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Nash equilibrium is the central solution concept in Game Theory. Since Nash's original paper in 1951, it has found countless applications in modeling strategic behavior of traders in markets, (human) drivers and (electronic) routers in congested networks, nations in nuclear disarmament negotiations, and more. A decade ago, the relevance of this solution concept was called into question by computer scientists~\cite{DGP, CDT}, who proved (under appropriate complexity assumptions) that {\em computing} a Nash equilibrium is an intractable problem. And if centralized, specially designed algorithms cannot find Nash equilibria, why should we expect distributed, selfish agents to converge to one? The remaining hope was that at least approximate Nash equilibria can be efficiently computed. Understanding whether there is an efficient algorithm for {\em approximate Nash equilibrium} has been the central open problem in this field for the past decade. In this thesis, we provide strong evidence that even finding an approximate Nash equilibrium is intractable. We prove several intractability theorems for different settings (two-player games and many-player games) and models (computational complexity, query complexity, and communication complexity). In particular, our main result is that under a plausible and natural complexity assumption (``Exponential Time Hypothesis for \PPAD''), there is no polynomial-time algorithm for finding an approximate Nash equilibrium in two-player games. The problem of approximate Nash equilibrium in a two-player game poses a unique technical challenge: it is a member of the class \PPAD, which captures the complexity of several fundamental total problems, i.e. problems that always have a solution; and it also admits a quasipolynomial ($\approx n^{\log n}$) time algorithm. Either property alone is believed to place this problem far below \NP-hard problems in the complexity hierarchy; having both simultaneously places it just above \P, at what can be called the frontier of intractability. Indeed, the tools we develop in this thesis to advance on this frontier are useful for proving hardness of approximation of several other important problems whose complexity lies between \P~and \NP: \begin{description} \item [Brouwer's fixed point] Given a continuous function $f$ mapping a compact convex set to itself, Brouwer's fixed point theorem guarantees that $f$ has a fixed point, i.e. $x$ such that $f(x) = x$. Our intractability result holds for the relaxed problem of finding an approximate fixed point, i.e. $x$ such that $f(x) \approx x$. \item [Market equilibrium] Market equilibrium is a vector of prices and allocations where the supply meets the demand for each good. %We consider the Arrow-Debreu model where agents are both sellers and buyers of goods. Our intractability result holds for the relaxed problem of finding an approximate market equilibrium, where the supply of each good approximately meets the demand. \item [CourseMatch (A-CEEI)] Approximate Competitive Equilibrium from Equal Income (A-CEEI) is the economic principle underlying CourseMatch, a system for fair allocation of classes to students (currently in use at Wharton, University of Pennsylvania). \item [Densest $k$-subgraph] Our intractability result holds for the following relaxation of the $k$-Clique problem: given a graph containing a $k$-clique, the algorithm has to find a subgraph over $k$ vertices that is ``almost a clique'', i.e. most of the edges are present. \item [Community detection] We consider a well-studied model of communities in social networks, where each member of the community is friends with a large fraction of the community, and each non-member is only friends with a small fraction of the community. \item [VC dimension and Littlestone dimension] The Vapnik-Chervonenkis (VC) dimension is a fundamental measure in learning theory that captures the complexity of a binary concept class. Similarly, the Littlestone dimension is a measure of complexity of online learning. \item [Signaling in zero-sum games] We consider a fundamental problem in signaling, where an informed signaler reveals private information about the payoffs in a two-player zero-sum game, with the goal of helping one of the players.
