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Robust game theory

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Abstract

We present a distribution-free model of incomplete-information games, both with and without private information, in which the players use a robust optimization approach to contend with payoff uncertainty. Our ``robust game'' model relaxes the assumptions of Harsanyi's Bayesian game model, and provides an alternative distribution-free equilibrium concept, which we call ``robust-optimization equilibrium,'' to that of the ex post equilibrium. We prove that the robust-optimization equilibria of an incomplete-information game subsume the ex post equilibria of the game and are, unlike the latter, guaranteed to exist when the game is finite and has bounded payoff uncertainty set. For arbitrary robust finite games with bounded polyhedral payoff uncertainty sets, we show that we can compute a robust-optimization equilibrium by methods analogous to those for identifying a Nash equilibrium of a finite game with complete information. In addition, we present computational results.

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References

  1. Ben-Tal, A., Nemirovski, A.: Robust convex optimization. Math. Oper. Res. 23 (4), 769–805 (1998)

    MathSciNet  Google Scholar 

  2. Ben-Tal, A., Nemirovski, A.: Robust solutions of uncertain linear programs. Oper. Res. Lett. 25 (1), 1–13 (1999)

    Article  MathSciNet  Google Scholar 

  3. Ben-Tal, A., Nemirovski, A.: Robust solutions of linear programming problems contaminated with uncertain data. Math. Prog., Ser. A 88, 411–424 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bertsekas, D.: Nonlinear Programming, 2nd edn. Athena Scientific, Belmont, MA, 1999

  5. Bertsimas, D., Pachamanova, D., Sim, M.: Robust linear optimization under general norms. Oper. Res. Lett. 32, 510–516 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bertsimas, D., Sim, M.: Robust discrete optimization and network flows. Math. Prog. 98, 49–71 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  7. Bertsimas, D., Sim, M.: The price of robustness. Oper. Res. 52 (1), 35–53 (2004)

    Article  MathSciNet  Google Scholar 

  8. Bohnenblust, H., Karlin, S.: On a theorem of Ville. In: H. Kuhn, A. Tucker (eds.), Contributions to the theory of games, vol. 1, Princeton UP, Princeton, 1950, pp. 155–160

  9. Borodin, A., El-Yaniv, R.: Online computation and competitive analysis. Cambridge UP, New York, 1998

  10. Brouwer, L.: Uber abbildung von mannigfaltikeiten. Mathematische Annalen 71, 97–115 (1912)

    Article  MATH  Google Scholar 

  11. Courant, R.: Variational methods for the solution of problems of equilibrium and vibrations. Bull. Am. Math. Soc. 49, 1–23 (1943)

    Article  MATH  MathSciNet  Google Scholar 

  12. Crémer, J., McLean, R.: Optimal selling strategies under uncertainty for a discriminating monopolist when demands are interdependent. Econometrica 53 (2), 345–362 (1985)

    Google Scholar 

  13. Datta, R.: Using computer algebra to find Nash equilibria. In: J. Senda (ed.), Proc. 2003 Intl. Symp. on Symb. and Alg. Comp., ACM Press, New York, 2003, pp. 74–79

  14. Debreu, G.: A social equilibrium existence theorem. Proc. Nat. Acad. Sci., USA 38, 886–893 (1952)

    MATH  MathSciNet  Google Scholar 

  15. Dow, J., Werlang, S.: Nash equilibrium under Knightian uncertainty: Breaking down backward induction. J. Econ. Theory 64, 305–324 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  16. El Ghaoui, L., Lebret, H.: Robust solutions to least-squares problems with uncertain data. SIAM J. Matrix Analysis and Applications 18 (4), 1035–1064 (1997)

    Article  Google Scholar 

  17. El Ghaoui, L., Oustry, F., Lebret, H.: Robust solutions to uncertain semidefinite programs. SIAM J. Optim. 9 (1), 33–52 (1999)

