Potentials in quadratic Cournot cross-holding games

Do firms in an oligopoly market behave “as if” they were maximizing a common fictitious objective function, as in perfect competition and monopoly? The answer is yes under certain mild technical conditions (Slade, 1994). That is, in terms of Monderer and Shapley (1996), the Cournot competition is a potential game. In this paper, we ask the same question for Cournot competition with quadratic payoff functions and cross-holdings, an important variant of the oligopoly market. We find that, for various potential functions, the question can be more easily understood from the structure of the influence network, which is constructed from the cross-holding network. Roughly, we find that the Cournot competition with cross-holdings is a potential game if and only if the influence network is symmetric in certain generalized sense. Extending the model to Cournot competition with both overlapping ownership and product differentiation, we find that the previous results still hold. We also provide two applications of our results.