698 research outputs found
Logarithmic Query Complexity for Approximate Nash Computation in Large Games
We investigate the problem of equilibrium computation for âlargeâ n-player games. Large
games have a Lipschitz-type property that no single playerâs utility is greatly affected by any
other individual playerâs actions. In this paper, we mostly focus on the case where any change of
strategy by a player causes other playersâ payoffs to change by at most 1
n
. We study algorithms
having query access to the gameâs payoff function, aiming to find Δ-Nash equilibria. We seek
algorithms that obtain Δ as small as possible, in time polynomial in n.
Our main result is a randomised algorithm that achieves Δ approaching 1
8
for 2-strategy games
in a completely uncoupled setting, where each player observes her own payoff to a query, and
adjusts her behaviour independently of other playersâ payoffs/actions. O(log n) rounds/queries
are required. We also show how to obtain a slight improvement over 1
8
, by introducing a small
amount of communication between the players.
Finally, we give extension of our results to large games with more than two strategies per
player, and alternative largeness parameters
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
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
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
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
Lower bounds for the query complexity of equilibria in Lipschitz games
Nearly a decade ago, Azrieli and Shmaya introduced the class of λ-Lipschitz games in which every player's payoff function is λ-Lipschitz with respect to the actions of the other players. They showed that such games admit Ï”-approximate pure Nash equilibria for certain settings of Ï” and λ. They left open, however, the question of how hard it is to find such an equilibrium. In this work, we develop a query-efficient reduction from more general games to Lipschitz games. We use this reduction to show a query lower bound for any randomized algorithm finding Ï”-approximate pure Nash equilibria of n-player, binary-action, λ-Lipschitz games that is exponential in nλ/Ï”. In addition, we introduce âMulti-Lipschitz games,â a generalization involving player-specific Lipschitz values, and provide a reduction from finding equilibria of these games to finding equilibria of Lipschitz games, showing that the value of interest is the average of the individual Lipschitz parameters. Finally, we provide an exponential lower bound on the deterministic query complexity of finding Ï”-approximate Nash equilibria of n-player, m-action, λ-Lipschitz games for strong values of Ï”, motivating the consideration of explicitly randomized algorithms in the above results
The complexity of solution concepts in Lipschitz games
Nearly a decade ago, Azrieli and Shmaya introduced the class of λ-Lipschitz games in which every
playerâs payoff function is λ-Lipschitz with respect to the actions of the other players. They showed
via the probabilistic method that n-player Lipschitz games with m strategies per player have pure
-approximate Nash equilibria, for ℠λâ8n log(2mn). They left open, however, the question of
how hard it is to find such an equilibrium. In this work, we develop an efficient reduction from more
general games to Lipschitz games. We use this reduction to study both the query and computational
complexity of algorithms finding λ-approximate pure Nash equilibria of λ-Lipschitz games and related
classes.
We show a query lower bound exponential in nλ/ against randomized algorithms finding -
approximatepure Nash equilibria of n-player, λ-Lipschitz games. We additionally present the first
PPAD-completeness result for finding pure Nash equilibria in a class of finite, non-Bayesian games
(we show this for λ-Lipschitz polymatrix games for suitable pairs of values and λ) in which both the
proof of PPAD-hardness and the proof of containment in PPAD require novel approaches (in fact,
our approach implies containment in PPAD for any class of Lipschitz games in which payoffs from
mixed-strategy profiles can be deterministically computed), and present a definition of ârandomized
PPADâ. We define and subsequently analyze the class of âMulti-Lipschitz gamesâ, a generalization of
Lipschitz games involving player-specific Lipschitz parameters in which the value of interest appears
to be the average of the individual Lipschitz parameters. We discuss a dichotomy of the deterministic
query complexity of finding -approximate Nash equilibria of general games and, subsequently, a query
lower bound for λ-Lipschitz games in which any non-trivial value of requires exponentially-many
queries to achieve. We examine which parts of this extend to the concepts of approximate correlated
and coarse correlated equilibria, and in the process generalize the edge-isoperimetric inequalities to
generalizations of the hypercube. Finally, we improve the block update algorithm presented by Goldberg
and Marmolejo to break the potential boundary of a 0.75-approximation factor, presenting a
randomized algorithm achieving a 0.7368-approximate Nash equilibrium making polynomially-many
profile queries of an n-player 1/nâ1 -Lipschitz game with an unbounded number of actions
- âŠ