Refinement of acyclic-and-asymmetric payoff aggregates of pure strategy efficient Nash equilibria in finite noncooperative games by maximultimin and superoptimality

  • Vadim Romanuke Polish Naval Academy, Gdynia, Poland
Keywords: Finite noncooperative games, efficient equilibria, refinement, maximultimin, superoptimality, metaequilibrium, uncertainty partial reduction.

Abstract

A theory of refining pure strategy efficient Nash equilibria in finite noncooperative games under uncertainty is outlined. The theory is based on guaranteeing the corresponding payoffs for the players by using maximultimin, which is an expanded version of maximin. If a product of the players’ maximultimin subsets contains more than one efficient Nash equilibrium, a superoptimality rule is attached wherein minimization is substituted with summation. The superoptimality rule stands like a backup plan, and it is involved if maximultimin fails to produce just a single refined efficient equilibrium (a metaequilibrium). The number of the refinement possible outcomes is 10. There are 3 single-metaequilibrium cases, 3 partial reduction cases, and 4 fail cases. Despite successfulness of refinement drops as the game gets bigger, pessimistic estimation of its part is above 54 % for games with no more than four players.

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References

Bajoori, E., Flesch, J., & Vermeulen, D. (2013). Perfect equilibrium in games with compact action spaces. Games and Economic Behavior, 82, 490–502.

Barelli, P., & Duggan, J. (2015). Purification of Bayes Nash equilibrium with correlated types and interdependent payoffs. Games and Economic Behavior, 94, 1–14.

Belhaiza, S., Audet, C., & Hansen, P. (2012). On proper refinement of Nash equilibria for bimatrix games. Automatica, 48 (2), 297–303.

Dopfer, K., & Potts, J. (2007). The General Theory of Economic Evolution. London: Routledge.

Fudenberg, D., & Tirole, J. (1991). Perfect Bayesian equilibrium and sequential equilibrium. Journal of Economic Theory, 53 (2), 236–260.

Gerardi, D., & Myerson, R. B. (2007). Sequential equilibria in Bayesian games with communication. Games and Economic Behavior, 60 (1), 104–134.

Harsanyi, J. C., & Selten, R. (1988). A general theory of equilibrium selection in games. Cambridge: The MIT Press.

Kohlberg, E., & Mertens, J.-F. (1986). On the strategic stability of equilibria. Econometrica, 54 (5), 1003–1037.

Kumano, T. (2017). Nash implementation of constrained efficient stable matchings under weak priorities. Games and Economic Behavior, 104, 230–240.

Leinfellner, W., & Köhler, E. (1998). Game theory, experience, rationality. foundations of social sciences, economics and ethics in honor of John C. Harsanyi. Netherlands: Springer.

Leyton-Brown, K., & Shoham, Y. (2008). Essentials of game theory: a concise, multidisciplinary introduction. San Rafael, CA: Morgan & Claypool Publishers.

Liu, S., & Forrest, J. Y. L. (2010). Grey Systems: Theory and Applications. Berlin: Springer.

Marden, J. R. (2017). Selecting efficient correlated equilibria through distributed learning. Games and Economic Behavior, 106, 114–133.

Mertens, J.-F. (1995). Two examples of strategic equilibrium. Games and Economic Behavior, 8 (2), 378–388.

Myerson, R. B. (1978). Refinements of the Nash equilibrium concept. International Journal of Game Theory, 7 (2), 73–80.

Myerson, R. B. (1997). Game theory: analysis of conflict. Harvard: Harvard University Press.

Osborne, M. J. (2003). An introduction to game theory. Oxford: Oxford University Press.

Romanuke, V. V. (2010). Determining and applying the set of superoptimal pure strategies in some antagonistic games with nonempty set of saddle points in pure strategies. Visnyk of Zaporizhzhya National University. Physical and Mathematical Sciences, 2, pp. 120–125.

Romanuke, V. V. (2010). Environment guard model as dyadic three-person game with the generalized fine for the reservoir pollution. Ecological Safety and Nature Management, 6, pp. 77–94.

Romanuke, V. V. (2016). Sampling individually fundamental simplexes as sets of players’ mixed strategies in finite noncooperative game for applicable approximate Nash equilibrium situations with possible concessions. Journal of Information and Organizational Sciences, 40 (1), 105–143.

Romanuke, V. V. (2016). Approximate equilibrium situations with possible concessions in finite noncooperative game by sampling irregularly fundamental simplexes as sets of players’ mixed strategies. Journal of Uncertain Systems, 10 (4), 269–281.

Romanuke, V. V. (2018). Pure strategy Nash equilibria refinement in bimatrix games by using domination efficiency along with maximin and the superoptimality rule. Research Bulletin of the National Technical University of Ukraine “Kyiv Polytechnic Institute”, 3, 42–52.

Romanuke, V. V. (2018). Acyclic-and-asymmetric payoff triplet refinement of pure strategy efficient Nash equilibria in trimatrix games by maximinimin and superoptimality. KPI Science News, 4, 38–53.

Selten, R. (1975). Reexamination of the perfectness concept for equilibrium points in extensive games. International Journal of Game Theory, 4 (1), 25–55.

Suh, S.-C. (2001). An algorithm for verifying double implementability in Nash and strong Nash equilibria. Mathematical Social Sciences, 41 (1), 103–110.

Tian, G. (2000). Implementation of balanced linear cost share equilibrium solution in Nash and strong Nash equilibria. Journal of Public Economics, 76 (2), 239–261.

Van Damme, E. (1984). A relation between perfect equilibria in extensive form games and proper equilibria in normal form games. International Journal of Game Theory, 13 (1), 1–13.

Vorob’yov, N. N. (1958). Situations of equilibrium in bimatrix games. Probability theory and its applications, 3, 318–331.

Vorob’yov, N. N. (1984). Game theory fundamentals. Noncooperative games. Moscow: Nauka.

Vorob’yov, N. N. (1985). Game theory for economist-cyberneticians. Moscow: Nauka.

Published
2021-07-12
How to Cite
Romanuke, V. (2021). Refinement of acyclic-and-asymmetric payoff aggregates of pure strategy efficient Nash equilibria in finite noncooperative games by maximultimin and superoptimality. Decision Making: Applications in Management and Engineering, 4(2), 178-199. https://doi.org/10.31181/dmame210402178r