Huang Xian, Hong Jia


The duopoly market research has a long history. Due to such reasons as material supply, product pa-tent right and concession of the government, development of many economic industries is similar to the process of duopoly. In game theory, the Bertrand model which considers price to be a strategic variable is closer to reality and provides the market with more references, especially for retail market and electricity market, as the competitive world develops.
Firstly, we analyze the classical Bertrand model and the Nash equilibrium in the model.
Secondly, multi-agent technology is applied and the Bertrand duopoly game bidding process is con-ducted; meanwhile, in order to help agents find the optimal solutions, genetic algorithm based on multi-agent Bertrand model is chosen as the main algorithm for the research; and we finish with software im-plementation of the algorithm and with example analysis. In the end, oligopoly market bidding is also modelled in MATLAB simulation, which provides us with more accuracies and flexibilities.
It is evidently shown in the model that when none of the two companies are able to meet all the de-mands in the market, the bigger the price gap, the more oscillated it is in the process; thus, the pure stra-tegic Nash equilibrium doesn’t exist. However, when one of the two can offer the demands independent-ly, Nash equilibrium appears and is shown as the calculated results in Bertrand-Edgeworth model where the equilibrium reaches the cost price. Further, the reason for no pure strategic Nash Equilibrium is also discussed

Ключевые слова

Bertrand model, multi-agent, genetic algorithm, nash Equilibrium

Полный текст:



Hirata D., Matsumura T. On the Uniqueness of Bertrand Equilibrium. Operations Research Letters, 2010, 38(6):533–535. DOI: 10.1016/j.orl.2010.08.010

Dastidar K.G. On the existence of pure strategy Bertrand equilibrium. Economic Theory 5 (1) (1995) 19–32. DOI: 10.1007/BF01213642

Liang Xiaoying, Xie, et al. Bertrand competition with intermediation. Economics Letters, 2012, 116(1):112–114. DOI: 10.1016/j.econlet.2012.01.019

Nijs R.D. Further results on the Bertrand game with different marginal costs. Economics Letters, 2012, 116(3):502–503. DOI: 10.1016/j.econlet.2012.04.055

Anderson S.P., de Palma A., Thisse J.-F., 1997. Privatization and efficiency in a differentiated industry. Eur. Econ. Rev. 41 (9), 1635–1654. DOI: 10.1016/S0014-2921(97)00086-X

Froeba L., Tschantzb S., Crookeb P., 2013. Bertrand competition with capacity constraints: mergers among parking lots. Journal of Econometrics, 113(1):49–67. DOI: 10.1016/S0304-4076(02)00166-5

Palmer I. Coalbed methane completions: a world view. International Journal of Coal Geology, 2010, vol. 82, no. 3, pp. 184–195. DOI: 10.1016/j.coal.2009.12.010

Andrés J., Burriel P. Inflation and optimal monetary policy in a model with firm heterogeneity and Ber-trand competition. European Economic Review, 2018, 103:18–38. DOI: 10.1016/j.euroecorev.2017.12.009

Amir R., Evstigneev I.V. A new look at the classical Bertrand duopoly. Games & Economic Behavior, 2018, 109:99–103. DOI: 10.1016/j.geb.2017.12.010

Baye, M., Kovenock, D., 2008. In: Durlauf, Steven, Blume, Lawrence (Eds.). Bertrand competition, The New Palgrave Dictionary of Economics, 2nd ed. Palgrave Macmillan. DOI: 10.1057/978-1-349-95121-5_2462-1

Łukasz Balbus, Reffett K, Łukasz Woźny. A Constructive Study of Markov Equilibria in Stochastic Games with Strategic Complementarities. Journal of Economic Theory, 2010, 150(1). DOI: 10.2139/ssrn.1723038

Algarvio H, Lopes F, Sousa J, et al. Multi-agent electricity markets: Retailer portfolio optimization us-ing Markowitz theory. Electric Power Systems Research, 2017, 148:282–294. DOI: 10.1016/j.epsr.2017.02.031

Cellini R, Lambertini L. R&D Incentives Under Bertrand Competition: A Differential Game. Japanese Economic Review, 2011, 62(3):387–400. DOI: 10.1111/j.1468-5876.2011.00541.x

Amir R., Erickson P., Jin J., 2017. On the microeconomic foundations of linear demand for differentiat-ed products. J. Econ. Theory, 169, 641–665. DOI: 10.1016/j.jet.2017.03.005

Soleymani S. Bidding strategy of generation companies using PSO combined with SA method in the pay as bid markets. Int J Electr Power Energy Syst, 2011; 33(7): 1272–8. DOI: 10.1016/j.ijepes.2011.05.003

Ladjici A.A., Boudour M. Nash cournot equilibrium of a deregulated electricity market using competi-tive coevolutionary algorithms. Electr Power Syst Res, 2010;81(4):958–66. DOI: 10.1016/j.epsr.2010.11.016

Chen H., et al. Analyzing oligopolistic electricity market using coevolutionary computation. IEEE Transactions On Power Systems, 21 (1) (2006) 143–152. DOI: 10.1109/TPWRS.2005.862005

Gwartney J.D., Stroup R., Clark J.R. Pure Competition And Monopoly. Essentials of Economics, 1985, pp. 307–332. DOI: 10.1016/B978-0-12-311035-0.50019-8

Vazhayil J.P., Balasubramanian R. Optimization of India’s electricity generation portfolio using intelli-gent Pareto-search genetic algorithm. International Journal of Electrical Power & Energy Systems, 2014, 55:13–20. DOI: 10.1016/j.ijepes.2013.08.024

Cai J., Kim D., Jaramillo R., et al. A general multi-agent control approach for building energy system optimization. Energy & Buildings, 2016, 127:337–351. DOI:


  • На текущий момент ссылки отсутствуют.