  We propose an analytical framework for studying parallel repetition, a basic
product operation for one-round two-player games. In this framework, we
consider a relaxation of the value of a game, $\mathrm{val}_+$, and prove that
for projection games, it is both multiplicative (under parallel repetition) and
a good approximation for the true value.
  These two properties imply a parallel repetition bound as $$
\mathrm{val}(G^{\otimes k}) \approx \mathrm{val}_+(G^{\otimes k}) =
\mathrm{val}_+(G)^{k} \approx \mathrm{val}(G)^{k}. $$
  Using this framework, we can also give a short proof for the NP-hardness of
Label-Cover$(1,\delta)$ for all $\delta>0$, starting from the basic PCP
theorem.
  We prove the following new results:
  - A parallel repetition bound for projection games with small soundness.
Previously, it was not known whether parallel repetition decreases the value of
such games. This result implies stronger inapproximability bounds for Set-Cover
and Label-Cover.
  - An improved bound for few parallel repetitions of projection games, showing
that Raz's counterexample is tight even for a small number of repetitions.
  Our techniques also allow us to bound the value of the direct product of
multiple games, namely, a bound on $\mathrm{val}(G_1\otimes ...\otimes G_k)$
for different projection games $G_1,...,G_k$.
