254 research outputs found
Smoothed Efficient Algorithms and Reductions for Network Coordination Games
Worst-case hardness results for most equilibrium computation problems have
raised the need for beyond-worst-case analysis. To this end, we study the
smoothed complexity of finding pure Nash equilibria in Network Coordination
Games, a PLS-complete problem in the worst case. This is a potential game where
the sequential-better-response algorithm is known to converge to a pure NE,
albeit in exponential time. First, we prove polynomial (resp. quasi-polynomial)
smoothed complexity when the underlying game graph is a complete (resp.
arbitrary) graph, and every player has constantly many strategies. We note that
the complete graph case is reminiscent of perturbing all parameters, a common
assumption in most known smoothed analysis results.
Second, we define a notion of smoothness-preserving reduction among search
problems, and obtain reductions from -strategy network coordination games to
local-max-cut, and from -strategy games (with arbitrary ) to
local-max-cut up to two flips. The former together with the recent result of
[BCC18] gives an alternate -time smoothed algorithm for the
-strategy case. This notion of reduction allows for the extension of
smoothed efficient algorithms from one problem to another.
For the first set of results, we develop techniques to bound the probability
that an (adversarial) better-response sequence makes slow improvements on the
potential. Our approach combines and generalizes the local-max-cut approaches
of [ER14,ABPW17] to handle the multi-strategy case: it requires a careful
definition of the matrix which captures the increase in potential, a tighter
union bound on adversarial sequences, and balancing it with good enough rank
bounds. We believe that the approach and notions developed herein could be of
interest in addressing the smoothed complexity of other potential and/or
congestion games
Improving the smoothed complexity of FLIP for max cut problems
Finding locally optimal solutions for max-cut and max--cut are well-known
PLS-complete problems. An instinctive approach to finding such a locally
optimum solution is the FLIP method. Even though FLIP requires exponential time
in worst-case instances, it tends to terminate quickly in practical instances.
To explain this discrepancy, the run-time of FLIP has been studied in the
smoothed complexity framework. Etscheid and R\"{o}glin showed that the smoothed
complexity of FLIP for max-cut in arbitrary graphs is quasi-polynomial. Angel,
Bubeck, Peres, and Wei showed that the smoothed complexity of FLIP for max-cut
in complete graphs is , where is an upper bound on
the random edge-weight density and is the number of vertices in the input
graph.
While Angel et al.'s result showed the first polynomial smoothed complexity,
they also conjectured that their run-time bound is far from optimal. In this
work, we make substantial progress towards improving the run-time bound. We
prove that the smoothed complexity of FLIP in complete graphs is . Our results are based on a carefully chosen matrix whose rank
captures the run-time of the method along with improved rank bounds for this
matrix and an improved union bound based on this matrix. In addition, our
techniques provide a general framework for analyzing FLIP in the smoothed
framework. We illustrate this general framework by showing that the smoothed
complexity of FLIP for max--cut in complete graphs is polynomial and for
max--cut in arbitrary graphs is quasi-polynomial. We believe that our
techniques should also be of interest towards addressing the smoothed
complexity of FLIP for max--cut in complete graphs for larger constants .Comment: 36 page
Some coordination problems are harder than others
In order to coordinate players in a game must first identify a target pattern
of behaviour. In this paper we investigate the difficulty of identifying
prominent outcomes in two kinds of binary action coordination problems in
social networks: pure coordination games and anti-coordination games. For both
environments, we determine the computational complexity of finding a strategy
profile that (i) maximises welfare, (ii) maximises welfare subject to being an
equilibrium, and (iii) maximises potential. We show that the complexity of
these objectives can vary with the type of coordination problem. Objectives (i)
and (iii) are tractable problems in pure coordination games, but for
anti-coordination games are NP-hard. Objective (ii), finding the best Nash
equilibrium, is NP-hard for both. Our results support the idea that
environments in which actions are strategic complements (e.g., technology
adoption) facilitate successful coordination more readily than those in which
actions are strategic substitutes (e.g., public good provision).Comment: arXiv admin note: text overlap with arXiv:2305.0712
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
Metastability of Asymptotically Well-Behaved Potential Games
One of the main criticisms to game theory concerns the assumption of full
rationality. Logit dynamics is a decentralized algorithm in which a level of
irrationality (a.k.a. "noise") is introduced in players' behavior. In this
context, the solution concept of interest becomes the logit equilibrium, as
opposed to Nash equilibria. Logit equilibria are distributions over strategy
profiles that possess several nice properties, including existence and
uniqueness. However, there are games in which their computation may take time
exponential in the number of players. We therefore look at an approximate
version of logit equilibria, called metastable distributions, introduced by
Auletta et al. [SODA 2012]. These are distributions that remain stable (i.e.,
players do not go too far from it) for a super-polynomial number of steps
(rather than forever, as for logit equilibria). The hope is that these
distributions exist and can be reached quickly by logit dynamics.
