Impact of a Non-linear Pay-Off Function on Pairwise Stable Collaborative Oligopolies

We extend the results of Goyal and Joshi (S. Goyal and S. Joshi. Networks of collaboration in oligopoly. Games and Economic behavior, 43(1):57-85, 2003), who first considered the problem of collaboration networks of oligopolies and showed that under certain linear assumptions network collaboration produced a stable complete graph through selfish competition. We show with nonlinear cost functions and player payoff alteration that stable collaboration graphs with an arbitrary degree sequence can result. We also show a generalized non-linear extension for the result in the aforementioned paper in which the complete graph is stable. An example is provided in which the complete set of pair wise stable collaborations is computed for an oligopoly consisting of five firms.