610 research outputs found
Query Complexity of Approximate Equilibria in Anonymous Games
We study the computation of equilibria of anonymous games, via algorithms
that may proceed via a sequence of adaptive queries to the game's payoff
function, assumed to be unknown initially. The general topic we consider is
\emph{query complexity}, that is, how many queries are necessary or sufficient
to compute an exact or approximate Nash equilibrium.
We show that exact equilibria cannot be found via query-efficient algorithms.
We also give an example of a 2-strategy, 3-player anonymous game that does not
have any exact Nash equilibrium in rational numbers. However, more positive
query-complexity bounds are attainable if either further symmetries of the
utility functions are assumed or we focus on approximate equilibria. We
investigate four sub-classes of anonymous games previously considered by
\cite{bfh09, dp14}.
Our main result is a new randomized query-efficient algorithm that finds a
-approximate Nash equilibrium querying
payoffs and runs in time . This improves on the running
time of pre-existing algorithms for approximate equilibria of anonymous games,
and is the first one to obtain an inverse polynomial approximation in
poly-time. We also show how this can be utilized as an efficient
polynomial-time approximation scheme (PTAS). Furthermore, we prove that
payoffs must be queried in order to find any
-well-supported Nash equilibrium, even by randomized algorithms
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
Learning Convex Partitions and Computing Game-theoretic Equilibria from Best Response Queries
Suppose that an -simplex is partitioned into convex regions having
disjoint interiors and distinct labels, and we may learn the label of any point
by querying it. The learning objective is to know, for any point in the
simplex, a label that occurs within some distance from that point.
We present two algorithms for this task: Constant-Dimension Generalised Binary
Search (CD-GBS), which for constant uses queries, and Constant-Region Generalised Binary
Search (CR-GBS), which uses CD-GBS as a subroutine and for constant uses
queries.
We show via Kakutani's fixed-point theorem that these algorithms provide
bounds on the best-response query complexity of computing approximate
well-supported equilibria of bimatrix games in which one of the players has a
constant number of pure strategies. We also partially extend our results to
games with multiple players, establishing further query complexity bounds for
computing approximate well-supported equilibria in this setting.Comment: 38 pages, 7 figures, second version strengthens lower bound in
Theorem 6, adds footnotes with additional comments and fixes typo
Privacy and Truthful Equilibrium Selection for Aggregative Games
We study a very general class of games --- multi-dimensional aggregative
games --- which in particular generalize both anonymous games and weighted
congestion games. For any such game that is also large, we solve the
equilibrium selection problem in a strong sense. In particular, we give an
efficient weak mediator: a mechanism which has only the power to listen to
reported types and provide non-binding suggested actions, such that (a) it is
an asymptotic Nash equilibrium for every player to truthfully report their type
to the mediator, and then follow its suggested action; and (b) that when
players do so, they end up coordinating on a particular asymptotic pure
strategy Nash equilibrium of the induced complete information game. In fact,
truthful reporting is an ex-post Nash equilibrium of the mediated game, so our
solution applies even in settings of incomplete information, and even when
player types are arbitrary or worst-case (i.e. not drawn from a common prior).
We achieve this by giving an efficient differentially private algorithm for
computing a Nash equilibrium in such games. The rates of convergence to
equilibrium in all of our results are inverse polynomial in the number of
players . We also apply our main results to a multi-dimensional market game.
Our results can be viewed as giving, for a rich class of games, a more robust
version of the Revelation Principle, in that we work with weaker informational
assumptions (no common prior), yet provide a stronger solution concept (ex-post
Nash versus Bayes Nash equilibrium). In comparison to previous work, our main
conceptual contribution is showing that weak mediators are a game theoretic
object that exist in a wide variety of games -- previously, they were only
known to exist in traffic routing games
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
Well-Supported vs. Approximate Nash Equilibria: Query Complexity of Large Games
In this paper we present a generic reduction from the problem of finding an epsilon-well-supported Nash equilibrium (WSNE) to that of finding an Theta(epsilon)-approximate Nash equilibrium (ANE), in large games with n players and a bounded number of strategies for each player.
Our reduction complements the existing literature on relations between WSNE and ANE, and can be applied to extend hardness results on WSNE to similar results on ANE.
This allows one to focus on WSNE first, which is in general easier to analyze and control in hardness constructions.
As an application we prove a 2^{Omega(n/log n)} lower bound on the randomized query complexity of finding an epsilon-ANE in binary-action n-player games, for some constant epsilon>0.
This answers an open problem posed by Hart and Nisan and Babichenko, and is very close to the trivial upper bound of 2^n.
Previously for WSNE, Babichenko showed a 2^{Omega(n)} lower bound on the randomized query complexity of finding an epsilon-WSNE for some constant epsilon>0.
Our result follows directly from combining Babichenko\u27s result and our new reduction from WSNE to ANE
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