### Book chapter

## Differential Game of Oil Market with Moving Informational Horizon and Non–Transferable Utility

Non-transferable utility game of oil market is considered. Special approach for defining solution is used. This approach enables to construct a real time models of conflicting processes. Connection between the solution in the game with moving information horizon and solutions on the truncated time intervals is shown.

### In book

Interval cooperative games are models of cooperative situation where only bounds for payoffs of coalitions are known with certainty. The extension of solutions of classical cooperative games to interval setting highly depends on their monotonicity properties. However. both the prenucleolus and the tau-value are not aggregate monotonic on the class of convex TU games Hokari (2000, 2001). Therefore, interval analogues of these solutions either should be defined by another manner, or perhaps they exist in some other class of interval games. Both approaches are used in the paper: the prenucleolus of a convex interval game is defined by lexicographical minimization of the lexmin relation on the set of joint excess vectors of lower and upper games. On the other hand, the tau-value is shown to satisfy extendability condition on a subclass of convex games -- on the class of totally positive convex games. The interval prenucleolus is determined , and the proof of non-emptiness of the interval \tau-value on the class of interval totally positive games is given.

**Importance** The paper is devoted to analysis of the effectiveness of economic integration of firms. By efficiency I mean the standard requirements for profitability of integration (non-decreasing of total profit) in microeconomics, theory of the firm and the theory of industrial organization, or non-negativity of synergy in the theory of corporate finance and business valuation theory.

**Objectives** The purpose of the article is to derive the fair value of companies within the economic integration. Its definition is necessary to take into account the effect of external interaction in the competitive environment on the value of the business, to determine the volume of shares exchanged in the merger, the synergy share to pay to the acquired company in the purchase price, for deciding about splitting the business.

**Methods **Each company aims to increase its own value, which indicates the conflicted nature of the interaction between agents. Thus, this paper proposes the use of tools of the cooperative game theory to determine the profitability of integration for each of the participating firms, taking into account non-decreasing its share in the fair value of the entire integration.

**Results **The paper formalizes the notion of profitability of economic integration and the fair value of companies with the tools of cooperative game theory. It proofs the interpretation of solution concept of cooperative game as a method of calculating the fair value of companies with regard to its external cooperative interaction with contractors or the purchase price of acquired companies in M&A deals. The paper provides an example of such an analysis for economic integration in the aviation industry.

**Conclusions and Relevance **The proposed approach to the analysis of economic integration extends the understanding of its nature, making it possible to estimate the contribution of each individual firm. The article is of practical importance for companies to jointly carry out R&D, supply chains, alliances, holdings, M&A deals or investment and consulting companies serving such transactions.

The most of solutions for games with non-transferable utilities (NTU) are NTU extensions of solution concepts defined for games with transferable utilities (TU). For example, there are three NTU versions of the Shapley value due to Aumann(1985), Kalai--Samet(1977), and Maschler--Owen(1992). The Shapley value is {\it standard} for two-person games. An NTU analog of standard solution is called the {\it symmetric proportional solution (SP)} (Kalai 1977), and the most of NTU solutions are SP solutions for two-person games. Another popular TU game solution which is not standard for two-person case is the {\it egalitarian Dutta-Ray solution (Dutta, Ray (1989), Dutta 1990). It was defined for the class of convex TU games and then extended to the class of all TU games (Branzei et al. 2006). . The DR solution for superadditive two-person TU games is the solution of constrained egalitarianism, it chooses the payoff vectors the closest to the diagonal of the space R^N. Its extension to superadditive two-person NTU games and then to n-person bargaining problems is the lexicographically maxmin solution}: for each game/bargaining problem it is the individually rational payoff vector which is maximal w.r.t. the lexmin relation. This solution if positively homogenous, but is not covariant w.r.t. shifts of individual payoffs. In the presentation this solution is extended to the class of NTU non-levelled games which are both ordinal and cardinal convex. Since convex TU games considered in NTU setting are ordinal and cardinal convex, the NTU DR solution is, in fact, an extension of the original TU version to the mentioned class of NTU games. It turns out that in this class the DR solution is single-valued and belongs to the core. A result similar to that of Dutta for TU convex games is obtained: the DR solution for the class of non-levelled ordinal and cardinal convex games is the single solution being the lexicographically maxmin solution for two-person games and consistent in (slightly modified) Peleg's definition (Peleg 1985) of the reduced games.