Game-Theoretic Models of Competition Between Producers with Random Product Yields Under Duopoly of Differentiated Goods
- PDF / 134,703 Bytes
- 10 Pages / 594 x 792 pts Page_size
- 40 Downloads / 223 Views
GAME-THEORETIC MODELS OF COMPETITION BETWEEN PRODUCERS WITH RANDOM PRODUCT YIELDS UNDER DUOPOLY OF DIFFERENTIATED GOODS K. V. Kosarevycha† and Ya. I. Yelejkoa‡
UDC 519.21
Abstract. Models of quantitative competition under duopoly of differentiated goods are described in which the controlled variable (product yield) of a producer is considered to be a random quantity. The class of distributions of random product yields is singled out that guarantees the existence of solutions for noncooperative games. Formulas for finding a “corrected” Nash equilibrium are obtained in explicit form for a duopoly with a random controlled variable of one and both producers. A method for estimating the risk level for a producer is proposed. Keywords: quantitative competition, duopoly, differentiated product, “corrected” Nash equilibrium, risk level.
Strategic decisions made by producers are often accompanied by the absence of exact information on market demand and product yield, for example, when a producer for the first time enters a market or introduces qualitatively new goods into it. Therefore, a demand arises for the creation of models of competition [1–3] that would take into account indefiniteness peculiar to producers. Under conditions of global inflation, the investigation of a market of not homogeneous but differentiated goods [4, 5] is especially topical. INVESTIGATION OF A DUOPOLY WITH RANDOM OUTPUT OF PRODUCTS BY ONE PRODUCER Let us consider the quantitative competitive interaction of two producers in a differentiated goods market. Let a product i be produced by the ith enterprise, i Î{1, 2}. Assume that one producer randomly makes his decision on his product yield and that the decision of the other producer is deterministic. Let the product yield q1 of the first enterprise be a random quantity with density of distribution f ( x; l ), where l > 0 is a variable distribution parameter (unknown at the moment of the beginning of interaction between the producers), let a function f ( x; l ) be such that Eq1 < ¥ and Eq12 < ¥ "l > 0, and let E( × ) be the expectation operator. Since the product yield of the first producer is random, he maximizes his expected profit making the following decision on the expected product yield: Eq1 = ò x f ( x; l )dx = j( l ). The second producer maximizes his expected profit by choosing a deterministic product yield q 2 . The quantitative competitive interaction between these two producers with a random product yield of one of them can be represented as a strategic-form game G1( 2 ) = ( I , {S i }, {Ep i ( Eq1 , q 2 )}, i Î I ), where I = {1, 2} is the set of players; S i is æ æ the set of admissible strategies of the ith player, i Î I ; Ep i ( Eq1 , q 2 ) = E ç q i Di-1 ç å q j ç ç jÎI è è
ö ÷ - c (q ÷ i i ø
ö ) ÷ is the payoff function ÷ ø
a
Ivan Franco National University of Lviv, Lviv, Ukraine, †[email protected]; ‡[email protected]. Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 127–136, July–August, 2015. Original article submitted April 7, 2015. 1060-0396/15/5104-0609
Data Loading...