We identify a class of potential games, called asymptotically well-behaved,
for which the behavior of the logit dynamics is not chaotic as the number of
players increases so to guarantee meaningful asymptotic results. We prove that
any such game admits distributions which are metastable no matter the level of
noise present in the system, and the starting profile of the dynamics. These
distributions can be quickly reached if the rationality level is not too big
when compared to the inverse of the maximum difference in potential. Our proofs
build on results which may be of independent interest, including some spectral
characterizations of the transition matrix defined by logit dynamics for
generic games and the relationship of several convergence measures for Markov
chains
Search and optimization with randomness in computational economics: equilibria, pricing, and decisions
In this thesis we study search and optimization problems from computational economics with primarily stochastic inputs. The results are grouped into two categories: First, we address the smoothed analysis of Nash equilibrium computation. Second, we address two pricing problems in mechanism design, and solve two economically motivated stochastic optimization problems.
Computing Nash equilibria is a central question in the game-theoretic study of economic systems of agent interactions. The worst-case analysis of this problem has been studied in depth, but little was known beyond the worst case. We study this problem in the framework of smoothed analysis, where adversarial inputs are randomly perturbed. We show that computing Nash equilibria is hard for 2-player games even when input perturbations are large. This is despite the existence of approximation algorithms in a similar regime. In doing so, our result disproves a conjecture relating approximation schemes to smoothed analysis. Despite the hardness results in general, we also present a special case of co-operative games, where we show that the natural greedy algorithm for finding equilibria has polynomial smoothed complexity. We also develop reductions which preserve smoothed analysis.
In the second part of the thesis, we consider optimization problems which are motivated by economic applications. We address two stochastic optimization problems. We begin by developing optimal methods to determine the best among binary classifiers, when the objective function is known only through pairwise comparisons, e.g. when the objective function is the subjective opinion of a client. Finally, we extend known algorithms in the Pandora's box problem --- a classic optimal search problem --- to an order-constrained setting which allows for richer modelling.
The remaining chapters address two pricing problems from mechanism design. First, we provide an approximately revenue-optimal pricing scheme for the problem of selling time on a server to jobs whose parameters are sampled i.i.d. from an unknown distribution. We then tackle the problem of fairly dividing chores among a collection of economic agents via a competitive equilibrium, which balances assigned tasks with payouts. We give efficient algorithms to compute such an equilibrium
On Tightness of the Tsaknakis-Spirakis Algorithm for Approximate Nash Equilibrium
Finding the minimum approximate ratio for Nash equilibrium of bi-matrix games
has derived a series of studies, started with 3/4, followed by 1/2, 0.38 and
0.36, finally the best approximate ratio of 0.3393 by Tsaknakis and Spirakis
(TS algorithm for short). Efforts to improve the results remain not successful
in the past 14 years. This work makes the first progress to show that the bound
of 0.3393 is indeed tight for the TS algorithm. Next, we characterize all
possible tight game instances for the TS algorithm. It allows us to conduct
extensive experiments to study the nature of the TS algorithm and to compare it
with other algorithms. We find that this lower bound is not smoothed for the TS
algorithm in that any perturbation on the initial point may deviate away from
this tight bound approximate solution. Other approximate algorithms such as
Fictitious Play and Regret Matching also find better approximate solutions.
However, the new distributed algorithm for approximate Nash equilibrium by
Czumaj et al. performs consistently at the same bound of 0.3393. This proves
our lower bound instances generated against the TS algorithm can serve as a
benchmark in design and analysis of approximate Nash equilibrium algorithms
Coordination problems on networks revisited: statics and dynamics
Simple binary-state coordination models are widely used to study collective
socio-economic phenomena such as the spread of innovations or the adoption of
products on social networks. The common trait of these systems is the
occurrence of large-scale coordination events taking place abruptly, in the
form of a cascade process, as a consequence of small perturbations of an
apparently stable state. The conditions for the occurrence of cascade
instabilities have been largely analysed in the literature, however for the
same coordination models no sufficient attention was given to the relation
between structural properties of (Nash) equilibria and possible outcomes of
dynamical equilibrium selection. Using methods from the statistical physics of
disordered systems, the present work investigates both analytically and
numerically, the statistical properties of such Nash equilibria on networks,
focusing mostly on random graphs. We provide an accurate description of these
properties, which is then exploited to shed light on the mechanisms behind the
onset of coordination/miscoordination on large networks. This is done studying
the most common processes of dynamical equilibrium selection, such as best
response, bounded-rational dynamics and learning processes. In particular, we
show that well beyond the instability region, full coordination is still
globally stochastically stable, however equilibrium selection processes with
low stochasticity (e.g. best response) or strong memory effects (e.g.
reinforcement learning) can be prevented from achieving full coordination by
being trapped into a large (exponentially in number of agents) set of locally
stable Nash equilibria at low/medium coordination (inefficient equilibria).
These results should be useful to allow a better understanding of general
coordination problems on complex networks.Comment: Revtex style, 56 pages, 21 figure
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