Convergence in the finite Cournot oligopoly with social and individual learning

Abstract

Convergence to the Nash equilibrium in a Cournot oligopoly is a question that recurrently arises as a subject of controversy in economics. The development of evolutionary game theory has provided an equilibrium concept more directly connected with adjustment dynamics, and the evolutionary stability of the equilibria of the Cournot game have been extensively studied in the literature. Several articles show that the Walrasian equilibrium is the stable ESS of the Cournot game. But no general result has been established for the difficult case of simultaneous heterogenous mutations. Authors propose specific selection dynamics to analyze this case. Vriend (2000) proposes using a genetic algorithm for studying learning dynamics in this game and obtains convergence to Cournot equilibrium with individual learning. The resulting convergence has been questioned by Arifovic and Maschek (2006). The aim of this article is to clarify this controversy. It analyzes the mechanisms that are behind these contradictory results and underlines the specific role of the spite effect. We show why social learning gives rise to the Walrasian equilibrium and why, in a general setup, individual learning can effectively yield convergence to the Cournot equilibrium. We also illustrate these general results by systematic computational experiments.

Introduction

The starting point of this article is the recent contradiction that has arisen in the literature about the convergence to equilibrium in Cournot oligopoly games, where boundedly rational agents’ learning is represented using genetic algorithms (GA). The interesting and contrasted results of Vriend (2000) have been questioned by more recent studies, in particular, by Arifovic and Maschek (2006). The main result of Vriend (2000) is convergence to Cournot equilibrium (CE) only when agents’ learning is individual, while social learning yields convergence to the Walrasian equilibrium (WE). Arifovic and Maschek (2006) confirm the results for social learning, but question the convergence results under individual learning. The aim of this article is to clarify both the initial contrasted results, and the contradiction between these two computational articles. This contradiction adds to a rather old debate in economics.

Convergence to the Nash equilibrium in a Cournot oligopoly is indeed a problem that recurrently arises as a subject of controversy in economics. Already in the 1960s, several articles in the Review of Economic Studies discussed the convergence under best reply dynamics (see, for example, Theocharis, 1960). This debate has led to the conclusion that best reply dynamics become unstable under standard assumptions about demand and cost conditions as soon as there are three or more firms in the oligopoly.

The development of evolutionary game theory (Maynard-Smith, 1982) has indirectly contributed to this debate by providing an equilibrium concept more directly connected with other types of adjustment dynamics (than best reply) which can arise when the learning of firms is taken into account. Several articles have studied the evolutionary stability of equilibria in the Cournot game. In a path-breaking article, Vega-Redondo (1997) shows that the Walrasian equilibrium (WE) is the unique evolutionary stable outcome in a quantity competition game with a homogenous product under certain specific assumptions about the interaction structure and strategy sets of the agents. The results obtained in the articles that followed Vega-Redondo (1997) cast more doubt on the potential convergence of adjustment dynamics to the Cournot equilibrium (CE). The nature of mutations and the selection dynamics are at the core of these results. These dimensions directly result from assumptions about the learning processes of boundedly rational firms with incomplete and imperfect information.

The evolutionary stability concept is very useful for analyzing the convergence to equilibrium with dynamics more naturally connected to social learning of firms through imitation of strategies and experimentation (random mutations). But this stability concept does not exclude other learning schemes, as long as they can be formulated as selection mechanisms operating at the level of firm population. Unfortunately, richer learning schemes generally imply more complex dynamics, and oligopoly is more naturally formulated as a “playing the field” game (each player playing against all the others in each period), than a pairwise matching process. The analysis of these dynamics is very difficult under general conditions and the existing literature proposes only partial results obtained under specific assumptions, as in Vega-Redondo (1997). Stegeman and Rhode (2004) or Bergin and Bernhardt (2004) consider relatively richer frameworks but they again obtain partial results that cannot be used to draw general conclusions about the consequences of different representations of firms’ learning. This difficulty has motivated other recent studies to resort to agent-based computational models to explore the question of convergence (see, for example, Kirman (1995) for a simulation-based analysis of three such specific learning rules).

Arifovic (1994) and Vriend (2000) develop models that adopt such an explorative strategy. They use a computational representation of learning that is quite commonly retained in the literature: genetic algorithms (see Dawid, 1999, Vallée and Yıldızoğlu, 2004). GA represents the learning of agents as an exploratory combination process based on random experiments and combination of already discovered strategies. Arifovic (1994) initially obtained a convergence to the WE with this type of learning mechanisms. Vriend (2000) advances that genetic algorithms (GA) can be used to demonstrate the “essential difference between individual and social learning” of firms in a Cournot oligopoly. A GA operating at the firm population level yields convergence, under social learning, to the WE, while a set of GAs each adjusting the strategies of an individual firm imply, under individual learning, convergence to the CE. These results apparently contrast with Arifovic (1994). They have also recently been questioned by Arifovic and Maschek (2006), who draw attention to the specific implementation adopted by Vriend in terms of the mechanisms of GA-based learning and the cost structure of firms. More precisely, Arifovic and Maschek confirm convergence to the WE under social learning, but they cast some doubt on the possibility of convergence to the CE under individual learning. Since all the results of these three articles are computational, it is difficult to understand why the convergence results under social learning are so robust and why the results for the individual learning are so controversial. It is important to verify if the difference between these two types of learning are really essential. If this is the case, then we must try to understand the fundamental mechanisms around which the debate about individual learning evolves.

The article is organized in two parts. The first part exposes the main results concerning imitation-based social learning. These results will be established by studying the evolutionary stability of equilibria in Cournot games (Section 2), and by focusing on the main mechanism behind these results (Section 3, on spite effect). These analytical convergence results will cover all general cases but one: the case with simultaneous and heterogenous mutations from equilibrium. Since this is the most interesting case for the Cournot game, it will be necessary to follow the analysis by combining partial analytical results with computational experiments. The second part (Section 4) will consequently analyze the convergence conditions when simultaneous heterogenous mutations are allowed. In order to clarify the contradiction cited above, the analysis will be restricted to the special selection mechanism used by genetic algorithm-based learning. When this is possible, we will rely on the general results of the first part in this analysis (in particular for the social learning case). Our results should also interest other domains where GA is used to represent individual or social learning (see, for example, Vallée and Başar, 1999, Yıldızoğlu, 2002). We will show why social learning cannot yield convergence to CE, and why individual learning can converge to CE if the interactions of the firms allow them to discover the decreasing relationship between the market price and their quantities. The last section concludes the article.