Pareto-efficient strategies in 2-person games in staircase-function continuous and finite spaces
A tractable method of solving noncooperative 2-person games in which strategies are staircase functions is suggested. The solution is meant to be Pareto-efficient. The method considers any 2-person staircase-function game as a succession of 2-person games in which strategies are constants. For a finite staircase-function game, each constant-strategy game is a bimatrix game whose size is sufficiently small to solve it in a reasonable time. It is proved that any staircase-function game has a single Pareto-efficient situation if every constant-strategy game has a single Pareto-efficient situation, and vice versa. Besides, it is proved that, whichever the staircase-function game continuity is, any Pareto-efficient situation of staircase function-strategies is a stack of successive Pareto-efficient situations in the constant-strategy games. If a staircase-function game has two or more Pareto-efficient situations, the best efficient situation is found by holding it the farthest from the pair of the most unprofitable payoffs.
Alva, S., & Manjunath, V. (2020). The impossibility of strategy-proof, Pareto efficient, and individually rational rules for fractional matching. Games and Economic Behavior, 119, 15–29.
Belhaiza, S., Audet, C., & Hansen, P. (2012). On proper refinement of Nash equilibria for bimatrix games. Automatica, 48 (2), 297–303. DOI: https://doi.org/10.1016/j.automatica.2011.07.013
Edwards, R. E. (1965). Functional Analysis: Theory and Applications. New York City, New York, USA: Holt, Rinehart and Winston.
Fu, H. (2021). On the existence of Pareto undominated mixed-strategy Nash equilibrium in normal-form games with infinite actions. Economics Letters, 201, 109771.
Harsanyi, J. C., & Selten, R. (1988). A general theory of equilibrium selection in games. Cambridge: The MIT Press.
Kayı, Ç., & Ramaekers, E. (2010). Characterizations of Pareto-efficient, fair, and strategy-proof allocation rules in queueing problems. Games and Economic Behavior, 68 (1), 220–232. DOI: https://doi.org/10.1016/j.geb.2009.07.003
Ke, J., Li, X., Yang, H., & Yin, Y. (2021). Pareto-efficient solutions and regulations of congested ride-sourcing markets with heterogeneous demand and supply. Transportation Research Part E: Logistics and Transportation Review, 154, 102483.
Kim, S., Lee, Y. R., & Kim, M. K. (2019). Flexible risk control strategy based on multi-stage corrective action with energy storage system. International Journal of Electrical Power & Energy Systems, 110, 679–695.
Kontogiannis, S. C., Panagopoulou, P. N., & Spirakis, P. G. (2009). Polynomial algorithms for approximating Nash equilibria of bimatrix games. Theoretical Computer Science, 410 (19), 1599–1606. DOI: https://doi.org/10.1016/j.tcs.2008.12.033
Leyton-Brown, K., & Shoham, Y. (2008). Essentials of game theory: a concise, multidisciplinary introduction. San Rafael, CA: Morgan & Claypool Publishers. DOI: https://doi.org/10.2200/S00108ED1V01Y200802AIM003
Li, Y., Li, K., Xie, Y., Liu, J., Fu, C., & Liu, B. (2020). Optimized charging of lithium-ion battery for electric vehicles: Adaptive multistage constant current–constant voltage charging strategy. Renewable Energy, 146, 2688–2699.
Loesche, F., & Ionescu, T. (2020). Mindset and Einstellung Effect. Encyclopedia of Creativity. Cambridge, Massachusetts, USA: Academic Press, 174–178.
Moulin, H. (1981). Théorie des jeux pour l’économie et la politique. Paris: Hermann Paris.
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. (2018). A couple of collective utility and minimum payoff parity loss rules for refining Nash equilibria in bimatrix games with asymmetric payoffs. Visnyk of Kremenchuk National University of Mykhaylo Ostrogradskyy, 1 (114), 38–43. DOI: https://doi.org/10.30929/1995-0519.2018.1.38-43
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., & Kamburg, V. G. (2016). Approximation of isomorphic infinite two-person noncooperative games via variously sampling the players’ payoff functions and reshaping payoff matrices into bimatrix game. Applied Computer Systems, 20, 5–14. DOI: https://doi.org/10.1515/acss-2016-0009
Romanuke, V. V. (2020). Finite approximation of continuous noncooperative two-person games on a product of linear strategy functional spaces. Journal of Mathematics and Applications, 43, 123–138.
Schelling, T. C. (1980). The Strategy of Conflict. Harvard: Harvard University Press.
Vorob’yov, N. N. (1958). Situations of equilibrium in bimatrix games. Probability theory and its applications, 3, 318–331. DOI: https://doi.org/10.1137/1103024
Vorob’yov, N. N. (1984). Game theory fundamentals. Noncooperative games. Moscow: Nauka.
Vorob’yov, N. N. (1985). Game theory for economist-cyberneticians. Moscow: Nauka.
Zheng, G., Fan, Q., Zhang, T., Zheng, W., Sun, J., Zhou, H., & Diao, Y. (2019). Multistage regulation strategy as a tool to control the vertical displacement of railway tracks placed over the building site of two overlapped shield tunnels. Tunnelling and Underground Space Technology, 83, 282–290.