The meta-theorem for the existence of equilibrium in secure strategies
- Authors: Iskakov M.B.1, Iskakov A.B.1
-
Affiliations:
- V.A. Trapeznikov Institute of Control Sciences of RAS
- Issue: No 111 (2024)
- Pages: 6-65
- Section: Systems analysis
- URL: https://bakhtiniada.ru/1819-2440/article/view/289114
- DOI: https://doi.org/10.25728/ubs.2024.111.1
- ID: 289114
Cite item
Full Text
Abstract
The paper is a continuation of the cycle of papers in 2018-2023 devoted to the theoretical justification of equilibrium in secure strategies (EinSS) as a concept of solving non-cooperative games in pure strategies.A method for constructing EinSS existence theorems from known Nash equilibrium (NE) existence theorems is presented. In particular, theorems for the existence of Nash equilibria are formulated in a standard form and are inserted as a condition into the meta-theorem for the existence of EinSS. According to this method, two theorems for the existence of EinSS are derived and proven based on the theorem of Reny (1999) on the existence of Nash equilibria. The general scheme of deriving existence theorems is as follows. Section 2 summarizes the theorems published in the author's previous papers. Section 3 presents two original theorems from Reny's paper. Section 4 gives a detailed interpretation of the conditions of Reni's theorems, compared to the conditions of Debre's theorem. Section 5 gives a detailed analysis of Reni's theorem. Using a number of examples, the condition of the theorem is interpreted as the condition that there are no jump points or points that guarantee the best answer. Section 6 constructs formally, by the method of meta-theorem, two existence criteria for EinSS that use the original NE existence theorems. In Sections 7 and 8, two theorems are formulated and proved, which are specifically refined for solving applied problems (Hotelling's spatial competition, Tullock's rent competition, Bertrand – Edgeworth oligopoly). All considered theorems are summarized in a final table.
About the authors
Mikhail Borisovich Iskakov
V.A. Trapeznikov Institute of Control Sciences of RAS
Email: mih_iskakov@mail.ru
Moscow
Alexey Borisovich Iskakov
V.A. Trapeznikov Institute of Control Sciences of RAS
Email: isk_alex@mail.ru
Moscow
References
- ИСКАКОВ М.Б. Равновесие в безопасных стратегиях // Управление большими системами. – 2004. – Вып. 9: «Лаборатория активных систем: 30 лет». – С. 145–157.
- ИСКАКОВ М.Б. Равновесие в безопасных стратегиях // Автоматика и телемеханика. – 2005. – №3. – С. 139–153. (ISKAKOV M.B. Equilibrium in Safe Strategies // Automation and Remote Control. – 2005. – Vol. 66, No.3. – P. 465-478).
- ИСКАКОВ М.Б. Теоремы существования равновесия Нэша и равновесия в безопасных стратегиях // Журнал новой экономической ассоциации. – 2022. – №4(56). – С. 12–27.
- ИСКАКОВ М.Б., ИСКАКОВ А.Б. Равновесие в без-опасных стратегиях как развитие концепции равнове-сия Нэша // Математическая теория игр и ее приложе-ния. – 2023. – Т. 15, Вып. 1. – С. 48–72.
- D’ASPREMONT C., GABSZEWICZ J., THISSE J.-F. On Hotelling’s "Stability in Competition’ // Econometrica. – 1979. – 47(5). – P. 1145–1150.
- BERTRAND J. Review of Cournot’s ’Rechercher sur la theoric mathematique de la richesse’ // Journal des Savants. – 1883. – P. 499-508.
- BAGH A., JOFRE A. Reciprocal upper semicontinuity and better reply secure games: a comment // Econometrica. – 2006. – Vol. 74. – P. 1715–1721.
- BICH P. Existence of pure Nash equilibriua in discontinu-ous and non quasiconcave games // Int. J. Game Theory. – 2009. – Vol. 38. – P. 395–410.
- DASGUPTA P., MASKIN E. The existence of equilibrium in discontinuous economic games, I: Theory // Rev. Econ. Stud. – 1986. – Vol. 53(1). – P. 1–26.
- DEBREU G. A social equilibrium existence theorem // Proc. Natl. Acad. Sci. USA. – 1952. – Vol. 38(10). – P. 886–893.
- EDGEWORTH F.M. Papers relating to Political Econo-my I. – London: Macmillan, 1925.
- ISKAKOV M., ISKAKOV A., D’ASPREMONT C. Games for cautious players: the Equilibrium in Secure Strategies // Games and Economic Behavior. – July 2018. – Vol. 110. – P. 58–70.
- HOTELLING H. Stability in Competition // Econ. J. – 1929. – Vol. 39(153). – P. 41–57.
- RENY P.J. On the existence of pure and mixed strategy Nash equilibria in discontinuous games // Econometrica. – 1999. – Vol. 67(5). – P. 1029–1056.
- ROTHSCHILD M., STGLITZ J.E. Equilibrium in competi-tive insurance markets: an essay on the economics of imper-fect information // Q. J. of Econ. – 1976. – Vol. 90. – P. 629-649.
- SIMON L. Games with Discontinuous Payoffs // Review of Economic Studies. – 1987. – Vol. 54. – P. 569–597
- SKAPERDAS S. Contest success functions // Economic Theory. – 1994. – Vol. 7. – P. 283–290.
- TULLOCK G. The welfare costs of tariffs, monopoly and theft // Western Econ. J. – 1967. – Vol. 5. – P. 224–232.
- TULLOCK G. Efficient rent seeking // BUCHANAN J.M., TOLLISON R.D., TULLOCK G. E. Toward a theory of the rent-seeking society – College Station, TX: Texas A&M University Press, 1980. – P. 97–112.
- WILSON C. A model of insurance markets with incomplete information // J. Econ. Theory. – 1977. – Vol. 16. – P. 167-207.
Supplementary files
