We continue the investigation of interactive proofs with bounded
communication, as initiated by Goldreich and Hastad (IPL 1998).
Let $L$ be a language that has an interactive proof in which the prover
sends few (say $b$) bits to the verifier.
We prove that the complement $\bar L$ has ... more >>>

Game theory has been used for a long time to study phenomena in evolutionary biology, beginning systematically with the seminal work of John Maynard Smith. A central concept in this connection has been the notion of an evolutionarily stable strategy (ESS) in a symmetric two-player strategic form game. A regular ... more >>>

Consider a system M of parallel machines, each with a strictly increasing and differentiable load dependent latency function. The users of such a system are of infinite number and act selfishly, routing their infinitesimally small portion of the total flow r they control, to machines of currently minimum delay. It ... more >>>

We address the fundamental question of whether the Nash equilibria of
a game can be computed in polynomial time. We describe certain
efficient reductions between this problem for
normal form games with a fixed number of players
and graphical games with fixed degree. Our main result is that ... more >>>

We resolve the question of the complexity of Nash equilibrium by
showing that the problem of computing a Nash equilibrium in a game
with 4 or more players is complete for the complexity class PPAD.
Our proof uses ideas from the recently-established equivalence
between polynomial-time solvability of normal-form games and
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A natural algorithmic scheme in online game playing is called `follow-the-leader', first proposed by Hannan in the 1950's. Simply stated, this method advocates the use of past history to make future predictions, by using the optimal strategy so far as the strategy for the next game iteration. Randomized variations on ... more >>>

Strategic games may exhibit symmetries in a variety of ways. A common aspect of symmetry, enabling the compact representation of games even when the number of players is unbounded, is that players cannot (or need not) distinguish between the other players. We define four classes of symmetric games by considering ... more >>>

We study multiplayer games in which the participants have access to
only limited randomness. This constrains both the algorithms used to
compute equilibria (they should use little or no randomness) as well
as the mixed strategies that the participants are capable of playing
(these should be sparse). We frame algorithmic ... more >>>

In this paper we propose a methodology for determining approximate Nash equilibria of non-cooperative bimatrix games and, based on that, we provide a polynomial time algorithm that computes $\frac{1}{3} + \frac{1}{p(n)} $ -approximate equilibria, where $p(n)$ is a polynomial controlled by our algorithm and proportional to its running time. The ... more >>>

The folk theorem suggests that finding Nash Equilibria
in repeated games should be easier than in one-shot games. In
contrast, we show that the problem of finding any (epsilon) Nash
equilibrium for a three-player infinitely-repeated game is
computationally intractable (even when all payoffs are in
{-1,0,-1}), unless all of PPAD ... more >>>

We study graphical games where the payoff function of each player satisfies one of four types of symmetries in the actions of his neighbors. We establish that deciding the existence of a pure Nash equilibrium is NP-hard in graphical games with each of the four types of symmetry. Using a ... more >>>

Given a set of observed economic choices, can one infer
preferences and/or utility functions for the players that are
consistent with the data? Questions of this type are called {\em
rationalization} or {\em revealed preference} problems in the
economic literature, and are the subject of a rich body of work.

Abstract. We study the decision theory of a maximally risk-averse investor | one whose objec-
tive, in the face of stochastic uncertainties, is to minimize the probability of ever going broke. With
a view to developing the mathematical basics of such a theory, we start with a very simple model
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This document collects the lecture notes from my mini-course "Complexity Theory, Game Theory, and Economics," taught at the Bellairs Research Institute of McGill University, Holetown, Barbados, February 19-23, 2017, as the 29th McGill Invitational Workshop on Computational Complexity.

The goal of this mini-course is twofold: (i) to explain how complexity ... more >>>