73,804 research outputs found
Graphical potential games
We study the class of potential games that are also graphical games with
respect to a given graph of connections between the players. We show that,
up to strategic equivalence, this class of games can be identified with the set
of Markov random fields on .
From this characterization, and from the Hammersley-Clifford theorem, it
follows that the potentials of such games can be decomposed to local
potentials. We use this decomposition to strongly bound the number of strategy
changes of a single player along a better response path. This result extends to
generalized graphical potential games, which are played on infinite graphs.Comment: Accepted to the Journal of Economic Theor
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
Learning the Structure and Parameters of Large-Population Graphical Games from Behavioral Data
We consider learning, from strictly behavioral data, the structure and
parameters of linear influence games (LIGs), a class of parametric graphical
games introduced by Irfan and Ortiz (2014). LIGs facilitate causal strategic
inference (CSI): Making inferences from causal interventions on stable behavior
in strategic settings. Applications include the identification of the most
influential individuals in large (social) networks. Such tasks can also support
policy-making analysis. Motivated by the computational work on LIGs, we cast
the learning problem as maximum-likelihood estimation (MLE) of a generative
model defined by pure-strategy Nash equilibria (PSNE). Our simple formulation
uncovers the fundamental interplay between goodness-of-fit and model
complexity: good models capture equilibrium behavior within the data while
controlling the true number of equilibria, including those unobserved. We
provide a generalization bound establishing the sample complexity for MLE in
our framework. We propose several algorithms including convex loss minimization
(CLM) and sigmoidal approximations. We prove that the number of exact PSNE in
LIGs is small, with high probability; thus, CLM is sound. We illustrate our
approach on synthetic data and real-world U.S. congressional voting records. We
briefly discuss our learning framework's generality and potential applicability
to general graphical games.Comment: Journal of Machine Learning Research. (accepted, pending
publication.) Last conference version: submitted March 30, 2012 to UAI 2012.
First conference version: entitled, Learning Influence Games, initially
submitted on June 1, 2010 to NIPS 201
A Robust Characterization of Nash Equilibrium
We give a robust characterization of Nash equilibrium by postulating coherent
behavior across varying games: Nash equilibrium is the only solution concept
that satisfies consequentialism, consistency, and rationality. As a
consequence, every equilibrium refinement violates at least one of these
properties. We moreover show that every solution concept that approximately
satisfies consequentialism, consistency, and rationality returns approximate
Nash equilibria. The latter approximation can be made arbitrarily good by
increasing the approximation of the axioms. This result extends to various
natural subclasses of games such as two-player zero-sum games, potential games,
and graphical games
Convergent learning algorithms for potential games with unknown noisy rewards
In this paper, we address the problem of convergence to Nash equilibria in games with rewards that are initially unknown and which must be estimated over time from noisy observations. These games arise in many real-world applications, whenever rewards for actions cannot be prespecified and must be learned on-line. Standard results in game theory, however, do not consider such settings. Specifically, using results from stochastic approximation and differential inclusions, we prove the convergence of variants of fictitious play and adaptive play to Nash equilibria in potential games and weakly acyclic games, respectively. These variants all use a multi-agent version of Q-learning to estimate the reward functions and a novel form of the e-greedy decision rule to select an action. Furthermore, we derive e-greedy decision rules that exploit the sparse interaction structure encoded in two compact graphical representations of games, known as graphical and hypergraphical normal form, to improve the convergence rate of the learning algorithms. The structure captured in these representations naturally occurs in many distributed optimisation and control applications. Finally, we demonstrate the efficacy of the algorithms in a simulated ad hoc wireless sensor network management problem
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
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