Autocratic strategies in Cournot oligopoly game

TL;DR

This paper demonstrates that zero-determinant (autocratic) strategies exist in the repeated Cournot oligopoly game, including a fair strategy that guarantees the opponents' average payoff. It provides a complete classification of two-point autocratic strategies: equalizers do not exist, while self-pinning, positively correlated, and negatively correlated strategies do under precise parameter regimes. Numerical experiments show that fair autocratic strategies can promote collusion in a two-player (duopoly) setting but fail to do so in a three-player (triopoly) setting, with adaptive learners sometimes undermining collusion. The work highlights the potential for unilateral payoff control to influence oligopolistic dynamics, including possible ZD alliances among opponents, and contrasts these findings with the classic prisoner's dilemma literature.

Abstract

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.

Figures (4)

• Figure 1: Linear relations between $\mathcal{S}_j$ and $\sum_{k\neq j} \mathcal{S}_k/(N-1)$ when player $j$ uses the two-point autocratic strategies with various $(\chi, \kappa)$ and all opponents repeat fixed actions.
• Figure 2: A linear relation between $\mathcal{S}_j$ and $\sum_{k\neq j} \mathcal{S}_k/(N-1)$ when player $j$ uses a two-point autocratic strategy with $(\chi, \kappa)=(1, 0)$ against random memory-one strategies.
• Figure 3: Numerical results for in the duopoly case ($N=2$). (a) Strategy adaptation of an adaptive memory-one player against the fair autocratic strategy. (b) Time-averaged payoffs of a fixed autocratic player and an adaptive memory-one player. Data points are plotted every 1000 time steps.
• Figure 4: Numerical results for in the triopoly case ($N=3$). (a) Strategy adaptation of two adaptive memory-one players against the fair autocratic strategy. (b) Time-averaged payoffs of a fixed autocratic player and two adaptive memory-one players. Data points are plotted every 1000 time steps.

Theorems & Definitions (7)

• Definition 1: McAHau2016
• Proposition 1: McAHau2016
• Proposition 2
• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4