An asymmetric Cournot-Nash equilibrium under uncertainty as a generalized Cournot-Stackelberg-Nash equilibrium
- PDF / 82,412 Bytes
- 7 Pages / 595 x 842 pts (A4) Page_size
- 96 Downloads / 210 Views
CYBERNETICS AN ASYMMETRIC COURNOT–NASH EQUILIBRIUM UNDER UNCERTAINTY AS A GENERALIZED COURNOT–STACKELBERG–NASH EQUILIBRIUM
UDC 519.8
V. M. Gorbachuk
The impact of a random production output on the expected commercial manufacture and price and also on the outputs and profits of companies is investigated. It is proved that the asymmetry of interactions between decision makers, in particular, the asymmetry of uncertainty can lead to the advantage of a leader and to generalized Cournot–Stackelberg–Nash equilibriums. Keywords: equilibrium, Cournot, Stackelberg, Nash, company, random output, competition, asymmetry.
In solving economic problems [1–5], game models and methods and, in particular, game models under uncertainty [6–12] are often used. The importance of uncertainty is obvious after the fall of stocks of Internet-companies in 2000, terrorist attack on September 11, 2001, oil price increase up to a threshold that was more than 70 dollars per barrel, and drastic climatic changes. Therefore, in the New Millennium, the need to further develop stochastic optimization methods [7–11] arises. In this article, the influence of the randomness of a production output on the expected commercial manufacture and price and also on outputs and profits of companies is investigated. We assume that a homogeneous product is manufactured only by two decision makers (DMs), namely, by the 1st company and the 2nd company. Let the inverse demand function for this product be specified by the formula P = a - bQ,
(1)
where P is a (positive) price of the product, Q is the number (volume) of product units manufactured (and sold) by the companies, and a and b are positive parameters. The ith company manufactures q i ³ 0 product units and its expenditures for a product unit equal c i , i = 1, 2 . It may be noted that we have Q = q1 + q 2 .
(2)
The situation in which the values of q1 and q 2 are deterministic (are not random) was investigated in detail in many works [1–4]. The randomness of one of these values can influence the strategy of companies [6]. Let the value of q 2 be deterministic, and let the value of q1 be uniformly distributed over a (nondegenerate) segment [ d1 , D1 ], i.e., the density f ( q1 ) = {1 / ( D1 - d1 ), q1 Î [ d1 , D1 ]; 0, q1 Ï [ d1 , D1 ]}. In particular, it is assumed in [6] that the oil price is uniformly distributed between 17 and 51 dollars per barrel. The 2nd company chooses the volume q 2 of its production to maximize the expected value of its profit E ( p 2 ) = E ( Pq 2 - c 2 q 2 ) , whence, in view of formulas (1) and (2), we have
Cybernetics Institute, National Academy of Sciences of Ukraine, Kiev, Ukraine, [email protected]. Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 3–10, July–August 2007. Original article submitted October 13, 2006. 1060-0396/07/4304-0471
©
2007 Springer Science+Business Media, Inc.
471
E ( p 2 ) = E {[ a - b ( q1 + q 2 )]q 2 - c 2 q 2 } = E {[ a - c 2 - b ( q1 + q 2 )]q 2 }= ( a - c 2 )q 2 - b ( q 2 ) 2 - bq 2 q1 ,
(3)
where the expected value of
Data Loading...
