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Nash equilibrium in games with incomplete preferences


  • Sophie Bade



This paper investigates Nash equilibrium under the possibility that preferences may be incomplete. I characterize the Nash-equilibrium-set of such a game as the union of the Nash-equilibrium-sets of certain derived games with complete preferences. These games with complete preferences can be derived from the original game by a simple linear procedure, provided that preferences admit a concave vector-representation. These theorems extend some results on finite games by Shapley and Aumann. The applicability of the theoretical results is illustrated with examples from oligopolistic theory, where firms are modelled to aim at maximizing both profits and sales (and thus have multiple objectives). Mixed strategy and trembling hand perfect equilibria are also discussed. Copyright Springer-Verlag Berlin/Heidelberg 2005

Suggested Citation

  • Sophie Bade, 2005. "Nash equilibrium in games with incomplete preferences," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 26(2), pages 309-332, August.
  • Handle: RePEc:spr:joecth:v:26:y:2005:i:2:p:309-332 DOI: 10.1007/s00199-004-0541-1

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    References listed on IDEAS

    1. Jun Iritani, 1981. "On Uniqueness of General Equilibrium," Review of Economic Studies, Oxford University Press, vol. 48(1), pages 167-171.
    2. Xavier Vives, 1987. "Small Income Effects: A Marshallian Theory of Consumer Surplus and Downward Sloping Demand," Review of Economic Studies, Oxford University Press, vol. 54(1), pages 87-103.
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    Cited by:

    1. Georgios Gerasimou, 2014. "Dominance Solvable Games with Multiple Payoff Criteria," Discussion Paper Series, Department of Economics 201406, Department of Economics, University of St. Andrews, revised 22 Jan 2018.
    2. Özgür Evren, 2012. "Scalarization Methods and Expected Multi-Utility Representations," Working Papers w0174, Center for Economic and Financial Research (CEFIR).
    3. Georgios, Gerasimou, 2013. "A Behavioural Model of Choice in the Presence of Decision Conflict," SIRE Discussion Papers 2013-25, Scottish Institute for Research in Economics (SIRE).
    4. repec:kap:theord:v:83:y:2017:i:3:d:10.1007_s11238-017-9595-y is not listed on IDEAS
    5. Bosi, Gianni & Herden, Gerhard, 2012. "Continuous multi-utility representations of preorders," Journal of Mathematical Economics, Elsevier, vol. 48(4), pages 212-218.
    6. repec:bla:metroe:v:68:y:2017:i:4:p:947-965 is not listed on IDEAS
    7. M. Caraballo & A. Mármol & L. Monroy & E. Buitrago, 2015. "Cournot competition under uncertainty: conservative and optimistic equilibria," Review of Economic Design, Springer;Society for Economic Design, vol. 19(2), pages 145-165, June.
    8. Evren, Özgür, 2014. "Scalarization methods and expected multi-utility representations," Journal of Economic Theory, Elsevier, vol. 151(C), pages 30-63.
    9. repec:ipg:wpaper:59 is not listed on IDEAS
    10. Andrikopoulos, Athanasios, 2009. "Szpilrajn-type theorems in economics," MPRA Paper 14345, University Library of Munich, Germany.
    11. Sophie Bade, 2016. "Divergent platforms," Theory and Decision, Springer, vol. 80(4), pages 561-580, April.
    12. Carlier, G. & Dana, R.-A., 2013. "Pareto optima and equilibria when preferences are incompletely known," Journal of Economic Theory, Elsevier, vol. 148(4), pages 1606-1623.
    13. G. Carlier & R.-A. Dana & R.-A. Dana, 2014. "Pareto optima and equilibria when preferences are incompletely known," Working Papers 2014-60, Department of Research, Ipag Business School.
    14. Kokkala, Juho & Poropudas, Jirka & Virtanen, Kai, 2015. "Rationalizable Strategies in Games With Incomplete Preferences," MPRA Paper 68331, University Library of Munich, Germany.
    15. Park, Jaeok, 2015. "Potential games with incomplete preferences," Journal of Mathematical Economics, Elsevier, vol. 61(C), pages 58-66.
    16. Athanasios Andrikopoulos, 2017. "Generalizations of Szpilrajn's Theorem in economic and game theories," Papers 1708.04711,
    17. repec:ipg:wpaper:2014-060 is not listed on IDEAS
    18. Eliaz, Kfir & Ok, Efe A., 2006. "Indifference or indecisiveness? Choice-theoretic foundations of incomplete preferences," Games and Economic Behavior, Elsevier, vol. 56(1), pages 61-86, July.
    19. Evren, Özgür & Ok, Efe A., 2011. "On the multi-utility representation of preference relations," Journal of Mathematical Economics, Elsevier, vol. 47(4-5), pages 554-563.


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