Formula omitted]-stable model and essential equilibria

Abstract

The aim of this paper is twofold. First, we extend the concepts of structural stability and robustness to ϵ-equilibria that is defined in Anderlini and Canning (2001), Yu and Yu (2006) and Yu et al. (2009) so that we investigate whether the introduction of a small amount of additional rationality into a model has little impact on the set of bounded rational equilibria. In this paper, we call a model (λ,ϵ)-robust if small deviations of rationality in a bounded rational environment result in only small changes in the set of boundedly rational equilibria. We also say that a model is (λ,ϵ)-stable if given bounded rational agents, the set of the bounded rational equilibria varies continuously with the pair of the parameter value and the degree of rationality in the model. Secondly, we discuss the relationship between (λ,ϵ)-stability and essential equilibria. A boundedly rational equilibrium action is said to be essential at some parameter value and some degree of rationality if each action nearby is also a boundedly rational equilibrium behavior at nearby parameter values and degrees of rationality, i.e. the essentiality is a kind of stability property of the set of boundedly rational equilibria against slight perturbations of the parameter value and the degree of rationality. Our main results are as follows:(1) (λ,ϵ)-stability implies (λ,ϵ)-robustness of the model, and as a corollary, (λ,ϵ)-robustness for “almost all” pairs of the parameter value and the degree of rationality is established, (2) under some mild conditions, a model is (λ,ϵ)-stable if and only if any element of the set of bounded rational equilibria is essential.