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A note on robust Nash equilibriawith uncertainties∗

Published online by Cambridge University Press:  30 April 2014

Vianney Perchet
Vianney Perchet*
Affiliation:
Laboratoire de Probabilités et de Modèles Aléatoires, UMR 7599, Université Paris Diderot, 8 place FM/13, 75013 Paris, France. vianney.perchet@normalesup.org
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Abstract

In this short note, we investigate the framework where agents or players have some uncertainties upon their payoffs or losses, the behavior (or the type, number or any other characteristics) of other players. More specifically, we introduce an extension of the concept of Nash equilibria that generalize different solution concepts called by their authors, and depending on the context, either as robust, ambiguous, partially specified or with uncertainty aversion. We provide a simple necessary and sufficient condition that guarantees its existence and we show that it is actually a selection of conjectural (or self-confirming) equilibria. We finally conclude by how this concept can and should be defined in games with partial monitoring in order to preserve existence properties.

Keywords

Robust gamesrobust Nash equilibriauncertaintiespartial monitoringconjectural equilibria
Type
Research Article
Information
RAIRO - Operations Research , Volume 48 , Issue 3 , July 2014 , pp. 365 - 371
Copyright
© EDP Sciences, ROADEF, SMAI, 2014

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References

M. Aghassi and D. Bertsimas, Robust game theory. Math. Program., Ser. B 107 (2006) 231–273.
Azrieli, Y., On pure conjectural equilibrium with non-manipulable information. Int. J. Game Theory 38 (2009) 209–219. Google Scholar
Bade, S., Ambiguous act equilibria. Games. Econ. Behav. 71 (2010) 246–260. Google Scholar
P. Battigalli and D. Guaitoli, Conjectural equilibrium. Mimeo (1988).
Ben-Tal, A. and Nemirovski, A., Robust convex optimization. Math. Oper. Res. 23 (1998) 769–805. Google Scholar
A. Ben-Tal, L. El Ghaoui and A. Nemirovski, Robust optimization. Princeton University Press (2009).
Ben-Tal, A., Golany, B. and Shtern, S., Robust multi echelon multi period inventory control. Eur. J. Oper. Res. 199 (2009) 922–935. Google Scholar
D. Bertsimas. and A. Thiele, Robust and data-driven optimization: modern decision-making under uncertainty, in Tutorials on Oper. Res. (2006) Chap. 4, 122–195.
Candia-Véjar, A., Álvarez-Miranda, E. and Maculan, N., Minmax regret combinatorial optimization problems: an algorithmic perspective. RAIRO – Oper. Res. 45 (2011) 101–129. Google Scholar
Ellsberg, D., Risk, ambiguity and the Savage axiom. Quat. J. Econom. 75 (1961) 643–669. Google Scholar
Fudenberg, D. and Levine, D.K., Self-confirming equilibrium. Econometrica 61 (1993) 523–545. Google Scholar
Gilboa, I. and Schmeidler, D., Maxmin expected utility with a non-unique prior. J. Math. Econ. 61 (1989) 141–153. Google Scholar
Glicksberg, I., A further generalization of the Kakutani fixed point theorem, with applications to Nash equilibrium points. Proc. Am. Math. Soc. 3 (1952) 170–174. Google Scholar
Hitch, C., Uncertainties in operations research. Oper. Res. 8 (1960) 437–445. Google Scholar
Kalai, E. and Lehrer, E., Subjective equilibrium in repeated games. Econometrica 61 (1993) 1231–1240. Google Scholar
P. Klibanoff, Uncertainty, decision and normal form games. Mimeo (1996).
Lehrer, E., Partially specified probabilities: decisions and games. Am. Econ. J. Micro. 4 (2012) 70–100. Google Scholar
Lugosi, G., Mannor, S. and Stoltz, G., Strategies for prediction under imperfect monitoring. Math. Oper. Res. 33 (2008) 513–528. Google Scholar
J.-F. Mertens, S. Sorin and S. Zamir, Repeated games, CORE discussion paper (1994) 9420–9422.
Nash, J.F., Equilibrium points in N-person games. Proc. Natl. Acad. Sci. USA 36 (1950) 48–49. CrossRefGoogle ScholarPubMed
Nash, J.F., Non-cooperative games. Ann. Math. 54 (1951) 286–295. Google Scholar
Perchet, V., Internal regret with partial monitoring, calibration-based optimal algorithms. J. Mach. Learn. Res. 12 (2011) 1893–1921. Google Scholar
F. Riedel and L. Sass, The strategic use of ambiguity. Mimeo (2011).
Rustichini, A., Minimizing regret: the general case. Games Econom. Behav. 29 (1999) 224–243. Google Scholar