953 research outputs found
Query Complexity of Approximate Nash Equilibria
We study the query complexity of approximate notions of Nash equilibrium in
games with a large number of players . Our main result states that for
-player binary-action games and for constant , the query
complexity of an -well-supported Nash equilibrium is exponential
in . One of the consequences of this result is an exponential lower bound on
the rate of convergence of adaptive dynamics to approxiamte Nash equilibrium
Complexity Theory, Game Theory, and Economics: The Barbados Lectures
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 theory
has helped illuminate several barriers in economics and game theory; and (ii)
to illustrate how game-theoretic questions have led to new and interesting
complexity theory, including recent several breakthroughs. It consists of two
five-lecture sequences: the Solar Lectures, focusing on the communication and
computational complexity of computing equilibria; and the Lunar Lectures,
focusing on applications of complexity theory in game theory and economics. No
background in game theory is assumed.Comment: Revised v2 from December 2019 corrects some errors in and adds some
recent citations to v1 Revised v3 corrects a few typos in v
Computational Complexity of Approximate Nash Equilibrium in Large Games
We prove that finding an epsilon-Nash equilibrium in a succinctly
representable game with many players is PPAD-hard for constant epsilon. Our
proof uses succinct games, i.e. games whose payoff function is represented by a
circuit. Our techniques build on a recent query complexity lower bound by
Babichenko.Comment: New version includes an addendum about subsequent work on the open
problems propose
On Communication Complexity of Fixed Point Computation
Brouwer's fixed point theorem states that any continuous function from a
compact convex space to itself has a fixed point. Roughgarden and Weinstein
(FOCS 2016) initiated the study of fixed point computation in the two-player
communication model, where each player gets a function from to
, and their goal is to find an approximate fixed point of the
composition of the two functions. They left it as an open question to show a
lower bound of for the (randomized) communication complexity of
this problem, in the range of parameters which make it a total search problem.
We answer this question affirmatively.
Additionally, we introduce two natural fixed point problems in the two-player
communication model.
Each player is given a function from to ,
and their goal is to find an approximate fixed point of the concatenation of
the functions.
Each player is given a function from to , and
their goal is to find an approximate fixed point of the interpolation of the
functions.
We show a randomized communication complexity lower bound of
for these problems (for some constant approximation factor).
Finally, we initiate the study of finding a panchromatic simplex in a
Sperner-coloring of a triangulation (guaranteed by Sperner's lemma) in the
two-player communication model: A triangulation of the -simplex is
publicly known and one player is given a set and a coloring
function from to , and the other player is given a set
and a coloring function from to ,
such that , and their goal is to find a panchromatic
simplex. We show a randomized communication complexity lower bound of
for the aforementioned problem as well (when is large)
Search for the end of a path in the d-dimensional grid and in other graphs
We consider the worst-case query complexity of some variants of certain
\cl{PPAD}-complete search problems. Suppose we are given a graph and a
vertex . We denote the directed graph obtained from by
directing all edges in both directions by . is a directed subgraph of
which is unknown to us, except that it consists of vertex-disjoint
directed paths and cycles and one of the paths originates in . Our goal is
to find an endvertex of a path by using as few queries as possible. A query
specifies a vertex , and the answer is the set of the edges of
incident to , together with their directions. We also show lower bounds for
the special case when consists of a single path. Our proofs use the theory
of graph separators. Finally, we consider the case when the graph is a grid
graph. In this case, using the connection with separators, we give
asymptotically tight bounds as a function of the size of the grid, if the
dimension of the grid is considered as fixed. In order to do this, we prove a
separator theorem about grid graphs, which is interesting on its own right
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