Ambiguity aversion as a route to randomness in a duopoly game

TL;DR

This paper explores how ambiguity aversion influences the stability and dynamics of a duopoly game, revealing that worst-case optimization leads to chaos and complex behaviors when firms face uncertainty about the market structure.

Contribution

It demonstrates that ambiguity aversion causes nonlinear dynamics and chaos in duopoly games, highlighting the impact of uncertainty on equilibrium stability and global market behavior.

Findings

Stable Cournot-Nash equilibrium loses stability under uncertainty.

Chaos and multiple attractors emerge when firms are ambiguity averse.

Chaotic dynamics prevent firms from being ambiguity averse in practice.

Abstract

The global dynamics is investigated for a duopoly game where the perfect foresight hypothesis is relaxed and firms are worst-case maximizers. Overlooking the degree of product substitutability as well as the sensitivity of price to quantity, the unique and globally stable Cournot-Nash equilibrium of the complete-information duopoly game, loses stability when firms are not aware if they are playing a duopoly game, as it is, or an oligopoly game with more than two competitors. This finding resembles Theocharis' condition for the stability of the Cournot-Nash equilibrium in oligopolies without uncertainty. As opposed to complete-information oligopoly games, coexisting attractors, disconnected basins of attractions and chaotic dynamics emerge when the Cournot-Nash equilibrium loses stability. This difference in the global dynamics is due to the nonlinearities introduced by the worst-case approach to uncertainty, which mirror in bimodal best-reply functions. Conducted with techniques that require a symmetric setting of the game, the investigation of the dynamics reveals that a chaotic regime prevents firms from being ambiguity averse, that is, firms are worst-case maximizers only in the quantity-expectation space. Therefore, chaotic dynamics are the result and at the same time the source of profit uncertainty.