We develop a new local characterization of the zero-error information complexity function for two-party communication problems, and use it to compute the exact internal and external information complexity of the 2−bit AN D function: IC(AND, 0) = C ∧ ≈ 1.4923 bits, and IC ext (AND, 0) = log 2 3 ≈ 1.5839 bits. This leads to a tight (upper and lower bound) characterization of the communication complexity of the set intersection problem on subsets of {1, . . . , n} (the player are required to compute the intersection of their sets), whose randomized communication complexity tends to C ∧ · n ± o(n) as the error tends to zero.The information-optimal protocol we present has an infinite number of rounds. We show this is necessary by proving that the rate of convergence of the r−round information cost of AN D to IC(AND, 0) = C ∧ behaves like Θ(1/r 2 ), i.e. that the r-round information complexity of AN D is C ∧ + Θ(1/r 2 ).We leverage the tight analysis obtained for the information complexity of AN D to calculate and prove the exact communication complexity of the set disjointness function Disj n(X, Y ) = ¬ ∨ n i=1 AN D(xi, yi) with error tending to 0, which turns out to be = C DISJ · n ± o(n), where C DISJ ≈ 0.4827. Our rate of convergence results imply that an asymptotically optimal protocol for set disjointness will have to use ω(1) rounds of communication, since every rround protocol will be sub-optimal by at least Ω(n/r 2 ) bits of communication. * In this extended abstract, we omit most proofs, and shorten others. The full version of this paper can be found at ECCC † Research supported in part by an Alfred P. Sloan Fellowship, an NSF CAREER award (CCF-1149888), and a Turing Centenary Fellowship.We also obtain the tight bound of 2 ln 2 k ± o(k) on the communication complexity of disjointness of sets of size ≤ k. An asymptotic bound of Θ(k) was previously shown by Håstad and Wigderson.

We continue the study of welfare maximization in unit-demand (matching) markets, in a distributed information model where agent's valuations are unknown to the central planner, and therefore communication is required to determine an efficient allocation. Dobzinski, Nisan and Oren (STOC'14) showed that if the market size is n, then r rounds of interaction (with logarithmic bandwidth) suffice to obtain an n 1/(r+1) -approximation to the optimal social welfare. In particular, this implies that such markets converge to a stable state (constant approximation) in time logarithmic in the market size.We obtain the first multi-round lower bound for this setup. We show that even if the allowable perround bandwidth of each agent is n ε(r) , the approximation ratio of any r-round (randomized) protocol is no better than Ω(n 1/5 r+1 ), implying an Ω(log log n) lower bound on the rate of convergence of the market to equilibrium. Our construction and technique may be of interest to round-communication tradeoffs in the more general setting of combinatorial auctions, for which the only known lower bound is for simultaneous (r = 1) protocols [DNO14].

We give exponentially small upper bounds on the success probability for computing the direct product of any function over any distribution using a communication protocol.

The celebrated PPAD hardness result for finding an exact Nash equilibrium in a two-player game initiated a quest for finding approximate Nash equilibria efficiently, and is one of the major open questions in algorithmic game theory. We study the computational complexity of finding an ε-approximate Nash equilibrium with good social welfare. Hazan and Krauthgamer and subsequent improvements showed that finding an ε-approximate Nash equilibrium with good social welfare in a two player game and many variants of this problem is at least as hard as finding a planted clique of size O(log n) in the random graph G(n, 1/2). We show that any polynomial time algorithm that finds an ε-approximate Nash equilibrium with good social welfare refutes (the worst-case) Exponential Time Hypothesis by Impagliazzo and Paturi. Specifically, it would imply a 2Õ (n 1/2) algorithm for SAT. Our lower bound matches the quasi-polynomial time algorithm by Lipton, Markakis and Mehta for solving the problem. Our key tool is a reduction from the PCP machinery to finding Nash equilibrium via free games, the framework introduced in the recent work by Aaronson, Impagliazzo and Moshkovitz. Techniques developed in the process may be useful for replacing planted clique hardness with ETH-hardness in other applications.