DOI: https://doi.org/10.15587/1729-4061.2019.187844

Optimizing the strategy of activities using numerical methods for determining equilibrium

Iryna Sievidova, Tamila Oliynik, Oleksandra Mandych, Tetyana Kvyatko, Iryna Romaniuk, Larisa Leshchenko, Serhiy Vynohradenko, Serhii Plyhun

Abstract


The paper considers issues on the theoretical substantiation of options for choosing an optimal strategy to integrate an agricultural enterprise into the wholesale market by using methodological tools of the non-cooperative game theory. We have proposed modeling the behavior of an agrarian enterprise in the market by achieving a Nash equilibrium under various scenarios of competitors’ activities and volumes of information on market conditions.

The methodology has been substantiated to apply the iterative algorithms to calculate equilibria in a general class of non-quadratic convex polyhedra in order to form the methodologies and construct algorithms for a behavior of agricultural enterprises in market activity. It was determined that decision-making occurs in parallel to the real conditions of activity of an agricultural enterprise in the wholesale market. The comprehensive application of numerical methods based on solving the optimization problems provides a smooth approach to the Nash equilibrium. A game can have multiple isolated Nash equilibria if players have non-quadratic payment functions when solving such problems. Based on the above, the results were determined of local convergence, since global results have strong constraints in non-quadratic problems. However, there is a connection with semi-global practical asymptotic stability if players have quadratic payoff functions. It has been shown that there is a shift in the convergence in proportion to the amplitudes of disturbance signals and the third derivative of payoff functions for non-quadratic payoff functions. This shift in the convergence corresponds to the shift in a numerical example.

It has been determined that the learning strategy developed in accordance with the main provisions of the theory of games remains attractive if one has partial information on the state of the market. Application of the indicated action strategy provides a company with a possibility to improve its initial position by measuring its own payoff values only and not using estimates of potentially uncertain parameters. It has been proposed to use applied tools from the game theory to determine an optimal action strategy for an agricultural enterprise for its integration into the wholesale market of vegetable products

Keywords


game theory; action strategy; agricultural enterprise; wholesale market; Nash equilibrium

References


Sievidova, I. A. (2017). Factors affecting the economic management efficiency of agricultural enterprises in Ukraine. Problems and Perspectives in Management, 15 (2), 204–211. doi: https://doi.org/10.21511/ppm.15(2-1).2017.04

Brown, G. W. (1951). Iterative solutions of games by fictitious play. Activity Analysis of Production and Allocation. Wiley, 374–376.

Cournot, A. (1938). Recherches sur les PrincipesMathématiques de la Théorie des Richesses. Paris, France: Hachette.

Shamma, J. S., Arslan, G. (2005). Dynamic fictitious play, dynamic gradient play, and distributed convergence to Nash equilibria. IEEE Transactions on Automatic Control, 50 (3), 312–327. doi: https://doi.org/10.1109/tac.2005.843878

Zhu, M., Martínez, S. (2010). Distributed coverage games for mobile visual sensor networks. SIAM J. Control Optim. Available at: https://arxiv.org/pdf/1002.0367.pdf

Babenko, V., Nazarenko, O., Nazarenko, I., Mandych, O., Krutko, M. (2018). Aspects of program control over technological innovations with consideration of risks. Eastern-European Journal of Enterprise Technologies, 3 (4 (93)), 6–14. doi: https://doi.org/10.15587/1729-4061.2018.133603

Li, J. (2018). Infinitely split Nash equilibrium problems in repeated games. Fixed Point Theory and Applications, 2018 (1). doi: https://doi.org/10.1186/s13663-018-0636-1

Duffy, J. (2015). Game Theory and Nash Equilibrium. A project submitted to the Department of Mathematical Sciences in conformity with the requirements for Math 4301 (Honours Seminar). Lakehead University, 37. Available at: https://www.lakeheadu.ca/sites/default/files/uploads/77/images/Duffy%20Jenny.pdf

Ye, M., Hu, G. (2017). Game Design and Analysis for Price-Based Demand Response: An Aggregate Game Approach. IEEE Transactions on Cybernetics, 47 (3), 720–730. doi: https://doi.org/10.1109/tcyb.2016.2524452

Zeng, X., Chen, J., Liang, S., Hong, Y. (2019). Generalized Nash equilibrium seeking strategy for distributed nonsmooth multi-cluster game. Automatica, 103, 20–26. doi: https://doi.org/10.1016/j.automatica.2019.01.025

Liang, S., Yi, P., Hong, Y. (2017). Distributed Nash equilibrium seeking for aggregative games with coupled constraints. Automatica, 85, 179–185. doi: https://doi.org/10.1016/j.automatica.2017.07.064

Hefti, A. (2017). Equilibria in symmetric games: Theory and applications. Theoretical Economics, 12 (3), 979–1002. doi: https://doi.org/10.3982/te2151

Zeng, J., Wang, Q., Liu, J., Chen, J., Chen, H. (2019). A Potential Game Approach to Distributed Operational Optimization for Microgrid Energy Management With Renewable Energy and Demand Response. IEEE Transactions on Industrial Electronics, 66 (6), 4479–4489. doi: https://doi.org/10.1109/tie.2018.2864714

