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We conjecture that PPAD has a PCP-like complete problem, seeking a nearequilibrium in which all but very few players have very little incentive todeviate. We show that, if one assumes that this problem requires exponentialtime, several open problems in this area are settled. The most importantimplication, proved via a "birthday repetition" reduction, is that the n^O(logn) approximation scheme of [LMM03] for the Nash equilibrium of two-player gamesis essentially optimum. Two other open problems in the area are resolved onceone assumes this conjecture, establishing that certain approximate equilibriaare PPAD-complete: Finding a relative approximation of two-player Nashequilibria (without the well-supported restriction of [Das13]), and anapproximate competitive equilibrium with equal incomes [Bud11] with smallclearing error and near-optimal Gini coefficient.

Can Almost Everybody be Almost Happy? PCP for PPAD and the Inapproximability of Nash