Non-existence of subgame-perfect ε-equilibrium in perfect information games with infinite horizon

Every finite extensive-form game with perfect information has a subgame-perfect equilibrium. In this note we settle to the negative an open problem regarding the existence of a subgame-perfect ε-equilibrium in perfect information games with infinite horizon and Borel measurable payoffs, by providing a counter-example. We also consider a refinement called strong subgame-perfect ε-equilibrium, and show by means of another counter-example, with a simpler structure than the previous one, that a game may have no strong subgame-perfect ε-equilibrium for sufficiently smallε>0, even though it admits a subgame-perfect ε-equilibrium for every ε>0.