On refinements of subgame perfect -equilibrium

The concept of subgame perfect -equilibrium (-SPE), where is an error-term, has in recent years emerged as a prominent solution concept for perfect information games of infinite duration. We propose two refinements of this concept: continuity -SPE and -tolerance equilibrium. A continuity -SPE is an -SPE in which, in any subgame, the induced play is a continuity point of the payoff functions. We prove that continuity -SPE exists for each if the payoff functions are bounded and lower semicontinuous. A loss tolerance function is a function that assigns to each history a positive real number . A strategy profile is said to be a -tolerance equilibrium if for each history it is a -equilibrium in the subgame starting at . We prove that, for each loss tolerance function , there exists a -tolerance equilibrium provided that the payoff functions are bounded and continuous. We give counterexamples to show the sharpness of the existence results.