On the power of unique 2-prover 1-round games

A 2-prover game is called unique if the answer of one prover uniquely determines the answer of the second prover and vice versa (we implicitly assume games to be one round games). The value of a 2-prover game is the maximum acceptance probability of the verifier over all the prover strategies. We make the following conjecture regarding the power of unique 2-prover games, which we call the Unique Games Conjecture:(MATH) The Unique Games Conjecture: For arbitrarily small constants $ \ \zeta, \ \delta > 0$, there exists a constant $k = k(\zeta,\delta)$ such that it is NP-hard to determine whether a unique 2-prover game with answers from a domain of size $k$ has value at least $1-\zeta$ or at most $\delta$.…