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 constraints, where n is the number of variables. For suficiently large alphabets, it improves an algorithm of Khot (STOC?02) that satisfies a constant fraction of constraints in unique games of value 1 - O(1/(k^{10} (\log k)^5 )), where k is the size of the alphabet. We also present a simpler algorithm for the special case of unique games with linear constraints.

Finally, we present a simple approximation algorithm for 2-to-1 games.