    Article  MathSciNet  Google Scholar 

  18. Fiacco, A., McCormick, G.: Nonlinear programming: Sequential unconstrained minimization techniques. John Wiley & Sons, New York, 1968

  19. Fudenberg, D., Levine, D.: The Theory of Learning in Games. Series on Economic Learning and Social Evolution. MIT Press, Cambridge, MA, 1998

  20. Fudenberg, D., Tirole, J.: Game Theory. MIT Press, Cambridge, MA, 1991

  21. Gilboa, I., Schmeidler, D.: Maxmin expected utility with a non-unique prior. J. Math. Econ. 18, 141–153 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  22. Goldberg, A., Wright, A., Karlin, A., Hartline, J., Saks, M.: Competitive auctions, 2002. Submitted to Games and Economic Behavior

  23. Govindan, S., Wilson, R.: A global Newton method to compute Nash equilibria. J. Econ. Theory 110, 65–86 (2003)

    MATH  MathSciNet  Google Scholar 

  24. Harsanyi, J.: Games with incomplete information played by `Bayesian' players, parts I–III. Mgmt. Sci. 14, 159–182,320–334,486–502 (1967, 1968)

    Google Scholar 

  25. Holmström, B., Myerson, R.: Efficient and durable decision rules with incomplete information. Econometrica 51, 1799–1820 (1983)

    MATH  Google Scholar 

  26. Holzman, R., Kfir-Dahav, N., Monderer, D., Tennenholtz, M.: Bundling equilibrium in combinatorial auctions. Games and Economic Behavior 47 (1), 104–123 (2004)

    Article  MathSciNet  Google Scholar 

  27. Hyafil, N., Boutilier, C.: Regret minimizing equilibria and mechanisms for games with strict type uncertainty. In: Proceedings of the 20th Conference on Uncertainty in Artificial Intelligence (AUAI '04), AUAI Press, Arlington, VA, USA, 2004, pp. 268–277

  28. Kakutani, S.: A generalization of Brouwer's fixed point theorem. Duke Math. J. 8, 457–459 (1941)

    Article  MATH  MathSciNet  Google Scholar 

  29. Klibanoff, P.: Uncertainty, decision, and normal form games, 1993. Manuscript, MIT

  30. Knight, F.: Risk, Uncertainty and Profit. Houghton Mifflin, Boston, 1921

  31. Lemke, C., Howson, J.: Equilibrium points of bimatrix games. SIAM J. Appl. Math. 12 (2), 413–423 (1964)

    Article  MathSciNet  Google Scholar 

  32. Lo, K.: Equilibrium in beliefs under uncertainty. J. Econ. Theory 71 (2), 443–484 (1996)

    Article  MathSciNet  Google Scholar 

  33. Ma, T.: Banach-Hilbert spaces, vector measures and group representations. World Scientific, New Jersey, 2002

  34. Marinacci, M.: Ambiguous games. Games and Economic Behavior 31 (2), 191–219 (2000)

    Article  MathSciNet  Google Scholar 

  35. McKelvey, R., McLennan, A.: Computation of equilibria in finite games. In: H. Amman, D. Kendrick, J. Rust (eds.), Handbook of computational economics, vol. I, Elsevier, 1996, pp. 87–142

  36. McKelvey, R., McLennan, A., Turocy, T.: Gambit: Software tools for game theory, version 0.97.0.6, 2004. http://econweb.tamu.edu/gambit/

  37. Mertens, J., Zamir, S.: Formulation of Bayesian analysis for games with incomplete information. Intl. J. Game Theory 14 (1), 1–29 (1985)

