2,320 research outputs found
A General Framework for Computing Optimal Correlated Equilibria in Compact Games
We analyze the problem of computing a correlated equilibrium that optimizes
some objective (e.g., social welfare). Papadimitriou and Roughgarden [2008]
gave a sufficient condition for the tractability of this problem; however, this
condition only applies to a subset of existing representations. We propose a
different algorithmic approach for the optimal CE problem that applies to all
compact representations, and give a sufficient condition that generalizes that
of Papadimitriou and Roughgarden. In particular, we reduce the optimal CE
problem to the deviation-adjusted social welfare problem, a combinatorial
optimization problem closely related to the optimal social welfare problem.
This framework allows us to identify new classes of games for which the optimal
CE problem is tractable; we show that graphical polymatrix games on tree graphs
are one example. We also study the problem of computing the optimal coarse
correlated equilibrium, a solution concept closely related to CE. Using a
similar approach we derive a sufficient condition for this problem, and use it
to prove that the problem is tractable for singleton congestion games.Comment: 14 pages. Short version to appear in WINE 201
Finding Any Nontrivial Coarse Correlated Equilibrium Is Hard
One of the most appealing aspects of the (coarse) correlated equilibrium
concept is that natural dynamics quickly arrive at approximations of such
equilibria, even in games with many players. In addition, there exist
polynomial-time algorithms that compute exact (coarse) correlated equilibria.
In light of these results, a natural question is how good are the (coarse)
correlated equilibria that can arise from any efficient algorithm or dynamics.
In this paper we address this question, and establish strong negative
results. In particular, we show that in multiplayer games that have a succinct
representation, it is NP-hard to compute any coarse correlated equilibrium (or
approximate coarse correlated equilibrium) with welfare strictly better than
the worst possible. The focus on succinct games ensures that the underlying
complexity question is interesting; many multiplayer games of interest are in
fact succinct. Our results imply that, while one can efficiently compute a
coarse correlated equilibrium, one cannot provide any nontrivial welfare
guarantee for the resulting equilibrium, unless P=NP. We show that analogous
hardness results hold for correlated equilibria, and persist under the
egalitarian objective or Pareto optimality.
To complement the hardness results, we develop an algorithmic framework that
identifies settings in which we can efficiently compute an approximate
correlated equilibrium with near-optimal welfare. We use this framework to
develop an efficient algorithm for computing an approximate correlated
equilibrium with near-optimal welfare in aggregative games.Comment: 21 page
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
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
Joint Channel Selection and Power Control in Infrastructureless Wireless Networks: A Multi-Player Multi-Armed Bandit Framework
This paper deals with the problem of efficient resource allocation in dynamic
infrastructureless wireless networks. Assuming a reactive interference-limited
scenario, each transmitter is allowed to select one frequency channel (from a
common pool) together with a power level at each transmission trial; hence, for
all transmitters, not only the fading gain, but also the number of interfering
transmissions and their transmit powers are varying over time. Due to the
absence of a central controller and time-varying network characteristics, it is
highly inefficient for transmitters to acquire global channel and network
knowledge. Therefore a reasonable assumption is that transmitters have no
knowledge of fading gains, interference, and network topology. Each
transmitting node selfishly aims at maximizing its average reward (or
minimizing its average cost), which is a function of the action of that
specific transmitter as well as those of all other transmitters. This scenario
is modeled as a multi-player multi-armed adversarial bandit game, in which
multiple players receive an a priori unknown reward with an arbitrarily
time-varying distribution by sequentially pulling an arm, selected from a known
and finite set of arms. Since players do not know the arm with the highest
average reward in advance, they attempt to minimize their so-called regret,
determined by the set of players' actions, while attempting to achieve
equilibrium in some sense. To this end, we design in this paper two joint power
level and channel selection strategies. We prove that the gap between the
average reward achieved by our approaches and that based on the best fixed
strategy converges to zero asymptotically. Moreover, the empirical joint
frequencies of the game converge to the set of correlated equilibria. We
further characterize this set for two special cases of our designed game
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