2,868 research outputs found
Convergence to Equilibrium of Logit Dynamics for Strategic Games
We present the first general bounds on the mixing time of the Markov chain
associated to the logit dynamics for wide classes of strategic games. The logit
dynamics with inverse noise beta describes the behavior of a complex system
whose individual components act selfishly and keep responding according to some
partial ("noisy") knowledge of the system, where the capacity of the agent to
know the system and compute her best move is measured by the inverse of the
parameter beta.
In particular, we prove nearly tight bounds for potential games and games
with dominant strategies. Our results show that, for potential games, the
mixing time is upper and lower bounded by an exponential in the inverse of the
noise and in the maximum potential difference. Instead, for games with dominant
strategies, the mixing time cannot grow arbitrarily with the inverse of the
noise.
Finally, we refine our analysis for a subclass of potential games called
graphical coordination games, a class of games that have been previously
studied in Physics and, more recently, in Computer Science in the context of
diffusion of new technologies. We give evidence that the mixing time of the
logit dynamics for these games strongly depends on the structure of the
underlying graph. We prove that the mixing time of the logit dynamics for these
games can be upper bounded by a function that is exponential in the cutwidth of
the underlying graph and in the inverse of noise. Moreover, we consider two
specific and popular network topologies, the clique and the ring. For games
played on a clique we prove an almost matching lower bound on the mixing time
of the logit dynamics that is exponential in the inverse of the noise and in
the maximum potential difference, while for games played on a ring we prove
that the time of convergence of the logit dynamics to its stationary
distribution is significantly shorter
Convergence to Equilibrium of Logit Dynamics for Strategic Games
We present the first general bounds on the mixing time of the Markov chain associated to the logit dynamics for wide classes of strategic games. The logit dynamics with inverse noise β describes the behavior of a complex system whose individual components act selfishly according to some partial (“noisy”) knowledge of the system, where the capacity of the agent to know the system and compute her best move is measured by parameter β. In particular, we prove nearly tight bounds for potential games and games with dominant strategies. Our results show that for potential games the mixing time is bounded by an exponential in β and in the maximum potential difference. Instead, for games with dominant strategies the mixing time cannot grow arbitrarily with β. Finally, we refine our analysis for a subclass of potential games called graphical coordination games, often used for modeling the diffusion of new technologies. We prove that the mixing time of the logit dynamics for these games can be upper bounded by a function that is exponential in the cutwidth of the underlying graph and in β. Moreover, we consider two specific and popular network topologies, the clique and the ring. For the clique, we prove an almost matching lower bound on the mixing time of the logit dynamics that is exponential in β and in the maximum potential difference, while for the ring we prove that the time of convergence of the logit dynamics to its stationary distribution is significantly shorter
Dynamics in Near-Potential Games
Except for special classes of games, there is no systematic framework for
analyzing the dynamical properties of multi-agent strategic interactions.
Potential games are one such special but restrictive class of games that allow
for tractable dynamic analysis. Intuitively, games that are "close" to a
potential game should share similar properties. In this paper, we formalize and
develop this idea by quantifying to what extent the dynamic features of
potential games extend to "near-potential" games. We study convergence of three
commonly studied classes of adaptive dynamics: discrete-time better/best
response, logit response, and discrete-time fictitious play dynamics. For
better/best response dynamics, we focus on the evolution of the sequence of
pure strategy profiles and show that this sequence converges to a (pure)
approximate equilibrium set, whose size is a function of the "distance" from a
close potential game. We then study logit response dynamics and provide a
characterization of the stationary distribution of this update rule in terms of
the distance of the game from a close potential game and the corresponding
potential function. We further show that the stochastically stable strategy
profiles are pure approximate equilibria. Finally, we turn attention to
fictitious play, and establish that the sequence of empirical frequencies of
player actions converges to a neighborhood of (mixed) equilibria of the game,
where the size of the neighborhood increases with distance of the game to a
potential game. Thus, our results suggest that games that are close to a
potential game inherit the dynamical properties of potential games. Since a
close potential game to a given game can be found by solving a convex
optimization problem, our approach also provides a systematic framework for
studying convergence behavior of adaptive learning dynamics in arbitrary finite
strategic form games.Comment: 42 pages, 8 figure
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
Decentralized Dynamics for Finite Opinion Games
Game theory studies situations in which strategic players can modify the state of a given system, in the absence of a central authority. Solution concepts, such as Nash equilibrium, have been defined in order to predict the outcome of such situations. In multi-player settings, it has been pointed out that to be realistic, a solution concept should be obtainable via processes that are decentralized and reasonably simple. Accordingly we look at the computation of solution concepts by means of decentralized dynamics. These are algorithms in which players move in turns to decrease their own cost and the hope is that the system reaches an “equilibrium” quickly.
