  We study the class of languages, denoted by $\MIP[k, 1-\epsilon, s]$, which
have $k$-prover games where each prover just sends a \emph{single} bit, with
completeness $1-\epsilon$ and soundness error $s$. For the case that $k=1$
(i.e., for the case of interactive proofs), Goldreich, Vadhan and Wigderson
({\em Computational Complexity'02}) demonstrate that $\SZK$ exactly
characterizes languages having 1-bit proof systems with"non-trivial" soundness
(i.e., $1/2 < s \leq 1-2\epsilon$). We demonstrate that for the case that
$k\geq 2$, 1-bit $k$-prover games exhibit a significantly richer structure:
  + (Folklore) When $s \leq \frac{1}{2^k} - \epsilon$, $\MIP[k, 1-\epsilon, s]
= \BPP$;
  + When $\frac{1}{2^k} + \epsilon \leq s < \frac{2}{2^k}-\epsilon$, $\MIP[k,
1-\epsilon, s] = \SZK$;
  + When $s \ge \frac{2}{2^k} + \epsilon$, $\AM \subseteq \MIP[k, 1-\epsilon,
s]$;
  + For $s \le 0.62 k/2^k$ and sufficiently large $k$, $\MIP[k, 1-\epsilon, s]
\subseteq \EXP$;
  + For $s \ge 2k/2^{k}$, $\MIP[k, 1, 1-\epsilon, s] = \NEXP$.
  As such, 1-bit $k$-prover games yield a natural "quantitative" approach to
relating complexity classes such as $\BPP$,$\SZK$,$\AM$, $\EXP$, and $\NEXP$.
We leave open the question of whether a more fine-grained hierarchy (between
$\AM$ and $\NEXP$) can be established for the case when $s \geq \frac{2}{2^k} +
\epsilon$.
