Probabilistic Approach in the Problem of International Competition of Manufacturers with Random Variables
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PROBABILISTIC APPROACH IN THE PROBLEM OF INTERNATIONAL COMPETITION OF MANUFACTURERS WITH RANDOM VARIABLES K. V. Kosarevych1† and Ya. I. Yelejko1‡
UDC 519.21
Abstract. Game-theoretic models of manufacturers’ competition in the international market of a homogeneous product are constructed provided that the strategic variables of the manufacturers are random. A class of distributions of random variables that guarantees the existence of a solution to non-cooperative games describing international trade is distinguished. Explicit formulas for the “corrected” Nash equilibrium are established in the constructed models. Keywords: quantitative competition, strategy, problem of international trade, game-theoretic model, “corrected” Nash equilibrium.
Economic decisions are mostly made under uncertainty. This also takes place in international trade [1]. Strategic decisions of manufacturers are formed under absence of exact information about market demand and production volume, in particular, if a manufacturer enters the market for the first time or presents the market of qualitatively new goods. In this connection, there is a need to create competition models that consider uncertainty inherent in manufacturers and as a consequence, random nature of decision making. Development and analysis of such models is an important problem since their purpose is to reflect adequately a real, not idealized economic process. QUANTITATIVE COMPETITION WITH EXPORTER’S RANDOM PRODUCTION VOLUME Let us consider a game-theoretic model of the problem of quantitative competition in the international market where equilibrium price for goods is established on the basis of manufacturers’ decisions about production volume, supply, and demand [1, 2]. We assume that the market of some homogeneous goods is presented by two manufacturers: foreign (exporter) and domestic (“own” manufacturer in country’s home market) enterprises. Both manufacturers tend to maximize their expected profit by making optimal strategic decisions [3, 4]. Assume that production volume of the foreign enterprise (in what follows, the first manufacturer) is random, and the domestic enterprise (in what follows, the second manufacturer) uses a deterministic strategy at the same time. Let production volume q1 of the first manufacturer be a random variable with distribution density f ( x ; l ) , where l > 0 is variable distribution parameter (unknown at the beginning of interaction of the manufacturers) and function f ( x ; l ) be such that Eq1 < ¥ , Eq12 < ¥ "l > 0 , E (×) is expectation operator. Production volume of the first manufacturer is random; therefore the enterprise maximizes the expected profit by choosing the expected production volume Eq1 as the behavior strategy, and Eq1 = ò x f ( x ; l ) dx = j ( l ) . Along with permanent manufacturing costs of the first manufacturer (which is a foreign enterprise), there are also tariff payments t for import of a commodity unit. 1
Ivan Franko National University of Lviv, Lviv, Ukraine, †[email protected]; ‡[email protected]. Trans
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