Parallel Repetition in Projection Games and a Concentration Bound
Anup Rao · SIAM Journal on Computing · 2008
A two-player game is played by cooperating players who are not allowed to communicate. A referee asks the players questions sampled from some known distribution and decides whether they win or not based on a known predicate of the questions and the players' answers. The parallel repetition of the game is the game in which the referee samples n independent pairs of questions and sends the corresponding questions to the players simultaneously. If the players cannot win the original game with probability better than $(1-\epsilon)$, what's the best they can do in the repeated game? We improve earlier results of [R. Raz, SIAM J. Comput., 27 (1998), pp. 763–803] and [T. Holenstein, Theory Comput., 5 (2009), pp. 141–172], who showed that the players cannot win all copies in the repeated game with probability better than $(1-\epsilon/2)^{\Omega(n\epsilon^2/c)}$ (here c is the length of the answers in the game), in the following ways: (i) We show that the probability of winning all copies is $(1-\epsilon/2)^{\Omega(\epsilon n)}$ as long as the game is a “projection game,” the type of game most commonly used in hardness of approximation results. (ii) We prove a concentration bound for parallel repetition (of general games) showing that for any constant $0 0$, there exists an alphabet size $M(\epsilon)$ for which it is NP-hard to distinguish a unique game with alphabet size M in which a $(1-\epsilon^2)$ fraction of the constraints can be satisfied from one in which a $(1-\epsilon f(1/\epsilon))$ fraction of the constraints can be satisfied.