A classic result in computational game theory states that there are infinitely repeated games where one player has a computable strategy that has a best response, but no computable best response. For games with discounted payoff, the result is known to hold for a specific class of games-essentially generalizations of Prisoner's Dilemma-but until now, no necessary and sufficient condition is known. To be of any value, the computable strategy having no computable best response must be part of a subgame-perfect equilibrium, as otherwise a rational, self-interested player would not play the strategy.We give the first necessary and sufficient conditions for a two-player repeated game to have such a computable strategy with no computable best response for all discount factors above some threshold. The conditions involve existence of a Nash equilibrium of the repeated game whose discounted payoffs satisfy certain conditions involving the min-max payoffs of the underlying stage game. We show that it is decidable in polynomial time in the size of the payoff matrix of whether it satisfies these conditions.