# Equilibrium stacks for a non-cooperative game defined on a product of staircase-function continuous and finite strategy spaces

Keywords:
game theory; payoff functional; staircase-function strategy; trimatrix game; approximate equilibrium consistency; equilibrium stacking

### Abstract

A method of finite uniform approximation of 3-person games played with staircase-function strategies is presented. A continuous staircase 3-person game is approximated to a staircase trimatrix game by sampling the player’s pure strategy value set. The set is sampled uniformly so that the resulting staircase trimatrix game is cubic. An equilibrium of the staircase trimatrix game is obtained by stacking the equilibria of the subinterval trimatrix games, each defined on an interval where the pure strategy value is constant. The stack is an approximate solution to the initial staircase game. The (weak) consistency, equivalent to the approximate solution acceptability, is studied by how much the players’ payoff and equilibrium strategy change as the sampling density minimally increases. The consistency includes the payoff, equilibrium strategy support cardinality, equilibrium strategy sampling density, and support probability consistency. The most important parts are the payoff consistency and equilibrium strategy support cardinality (weak) consistency, which are checked in the quickest and easiest way. However, it is practically reasonable to consider a relaxed payoff consistency, by which the player’s payoff change in an appropriate approximation may grow at most by epsilon as the sampling density minimally increases. The weak consistency itself is a relaxation to the consistency, where the minimal decrement of the sampling density is ignored. An example is presented to show how the approximation is fulfilled for a case of when every subinterval trimatrix game has pure strategy equilibria.### References

[1] S. Adlakha, R. Johari, and G. Y. Weintraub, Equilibria of dynamic games with many players: Existence, approximation, and market structure, Journal of Economic Theory, vol. 156,

pp. 269–316, 2015.

[2] J. P. Benoit and V. Krishna, Finitely repeated games, Econometrica, vol. 53, iss. 4, pp. 905–922, 1985.

[3] P. Bernhard and J. Shinar, On finite approximation of a game solution with mixed strategies, Applied Mathematics Letters, vol. 3, iss. 1, pp. 1–4, 1990.

[4] R. E. Edwards, Functional Analysis: Theory and Applications, Holt, Rinehart and Winston, 1965.

[5] J. C. Harsanyi and R. Selten, A General Theory of Equilibrium Selection in Games, The MIT Press, 1988.

[6] K. Leyton-Brown and Y. Shoham, Essentials of game theory: a concise, multidisciplinary introduction, Morgan & Claypool Publishers, 2008.

[7] G. J. Mailath and L. Samuelson, Repeated Games and Reputations: Long-Run Relationships, Oxford University Press, 2006.

[8] R. B. Myerson, Game theory: analysis of conflict, Harvard University Press, 1997.

[9] M. J. Osborne, An introduction to game theory, Oxford University Press, 2003.

[10] V. V. Romanuke, Theory of Antagonistic Games, New World — 2000, 2010.

[11] V. V. Romanuke, Uniform sampling of fundamental simplexes as sets of players’ mixed strategies in the finite noncooperative game for finding equilibrium situations with possible concessions, Journal of Automation and Information Sciences, vol. 47, iss. 9, pp. 76–85, 2015.

[12] V. V. Romanuke, Approximate equilibrium situations with possible concessions in finite noncooperative game by sampling irregularly fundamental simplexes as sets of players’ mixed strategies, Journal of Uncertain Systems, vol. 10, no. 4, pp. 269–281, 2016.

[13] V. V. Romanuke, 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, iss. 1 (115), pp. 38–43, 2018.

[14] V. V. Romanuke, Acyclic-and-asymmetric payoff triplet refinement of pure strategy efficient Nash equilibria in trimatrix games by maximinimin and superoptimality, KPI Science News, no. 4, pp. 38–53, 2018.

[15] V. V. Romanuke, Ecological-economic balance in fining environmental pollution subjects by a dyadic 3-person game model, Applied Ecology and Environmental Research, vol. 17, no. 2, pp. 1451–1474, 2019.

[16] V. V. Romanuke, Finite approximation of continuous noncooperative two-person games on a product of linear strategy functional spaces, Journal of Mathematics and Applications, vol. 43, pp. 123–138, 2020.

[17] N. N. Vorob’yov, Foundations of Game Theory. Noncooperative Games, Nauka, 1984.

[18] N. N. Vorob’yov, Game Theory for Economists-Cyberneticists, Nauka, 1985.

[19] K.-E. Wee and A. Iyer, Consolidating or non-consolidating queues: A game theoretic queueing model with holding costs, Operations Research Letters, vol. 39, iss. 1, pp. 4–10, 2011.

[20] 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, vol. 505, pp. 737–743, 2018.

[21] E. B. Yanovskaya, Antagonistic games played in function spaces, Lithuanian Mathematical Bulletin, no. 3, pp. 547–557, 1967.

