Compare Nash Equilibria and Berzh in a Model of Bertrand Duopoly.

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. The Golden rule of morality forms the basis of it: Do as you would be done by. Such approach obviously excludes “hard” measures of balancing of conflict — wars, conflicts, bloodshed. 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 $p_i = const ≥ 0~(i = 1, 2)$.
After that the situation (set price) is created — $vector ~ \vec p = (p_1, p_2)$. The demand on the product of $i$-player $(i ∈ \{1, 2\})$, 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 $i ∈ \{1, 2\}$) 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

\[Γ = \langle \{1, 2\}, \{P_i = (c, β]\}_{i=1,2} , \{fi(\vec p\}_{i=1,2} \rangle.\]

We note the following peculiarities of the game $Γ$:

  • first, it is supposed that maximal price $β$ 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 $Γ$ 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.

Keywords: Nash equilibrium, Berge equilibrium, mathematical game theory, mathematical model
of Bertran duopoly.