We study these dynamics for the class of opinion games, recently introduced by Bindel et al. [FOCS, 2011]. These are games, important in economics and sociology, that model the formation of an opinion in a social network. We study best-response dynamics and show upper and lower bounds on the convergence to Nash equilibria. We also study a noisy version of best-response dynamics, called logit dynamics, and prove a host of results about its convergence rate as the noise in the system varies. To get these results, we use a variety of techniques developed to bound the mixing time of Markov chains, including coupling, spectral characterizations and bottleneck ratio
Metastability of Logit Dynamics for Coordination Games
Logit Dynamics [Blume, Games and Economic Behavior, 1993] are randomized best
response dynamics for strategic games: at every time step a player is selected
uniformly at random and she chooses a new strategy according to a probability
distribution biased toward strategies promising higher payoffs. This process
defines an ergodic Markov chain, over the set of strategy profiles of the game,
whose unique stationary distribution is the long-term equilibrium concept for
the game. However, when the mixing time of the chain is large (e.g.,
exponential in the number of players), the stationary distribution loses its
appeal as equilibrium concept, and the transient phase of the Markov chain
becomes important. It can happen that the chain is "metastable", i.e., on a
time-scale shorter than the mixing time, it stays close to some probability
distribution over the state space, while in a time-scale multiple of the mixing
time it jumps from one distribution to another.
In this paper we give a quantitative definition of "metastable probability
distributions" for a Markov chain and we study the metastability of the logit
dynamics for some classes of coordination games. We first consider a pure
-player coordination game that highlights the distinctive features of our
metastability notion based on distributions. Then, we study coordination games
on the clique without a risk-dominant strategy (which are equivalent to the
well-known Glauber dynamics for the Curie-Weiss model) and coordination games
on a ring (both with and without risk-dominant strategy)
Stability of Mixed-Strategy-Based Iterative Logit Quantal Response Dynamics in Game Theory
Using the Logit quantal response form as the response function in each step,
the original definition of static quantal response equilibrium (QRE) is
extended into an iterative evolution process. QREs remain as the fixed points
of the dynamic process. However, depending on whether such fixed points are the
long-term solutions of the dynamic process, they can be classified into stable
(SQREs) and unstable (USQREs) equilibriums. This extension resembles the
extension from static Nash equilibriums (NEs) to evolutionary stable solutions
in the framework of evolutionary game theory. The relation between SQREs and
other solution concepts of games, including NEs and QREs, is discussed. Using
experimental data from other published papers, we perform a preliminary
comparison between SQREs, NEs, QREs and the observed behavioral outcomes of
those experiments. For certain games, we determine that SQREs have better
predictive power than QREs and NEs
Incentives-Based Mechanism for Efficient Demand Response Programs
In this work we investigate the inefficiency of the electricity system with
strategic agents. Specifically, we prove that without a proper control the
total demand of an inefficient system is at most twice the total demand of the
optimal outcome. We propose an incentives scheme that promotes optimal outcomes
in the inefficient electricity market. The economic incentives can be seen as
an indirect revelation mechanism that allocates resources using a
one-dimensional message space per resource to be allocated. The mechanism does
not request private information from users and is valid for any concave
customer's valuation function. We propose a distributed implementation of the
mechanism using population games and evaluate the performance of four popular
dynamics methods in terms of the cost to implement the mechanism. We find that
the achievement of efficiency in strategic environments might be achieved at a
cost, which is dependent on both the users' preferences and the dynamic
evolution of the system. Some simulation results illustrate the ideas presented
throughout the paper.Comment: 38 pages, 9 figures, submitted to journa
Multiple Steady States, Limit Cycles and Chaotic Attractors in Evolutionary Games with Logit Dynamics
This paper investigates, by means of simple, three and four strategy games, the occurrence of periodic and chaotic behaviour in a smooth version of the Best Response Dynamics, the Logit Dynamics. The main finding is that, unlike Replicator Dynamics, generic Hopf bifurcation and thus, stable limit cycles, do occur under the Logit Dynamics, even for three strategy games. Moreover, we show that the Logit Dynamics displays another bifurcation which cannot to occur under the Replicator Dynamics: the fold catastrophe. Finally, we find, in a four strategy game, a period-doubling route to chaotic dynamics under a 'weighted' version of the Logit Dynamics.
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