# Berge equilibrium in Bertrand oligopoly with import

In recent years, there is an active formation of the mathematical theory of Berge equilibrium, proposed in 1994, by Russian mathematician, K.S. Vaisman (who died in 1998, did not live up to 36 years). However, the use of this balance is not beyond the scope of matrix games of two persons. This article, apparently for the first time, breaks this tradition. The article by using dynamic programming shows explicit form of strongly guaranteed Berge equilibrium in the mathematical model taking into account the Bertrand oligopoly under uncertainty (imports suddenly appeared on the market).

Considering in 1836 the interaction processes between companies, Cournot suggested that an oligopoly only determine the amount of product, the price is formed as a result of the balance between supply and demand. The market price is set at a level at which buyers will be present demand for all the goods on the market. However, a more natural behavior seller is the direct appointment of the prices (« Price — Value, plus a reasonable sum for the wear and tear of conscience in demanding it.», says American writer Ambrose Bierce (1842-1914)). This is the approach suggested by Joseph Bertrand in published in 1883. In this his model the companies

set the price for their goods themselves, the volume of the goods offered by them are formed so as to completely satisfy caused demand at given prices.

In the presented article for controlled model of Bertrand oligopoly the explicit form of guaranteed Berge equilibrium is found by the dynamic programming method.

Key words: non-cooperative game, Bertrand oligopoly, dynamic programming, Berge equilibrium, uncertainty.