On the supermodularity of homogeneous oligopoly games

The main purpose is to prove the supermodularity (convexity) property of a cooperative game arising from an economical situation. The underlying oligopoly situation is based on a linear inverse demand function as well as linear cost functions for the participating firms. The characteristic function of the so-called oligopoly game is determined by maximizing, for any cartel of firms, the net profit function over the feasible production levels of the firms in the cartel, taking into account their individual capacities of production and production technologies. The (rather effective) proof of the supermodularity of the characteristic function of the oligopoly game relies on the use of maximizers for the relevant maximization problems. A similar proof technique will be reviewed for a related cooperative oligopoly game arising from a slightly modified oligopoly situation where the production technology of the cartel is determined by the most efficient member firm.