«THE BULLETIN OF IRKUTSK STATE UNIVERSITY». SERIES «MATHEMATICS»
«IZVESTIYA IRKUTSKOGO GOSUDARSTVENNOGO UNIVERSITETA». SERIYA «MATEMATIKA»
ISSN 1997-7670 (Print)
ISSN 2541-8785 (Online)

Use of Branch and Bound Method for Search of an Equilibrium in Potential Cournot Model

Author(s)
I. M. Minarchenko
Abstract

It is known that Nash equilibria of potential game belong to the set of stationary points of the potential, moreover only the potential’s global maximum is an equilibrium in general case. In the paper, we consider Cournot oligopoly model with linear inverse demand function and S-shape players’ costs functions determined by cubical polynomials. S-shape form of a function means changing of function’s concavity by its convexity. Costs function of such form reflects changing of increasing return of the scale by decreasing return of the scale, what may be explained as a stage of introduction of a new capacities that is changed by a stage of its normal operation. Such a model is a potential game due to linearity of inverse demand function. The potential is constructed and it has a form of cubical polynomial. Nonconcavity of potential leads to non-uniqueness of equilibrium in general case. Author investigated in other papers the local search of stationary points with multi-start approach and with the following check of the point whether it is an equilibrium. That paper is concerned with an adaptation of branch and bound method for search of the potential’s global maximum which is always an equilibrium point. In the paper, the method is described and numerical experiment results are given.

Keywords
Cournot model, potential games, Nash equilibrium, branch and bound method, d.c.-decomposition
UDC
519.833.2
References

1. Bredikhin S.V., Tiunova E.M., Khutoretskiy A.B. Price Coordination of Demand and Supply with Capacity Distribution of Multiprocessor System (in Russian). Sibirskiy Zhurn. Industr. Matem., 2007, vol. 10, no. 3(31), pp. 20–28.

2. Busygin V.P., Zhelobod’ko E.V., Kokovin S.G., Tsyplakov A.A. Microeconomic Analysis of Imperfect Markets (in Russian). Novosibirsk, NSU, 1999. 133 p.

3. Gal’perin V. M., Ignat’ev S.M., Morgunov V. I. Microeconomics (in Russian). Ekonomicheskaya Shkola, 1994, vol. 1. 349 p.

4. Gorelov M.A., Kononenko A. F. Games with Forbidden Situations. Models with Soft Constraints. Autom. Remote Control, 2010, vol. 71, no. 5, pp. 826–836.

5. Minarchenko I.M. On Potential and Non-potential Equilibrium Problems in Cournot Model (in Russian). Tr. XV Baykalskoy mezhdunarodnoy shkoly-seminara «Metody optimizatsii i ikh prilozheniya», vol. 6 (Math. Econ.), Irkutsk, 2011,pp. 197–202.

6. Minarchenko I.M. Numerical Search of an Equilibrium in Cournot Model with Sshape Сosts Functions (in Russian). Diskretn. Analiz i Issled. Oper., 2014, vol. 21, no. 5, pp. 40–53.

7. Petrosyan L.A., Zenkevich N.A., Semina E. A. Game Theory (in Russian). Moscow, Vyshaya Shkola, 1998. 304 p.

8. Podkoval’nikov S.V., Khamisov O.V. Imperfect Electric Power Markets: Modelling and Investigation of Development of Capacities (in Russian). Izvestiya Akademii Nauk. Energetika, 2011, no. 2, pp. 57–76.

9. Popov L.D. Introduction to the Theory, Methods and Economical Applications of Complementarity Problems (in Russian). Ekaterinburg, Izd. Ural. Univ., 2001. 124 p.

10. Sukharev A.G., Timokhov A. V., Fedorov V. V. Course of Methods of Optimization (in Russian). Moscow, Fizmatlit, 2005. 368 p.

11. Tarasevich L. S. Grebennikov P. I., Leusskiy A. I. Microeconomics (in Russian). Moscow, Yurayt-Izdat, 2006. 374 p.

12. Tokarev V.V. Guaranteed Results in Games with Forbidden Situations. Autom. Remote Control, 2009, vol. 70, no. 6, pp. 1026–1042.

13. Tokarev V.V. Peculiarities of Equilibria in the Forbidden-Situation Games. Autom. Remote Control, 2009, vol. 70, no. 7, pp. 1206–1216.

14. Badri A., Rashidinejad M. Security Constrained Optimal Bidding Strategy of GenCos in Day Ahead Oligopolistic Power Markets: a Cournot-Based Model. Electrical Engineering, 2013, vol. 95. pp. 63–72.

15. Bagwell K., Staiger R.W. The economics of trade agreements in the linear Cournot delocation model. Journal of Inernational Economics, 2012, vol. 88, pp. 32–46.

16. Bischi G.-I., Chiarella C., Kopel M., Szidarovszky F. Nonlinear Oligopolies. Berlin, Springer-Verl., 2010. 334 p.

17. Botterud A., Ilic M.D., Wangensteen I. Optimal Investments in Power Generation under Centralized and Decentralized Decision Making. Power Systems, IEEE Transactions on, 2005, vol. 20, no. 1, pp. 254–263.

18. Chen H., Wong K.P., Nguyen D.H. M., Chung C.Y. Analyzing Oligopolistic Electricity Market Using Coevolutionary Computation. Power Systems, IEEE Transactions on, 2006, vol. 21, no. 1, pp. 143–152.

19. Ewerhart C. Cournot games with biconcave demand. Games and Economic Behavior, 2014, vol. 85, pp. 37–47.

20. Horst R., Tuy H. Global Optimization: Deterministic Approaches. Berlin, Springer-Verl., 1996, P. 730.

21. Metzler C. Nash-Cournot Equilibria in Power Markets on a Linearized DC Network with Arbitrage: Formulations and Properties. Networks and Spatial Economics, 2003, vol. 3, no. 2, pp. 123–150.

22. Monderer D., Shapley L. S. Potential Games. Games and Economic Behavior, 1996, no. 14, pp. 124–143.

23. Peters H. Game Theory: A Multi-Leveled Approach. Berlin, Springer-Verl., 2008, 366 p.

24. Puu T. Oligopoly: Old Ends — New Means. Berlin, Springer-Verl., 2011. 172 p.

25. Ryan J. K., Daewon S., Xuying Z. Coordinating a Supply Chain With a Manufacturer-Owned Online Channel: A Dual Channel Model under Price Competition. Engineering Management, IEEE Transactions on, 2013, vol. 60, no. 2, pp. 247–259.

26. Ryan S. M., Downward A., Philpott A.B., Zakeri G. Welfare Effects of Expansions in Equilibrium Models of an Electricity Market with Fuel Network. Power Systems, IEEE Transactions on, 2010, vol. 25, no. 3, pp. 1337–1349.

27. Shan J., Botterud A., Ryan S. M. Impact of Demand Response on Thermal Generation Investment with High Wind Penetration. Smart Grid, IEEE Transactions on, 2013, vol. 4, no. 4, pp. 2374–2383.

28. Slade M.E. What Does an Oligopoly Maximize? The Journal of Industrial Economics, 1994, vol. 42, no. 1, pp. 45–61.

29. Vallee T., Yildizoglu M. Can They Beat the Cournot Equilibrium? Learning with Memory and Convergence to Equilibria in a Cournot Oligopoly. Computational Economics, 2013, vol. 41, pp. 493-516.

30. Wang R., Li Y., Zhang S. Analysis of Nash-Cournot Equilibrium for Electricity Markets Considering Option Contracts. Journal of Shanghai University (Eng. Edition), 2008, vol. 12, no. 6, pp. 542–547.