A new tensor approach of computing pure and mixed Unilateral Support Equilibria

Safatly, Elias; Abdou, Joanna E.

RAIRO. Operations Research, Tome 55 (2021) no. 2, pp. 395-413

In this paper, pure unilateral support equilibrium (USE) is located among pure Nash and pure Berge equilibrium using tensors. The differences between these equilibria are shown using tensor form of a game and are illustrated with numerical examples. Tensors will help specify the location of each equilibrium using a system of coordinates that brings a solid mathematical foundation of all equilibria and provides the possibility to solve high dimensional problems. A numerical example with a 15-player game is studied to demonstrate the efficiency. Besides, we extend the notion of pure USE to mixed USE when the sets of strategies of all players are finite. We prove a lemma dedicated to inaugurate a method of computing mixed USE profiles. We write corresponding formulas using tensors and their operations, and then we illustrate the new method and lemma method by a numerical example of a 7-player game.

Classification : 91A99
Keywords: Game theory, Pure unilateral support equilibrium, mixed unilateral support equilibrium, Berge equilibrium, Nash equilibrium, tensors

[1] K. Abalo and M. Kostreva, Some existence theorems of nash and berge equilibria. Appl. Math. Lett. 17 (2004) 569–573. | MR | Zbl | DOI

[2] K. Abalo and M. Kostreva, Berge equilibrium: some recent results from fixed-point theorems. Appl. Math. Comput. 169 (2005) 624–638. | MR | Zbl

[3] J. E. Abdou, E. Safatly, B. Nakhle and A. El Khoury, High-dimensional nash equilibria problems and tensors applications. Int. Game Theory Rev. 19 (2017) 1750015. | MR | Zbl | DOI

[4] B. W. Bader, T. G. Kolda, et al. Matlab tensor toolbox version 2.6. Available online (2015).

[5] C. Berge, Théorie générale des jeux à $n$ personnes. In Vol. 138 of Mémorial des sciences mathématiques. Gauthier-Villars (1957). | MR | Zbl | Numdam

[6] R. Bishop and S. Goldberg, Tensor analysis on manifolds. Dover Books on Mathematics. Dover Publications (1968). | Zbl

[7] B. R. Cobb and T. Sen, Finding mixed strategy nash equilibria with decision trees. Int. Rev. Econ. Edu. 15 (2014) 43–50. | DOI

[8] A. M. Colman, T. W. Körner, O. Musy and T. Tazdaït, Mutual support in games: some properties of berge equilibria. J. Math. Psychol. 55 (2011) 166–175. | MR | Zbl | DOI

[9] H. W. Corley, A mixed cooperative dual to the nash equilibrium. Game Theory 2015 (2015) 7. | MR | Zbl | DOI

[10] P. Courtois, R. Nessah and T. Tazdaït, How to play games? Nash versus berge behaviour rules. Econ. Philos. 31 (2015) 123–139. | DOI

[11] P. Courtois, R. Nessah and T. Tazdaït, Existence and computation of berge equilibrium and of two refinements. J. Math. Econ. 72 (2017) 7–15. | MR | Zbl | DOI

[12] B. Crettez, A new sufficient condition for a berge equilibrium to be a Berge-Vaisman equilibrium. J. Quant. Econ. 15 (2017) 451–459. | DOI

[13] B. Crettez, On sugden’s ``mutually beneficial practice’’ and berge equilibrium. Int. Rev. Econ. 64 (2017) 357–366. | DOI

[14] B. Crettez, Unilateral support equilibrium, berge equilibrium, and team problems solutions. J. Quant. Econ. 17 (2019) 727–739. | DOI

[15] B. Crettez and R. Nessah, On the existence of unilateral support equilibrium. Math. Soc. Sci. 105 (2020) 41–47. | MR | Zbl | DOI

[16] D. Fudenberg and J. Tirole, Game Theory. MIT Press, Cambridge, MA, USA (1991). | MR

[17] R. Gibbons, Game Theory for Applied Economists. Princeton University Press, Princeton, NJ, USA (1992).

[18] Z.-H. Huang and L. Qi, Formulating an $n$-person noncooperative game as a tensor complementarity problem. Comput. Optim. Appl. 66 (2017) 557–576. | MR | Zbl | DOI

[19] H. A. L. Kiers, Towards a standardized notation and terminology in multiway analysis. J. Chemom. 14 (2000) 105–122. | DOI

[20] T. G. Kolda and B. W. Bader, Tensor decompositions and applications. SIAM Rev. 51 (2009) 455–500. | MR | Zbl | DOI

[21] M. Larbani and R. Nessah, A note on the existence of berge and berge-nash equilibria. Math. Soc. Sci. 55 (2008) 258–271. | MR | Zbl | DOI

[22] Y. Li, Centering, trust region, reflective techniques for nonlinear minimization subject to bounds. Technical report, Ithaca, NY, USA (1993).

[23] O. Musy, A. Pottier and T. Tazdaït, A new theorem to find berge equilibria. Int. Game Theory Rev. 14 (2012) 1250005. | MR | Zbl | DOI

[24] J. F. Nash, Equilibrium points in $n$-person games. Proc. Nat. Acad. Sci. 36 (1950) 48–49. | MR | Zbl | DOI

[25] R. Nessah and M. Larbani, Berge-Zhukovskii equilibria: existence and characterization. Int. Game Theory Rev. 16 (2014) 1450012. | MR | Zbl | DOI

[26] R. Nessah, M. Larbani and T. Tazdaït, A note on berge equilibrium. Appl. Math. Lett. 20 (2007) 926–932. | MR | Zbl | DOI

[27] J. V. Neumann and O. Morgenstern, Theory of Games and Economic Behavior. Princeton University Press, Princeton, NJ, USA (1944). | MR | Zbl

[28] M. J. Osborne and A. Rubinstein, A Course in Game Theory. MIT Press, Cambridge, MA, USA (1994). | MR | Zbl

[29] E. Safatly and J. E. Abdou, Locating pure and mixed berge equilibria using tensor form. Submitted (2019).

[30] J. Schouten, P. Borm and R. Hendrickx, Unilateral support equilibria. J. Math. Psychol. 93 (2019). | MR | Zbl | DOI

[31] C. Semay and B. Silvestre-Brac, Introduction au calcul tensoriel: Applications à la physique. Sciences sup. Dunod (2007).

[32] R. L. Trivers, The evolution of reciprocal altruism. Q. Rev. Biol. 46 (1971) 35–57. | DOI

[33] V. I. Zhukovskii, Some problems of nonantagonistic differential games. In Matematiceskie Metody v Issledovanii Operacij [Mathematical Methods in Operations Research] Edited by P. Kenderov. Bulgarian Academy of Sciences (1985) 103–195.