Refining the equilibrium concept for bimatrix games via control costs

The perfectness and the properness concept are two refinements of the Nash equilibrium concept. Both are based on the idea that with a small probability a player will make mistakes. In this paper, we elaborate the idea that a player can control his mistake probability. However, the more a player wants to reduce this probability, the harder he has to work for it, and hence, the more costs he incurs. We formalize this idea by introducing bimatrix games with control costs. We investigate which equilibrium points of an ordinary bimatrix game are approximated by equilibrium points of bimatrix games with control costs as the control costs go to zero. We prove that, if the control costs satisfy certain conditions, then the set of approximable equilibria is a (possibly proper) subset of the set of the perfect equilibria, which may be disjoint with the set of the proper equilibria. Furthermore, we show that the basic assumption underlying the properness concept, viz. that more costly mistakes are made with an infinitely less probability, cannot be justified by our approach. In general, the set of approachable equilibria may depend on the control costs. However, we show that regular equilibria are approximated for all choices of the control costs.