1,344 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
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
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
Equilibria in the Tangle
We analyse the Tangle --- a DAG-valued stochastic process where new vertices
get attached to the graph at Poissonian times, and the attachment's locations
are chosen by means of random walks on that graph. These new vertices, also
thought of as "transactions", are issued by many players (which are the nodes
of the network), independently. The main application of this model is that it
is used as a base for the IOTA cryptocurrency system (www.iota.org). We prove
existence of "almost symmetric" Nash equilibria for the system where a part of
players tries to optimize their attachment strategies. Then, we also present
simulations that show that the "selfish" players will nevertheless cooperate
with the network by choosing attachment strategies that are similar to the
"recommended" one.Comment: 33 pages, 11 figure
Discretized Multinomial Distributions and Nash Equilibria in Anonymous Games
We show that there is a polynomial-time approximation scheme for computing
Nash equilibria in anonymous games with any fixed number of strategies (a very
broad and important class of games), extending the two-strategy result of
Daskalakis and Papadimitriou 2007. The approximation guarantee follows from a
probabilistic result of more general interest: The distribution of the sum of n
independent unit vectors with values ranging over {e1, e2, ...,ek}, where ei is
the unit vector along dimension i of the k-dimensional Euclidean space, can be
approximated by the distribution of the sum of another set of independent unit
vectors whose probabilities of obtaining each value are multiples of 1/z for
some integer z, and so that the variational distance of the two distributions
is at most eps, where eps is bounded by an inverse polynomial in z and a
function of k, but with no dependence on n. Our probabilistic result specifies
the construction of a surprisingly sparse eps-cover -- under the total
variation distance -- of the set of distributions of sums of independent unit
vectors, which is of interest on its own right.Comment: In the 49th Annual IEEE Symposium on Foundations of Computer Science,
FOCS 200
On the Complexity of Nash Equilibria of Action-Graph Games
We consider the problem of computing Nash Equilibria of action-graph games
(AGGs). AGGs, introduced by Bhat and Leyton-Brown, is a succinct representation
of games that encapsulates both "local" dependencies as in graphical games, and
partial indifference to other agents' identities as in anonymous games, which
occur in many natural settings. This is achieved by specifying a graph on the
set of actions, so that the payoff of an agent for selecting a strategy depends
only on the number of agents playing each of the neighboring strategies in the
action graph. We present a Polynomial Time Approximation Scheme for computing
mixed Nash equilibria of AGGs with constant treewidth and a constant number of
agent types (and an arbitrary number of strategies), together with hardness
results for the cases when either the treewidth or the number of agent types is
unconstrained. In particular, we show that even if the action graph is a tree,
but the number of agent-types is unconstrained, it is NP-complete to decide the
existence of a pure-strategy Nash equilibrium and PPAD-complete to compute a
mixed Nash equilibrium (even an approximate one); similarly for symmetric AGGs
(all agents belong to a single type), if we allow arbitrary treewidth. These
hardness results suggest that, in some sense, our PTAS is as strong of a
positive result as one can expect
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