Zhou, W., Koptyug, N., Ye, S., Jia, Y., Lu, X. (2016). An Extended N-Player Network Game and Simulation of Four Investment Strategies on a Complex Innovation Network. PLOS ONE, 11 (1), e0145407. doi: https://doi.org/10.1371/journal.pone.0145407

Gordji, M. E., Askari, G. (2018). Hyper-Rational Choice and Economic Behaviour. Advances in mathematical finance & applications, 3 (3), 69–76. doi: http://doi.org/10.22034/amfa.2018.544950

Caruso, F., Ceparano, M. C., Morgan, J. (2018). Uniqueness of Nash equilibrium in continuous two-player weighted potential games. Journal of Mathematical Analysis and Applications, 459 (2), 1208–1221. doi: https://doi.org/10.1016/j.jmaa.2017.11.031

Li, X. (2018). Existence of Generalized Nash Equilibrium in n-Person Noncooperative Games under Incomplete Preference. Journal of Function Spaces, 2018, 1–5. doi: https://doi.org/10.1155/2018/3737253

Kreuzberg, F., Hein, N., Rodrigues Junior, M. M. (2015). Teoria dos Jogos: Identificação do Ponto de Equilíbrio de Nash em Jogos Bimatriciais em Indicadores Econômicos e Sociais. Future Studies Research Journal: Trends and Strategies, 7 (2), 42. doi: https://doi.org/10.24023/futurejournal/2175-5825/2015.v7i2.196

Häfner, S., Nöldeke, G. (2016). Payoff Shares in Two-Player Contests. Games, 7 (3), 25. doi: https://doi.org/10.3390/g7030025

Wu, F., Ma, J. (2014). The Chaos Dynamic of Multiproduct Cournot Duopoly Game with Managerial Delegation. Discrete Dynamics in Nature and Society, 2014, 1–10. doi: https://doi.org/10.1155/2014/206961

Madandar ,F., Haghayeghi, S., S. Vaezpour, M. (2018). Characterization of Nash Equilibrium Strategy for Heptagonal Fuzzy Games. International Journal of Analysis and Applications, 16 (3), 353–367. doi: https://doi.org/10.28924/2291-8639-16-2018-353

Chattopadhyay, S., Mitka, M. M. (2019). Nash equilibrium in tariffs in a multi-country trade model. Journal of Mathematical Economics, 84, 225–242. doi: https://doi.org/10.1016/j.jmateco.2019.07.011

Christodoulou, G., Gairing, M., Giannakopoulos, Y., Spirakis, P. G. (2019). The Price of Stability of Weighted Congestion Games. SIAM Journal on Computing, 48 (5), 1544–1582. doi: https://doi.org/10.1137/18m1207880

Rosenthal, R. W. (1973). A class of games possessing pure-strategy Nash equilibria. International Journal of Game Theory, 2 (1), 65–67. doi: https://doi.org/10.1007/bf01737559

Hou, F., Zhai, Y., You, X. (2020). An equilibrium in group decision and its association with the Nash equilibrium in game theory. Computers & Industrial Engineering, 139, 106138. doi: https://doi.org/10.1016/j.cie.2019.106138

Babichenko, Y. (2014). Query complexity of approximate nash equilibria. Available at: https://arxiv.org/pdf/1306.6686v3.pdf

Nisan, N., Roughgarden, T., Tardos, E., Vazirani, V. V. (Eds.) (2007). Algorithmic Game Theory. Cambridge University Press. doi: https://doi.org/10.1017/cbo9780511800481

Frihauf, P., Krstic, M., Basar, T. (2012). Nash Equilibrium Seeking in Noncooperative Games. IEEE Transactions on Automatic Control, 57 (5), 1192–1207. doi: https://doi.org/10.1109/tac.2011.2173412

Fisher, R. (1991). Getting to yes: negotiating agreement without giving in. Boston: Houghton Mifflin, 200.

Mak-Kinsi, Dzh. (1960). Vvedenie v teoriyu igr. Moscow: Gos. izd-vo fiz-mat literatury, 420.

Collard-Wexler, A., Gowrisankaran, G., Lee, R. S. (2019). “Nash-in-Nash” Bargaining: A Microfoundation for Applied Work. Journal of Political Economy, 127 (1), 163–195. doi: https://doi.org/10.1086/700729

Губко, М. В., Новиков, Д. А. (2005). Теория игр в управлении организационными системами. М., 168.

Nash, J. F., Shapley, L. S. (1950). A Simple Three-Person Poker Game. Princeton University Press.

Sarychev, A. V. (2001). Lie- and chronologico-algebraic tools for studying stability of time-varying systems. Systems & Control Letters, 43 (1), 59–76. doi: https://doi.org/10.1016/s0167-6911(01)00090-1

Tan, Y., Nešić, D., Mareels, I. (2006). On non-local stability properties of extremum seeking control. Automatica, 42 (6), 889–903. doi: https://doi.org/10.1016/j.automatica.2006.01.014

Krstic, M., Kanellakopoulos, I., Kokotovic, P. (1995). Nonlinear and Adaptive Control Design. Wiley-Interscience, 576.


GOST Style Citations








Copyright (c) 2019 Iryna Sievidova, Tamila Oliynik, Oleksandra Mandych, Tetyana Kvyatko, Iryna Romaniuk, Larisa Leshchenko, Serhiy Vynohradenko, Serhii Plyhun

Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.

ISSN (print) 1729-3774, ISSN (on-line) 1729-4061