  It is well-known that for infinitely repeated games, there are computable
strategies that have best responses, but no computable best responses. These
results were originally proved for either specific games (e.g., Prisoner's
dilemma), or for classes of games satisfying certain conditions not known to be
both necessary and sufficient.
  We derive a complete characterization in the form of simple necessary and
sufficient conditions for the existence of a computable strategy without a
computable best response under limit-of-means payoff. We further refine the
characterization by requiring the strategy profiles to be Nash equilibria or
subgame-perfect equilibria, and we show how the characterizations entail that
it is efficiently decidable whether an infinitely repeated game has a
computable strategy without a computable best response.
