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Strong robustness to incomplete information and the uniqueness of a correlated equilibrium

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Abstract

We define and characterize the notion of strong robustness to incomplete information, whereby a Nash equilibrium in a game \(\mathbf{u}\) is strongly robust if, given that each player knows that his payoffs are those in \(\mathbf{u}\) with high probability, all Bayesian–Nash equilibria in the corresponding incomplete-information game are close—in terms of action distribution—to that equilibrium of \(\mathbf{u}\). We prove, under some continuity requirements on payoffs, that a Nash equilibrium is strongly robust if and only if it is the unique correlated equilibrium. We then review and extend the conditions that guarantee the existence of a unique correlated equilibrium in games with a continuum of actions. The existence of a strongly robust Nash equilibrium is thereby established for several domains of games, including those that arise in economic environments as diverse as Tullock contests, all-pay auctions, Cournot and Bertrand competitions, network games, patent races, voting problems and location games.

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References

  • Amir, R.: Cournot oligopoly and the theory of supermodular games. Games Econ. Behav. 15, 132–148 (1996)

    Article  Google Scholar 

  • Aumann, R.J.: Subjectivity and correlation in randomized strategies. J. Math. Econ. 1, 67–96 (1974)

    Article  Google Scholar 

  • Ballester, C., Calvó-Armengol, A., Zenou, Y.: Who’s who in networks. Wanted: the key player. Econometrica 74, 1403–1417 (2006)

    Article  Google Scholar 

  • Baye, M.R., Hoppe, H.C.: The strategic equivalence of rent-seeking, innovation, and patent-race games. Games Econ. Behav. 44, 217–226 (2003)

    Article  Google Scholar 

  • Ben-Porat, O., Tennenholtz, M.: Multi-unit facility location games. In: WINE 2016: The 12th Conference on Web and Internet Economics (2016)

  • Bertsekas, D.P., Shreve, S.E.: Stochastic Optimal Control: The Discrete Time Case. Athena Scientific, New York (2004)

    Google Scholar 

  • Borkar, V.S.: Probability Theory: An Advanced Course. Springer, Berlin (1995)

    Book  Google Scholar 

  • Bramoullé, Y., Kranton, R., D’Amours, M.: Strategic interaction and networks. Am. Econ. Rev. 104, 898–930 (2014)

    Article  Google Scholar 

  • Carlsson, H., van Damme, E.: Global games and equilibrium selection. Econometrica 61, 989–1018 (1993)

    Article  Google Scholar 

  • Dasgupta, P., Maskin, E.: The existence of equilibrium in discontinuous economic games, Part I (theory). Rev. Econ. Stud. 53, 1–26 (1986)

  • Dasgupta, P., Stiglitz, J.: Uncertainty, industrial structure, and the speed of R&D. Bell J. Econ. 11, 1–28 (1980)

    Article  Google Scholar 

  • Dekel, E., Fudenberg, D.: Rational behavior with payoff uncertainty. J. Econ. Theory 52, 243–267 (1990)

    Article  Google Scholar 

  • Dixit, A.: Strategic behavior in contests. Am. Econ. Rev. 77, 891–898 (1987)

    Google Scholar 

  • Dütting, P., Kesselheim, T., Tardos, É.: Mechanism with unique learnable equilibria. In: Proceedings of the Fifteenth ACM Conference on Economics and Computation—EC’14, pp. 877–894. ACM Press, New York (2014)

  • Even-dar, E., Mansour, Y., Nadav, U.: On the convergence of regret minimization dynamics in concave games. In: Proceedings of the 41st Annual ACM Symposium on Theory of Computing—STOC’09, pp. 523–532. ACM Press, New York (2009)

  • Ewerhart, C., Quartieri, F.: Unique equilibrium in contests with incomplete information. Econ. Theory 70, 243–271 (2020)

    Article  Google Scholar 

  • Foster, D.P., Vohra, R.V.: Calibrated learning and correlated equilibrium. Games Econ. Behav. 21, 40–55 (1997)

    Article  Google Scholar 

  • Fudenberg, D., Kreps, D.M., Levine, D.K.: On the robustness of equilibrium refinements. J. Econ. Theory 44, 354–380 (1988)

    Article  Google Scholar 

  • Gale, D., Klee, V., Rockafellar, R.T.: Convex functions on convex polytopes. Proc. Am. Math. Soc. 19, 867–873 (1968)

    Article  Google Scholar 

  • Goodman, J.C.: Note on existence and uniqueness of equilibrium points for concave N-Person games. Econometrica 48, 251–251 (1980)

