Autocratic strategies in Cournot oligopoly game

An oligopoly is a market in which the price of goods is controlled by a few firms. Cournot introduced the simplest game-theoretic model of oligopoly, where profit-maximizing behavior of each firm results in market failure. Furthermore, when the Cournot oligopoly game is infinitely repeated, firms can tacitly collude to monopolize the market. Such tacit collusion is realized by the same mechanism as direct reciprocity in the repeated prisoner’s dilemma game, where mutual cooperation can be realized whereas defection is favorable for both prisoners in a one-shot game. Recently, in the repeated prisoner’s dilemma game, a class of strategies called zero-determinant strategies attracts much attention in the context of direct reciprocity. Zero-determinant strategies are autocratic strategies which unilaterally control payoffs of players by enforcing linear relationships between payoffs. There were many attempts to find zero-determinant strategies in other games and to extend them so as to apply them to broader situations. In this paper, first, we show that zero-determinant strategies exist even in the repeated Cournot oligopoly game, and that they are quite different from those in the repeated prisoner’s dilemma game. Especially, we prove that a fair zero-determinant strategy exists, which is guaranteed to obtain the average payoff of the opponents. Second, we numerically show that the fair zero-determinant strategy can be used to promote collusion when it is used against an adaptively learning player, whereas it cannot promote collusion when it is used against two adaptively learning players. Our findings elucidate some negative impact of zero-determinant strategies in the oligopoly market.Author summary: Repeated games have been used to analyze the rational decision-making of multiple agents in a long-term interdependent relationship. Recently, a class of autocratic strategies, called zero-determinant strategies, was discovered in repeated games, which unilaterally controls payoffs of players via enforcing linear relations between payoffs. So far, properties of zero-determinant strategies in social dilemma situations have extensively been investigated, and it has been shown that some zero-determinant strategies promote cooperation. Moreover, zero-determinant strategies have been found in several games. However, it has not been known whether zero-determinant strategies exist in oligopoly games. In this paper, we investigate zero-determinant strategies in the repeated Cournot oligopoly game, which is the simplest mathematical model of oligopoly. We prove the existence of zero-determinant strategies which unilaterally enforce linear relations between the payoff of the player and the average payoff of the opponents. Furthermore, we numerically show that some zero-determinant strategy can promote collusion in a duopoly case, although it cannot promote collusion in a triopoly case. Our results imply that zero-determinant strategies can be used to promote cooperation between firms even in an oligopoly market.