بازار در تعادل با شرکت های خارج از تعادل: مطالعه شبیه سازی

We explore the effect of the limited ability to process information on the convergence of firms toward equilibrium. In the context of a Cournot oligopoly with a unique and symmetric Nash equilibrium, firms are modeled as adaptive economic agents through a genetic algorithm. Computational experiments show that while market production is close to equilibrium, firm production is relatively far from the individual equilibrium level. This pattern of firm heterogeneity is not an artifact of random elements built into the decisional process. Instead, it comes from the market interaction of firms with cognitive limitations.

Many experimental studies have documented that when subjects are given identical monetary incentives they often choose different actions. We mention three instances among many. Bossaert and Plott (2000) study financial markets where individual portfolio holdings are predicted to be identical and report persistent individual differences. In the appropriation of a common-pool resource, individual choices are remarkably different one from another. Ostrom et al. (1994) carefully document this pattern. While reviewing voluntary public good contribution experiments, Ledyard (1995) poses as a puzzle the wide heterogeneity of individual contributions.1 We claim that a powerful source of such individual heterogeneity could be the limited ability of agents to process information. To support this claim, simulation results are presented in a Cournot oligopoly where firms are modeled as adaptive economic agents with limited knowledge of the task and limited memory. These firms experiment with new strategies, and they learn from experience. We implement an evolutionary approach through a genetic algorithm, where firms are identical in their level of bounded rationality and where the equilibrium discovery process has a random element. The strategic environment adopted is simple and exhibits a unique symmetric equilibrium that makes it hard for firm heterogeneity to persist, yet, simulation results show that boundedly rational although identical firms are heterogeneous in their strategy choice. In order to understand the forces that generate this pattern, an extensive sensitivity analysis was performed in two dimensions: degree of noise and rationality. Contrary to what one might expect, we show that the heterogeneity result is not simply a consequence of the random elements contained in the genetic algorithm. Moreover, with a rise in the memory capabilities and in the ability to evaluate potential strategies, individual differences decline. In the limit case of full rationality, there is a convergence toward the canonical result of uniform individual behavior. The main goal of the paper is to use genetic algorithm firms to replicate some qualitative features in the experimental literature. In addition, an in-depth analysis of genetic algorithms as economic models is provided.2 Genetic algorithms have been used in economics as a black box to model boundedly rational agents. This paper goes beyond that by assessing the impact of several key parameters in the model and showing the interplay between random search and degree of rationality. A major point is that for a large class of genetic algorithm firms, the discovery of the market equilibrium is much easier than the discovery of the individual equilibrium strategy. The paper is organized as follows. The Cournot model and simulations parameters are outlined in Section 1. The decision-making process of the individual-learning genetic algorithm is explained in Section 2. The main result regarding individual heterogeneity is in Section 3, along with the discussion on the random element. In Sections 4 and 5 we explore changes in rationality levels with respect to pre-play evaluation of new strategies and by varying working memory constraints. Conclusions are in Section 6.

We use simulations to study the impact of bounded rationality on the convergence of identical firms toward equilibrium in a Cournot oligopoly. A major result is that the interaction of firms with limited information processing capabilities and limited working memory generates outcomes that are close to the market equilibrium while being relatively far from the individual Nash equilibrium level. A sensitivity analysis confirms that firms exhibit a persistent heterogeneity in individual behavior for a wide set of parameter values (Result 1). This result reproduces a qualitative feature found in the experimental economic literature. Under identical incentives subjects behave in a diverse fashion (Bossaert and Plott, 2000, Ostrom et al., 1994 and Ledyard, 1995). The model employed in the simulations is ordinal in the payoff function and has neither firm-specific goals nor skills. We used an individual learning genetic algorithm, where, over time, the best-performing strategies gain a higher probability of being played. New strategies are randomly generated and introduced into the set of available strategies. Even though one might suspect that the outcome originates from the stochastic nature of some genetic algorithm operators (hence it is built-in by construction), we show that this is not the case (Result 2). Moreover, we identify conditions under which firm heterogeneity becomes minimal. Pointing at those conditions sheds light on the forces that generated firm heterogeneity in the first place. First, as the level of agent rationality increases, individual heterogeneity fades away, yielding the canonical prediction that fully rational agents have uniform behavior. In particular, the introduction of a pre-evaluation of new potential strategies (election operator) has the effect of lowering firms’ heterogeneity (Result 3). A similar effect comes from relaxing memory constraints (Result 4). Raising memory skills and firm rationality in general generates an immediately tighter fit of aggregate outcome to the market equilibrium while it produces a slow but steady decrement in firm heterogeneity. These comparative static results suggest that the discovery of the aggregate Nash equilibrium is easier than that of the individual Nash equilibrium. Moreover, they make clear that calibrating a bounded rationality model only on its aggregate convergence to equilibrium may not be enough. The model performance in terms of individual convergence to equilibrium provides a richer and more challenging set of parameter restrictions. Second, an alternative way to reduce firm heterogeneity is to lower the rate of random search. The simple adoption of a low experimentation rate, however, is not a sufficient condition to attain low firm heterogeneity (Fig. 1). That goal can be achieved only if, in addition, firms are initialized at the equilibrium strategy. Equilibrium initialization keeps firm homogeneity as long as this situation is “frozen” through zero or a very low experimentation rate. Irrespectively of their impact on heterogeneity, these model changes have little effect on aggregate results. Again, this finding points out that coordinating on the individual Nash equilibrium requires a level of skills considerably higher than discovering the market equilibrium. This work is not a statement that any form of bounded rationality will lead to individual heterogeneity in behavior. In fact, in the context that we have analyzed, only heavy bounds to rationality have produced it. The open issue is then how to calibrate these models to the actual cognitive limitations of people in order to understand if and how much of the individual heterogeneity observed in experimental data is due to bounded rationality. In conclusion, we claim that a significant source of individual heterogeneity in human behavior could be due to the limited ability of agents to process information. This explanation has the advantage of not building individual-specific variations into a model in order to explain the empirical individual diversity. The simulations presented suggest on one side the existence of an inverse correlation between levels of rationality and levels of individual heterogeneity and on the other side that the Nash equilibrium is a more robust predictor of aggregate behavior than of individual behavior.