The Experimetrics of Depth-of-Reasoning Models

The theme of this chapter is the parametric estimation of depth-of-reasoning models. This sort of model has been applied to many game theoretic settings, including various 2-player games, auction experiments, and the well-known beauty contest game. The most popular depth-of-reasoning models currently appearing in the literature are variants of the Level-k (LK) model and the cognitive hierarchy (CH) model. After describing these models, previous attempts at econometric estimation of them are surveyed. The finite mixture modelling framework plays a key role here. Under this approach, agents at different levels of reasoning are treated as different “types”, with theory dictating a “best-response” for each type. Models typically include a finite number of low types, e.g. reasoning levels 0-3, and sometimes in addition a “Nash” type corresponding to level . Data on choices are then used to estimate a set of “mixing proportions”, which indicate the proportion of the population who are at each level of reasoning. The model sometimes contains other parameters as well, for example, one or more representing the extent of computational errors. A number of estimation issues are addressed. Firstly, identification problems that arise when applying the models to certain 2-player games are discussed. This leads into a discussion of the applications to situations in which the strategy space is continuous, in which identification problems are far less severe. The most popular application in this class is the beauty contest game. In this context, a number of other econometric issues are addressed, including dealing with censoring at the extremes of the strategy space. A number of post-estimation tools are then illustrated, such as the derivation of type-conditional densities and posterior-type probabilities. We will also consider the key assumption underlying levels of reasoning models that agents actually form specific beliefs, and best-respond to them. We will consider methods for testing this assumption in the context of a simple auction experiment.