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The equivalence between two-person symmetric games and decision problems

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  • Ismail, M.S.

    (Microeconomics & Public Economics)

Abstract

We illustrate an equivalence between the class of two-person symmetric games and the class of decision problems with a complete preference relation. Moreover, we show that a strategy is an optimal threat strategy (Nash, 1953) in a two-person symmetric game if and only if it is a maximal element in its equivalent decision problem. In particular, a Nash equilibrium in a two-person symmetric zero-sum game and a pair of maximal elements in its equivalent decision problem coincide. In addition, we show that a two-person symmetric zero-sum game can be extended to its von Neumann-Morgenstern (vN-M) mixed extension if and only if the extended decision problem satisfies the SSB utility (Fishburn, 1982) axioms. Furthermore, we demonstrate that a decision problem satisfies vN-M utility if and only if its equivalent symmetric game is a potential game. Accordingly, we provide a formula for the number of linearly independent equations in order for the independence axiom to be satisfied which grows quadratically as the number of alternatives increase.

Suggested Citation

  • Ismail, M.S., 2014. "The equivalence between two-person symmetric games and decision problems," Research Memorandum 023, Maastricht University, Graduate School of Business and Economics (GSBE).
    • Handle: RePEc:unm:umagsb:2014023
    DOI: 10.26481/umagsb.2014023
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    References listed on IDEAS

    as
    1. Nash, John, 1953. "Two-Person Cooperative Games," Econometrica, Econometric Society, vol. 21(1), pages 128-140, April.
    2. David K. Levine & Drew Fudenberg, 2006. "A Dual-Self Model of Impulse Control," American Economic Review, American Economic Association, vol. 96(5), pages 1449-1476, December.
    3. Peter Duersch & Jörg Oechssler & Burkhard Schipper, 2012. "Pure strategy equilibria in symmetric two-player zero-sum games," International Journal of Game Theory, Springer;Game Theory Society, vol. 41(3), pages 553-564, August.
    4. Fishburn, P. C., 1984. "Dominance in SSB utility theory," Journal of Economic Theory, Elsevier, vol. 34(1), pages 130-148, October.
    5. Rosenthal, Robert W., 1981. "Games of perfect information, predatory pricing and the chain-store paradox," Journal of Economic Theory, Elsevier, vol. 25(1), pages 92-100, August.
    6. Roth, Alvin, 2012. "The Shapley Value as a von Neumann-Morgenstern Utility," Ekonomicheskaya Politika / Economic Policy, Russian Presidential Academy of National Economy and Public Administration, vol. 6, pages 1-9.
    7. Gordon Tullock, 1964. "The Irrationality Of Intransitivity," Oxford Economic Papers, Oxford University Press, vol. 16(3), pages 401-406.
    8. Monderer, Dov & Shapley, Lloyd S., 1996. "Potential Games," Games and Economic Behavior, Elsevier, vol. 14(1), pages 124-143, May.
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