The Berge and Nash equilibrium in the Bertrand duopoly

V. I. Zhukovskii   ; L. V. Zhukovskaya   ; Yu. A. Bel'skikh   ; M. I. Vysokos

Taurida Journal of Computer Science Theory and Mathematics, no. 2 (2024), pp. 14-25

Résumé

One specialist in the field of operations research decided to use optimization methods to solve a very important problem for him — getting rid of extra fullness. He formalized this problem according to established rules just chosen the target function — weight, restrictions — the minimum of proteins, carbohydrates, fats, etc., then he solved the mathematical problem of minimization under restrictions. The optimal solution was to restrict the ration of food by using vinegar more than three hundred liters per day (the American philosopher Ch. Hitch). In present paper for excepting such «vinegar» solutions the relations between initial demand, cost prise and coefficients of elasticity are revealed. In mathematical game theory, recent years are characterized by active studying of the concept of Berge equilibrium as antipode to widely used Nash equilibrium. Difference is in the fact that the concept of Nash equilibrium has «egoistic character» — every player tries to increase his payoff only. On the contrary, Berge equilibrium has altruistic character: its goal is to increase payoffs of all other players. In the article offered to the reader we prove that the Berge equilibrium can be used in economics. So, let two firms producing the same product function on the market. The strategy of the firm (player) let be the price fixed by the firm for its product. Thus we consider that every firm declares its price ${p_i=const\geqslant 0}$ ${(i=1,2)}$. After that the situation (set price) is created — vector $\vec{p}=(p_1,p_2)$. The demand on the product of $i$-player $(i\in \{1,2\})$, appeared on the market we offer as linear concerning declared prices, namely $$ Q_1(\vec{p})=q-l_1p_1+l_2p_2, $$ $$ Q_2(\vec{p})=q-l_1p_2+l_2p_1. $$ Here $q$ — the initial demand, the coefficient of elasticity $l_1=const > 0$ shows how much the demand on the offered product under raising of the price per unit is reduced. In turn, coefficient of elasticity $l_2=const > 0$ shows how much the demand under extention per unit of the price of substitute goods is increased. If we set the cost price of unit of the product by $c > 0$, so the profit of $i$-firm (called payoff function of $i$-player $i\in \{1,2\}$) will be $$ f_1(\vec p)= [q-l_1p_1+l_2p_2](p_1-c), $$ $$ f_2(\vec p)= [q-l_1p_2+l_2p_1](p_2-c). $$ As a result the mathematical model of interaction among firms-sellers described above one can suppose as ordered triple $$ \Gamma=\langle \{1,2\},\;\{P_i=(c,\beta]\}_{i=1,2},\; \{f_i(\vec p)\}_{i=1,2} \rangle. $$ We note the following peculiarities of the game $\Gamma$: first, it is supposed that maximal price $\beta$ and the cost price $c$ for both players are equal (it’s naturally for the market of one product); secondly, the coalition $\{1,2\}$ is prohibited by the rules of the game (in particular the «noncooperative character» of the game is appeared in that); thirdly, the price ${p_i>c}$ ${(i=1,2)}$ for otherwise the $i$-player hardly may appear on the market. For the game $\Gamma$ Berge and Nash equilibriums are formalized. The relations among coefficients are found, under the realization of them the Berge equilibrium delivers the players more payoffs than the Nash equilibrium.

Détail

Keywords: Berge equilibrium, Nash equilibrium, strategies, non-cooperative two-player game, Bertrand duopoly, Cournot duopoly.