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 e e\varepsilon -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 e e\varepsilon -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 e e\varepsilon -equilibrium for sufficiently small e>0 e>0\varepsilon >0, even though it admits a subgame-perfect e e\varepsilon -equilibrium for every e>0 e>0\varepsilon >0.