Geodesic
One Guaranteed Equilibrium in Bertrand Duopoly under Uncertainty
Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, Tome 6 (2013) no. 4, pp. 48-54

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This paper considers Bertrand duopoly on a market of a differentiated product taking into account possible import. The price which is assigned for importers is nonstochastic uncertainty. The model of the duopoly is formalized as a non-cooperative two-person game under uncertainty. When the players choose their strategies, they tend to increase the price but they are guided by the value of uncertainty. The decision of the game is given as a strongly guaranteed equilibrium. It is based on the concept of an analog of a vector maximin. In the first stage (the analog of the interior minimum in the maximin) a continuous function is constructed for each player. This function is connected with the worst uncertainty. In the second stage (the analog of the exterior maximum in the maximin) Nash equilibrium is seen in «Guarantees game». «Guarantees game» is obtained after substitution uncertainties found earlier in the payoff functions. The strongly guaranteed equilibrium is built in an explicit form. The sufficient conditions for the existence of such decision are defined.
Keywords: guaranteed equilibrium; non-cooperative game; game under uncertainty; Bertrand duopoly. 
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     title = {One {Guaranteed} {Equilibrium} in {Bertrand} {Duopoly} under {Uncertainty}},
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A. A. Mansurova; I. S. Stabulit; S. A. Shunaylova. One Guaranteed Equilibrium in Bertrand Duopoly under Uncertainty. Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, Tome 6 (2013) no. 4, pp. 48-54. http://geodesic.mathdoc.fr/item/VYURU_2013_6_4_a4/