Convexity and the Shapley value of Bertrand oligopoly TU-games in β-characteristic function form

The Bertrand oligopoly situation with Shubik's demand functions is modeled as a cooperative transferable utility game in beta-characteristic function form. To achieve this, two sequential optimization problems are solved to describe the worth of each coalition in the associated Bertrand oligopoly transferable utility game. First, we show that these games are convex, indicating strong incentives for large-scale cooperation between firms. Second, the Shapley value of these games is fully determined by applying the linearity to a decomposition that involves the difference between two convex games and two non-essential games.