A Survey of Parallel Repetition and Projection Games

Parallel Repetition governs the decay of the verification probability in a repeated 2-prover interactive protocol. We will first analyze Feige’s game which shows the naive exponential bound does not hold. Then we will state Raz’s original bound[6] and observe some interesting properties of it. Though the original proof by Raz is very involved, we will give a high level proof idea and analyze major steps involved in Holenstein’s simplification as discussed by Rao[4]. Afterwards we will look at a variant of the bound recently discovered by Dinur and Steurer[2] when applied to projection games and compare it with that discovered by Rao. Critical definitions and lemmas are analyzed so that the reader can attain an intuitive reasoning for the derivation of these bounds. This is especially important in the case of Dinur and Steurer since they take a spectral graph theory approach but still utilize the notion of correlated sampling observed in Holenstein’s simplification. Afterwards we discuss applications and tightness of some parallel repetition bounds.