Abstract:
In a two player game, a referee asks two cooperating players (who are not allowed to communicate) questions sampled from some distribution and decides whether they win or not based on some predicate of the questions and their answers. The parallel repetition of the game is the game in which the referee samples n independent pairs of questions and sends corresponding questions to the players simultaneously. If the players cannot win the original game with probability better than (1-e), what's the best they can do in the repeated game? We improve earlier results of Raz and Holenstein, which showed that the players cannot win all copies in the repeated game with probability better than (1-e^3)^{\Omega(n/c)} (here c is the length of the answers in the game), in the following ways: We prove the bound (1-\e^2)^{Omega(n)} as long as n = Omega(log(1/e)/e^2) and the game is a ``projection game'', the type of game most commonly used in hardness of approximation results. Our bound is independent of the answer length and has a better dependence on e. By the recent work of Raz, this bound is essentially tight (we might still hope to get rid of the restriction on n). A consequence of this bound is that the Unique Games Conjecture of Khot is equivalent to: [Unique Games Conjecture] For every delta,e > 0, there exists an alphabet size M(e) such that it is NP-hard to distinguish a Unique Game with alphabet size M for which a 1-e^2 fraction of the constraints can be satisfied from one in which a 1-e^{1-delta} fraction of the constraints can be satisfied. We prove a concentration bound for parallel repetition (of general games) showing that for any constant 0

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