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Nash and Stackelberg Equilibria in Games with Pay-Off Functions Constructed by Minimum Convolutions of Antagonistic and Private Criteria

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Optimization and Applications (OPTIMA 2022)

Abstract

The paper proposes game models with pay-off functions being convolutions by the operation of taking minimum of two criteria one of which describes competition of players in some common (external) sphere of activity and the other describes private achievements of each player (in internal sphere). Strategies of players are distributions of resources between external and internal spheres. The first criterion of each player depends on strategies of all players; the second depends only on the strategy of given player. It is shown that under some natural assumptions of monotony of criteria such n-person games have good properties, namely, Nash equilibrium exists, is strong, stable and Pareto optimal. For two-person games, in Stackelberg equilibrium both the leader and the follower gain no less than in the best Nash equilibrium and the last belongs to \(\gamma \)-core.

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References

  1. Germeier, Y.B., Vatel, I.A.: Games with hierarchical vector of interests. Isv. AN SSSR. Teknicheskaya kibernetika 1(7), 54–69 (1974)

    Google Scholar 

  2. Kukushkin, N.S.: Strong Nash equilibrium in games with common and complementary local utilities. J. Math. Econ. 68(1), 1–12 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  3. Dixit, A.K., Nalebuff, B.J.: The Art of Strategy: A Game Theorist’s Guide to Success in Business and Life. W.W. Norton Company, New York (2010)

    Google Scholar 

  4. Baliga, S., Maskin, E.: Mechanism design for the environment. In: Handbook of Environmental Economics, vol. 1, pp. 305–324. Elsevier, Amsterdam (2003)

    Google Scholar 

  5. Sefton, M., Shupp, R., Walker, J.M.: The effect of rewards and sanctions in provision of Public Good. Econ. Inq. 45(4), 671–690 (2007)

    Article  Google Scholar 

  6. Fehr, E., Gachter, S.: Cooperation and punishment in public goods experiments. Am. Econ. Rev. 90(4), 980–994 (2000)

    Article  Google Scholar 

  7. Hauert, C., Holmes, M., Doebeli, M.: Evolutionary games and population dynamics: maintenance of cooperation in public goods games. Proc. R. Soc. Lond. B Biol. Sci. 273(1600), 2565–2571 (2006)

    Google Scholar 

  8. Zhang, J., Zhang, C., Cao, M.: How insurance affects altruistic provision in threshold public goods games. Sci. Rep. 5, Article number: 9098 (2015)

    Google Scholar 

  9. Mu, Y., Guo, L.: Towards a theory of game-based non-equilibrium control systems. J. Syst. Sci. Complex. 1(25), 209–226 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  10. Mu, Y.F., Guo, L.: How cooperation arises from rational players? SCIENCE CHINA Inf. Sci. 56(11), 1–9 (2013). https://doi.org/10.1007/s11432-013-4857-y

    Article  MathSciNet  MATH  Google Scholar 

  11. Mu, Y.: Inverse Stackelberg Public Goods Game with multiple hierarchies under global and local information structures. J. Optim. Theory Appl. 163(1), 332–350 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  12. Von, S.H.: Market Structure and Equilibrium. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-12586-7. Marktform und Gleichgewicht, Vienna, 1934. Translated by D. Bazin, L. Urch, R. Hill. (in English)

    Book  Google Scholar 

  13. Basar T., Olsder, G.J.: Dynamic noncooperative game theory. In: The Society for Industrial Applied Mathematics. Academic Press, New York (1999)

    Google Scholar 

  14. Olsder, G.J.: Phenomena in inverse Stackleberg games, Part I: static problems. J. Optim. Theory Appl. 143(3), 589–600 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Pang, J.S., Fukushima, M.: Quasi-variational inequalities, generalized Nash equilibria, and multi-leader-follower games. Comput. Manag. Sci. 2(1), 21–56 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  16. Luo, Z.Q., Pang, J.S., Ralph, D.: Mathematical Programs with Equilibrium Constraints. Cambridge University Press, Cambridge (1996)

    Book  MATH  Google Scholar 

  17. Ye, J., Zhu, D.: New necessary optimality conditions for bilevel programs by combining the MPEC and value function approaches. SIAM J. Optim. 20(4), 1885–1905 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  18. Su, C.L.: Equilibrium problems with equilibrium constraints: stationarities, algorithms and applications. Ph.D. thesis, Stanford University, Stanford (2005)

    Google Scholar 

  19. Shen, H., Basar, T.: Incentive-based pricing for network games with complete and incomplete information. Adv. Dyn. Game Theory 9, 431–458 (2007)

    MathSciNet  MATH  Google Scholar 

  20. Staňková, K, Olsder, G.J., Bliemer, M.C.J.: Bi-level optimal toll design problem solved by the inverse Stackelberg games approach. WIT Trans. Built Environ. 89 (2006)

    Google Scholar 

  21. Staňková, K., Olsder, G.J., De Schutter, B.: On European electricity market liberalization: a game-theoretic approach. Inf. Syst. Oper. Res. 48(4), 267–280 (2010)

    MathSciNet  Google Scholar 

  22. Groot, N., Schutter, B.D., Hellendoorn, H.: On systematic computation of optimal nonlinear solutions for the reverse Stackelberg game. IEEE Trans. Syst. Man Cybern. Syst. 44(10), 1315–1327 (2014)

    Article  Google Scholar 

  23. Groot, N., Schutter, B.D., Hellendoorn, H.: Optimal affine leader functions in reverse Stackelberg games. J. Optim. Theory Appl. 168(1), 348–374 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  24. Gorelik, V., Zolotova, T.: Stackelberg and Nash equilibria in games with linear-quadratic payoff functions as models of public goods. In: Olenev, N.N., Evtushenko, Y.G., Jaćimović, M., Khachay, M., Malkova, V. (eds.) OPTIMA 2021. LNCS, vol. 13078, pp. 275–287. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-91059-4_20

    Chapter  Google Scholar 

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Correspondence to Tatiana Zolotova .

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Gorelik, V., Zolotova, T. (2022). Nash and Stackelberg Equilibria in Games with Pay-Off Functions Constructed by Minimum Convolutions of Antagonistic and Private Criteria. In: Olenev, N., Evtushenko, Y., Jaćimović, M., Khachay, M., Malkova, V., Pospelov, I. (eds) Optimization and Applications. OPTIMA 2022. Lecture Notes in Computer Science, vol 13781. Springer, Cham. https://doi.org/10.1007/978-3-031-22543-7_13

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  • DOI: https://doi.org/10.1007/978-3-031-22543-7_13

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