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

We conjecture that PPAD has a PCP-like complete problem, seeking a near equilibrium in which all but very few players have very little incentive to deviate. We show that, if one assumes that this problem requires exponential time, several open problems in this area are settled. The most important implication, proved via a "birthday repetition" reduction, is that the n^O(log n) approximation scheme of [LMM03] for the Nash equilibrium of two-player games is essentially optimum. Two other open problems in the area are resolved once one assumes this conjecture, establishing that certain approximate equilibria are PPAD-complete: Finding a relative approximation of two-player Nash equilibria (without the well-supported restriction of [Das13]), and an approximate competitive equilibrium with equal incomes [Bud11] with small clearing error and near-optimal Gini coefficient.Comment: Revision 2 derandomizes the main reductio

Well-Supported versus Approximate Nash Equilibria: Query Complexity of Large Games

We study the randomized query complexity of approximate Nash equilibria (ANE) in large games. We prove that, for some constant $\epsilon>0$, any randomized oracle algorithm that computes an $\epsilon$-ANE in a binary-action, $n$-player game must make $2^{\Omega(n/\log n)}$ payoff queries. For the stronger solution concept of well-supported Nash equilibria (WSNE), Babichenko previously gave an exponential $2^{\Omega(n)}$ lower bound for the randomized query complexity of $\epsilon$-WSNE, for some constant $\epsilon>0$; the same lower bound was shown to hold for $\epsilon$-ANE, but only when $\epsilon=O(1/n)$. Our result answers an open problem posed by Hart and Nisan and Babichenko and is very close to the trivial upper bound of $2^n$. Our proof relies on a generic reduction from the problem of finding an $\epsilon$-WSNE to the problem of finding an $\epsilon/(4\alpha)$-ANE, in large games with $\alpha$ actions, which might be of independent interest.Comment: 10 page

Public Bayesian Persuasion: Being Almost Optimal and Almost Persuasive

Persuasion studies how an informed principal may influence the behavior of agents by the strategic provision of payoff-relevant information. We focus on the fundamental multi-receiver model by Arieli and Babichenko (2019), in which there are no inter-agent externalities. Unlike prior works on this problem, we study the public persuasion problem in the general setting with: (i) arbitrary state spaces; (ii) arbitrary action spaces; (iii) arbitrary sender's utility functions. We fully characterize the computational complexity of computing a bi-criteria approximation of an optimal public signaling scheme. In particular, we show, in a voting setting of independent interest, that solving this problem requires at least a quasi-polynomial number of steps even in settings with a binary action space, assuming the Exponential Time Hypothesis. In doing so, we prove that a relaxed version of the Maximum Feasible Subsystem of Linear Inequalities problem requires at least quasi-polynomial time to be solved. Finally, we close the gap by providing a quasi-polynomial time bi-criteria approximation algorithm for arbitrary public persuasion problems that, in specific settings, yields a QPTAS