3,504 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
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
A Comprehensive Survey of Potential Game Approaches to Wireless Networks
Potential games form a class of non-cooperative games where unilateral
improvement dynamics are guaranteed to converge in many practical cases. The
potential game approach has been applied to a wide range of wireless network
problems, particularly to a variety of channel assignment problems. In this
paper, the properties of potential games are introduced, and games in wireless
networks that have been proven to be potential games are comprehensively
discussed.Comment: 44 pages, 6 figures, to appear in IEICE Transactions on
Communications, vol. E98-B, no. 9, Sept. 201
- …