# Import accounting in Kurno duopoly

The competition of two firms on the market of one product with regard to import is considered. The particular volume of product supplied to the market by the importer is unknown to both producers. They know only restrictions about volume of import defined by the market. Non-cooperative game of two persons under uncertainty will serve as a mathematical model where "the role"of uncertainty "plays"the quantity of export goods, "the role"of players strategy - the quantity of goods supplied by them on the sale, payoff function - the income of the player. Two notions of guaranteed decision of players - strictly guaranteed (Nash) equilibrium and (Pareto) quaranteed equilibrium are used. The first one is on the boundary of the concept of maximin and Nash equilibrium, the second one - of Pareto minimum and Nash equilibrium decision. In this paper for stated mathematical Kurno model with regard to import the explicit form of both guaranteed equilibrium is found. Let us imagine that two companies (designated I and II respectively) compete in the market of the product. The volume of produced by them during a certain (specified a priori) period of time output is denoted by $x_1$ and $x_2$ respectively. At the same time there appears an importing company on the market, the managers of the companies I and II do not have any information about purposes and volume of goods produced by it. They can only consider that the volume of goods produced by the importer is nonnegative quantity $y ∈ [0, +∞)$. The i-production costs are assumed to be linearly dependent on the amount of output $x_i (i = 1, 2)$ and can be represented as $cx_i + d, c$ and $d$ here are respectively variable and constant costs (for example, variable costs include the costs to workers wages, the purchase of raw materials, depreciation of equipment, constant costs – rent of premises, land, machinery, licenses etc.). Depending on the demand the price of product is determined on the market, this price we also consider as linearly dependent on the amount of $\overline{x} = x_1 + x_2 + y$ goods entered on sale. The price of goods we represent in the form $p(\overline{x}) = a − b\overline{x}$, where $a = const > 0$ - the initial price of goods, and constant positive elasticity coefficient $b > 0$ shows how much the price is "falling"when a unit of production is on sale. Suppose that the price is determined so that to equalize supply and demand. Let each of the firms sells all that it produces, then the proceeds of the first company is $p(\overline{x})x_1 = (a − b\overline{x})x = [a − b(x_1 + x_2 + y)]x_1$, and its profit (proceeds minus costs) is $ψ_1(x_1, x_2, y) = [a−b(x_1 +x_2 +y)]x_1 −(cx_1 +d) = ax_1 −bx^2_1 −bx_1x_2 −byx_1 −cx_1 −d$, at the same time the second company’s profit is $ψ_2(x_1, x_2, y) = [a − b(x_1 + x_2 + y)]x_2 − (cx_2 + d) = ax_2 − bx_1x_2 − bx_2^2 − byx_2 − cx_2 − d.$

By determining the amount of production, the management of the manufacturing company is forced to rely not only on the "rational"choice of the competitor, but also on the possibility of implementing any in advance unpredictable uncertainty values - y volume supplied to the import market. Following the principle guaranteed result by Germeier Yu. B., we will assume that when choosing a production volume $x_i (i = 1, 2) i$-manufacturer is focused on maximizing the function $F_i(x_1, x_2, y) = φi(x_1, x_2, y)+by^2$. In this case the first term is a function of profit, and the second "compels"when choosing decision to focus on "maximum resistance to uncertainty".

Key words: Nash equilibrium, Pareto minimality, guaranteed decisions.