A unique game is a type of constraint satisfaction problem with two variables per constraint. The value of a unique game is the fraction of the constraints satisfied by an optimal solution. Khot (STOC'02) conjectured that for arbitrarily small $\gamma, \epsilon >0$ it is NP-hard to distinguish games of value smaller than $\gamma$ from games of value larger than $1-\epsilon$. Several recent inapproximability results rely on Khot's conjecture.

Considering the case of sub-constant $\epsilon$, Khot (STOC'02) analyzes an algorithm based on semidefinite programming that satisfies a constant fraction of the constraints in unique games of value $1-O(k^{-10}\cdot (\log k)^{-5})$, where $k$ is the size of the domain of the variables.

We present a polynomial time algorithm based on semidefinite programming that, given a unique game of value $1-O(1/\log n)$, satisfies a constant fraction of the constraints, where $n$ is the number of variables. This is an improvement over Khot's algorithm if the domain is sufficiently large.

We also present a simpler algorithm for the special case of unique games with linear constraints, and a simple approximation algorithm for the more general class of 2-to-1 games.