On refinements of subgame perfect $$\epsilon $$ϵ-equilibrium

The concept of subgame perfect $$\epsilon $$ϵ-equilibrium ($$\epsilon $$ϵ-SPE), where $$\epsilon $$ϵ 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 $$\epsilon $$ϵ-SPE and $$\phi $$ϕ-tolerance equilibrium. A continuity $$\epsilon $$ϵ-SPE is an $$\epsilon $$ϵ-SPE in which, in any subgame, the induced play is a continuity point of the payoff functions. We prove that continuity $$\epsilon $$ϵ-SPE exists for each $$\epsilon > 0$$ϵ>0 if the payoff functions are bounded and lower semicontinuous. A loss tolerance function $$\phi $$ϕ is a function that assigns to each history $$h$$h a positive real number $$\phi (h)$$ϕ(h). A strategy profile is said to be a $$\phi $$ϕ-tolerance equilibrium if for each history $$h$$h it is a $$\phi (h)$$ϕ(h)-equilibrium in the subgame starting at $$h$$h. We prove that, for each loss tolerance function $$\phi $$ϕ, there exists a $$\phi $$ϕ-tolerance equilibrium provided that the payoff functions are bounded and continuous. We give counterexamples to show the sharpness of the existence results.