Convexity of Bertrand oligopoly TU-games with differentiated products

In this article we consider Bertrand oligopoly TU-games with differentiated products. We assume that the demand system is Shubik's (1980) and that firms operate at a constant and identical marginal and average cost. First, we show that the alpha and beta- characteristic functions (Aumann 1959) lead to the same class of Bertrand oligopoly TU-games and we prove that the convexity property holds for this class of games. Then, following Chander and Tulkens (1997) we consider the gamma-characteristic function where firms react to a deviating coalition by choosing individual best reply strategies. For this class of games, we show that the Equal Division Solution belongs to the core and we provide a sufficient condition under which such games are convex.