What Model Best Describes Initial Choices in a Cournot Duopoly Experiment?

Introduction

This paper estimates several models of bounded rationality to determine which one best fit the data from initial choices in a Cournot duopoly experiment. Nash Equilibrium is the usual solution concept used in standard static duopoly models. However, this concept requires the mutual consistency of actions and beliefs, the quantity a firm chooses must be optimal given its beliefs about the quantity produced by the rival and those beliefs must also be consistent with its action. The cognitive requirements of the Nash Equilibrium concept are quite high, and in many static games experimental subjects fail to act as predicted by the theory, especially when they do not have the possibility of learning, see Kagel and Roth(1995) and Camerer (2003) for a survey. In many settings, managers do not have this possibility either, this may be due to the fact that the situation is new or that the nature of the game to be played is quite different from previous experiences. Cournot competition can also be interpreted as a choice of capacity, see Kreps and Scheinkman (1983), giving limited room for change after the choice has been made, thus making convergence to equilibrium slow or impossible. Independently of the reason, it is interesting to analyze initial choices to determine what model best explain behavior prior to learning.

In this paper we consider four models of bounded rationality. The first is the Level-k model. In a Level-k model there are different levels representing depth of reasoning. The lowest, Level-0, describes an agent with little understanding of the situation. It does not respond to any belief and chooses the quantity to be produced randomly. A Level-1 player thinks the other is Level-0 and chooses optimally given this belief. In general, a Level-k player believes the other is one step below and chooses to maximize expected payoff. This might be due to cognitive limitations (people have difficulties with higher order reasoning) and/or overconfidence (individuals think they are on average smarter than the rest). In this model actions are optimal given beliefs but beliefs are inconsistent with actions, thus the mutual consistency characteristic of Nash Equilibrium is broken. The Level-k model is a popular alternative to Nash Equilibrium to describe the choice of individuals in dominance solvable games who have little or no possibility of learning from previous experience.

The second model of bounded rationality is Quantal Response Equilibrium. QRE is an equilibrium concept and as such there is consistency between actions and beliefs. However, unlike Nash Equilibrium, the players “better” respond given their beliefs.¹ In a QRE players choose actions that generate higher expected payoffs with higher probability according to some function mapping beliefs to actions. In most practical applications a logit function is used to capture that relationship. In fact, this concept is a generalization of Nash Equilibrium since as the sensitivity of choice probabilities to expected payoffs increases the QRE converges to the NE solution.² One attractive feature of the QRE is its parsimony since it only depends on one parameter. Even though this is an equilibrium concept and thus particularly not well-suited to explain initial choices, we include it in the analysis to test the hypothesis that all other non-equilibrium models will do a better job explaining the data we analyze in terms of Bayesian Information Criterion (BIC).

The third model is Noisy Introspection. This is not an equilibrium concept but a generalization of rationalizability. The actual choice of a player depends on the belief about the action of the other player, this is her first order belief. The first order belief depends on the belief the other player has about the choice probability of the player, this is the second order belief. In general, the k order belief depends on the k+1 order belief, essentially Noisy Introspection consists of layers of beliefs that become more imprecise as we consider higher orders. In practice the link between higher order beliefs and lower order beliefs is also through the logit function as we will show in the next section when all four models are described in more detail. In this model bounded rationality is incorporated by breaking the consistency between actions and beliefs that is the signature of the Nash Equilibrium concept.