliable indicators of actual future market behavior or future performance of any particular investment which may differ materially, and should not be relied upon as such. Diversification does not eliminate the risk of experiencing investment losses.

The information in this paper may contain projections or other forward-looking statements regarding future events, targets, forecasts or expectations regarding the strategies described herein, and is only current as of the date indicated. There is no assurance that such events or targets will be achieved, and may be significantly different from that shown here. The information in this document, including statements concerning financial market trends, is based on current market conditions, which will fluctuate and may be superseded by subsequent market events or for other reasons.