    Article  MathSciNet  Google Scholar 

  38. Morris, S.: The common prior assumption in economic theory. Econ. Philosophy 11, 227–253 (1995)

    Article  Google Scholar 

  39. Nash, J.: Equilibrium points in N-person games. Proc. Nat. Acad. Sci., USA 36, 48–49 (1950)

    MATH  MathSciNet  Google Scholar 

  40. Nash, J.: Non-cooperative games. Ann. Math. 54 (2), 286–295 (1951)

    Article  MathSciNet  Google Scholar 

  41. Osborne, M., Rubenstein, A.: A Course in Game Theory. MIT Press, Cambridge, MA, 1994

  42. Papadimitriou, C.: Algorithms, games, and the internet. In: STOC '01: Proc. 33rd Annual ACM Symposium on the Theory of Computing, ACM Press, New York, 2001, pp. 749–753

  43. Porter, R., Nudelman, E., Shoham, Y.: Simple search methods for finding a Nash equilibrium. In: Proceedings of the 19th National Conference on Artificial Intelligence (AAAI-2004), 2004, pp. 664–669

  44. Scarf, H.: The approximation of fixed points of a continuous mapping. SIAM J. Appl. Math. 15 (5), 1328–1343 (1967)

    Article  MathSciNet  Google Scholar 

  45. Scarf, H.: The Computation of Economic Equilibria. Yale UP, New Haven, CT, 1973. In collaboration with T. Hansen

  46. Smart, D.: Fixed Point Theorems. Cambridge UP, New York, 1974

  47. Soyster, A.: Convex programming with set-inclusive constraints and applications to inexact linear programming. Oper. Res. 21, 1154–1157 (1973)

    MATH  MathSciNet  Google Scholar 

  48. Sturmfels, B.: Solving systems of polynomial equations. American Mathematical Society, Providence, RI, 2002

  49. van der Laan, G., Talman, A.: On the computation of fixed points in the product space of unit simplices and an application to noncooperative N person games. Math. Oper. Res. 7, 1–13 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  50. van der Laan, G., Talman, A., van der Heyden, L.: Simplicial variable dimension algorithms for solving the nonlinear complementarity problem on a product of unit simplices using a general labeling. Math. Oper. Res. 12, 377–397 (1987)

    MATH  MathSciNet  Google Scholar 

  51. Verschelde, J.: Algorithm 795: PHCpack: a general-purpose solver for polynomial systems by homotopy continuation. ACM Trans. Math. Softw. 25 (2), 251–276 (1999)

    Article  MathSciNet  Google Scholar 

  52. von Neumann, J., Morgenstern, O.: Theory of Games and Economic Behaviour. Princeton UP, 1944

  53. von Stengel, B.: Computing equilibria for two-person games. In: R. Aumann, S. Hart (eds.), Handbook of game theory with economic applications, vol. 3, chap. 45, Elsevier, 2002, pp. 1723–1759

  54. Wardrop, J.: Some theoretical aspects of road traffic research. In: Proceedings of the Institute of Civil Engineers, vol. 1, part II, 1952, pp. 325–78

  55. Wilson, R.: Game-theoretic approaches to trading processes. In: T. Bewley (ed.), Advances in Economic Theory: Fifth World Congress, chap. 2, Cambridge UP, New York, 1987, pp. 33–77

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Authors and Affiliations

  1. Operations Research Center, Massachusetts Institute of Technology, Cambridge, MA, 02139, USA

    Michele Aghassi

  2. Boeing Professor of Operations Research, Sloan School of Management and Operations Research Center, Massachusetts Institute of Technology, E53-363, Cambridge, MA, 02139, USA

    Dimitris Bertsimas

Authors
  1. Michele Aghassi
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  2. Dimitris Bertsimas
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Correspondence to Dimitris Bertsimas.

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The research of the author was partially supported by a National Science Foundation Graduate Research Fellowship and by the Singapore-MIT Alliance.

The research of the author was partially supported by the Singapore-MIT Alliance.

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Aghassi, M., Bertsimas, D. Robust game theory. Math. Program. 107, 231–273 (2006). https://doi.org/10.1007/s10107-005-0686-0

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