1,538 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
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
Finding Approximate Nash Equilibria of Bimatrix Games via Payoff Queries
We study the deterministic and randomized query complexity of finding approximate equilibria in a k × k bimatrix game. We show that the deterministic query complexity of finding an ϵ-Nash equilibrium when ϵ < ½ is Ω(k2), even in zero-one constant-sum games. In combination with previous results [Fearnley et al. 2013], this provides a complete characterization of the deterministic query complexity of approximate Nash equilibria. We also study randomized querying algorithms. We give a randomized algorithm for finding a (3-√5/2 + ϵ)-Nash equilibrium using O(k.log k/ϵ2) payoff queries, which shows that the ½ barrier for deterministic algorithms can be broken by randomization. For well-supported Nash equilibria (WSNE), we first give a randomized algorithm for finding an ϵ-WSNE of a zero-sum bimatrix game using O(k.log k/ϵ4) payoff queries, and we then use this to obtain a randomized algorithm for finding a (⅔ + ϵ)-WSNE in a general bimatrix game using O(k.log k/ϵ4) payoff queries. Finally, we initiate the study of lower bounds against randomized algorithms in the context of bimatrix games, by showing that randomized algorithms require Ω(k2) payoff queries in order to find an ϵ-Nash equilibrium with ϵ < 1/4k, even in zero-one constant-sum games. In particular, this rules out query-efficient randomized algorithms for finding exact Nash equilibria
Distributed Methods for Computing Approximate Equilibria
We present a new, distributed method to compute approximate Nash equilibria
in bimatrix games. In contrast to previous approaches that analyze the two
payoff matrices at the same time (for example, by solving a single LP that
combines the two players payoffs), our algorithm first solves two independent
LPs, each of which is derived from one of the two payoff matrices, and then
compute approximate Nash equilibria using only limited communication between
the players.
Our method has several applications for improved bounds for efficient
computations of approximate Nash equilibria in bimatrix games. First, it yields
a best polynomial-time algorithm for computing \emph{approximate well-supported
Nash equilibria (WSNE)}, which guarantees to find a 0.6528-WSNE in polynomial
time. Furthermore, since our algorithm solves the two LPs separately, it can be
used to improve upon the best known algorithms in the limited communication
setting: the algorithm can be implemented to obtain a randomized
expected-polynomial-time algorithm that uses poly-logarithmic communication and
finds a 0.6528-WSNE. The algorithm can also be carried out to beat the best
known bound in the query complexity setting, requiring payoff
queries to compute a 0.6528-WSNE. Finally, our approach can also be adapted to
provide the best known communication efficient algorithm for computing
\emph{approximate Nash equilibria}: it uses poly-logarithmic communication to
find a 0.382-approximate Nash equilibrium
The Query Complexity of Correlated Equilibria
We consider the complexity of finding a correlated equilibrium of an
-player game in a model that allows the algorithm to make queries on
players' payoffs at pure strategy profiles. Randomized regret-based dynamics
are known to yield an approximate correlated equilibrium efficiently, namely,
in time that is polynomial in the number of players . Here we show that both
randomization and approximation are necessary: no efficient deterministic
algorithm can reach even an approximate correlated equilibrium, and no
efficient randomized algorithm can reach an exact correlated equilibrium. The
results are obtained by bounding from below the number of payoff queries that
are needed
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
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
Towards Characterizing the First-order Query Complexity of Learning (Approximate) Nash Equilibria in Zero-sum Matrix Games
In the first-order query model for zero-sum matrix games, players
observe the expected pay-offs for all their possible actions under the
randomized action played by their opponent. This classical model has received
renewed interest after the discovery by Rakhlin and Sridharan that
-approximate Nash equilibria can be computed efficiently from
instead of queries.
Surprisingly, the optimal number of such queries, as a function of both
and , is not known. We make progress on this question on two
fronts. First, we fully characterise the query complexity of learning exact
equilibria (), by showing that they require a number of queries
that is linear in , which means that it is essentially as hard as querying
the whole matrix, which can also be done with queries. Second, for
, the current query complexity upper bound stands at
. We argue that, unfortunately, obtaining
a matching lower bound is not possible with existing techniques: we prove that
no lower bound can be derived by constructing hard matrices whose entries take
values in a known countable set, because such matrices can be fully identified
by a single query. This rules out, for instance, reducing to an optimization
problem over the hypercube by encoding it as a binary payoff matrix. We then
introduce a new technique for lower bounds, which allows us to obtain lower
bounds of order for any , where is a constant independent of . We further discuss
possible future directions to improve on our techniques in order to close the
gap with the upper bounds
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