Noncooperative oligopoly equilibrium in markets with hierarchical competition

Abstract

In this paper we study a non-cooperative sequential equilibrium concept, namely the Stackelberg–Nash equilibrium, in a game in which heterogeneous atomic traders interact in interrelated markets. To this end, we consider a two-stage quantity setting strategic market game with a finite number of traders. Within this framework, we define a Stackelberg–Nash equilibrium. Then, we show existence and local uniqueness of a Stackelberg–Nash equilibrium with trade. To this end, we use a differentiable approach: the vector mapping which determines the strategies of followers is a smooth local diffeomorphism, and the set of Stackelberg–Nash equilibria with trade is discrete, i.e., the interior equilibria of the game are locally unique. We also compare through examples the sequential and the simultaneous moves games. A striking difference is that exchange can take place in one subgame while autarky can hold in another subgame, in which case only leaders (followers) make trade.