    Article  Google Scholar 

  • Hart, S., Mas-Colell, A.: A simple adaptive procedure leading to correlated equilibrium. Econometrica 68, 1127–1150 (2000)

    Article  Google Scholar 

  • Hart, S., Mas-Colell, A.: Markets, correlation, and regret-matching. Games Econ. Behav. 93, 42–58 (2015)

    Article  Google Scholar 

  • Hart, S., Schmeidler, D.: Existence of correlated equilibria. Math. Oper. Res. 14, 18–25 (1989)

    Article  Google Scholar 

  • Hillman, A.L., Riley, J.G.: Politically contestable rents and transfers. Econ. Polit. 1, 17–39 (1989)

    Article  Google Scholar 

  • Hotelling, H.: Stability in competition. Econ. J. 39, 41 (1929)

    Article  Google Scholar 

  • Kajii, A., Morris, S.: The robustness of equilibria to incomplete information. Econometrica 65, 1283–1309 (1997)

    Article  Google Scholar 

  • Liu, L.: Correlated equilibrium of Cournot oligopoly competition. J. Econ. Theory 68, 544–548 (1996)

    Article  Google Scholar 

  • Loury, G.C.: Market structure and innovation. Q. J. Econ. 93, 395 (1979)

    Article  Google Scholar 

  • Milgrom, P.R., Roberts, J.: Rationalizability, learning, and equilibrium in games with strategic complementarities. Econometrica 58, 1255–1277 (1990)

    Article  Google Scholar 

  • Milgrom, P., Shannon, C.: Monotone comparative statics. Econometrica 62, 157 (1994)

    Article  Google Scholar 

  • Monderer, D., Shapley, L.S.: Potential games. Games Econ. Behav. 14, 124–143 (1996)

    Article  Google Scholar 

  • Neyman, A.: Correlated equilibrium and potential games. Int. J. Game Theory 26, 223–227 (1997)

    Article  Google Scholar 

  • Papadimitriou, C.H., Roughgarden, T.: Computing correlated equilibria in multi-player games. J. ACM 55, 1–29 (2008)

    Article  Google Scholar 

  • Pavlov, G.: Correlated Equilibria and Communication Equilibria in All-Pay Auctions. Research Report # 2013-2 of Western University (2013). https://ir.lib.uwo.ca/cgi/viewcontent.cgi?article=1255&context=economicsresrpt

  • Persson, T., Tabellini, G.E.: Political Economics: Explaining Economic Policy. MIT Press, Cambridge (2000)

    Google Scholar 

  • Reny, P.J.: On the existence of pure and mixed strategy Nash equilibria in discontinuous games. Econometrica 67, 1029–1056 (1999)

    Article  Google Scholar 

  • Rosen, J.B.: Existence and uniqueness of equilibrium points for concave N-Person games. Econometrica 33, 520–534 (1965)

    Article  Google Scholar 

  • Skaperdas, S.: Contest success functions. Econ Theory 7, 283–290 (1996)

    Article  Google Scholar 

  • Szidarovszky, F., Okuguchi, K.: On the existence and uniqueness of pure Nash equilibrium in rent-seeking games. Games Econ. Behav. 18, 135–140 (1997)

    Article  Google Scholar 

  • Tullock, G.: Efficient rent seeking. In: Efficient Rent-Seeking, pp. 3–16. Springer, Boston (2001)

  • Ui, T.: Correlated equilibrium and concave games. Int. J. Game Theory 37, 1–13 (2008)

    Article  Google Scholar 

  • Ui, T.: Bayesian Nash equilibrium and variational inequalities. J. Math. Econ. 63, 139–146 (2016)

    Article  Google Scholar 

  • Viossat, Y.: Is having a unique equilibrium robust? J. Math. Econ. 44, 1152–1160 (2008)

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Economics, Ben-Gurion University of the Negev, Beer Sheva, Israel

    Ezra Einy, Ori Haimanko & David Lagziel

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  1. Ezra Einy
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  2. Ori Haimanko
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  3. David Lagziel
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Correspondence to David Lagziel.

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The authors wish to thank Eddie Dekel, Ehud Lehrer, Yehuda John Levy, Daisuke Oyama, David Schmeidler, Aner Sela, and the participants of SWET 2019 workshop held in Otaru, Japan, for their valuable comments. The authors’ special grattitude goes to Atsushi Kajii, whose encouragement to put the first basic ideas into writing led to this work.

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Einy, E., Haimanko, O. & Lagziel, D. Strong robustness to incomplete information and the uniqueness of a correlated equilibrium. Econ Theory 73, 91–119 (2022). https://doi.org/10.1007/s00199-020-01327-4

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