77,027 research outputs found
An Evolutionary Approach to Congestion
Using techniques from evolutionary game theory, we analyze potential games with continuous player sets, a class of games which includes a general model of network congestion as a special case. We concisely characterize both the complete set of Nash equilibria and the set of equilibria which are robust against small disturbances of aggregate behavior. We provide a strong evolutionary justification of why equilibria must arise. We characterize situations in which stable equilibria are socially efficient, and show that in such cases, evolution always increases aggregate efficiency. Applying these results, we construct a parameterized class of congestion tolls under which evolution yields socially optimal play. Finally, we characterize potential games with continuous player sets by establishing that a generalization of these games is precisely the limiting version of finite player potential games (Monderer and Shapley (1996)) which satisfy an anonymity condition.
Game-theoretical control with continuous action sets
Motivated by the recent applications of game-theoretical learning techniques
to the design of distributed control systems, we study a class of control
problems that can be formulated as potential games with continuous action sets,
and we propose an actor-critic reinforcement learning algorithm that provably
converges to equilibrium in this class of problems. The method employed is to
analyse the learning process under study through a mean-field dynamical system
that evolves in an infinite-dimensional function space (the space of
probability distributions over the players' continuous controls). To do so, we
extend the theory of finite-dimensional two-timescale stochastic approximation
to an infinite-dimensional, Banach space setting, and we prove that the
continuous dynamics of the process converge to equilibrium in the case of
potential games. These results combine to give a provably-convergent learning
algorithm in which players do not need to keep track of the controls selected
by the other agents.Comment: 19 page
No-regret Dynamics and Fictitious Play
Potential based no-regret dynamics are shown to be related to fictitious
play. Roughly, these are epsilon-best reply dynamics where epsilon is the
maximal regret, which vanishes with time. This allows for alternative and
sometimes much shorter proofs of known results on convergence of no-regret
dynamics to the set of Nash equilibria
On Robustness Properties in Empirical Centroid Fictitious Play
Empirical Centroid Fictitious Play (ECFP) is a generalization of the
well-known Fictitious Play (FP) algorithm designed for implementation in
large-scale games. In ECFP, the set of players is subdivided into equivalence
classes with players in the same class possessing similar properties. Players
choose a next-stage action by tracking and responding to aggregate statistics
related to each equivalence class. This setup alleviates the difficult task of
tracking and responding to the statistical behavior of every individual player,
as is the case in traditional FP. Aside from ECFP, many useful modifications
have been proposed to classical FP, e.g., rules allowing for network-based
implementation, increased computational efficiency, and stronger forms of
learning. Such modifications tend to be of great practical value; however,
their effectiveness relies heavily on two fundamental properties of FP:
robustness to alterations in the empirical distribution step size process, and
robustness to best-response perturbations. The main contribution of the paper
is to show that similar robustness properties also hold for the ECFP algorithm.
This result serves as a first step in enabling practical modifications to ECFP,
similar to those already developed for FP.Comment: Submitted for publication. Initial Submission: Mar. 201
On an unified framework for approachability in games with or without signals
We unify standard frameworks for approachability both in full or partial
monitoring by defining a new abstract game, called the "purely informative
game", where the outcome at each stage is the maximal information players can
obtain, represented as some probability measure. Objectives of players can be
rewritten as the convergence (to some given set) of sequences of averages of
these probability measures. We obtain new results extending the approachability
theory developed by Blackwell moreover this new abstract framework enables us
to characterize approachable sets with, as usual, a remarkably simple and clear
reformulation for convex sets. Translated into the original games, those
results become the first necessary and sufficient condition under which an
arbitrary set is approachable and they cover and extend previous known results
for convex sets. We also investigate a specific class of games where, thanks to
some unusual definition of averages and convexity, we again obtain a complete
characterization of approachable sets along with rates of convergence
Strong Nash Equilibria in Games with the Lexicographical Improvement Property
We introduce a class of finite strategic games with the property that every
deviation of a coalition of players that is profitable to each of its members
strictly decreases the lexicographical order of a certain function defined on
the set of strategy profiles. We call this property the Lexicographical
Improvement Property (LIP) and show that it implies the existence of a
generalized strong ordinal potential function. We use this characterization to
derive existence, efficiency and fairness properties of strong Nash equilibria.
We then study a class of games that generalizes congestion games with
bottleneck objectives that we call bottleneck congestion games. We show that
these games possess the LIP and thus the above mentioned properties. For
bottleneck congestion games in networks, we identify cases in which the
potential function associated with the LIP leads to polynomial time algorithms
computing a strong Nash equilibrium. Finally, we investigate the LIP for
infinite games. We show that the LIP does not imply the existence of a
generalized strong ordinal potential, thus, the existence of SNE does not
follow. Assuming that the function associated with the LIP is continuous,
however, we prove existence of SNE. As a consequence, we prove that bottleneck
congestion games with infinite strategy spaces and continuous cost functions
possess a strong Nash equilibrium
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