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

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