Pareto-efficient strategies in 3-person games played with staircase-function strategies

Document Type : Original paper


Faculty of Mechanical and Electrical Engineering, Polish Naval Academy, Gdynia, Poland


A tractable method of solving 3-person games in which players’ pure strategies are staircase functions is suggested. The solution is meant to be Pareto-efficient. The method considers any 3-person staircase-function game as a succession of 3-person games in which strategies are constants. For a finite staircase-function game, each constant-strategy game is a trimatrix game whose size is likely to be relatively 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 one which is the farthest from the triple of the most unprofitable payoffs. In terms of 0-1-standardization, the best efficient situation is the farthest from the triple of zero payoffs.


Main Subjects

[1] A. S. Belenky, A 3-person game on polyhedral sets, Computers & Mathematics with Applications 28 (1994), no. 5, 53–56.
[2] E. Boros, V. Gurvich, M. Milanič, V. Oudalov, and J. Vičič, A three-person deterministic graphical game without Nash equilibria, Discrete Appl. Math. 243 (2018), 21–38.
[3] R. E. Edwards, Functional Analysis: Theory and Applications, Holt, Rinehart and Winston, 1965.
[4] G. Fandel and J. Trockel, Avoiding non-optimal management decisions by applying a three-person inspection game, European Journal of Operational Research 226 (2013), no. 1, 85–93.
[5] J. C. Harsanyi and R. Selten, A General Theory of Equilibrium Selection in Games, The MIT Press, 1988.
[6] D. Hirshleifer, D. Jiang, and Y. M. DiGiovanni, Mood beta and seasonalities in stock returns, Journal of Financial Economics 137 (2020), no. 1, 272–295.
[7] J. Ke, X. Li, H. Yang, and Y. Yin, Pareto-efficient solutions and regulations of congested ride-sourcing markets with heterogeneous demand and supply, Transportation Research Part E: Logistics and Transportation Review 154 (2021), Article ID: 102483.
[8] A. F. Kleimenov and M. A. Schneider, Cooperative dynamics in a repeated three-person game with finite number of strategies, IFAC Proceedings Volumes 38 (2005), no. 1, 171–175.
[9] Y. Lin and W. Zhang, Pareto efficiency in the infinite horizon mean-field type co-operative stochastic differential game, Journal of the Franklin Institute 358 (2021), no. 10, 5532–5551.
[10] F. Loesche and T. Ionescu, Mindset and Einstellung Effect, in: Encyclopedia of Creativity, Academic Press (2020), 174–178.
[11] H. Moulin, Théorie des jeux pour léconomie et la politique, Hermann, 1981. 
[12] N. Nisan, T. Roughgarden, É. Tardos, and V. V. Vazirani, Algorithmic Game Theory, Cambridge University Press, 2007.
[13] V. V. Romanuke, Convergence and estimation of the process of computer implementation of the optimality principle in matrix games with apparent play horizon, Journal of Automation and Information Sciences 45 (2013), no. 10, 49–56.
[14] V. V. Romanuke and V. G. Kamburg, 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 (2016), 5–14.
[15] V. V. Romanuke, Maximization of collective utility and minimization of payoff parity losses for ordering and scaling efficient Nash equilibria in trimatrix games with asymmetric payoffs, Herald of Khmelnytskyi national university. Technical
sciences 3 (2018), 279–281.
[16] V. V. Romanuke, Ecological-economic balance in fining environmental pollution subjects by a dyadic 3-person game model, Applied Ecology and Environmental Research 17 (2019), no. 2, 1451–1474.
[17] V. V. Romanuke, Adaptive finite approximation of continuous noncooperative games, Journal of Automation and Information Sciences 52 (2020), no. 10, 31–41.
[18] V. V. Romanuke, Finite approximation of continuous noncooperative two-person games on a product of linear strategy functional spaces, Journal of Mathematics and Applications 43 (2020), 123–138.
[19] L. Sääksvuori and A. Ramalingam, Bargaining under surveillance: Evidence from a three-person ultimatum game, Journal of Economic Psychology 51 (2015), 66–78.
[20] N. N. Vorob’yov, Game theory fundamentals. Noncooperative games, Nauka, 1984.
[21] N. N. Vorob’yov, Game theory for economists-cyberneticists, Nauka, 1985. 
[22] J. Yang, Y.-S. Chen, Y. Sun, H.-X. Yang, and Y. Liu, Group formation in the spatial public goods game with continuous strategies, Physica A: Statistical Mechanics and its Applications 505 (2018), 737–743.