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In theoretical computer science, parallel repetition is a direct product for probabilistic refereed games. It is one approach for amplifying completeness-soundness gaps in constraint satisfaction problems (with applications to proving hardness of approximation results) and in interactive proof systems. The question of how much parallel repetition decreases the value of a game as a function of the number of repetitions has been the focus of a large body of research.

= Parallel Repetition = A probabilistic refereed game is an interaction in which a player (or multiple cooperating players) exchange messages with a referee, who decides whether the players win or lose. One example is the (2-player) CHSH game, in which the referee sends a uniformly random bit to each player, and each player responds with a bit. If the $$i^{th}$$ player receives bit $$q_i$$ and outputs $$a_i$$, then the players are said to win if $$a_1 \oplus a_2 = q_1 \cdot q_2$$

That is, for any natural number $$n$$and any game $$\mathcal{G}$$, the $$n$$-wise repetition of $$\mathcal{G}$$, denoted $$\mathcal{G}^n$$, is a game in which each player simultaneously participates in $$n$$ independent instances of $$\mathcal{G}$$. A common intuition is that this operation decreases the game's value exponentially (i.e., if players can win $$\mathcal{G}$$ with probability $$p$$, they should be able to win $$\mathcal{G}^n$$ with probability at most $$p^n$$). While this intuition was initially asserted as fact, it is actually unfounded and false. There is a 2-player game $$\mathcal{G}$$ with value $$1/2$$ such that the value of $$\mathcal{G}^2$$ is also $$1/2$$.

Still, it has been proved that for all 2-player games $$\mathcal{G}$$, the value of $$\mathcal{G}^n$$ asymptotically decreases exponentially with $$n$$.

Parallel Repetition of Interactive Arguments
Todo: cite work by Haitner et al. that shows counterexamples and how to fix

Parallel Repetition of Entangled Games
Todo: cite Yuen, Bavarian, etc.

Parallel Repetition of Non-Signaling Games
Todo: should this section even exist?

Category:Theoretical computer science Category:Mathematics of computing Category:Computational complexity theory