  We study the amount of entropy players asymptotically need to play a repeated
normal-form game in a Nash equilibrium. Hub\'a\v{c}ek, Naor, and Ullman
(SAGT'15, TCSys'16) gave sufficient conditions on a game for the minimal amount
of randomness required to be $O(1)$ or $\Omega(n)$ for all players, where $n$
is the number of repetitions. We provide a complete characterization of games
in which there exists Nash equilibria of the repeated game using $O(1)$
randomness, closing an open question posed by Budinich and Fortnow (EC'11) and
Hub\'a\v{c}ek, Naor, and Ullman. Moreover, we show a 0--1 law for randomness in
repeated games, showing that any repeated game either has $O(1)$-randomness
Nash equilibria, or all of its Nash equilibria require $\Omega(n)$ randomness.
Our techniques are general and naturally characterize the payoff space of
sublinear-entropy equilibria, and could be of independent interest to the study
of players with other bounded capabilities in repeated games.
