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Consistency of equilibrium stacks in finite uniform approximation of a noncooperative game played with staircase-function strategies

The author is technically supported by the Faculty of Mechanical and Electrical Engineering, Polish Naval Academy

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  • A method of finite uniform approximation of an $ N $-person noncooperative game played with staircase-function strategies is presented. A continuous staircase $ N $-person game is approximated to a staircase $ N $-dimensional-matrix game by sampling the player's pure strategy value set. The set is sampled uniformly so that the resulting staircase $ N $-dimensional-matrix game is hypercubic. An equilibrium of the staircase $ N $-dimensional-matrix game is obtained by stacking the equilibria of the subinterval $ N $-dimensional-matrix games, each defined on a subinterval where the pure strategy value is constant. The stack is an approximate solution to the initial staircase game. The (weak) consistency of the approximate solution is studied by how much the players' payoff and equilibrium strategy change as the sampling density minimally increases. The consistency is equivalent to the approximate solution acceptability. An example of a 4-person noncooperative game is presented to show how the approximation is fulfilled for a case of when every subinterval quadmatrix game has pure strategy equilibria.

    Mathematics Subject Classification: Primary: 91A10, 90C59; Secondary: 62C99.


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  • Figure 1.  The players' pure-strategy equilibrium stacks in quadmatrix games (67) with (68)–(75) which approximate the initial staircase-function game with (53)–(62) by $ S = \overline {2,\,\,32} $

    Figure 2.  The players' best pure-strategy equilibrium stacks by $ S = \overline {2,\,\,32} $ and circle-highlighted equilibrium stacks by $ S = \overline {28,\,\,32} $

    Figure 3.  The player's payoff (from equilibria in Figure 2 by which the sum of the players' payoffs is maximal) versus $ S = \overline {2,\,\,32} $ in the 4-person staircase game (approximated by the sampling)

  • [1] E. BorosV. GurvichM. MilaničV. Oudalov and J. Vičič, A three-person deterministic graphical game without Nash equilibria, Discrete Applied Mathematics, 243 (2018), 21-38.  doi: 10.1016/j.dam.2018.01.008.
    [2] R. E. Edwards, Functional Analysis: Theory and Applications, Holt, Rinehart and Winston, New York City, New York, 1965.
    [3] D. Friedman, On economic applications of evolutionary game theory, Journal of Evolutionary Economics, 8 (1998), 15-43. 
    [4] D. Friedman and D. N. Ostrov, Evolutionary dynamics over continuous action spaces for population games that arise from symmetric two-player games, Journal of Economic Theory, 148 (2013), 743-777.  doi: 10.1016/j.jet.2012.07.004.
    [5] J. C. Harsanyi and  R. SeltenA General Theory of Equilibrium Selection in Games, The MIT Press, Cambridge, Massachusetts, 1988. 
    [6] 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), 171-175. 
    [7] R. Kneusel, Random Numbers and Computers, Springer International Publishing, 2018. doi: 10.1007/978-3-319-77697-2.
    [8] Y. Lin and W. Zhang, Pareto efficiency in the infinite horizon mean-field type cooperative stochastic differential game, Journal of the Franklin Institute, 358 (2021), 5532-5551.  doi: 10.1016/j.jfranklin.2021.05.013.
    [9] H. Moulin, Théorie des jeux pour l'économie et la politique, Hermann, Paris, 1981.
    [10] N. NisanT. RoughgardenÉ. Tardos and  V. V. VaziraniAlgorithmic Game Theory, Cambridge University Press, Cambridge, UK, 2007.  doi: 10.1017/CBO9781316779309.
    [11] M. J. OsborneAn Introduction to Game Theory, Oxford University Press, Oxford, UK, 2003. 
    [12] V. V. Romanuke, Uniform sampling of the infinite noncooperative game on unit hypercube and reshaping ultimately multidimensional matrices of players' payoff values, Electrical, Control and Communication Engineering, 8 (2015), 13-19. 
    [13] 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. 
    [14] 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), 1451-1474. 
    [15] 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.  doi: 10.1007/s40840-018-0666-1.
    [16] J. Scheffran, The dynamic interaction between economy and ecology: Cooperation, stability and sustainability for a dynamic-game model of resource conflicts, Mathematics and Computers in Simulation, 53 (2000), 371-380.  doi: 10.1016/S0378-4754(00)00229-9.
    [17] N. N. Vorob'yov, Foundations of Game Theory. Noncooperative Games, Nauka, Moscow, 1984.
    [18] N. N. Vorob'yov, Game Theory for Economists-Cyberneticists, Nauka, Moscow, 1985.
    [19] J. YangY.-S. ChenY. SunH.-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.  doi: 10.1016/j.physa.2017.12.114.
    [20] E. B. Yanovskaya, Antagonistic games played in function spaces, Lithuanian Mathematical Bulletin, 3 (1967), 547-557. 
    [21] D. W. K. Yeung and L. A. Petrosyan, Generalized dynamic games with durable strategies under uncertain planning horizon, Journal of Computational and Applied Mathematics, 395 (2021), Article No. 113595. doi: 10.1016/j.cam.2021.113595.
    [22] R. ZhaoG. NeighbourJ. HanM. McGuire and P. Deutz, Using game theory to describe strategy selection for environmental risk and carbon emissions reduction in the green supply chain, Journal of Loss Prevention in the Process Industries, 25 (2012), 927-936. 
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