An oligopoly is a market in which the price of a 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 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. 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. Especially, we prove that an averagely unbeatable zero-determinant strategy exists, which is guaranteed to obtain the average payoff of the opponents. Second, we numerically show that the averagely unbeatable 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 oligopoly market.

翻译：寡头垄断是指少数几家公司控制商品价格的市场。古诺提出了最简单的寡头垄断博弈论模型，其中各公司追求利润最大化的行为会导致市场失灵。此外，当古诺寡头博弈无限重复进行时，公司之间可能通过默契合谋来垄断市场。这种默契合谋的实现机制与重复囚徒困境博弈中的直接互惠相同：在单次博弈中，双方背叛对各自更有利；而在重复博弈中，双方合作却可能实现。近年来，在重复囚徒困境博弈中，一类称为零行列式策略的策略在直接互惠研究领域备受关注。零行列式策略是一种专制策略，能够单方面控制博弈参与者的收益。学界已多次尝试在其他博弈中寻找零行列式策略，并对其进行扩展以应用于更广泛的情境。本文首先证明，在重复古诺寡头博弈中同样存在零行列式策略。特别地，我们证明存在平均不可战胜的零行列式策略，该策略能确保获得对手的平均收益。其次，我们通过数值模拟表明：当该策略应用于对抗自适应学习型参与者时，能够促进合谋；但当同时对抗两个自适应学习型参与者时，则无法促进合谋。我们的研究结果揭示了零行列式策略在寡头垄断市场中可能产生的某些负面影响。