Cournot competition, introduced in 1838 by Antoine Augustin Cournot, is a fundamental economic model that represents firms competing in a single market of a homogeneous good. Each firm tries to maximize its utility-naturally a function of the production cost as well as market price of the product-by deciding on the amount of production. This problem has been studied comprehensively in Economics and Game Theory; however, in today's dynamic and diverse economy, many firms often compete in more than one market simultaneously, i.e., each market might be shared among a subset of these firms. In this situation, a bipartite graph models the access restriction where firms are on one side, markets are on the other side, and edges demonstrate whether a firm has access to a market or not. We call this game Network Cournot Competition (NCC). Computation of equilibrium, taking into account a network of markets and firms and the different forms of cost and price functions, makes challenging and interesting new problems. In this paper, we propose algorithms for finding pure Nash equilibria of NCC games in different situations. First, we carefully design a potential function for NCC, when the price function for each market is a linear function of it total production. This result lets us leverage optimization techniques for a single function rather than multiple utility functions of many firms. However, for nonlinear price functions, this approach is not feasible-there is indeed no single potential function that captures the utilities of all firms for the case of nonlinear price functions. We model the problem as a nonlinear complementarity problem in this case, and design a polynomial-time algorithm that finds an equilibrium of the game for strongly convex cost functions and strongly monotone revenue functions. We also explore the class of price functions that ensures strong monotonicity of the revenue function, and show it consists of a broad class of functions. Moreover, we discuss the uniqueness of equilibria in both these cases: our algorithms find the unique equilibria of the games. Last but not least, when the cost of production in one market is independent from the cost of production in other markets for all firms, the problem can be separated into several independent classical Cournot Oligopoly problems in which the firms compete over a single market. We give the first combinatorial algorithm for this widely studied problem. Interestingly, our algorithm is much simpler and faster than previous optimization-based approaches.
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