Pii: S0895-7177(02)00276-5

K e y w o r d s A n t a g o n i s t i c game, Coopera t ive game, Core of a coopera t ive game, Noncoopera t ive game wi th cons t an t sum, T h e Shapley vector. 1. I N T R O D U C T I O N I t is well known tha t every firm acting in a marketplace by supplying goods or services (or both) tries to do it in a manner which allows it to gain the best possible (in a particular sense) result or at least to receive its fair share of the market. To this end, such a firm should s tar t with evaluating its potential to compete in the marketplace and the fair share tha t can be at tained in the framework of agreements made with the firm's suppliers, complementers, and possibly, buyers. However, knowing its potential to compete in the marketplace and having made appropriate agreements with the suppliers, complementers, and some of the buyers, the firm should analyze if its guaranteed share of the market welfare can be increased on account of a reasonable cooperation with other firms acting in the same marketplace, and the result of such an analysis should be compared with the share tha t the firm can secure by only restructuring its own business by any means and acting separately from the other firms. The lat ter problem 0895-7177/02 /$ see front m a t t e r (~) 2002 Elsevier Science Ltd. All r ights reserved. T y p e s e t by .A,A/~-TEX PII: S0895-7177(02)00276-5 1280 A . S . BELENKY is mostly associated with identifying the ways of the firm's restructuring and with evaluating the firm's potential for each new structure and can be solved, for instance, using the approach described in [1]. In most instances, the major reason for the firm to seek cooperation with the other firms acting in the same marketplace is dictated by either the firm's intentions to increase its market share or by those to save the existing share which can be decreased if some (or all) of the other firms form a coalition which may act against the firm more effectively than it takes place when all the firms act separately. In the first case, a coalition can be formed by a group of firms acting in the marketplace (that includes the firm) if each of them possesses certain unique values the integrating of which leads to increasing the total share of the coalition acting in the marketplace as a unified player against the others, and the coalition members can agree on a manner of imputing this share increase seeming fair to all of them. In the second case, the firm anticipating unfavorable situations may agree to join a particular coalition even when this act only protects its existing share, whereas the other coalition participants may increase their shares; certainly, this may happen only if there are no other ways of cooperation for the firm that lead to increasing its market share. It is clear that the firm should analyze benefits to be attained in each potential coalition which it can form or join; if m is the number of firms competing in the same marketplace, the number of potential coalitions that can be formed with the participation of the firm equals 2 ~ 1 as such coalitions of two participants can be chosen by m 1 ways, those of three participants 2 m 2 can be chosen by C,~ 1 ways, etc., and coalitions of m 1 participants can be chosen by C,.~_ 1 ways. Although the number of coalitions of the above-mentioned kind that should be analyzed by the firm equals 2 ml , for certain purposes, this number can be substantially reduced if, for instance, the firm does not cooperate with certain other firms acting in the marketplace in virtue of strategic of whatever other reasons. Each such a coalition should be considered a unified player acting against the other firms, and it is natural to assume that the coalition members uni~their resources (all or some), skills and knowledge to achieve the best possible result (the maximal guaranteed share of the market ' welfare), and the best guaranteed result that can be attained by the coalition is that to be attained under the worst possible scenario, which incurs when the firms not joining (or not invited to join) the coalition also unify their potentials (resources, skills, knowledge, etc.) and act as a unified player against the coalition [2]. As well known [2]. it is one of concepts that game theory presents for the analysis of coalitions: namely, for each finite number of players (firms competing in the marketplace), for any coalition consisting of A" players among T players, another coalition containing remaining T \ K players is considered in such a manner that the interaction of the first coalition of K players and the second one of T \ K players (this second coalition is sometimes called the surrounding for the first coalition) is viewed as a two-person (Coalition 1 and Coalition 2) zero-sum game, and the best guaranteed result in this game (which may, however, not have a Nash equilibrium (saddle) point) for tile first person is considered the best guaranteed result for Coalition 1. If such a guaranteed result (gain of the coalition) can be imputed among the coalition participants in such a manner that a share of the gain to be received by each coalition member under this imputation is not less than the guaranteed result that can be secured by each such a member acting individually in the marketplace against the other firms, the coalition makes sense and may eventually be formed. Practical implementation of this approach is associated with considering two major problems: (a) how the firm can evaluate the potential of any coalition to get the market share not less than the sum of shares of its participants acting individually that can be secured by each of them, and (b) how the participants can work out a fair imputation mechanism to satisfy all the coalition members to an extent making unreasonable the idea of leaving the coalition for any other