Complete study of a coopetitive R&D Alliance

The paper proposes a mathematical model of coopetitive game that analyzes general asymmetric R&D alliances. The coopetitive point of view, which considers both collaboration and competition together, allows to analyze the functioning of alliances that arise between small and large firms. Starting from the economic models developed in managerial doctrine and from the model of coopetitive game introduced by David Carfì, we adopt a mathematical analysis, paying attention to some of the most debated quantitative questions and some analytical topics scarcely covered in the literature: a mathematical model of coopetitive game, particularly suitable for exploring a complex type of asymmetric R&D alliances. We propose a formal coopetitive approach, with corresponding numerical example, where the coopetitive variable of the model is a real variable. A cooperative effort is suggested even if partners are potentially competitors in the marketplace and they could shape the payoff space by the coopetitive approach. To maximize profits, we suggested: first of all, a complete Pareto analysis (introduced by David Carfì), secondly - to share conveniently and fairly the utilities - we propose a Kalai-Smorodinsky solution of the bargaining decision problem, in which the decisional constraint is the Pareto boundary of maximum collective utility.