[22] E. B. Yanovskaya, Minimax theorems for games on the unit square, Probability theory and its applications, no. 9 (3), pp. 554–555, 1964.

[23] D. Ye and J. Chen, Non-cooperative games on multidimensional resource allocation, Future Generation Computer Systems, vol. 29, iss. 7, pp. 1345–1352, 2013.

[24] P. Young and S. Zamir, Handbook of Game Theory. Volume 4, North Holland, 2015.

[25] R. Zhao, G. Neighbour, J. Han, M. 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, vol. 25, iss. 6, pp. 927–936, 2012.

[26] Z. Zhou and Z. Jin, Optimal equilibrium barrier strategies for time-inconsistent dividend problems in discrete time, Insurance: Mathematics and Economics, vol. 94, pp. 100–108, 2020.

pp. 269–316, 2015.

[2] J. P. Benoit and V. Krishna, Finitely repeated games, Econometrica, vol. 53, iss. 4, pp. 905–922, 1985.

[3] P. Bernhard and J. Shinar, On finite approximation of a game solution with mixed strategies, Applied Mathematics Letters, vol. 3, iss. 1, pp. 1–4, 1990.

[4] R. E. Edwards, Functional Analysis: Theory and Applications, Holt, Rinehart and Winston, 1965.

[5] J. C. Harsanyi and R. Selten, A General Theory of Equilibrium Selection in Games, The MIT Press, 1988.

[6] K. Leyton-Brown and Y. Shoham, Essentials of game theory: a concise, multidisciplinary introduction, Morgan & Claypool Publishers, 2008.

[7] G. J. Mailath and L. Samuelson, Repeated Games and Reputations: Long-Run Relationships, Oxford University Press, 2006.

[8] R. B. Myerson, Game theory: analysis of conflict, Harvard University Press, 1997.

[9] M. J. Osborne, An introduction to game theory, Oxford University Press, 2003.

[10] V. V. Romanuke, Theory of Antagonistic Games, New World — 2000, 2010.

[11] V. V. Romanuke, Uniform sampling of fundamental simplexes as sets of players’ mixed strategies in the finite noncooperative game for finding equilibrium situations with possible concessions, Journal of Automation and Information Sciences, vol. 47, iss. 9, pp. 76–85, 2015.

[12] V. V. Romanuke, Approximate equilibrium situations with possible concessions in finite noncooperative game by sampling irregularly fundamental simplexes as sets of players’ mixed strategies, Journal of Uncertain Systems, vol. 10, no. 4, pp. 269–281, 2016.

[13] V. V. Romanuke, 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, iss. 1 (115), pp. 38–43, 2018.

[14] V. V. Romanuke, Acyclic-and-asymmetric payoff triplet refinement of pure strategy efficient Nash equilibria in trimatrix games by maximinimin and superoptimality, KPI Science News, no. 4, pp. 38–53, 2018.

[15] V. V. Romanuke, Ecological-economic balance in fining environmental pollution subjects by a dyadic 3-person game model, Applied Ecology and Environmental Research, vol. 17, no. 2, pp. 1451–1474, 2019.

[16] V. V. Romanuke, Finite approximation of continuous noncooperative two-person games on a product of linear strategy functional spaces, Journal of Mathematics and Applications, vol. 43, pp. 123–138, 2020.

[17] N. N. Vorob’yov, Foundations of Game Theory. Noncooperative Games, Nauka, 1984.

[18] N. N. Vorob’yov, Game Theory for Economists-Cyberneticists, Nauka, 1985.

[19] K.-E. Wee and A. Iyer, Consolidating or non-consolidating queues: A game theoretic queueing model with holding costs, Operations Research Letters, vol. 39, iss. 1, pp. 4–10, 2011.

[20] 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, vol. 505, pp. 737–743, 2018.

[21] E. B. Yanovskaya, Antagonistic games played in function spaces, Lithuanian Mathematical Bulletin, no. 3, pp. 547–557, 1967.

[22] E. B. Yanovskaya, Minimax theorems for games on the unit square, Probability theory and its applications, no. 9 (3), pp. 554–555, 1964.

[23] D. Ye and J. Chen, Non-cooperative games on multidimensional resource allocation, Future Generation Computer Systems, vol. 29, iss. 7, pp. 1345–1352, 2013.

[24] P. Young and S. Zamir, Handbook of Game Theory. Volume 4, North Holland, 2015.

[25] R. Zhao, G. Neighbour, J. Han, M. 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, vol. 25, iss. 6, pp. 927–936, 2012.

[26] Z. Zhou and Z. Jin, Optimal equilibrium barrier strategies for time-inconsistent dividend problems in discrete time, Insurance: Mathematics and Economics, vol. 94, pp. 100–108, 2020.

Published

2023-10-27

How to Cite

*Statistics, Optimization & Information Computing*,

*12*(1), 45-74. https://doi.org/10.19139/soic-2310-5070-1356

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Research Articles

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