The Shapley value of a patent licensing game: The Asymptotic equivalence to non-cooperative results

We consider a game in characteristic form played by firms and an outside patent holder of a cost-reducing innovation. The worth of a coalition of players is the total Cournot profit the coalition can guarantee to obtain when it operates an optimal number of its firms while the complement operates any number of its firms as to minimize the profit of the coalition. Only firms in a coalition with the patent holder are allowed to use the efficient technology. We prove that when the number of firms is large, the Shapley value of the patent holder approximates the payoff he obtains in the non-cooperative auction game traditionally studied in the literature.