On the implications of integrating linear tracing procedure with imprecise probabilities

This paper explores the implications of integrating the so-called linear tracing procedure with uncertainty modeling using sets of probabilities for equilibrium refinements under strategic uncertainty. We first reexamine the linear tracing procedure [10] by studying the relationship between priors and Nash equilibria. A prior belongs to the source set of a Nash equilibrium if the linear tracing procedure based on this prior leads to that equilibrium. We show that the source set of any Nash equilibrium is always nonempty and closed, but not generally convex. We then motivate the idea of iteratively applying this procedure to the auxiliary games that are used to model hypothetical reasoning under the procedure. Based on this idea, we propose a notion of robustness for equilibria that allows for modeling of players' initial expectations in the linear tracing procedure through sets of probability distributions. By considering ϵ-contaminated classes for modeling uncertainty, we illustrate this concept with two examples, and then discuss the role of strategic uncertainty in coordination failure.