37,156 research outputs found
Solving generic nonarchimedean semidefinite programs using stochastic game algorithms
A general issue in computational optimization is to develop combinatorial
algorithms for semidefinite programming. We address this issue when the base
field is nonarchimedean. We provide a solution for a class of semidefinite
feasibility problems given by generic matrices. Our approach is based on
tropical geometry. It relies on tropical spectrahedra, which are defined as the
images by the valuation of nonarchimedean spectrahedra. We establish a
correspondence between generic tropical spectrahedra and zero-sum stochastic
games with perfect information. The latter have been well studied in
algorithmic game theory. This allows us to solve nonarchimedean semidefinite
feasibility problems using algorithms for stochastic games. These algorithms
are of a combinatorial nature and work for large instances.Comment: v1: 25 pages, 4 figures; v2: 27 pages, 4 figures, minor revisions +
benchmarks added; v3: 30 pages, 6 figures, generalization to non-Metzler sign
patterns + some results have been replaced by references to the companion
work arXiv:1610.0674
Model and Reinforcement Learning for Markov Games with Risk Preferences
We motivate and propose a new model for non-cooperative Markov game which
considers the interactions of risk-aware players. This model characterizes the
time-consistent dynamic "risk" from both stochastic state transitions (inherent
to the game) and randomized mixed strategies (due to all other players). An
appropriate risk-aware equilibrium concept is proposed and the existence of
such equilibria is demonstrated in stationary strategies by an application of
Kakutani's fixed point theorem. We further propose a simulation-based
Q-learning type algorithm for risk-aware equilibrium computation. This
algorithm works with a special form of minimax risk measures which can
naturally be written as saddle-point stochastic optimization problems, and
covers many widely investigated risk measures. Finally, the almost sure
convergence of this simulation-based algorithm to an equilibrium is
demonstrated under some mild conditions. Our numerical experiments on a two
player queuing game validate the properties of our model and algorithm, and
demonstrate their worth and applicability in real life competitive
decision-making.Comment: 38 pages, 6 tables, 5 figure
An Accretive Operator Approach to Ergodic Problems for Zero-Sum Games
Mean payoff stochastic games can be studied by means of a nonlinear spectral
problem involving the Shapley operator: the ergodic equation. A solution
consists in a scalar, called the ergodic constant, and a vector, called bias.
The existence of such a pair entails that the mean payoff per time unit is
equal to the ergodic constant for any initial state, and the bias gives
stationary strategies. By exploiting two fundamental properties of Shapley
operators, monotonicity and additive homogeneity, we give a necessary and
sufficient condition for the solvability of the ergodic equation for all the
Shapley operators obtained by perturbation of the transition payments of a
given stochastic game with finite state space. If the latter condition is
satisfied, we establish that the bias is unique (up to an additive constant)
for a generic perturbation of the transition payments. To show these results,
we use the theory of accretive operators, and prove in particular some
surjectivity condition.Comment: 4 pages, 1 figure, to appear in Proc. 22nd International Symposium on
Mathematical Theory of Networks and Systems (MTNS 2016
Information Theory - The Bridge Connecting Bounded Rational Game Theory and Statistical Physics
A long-running difficulty with conventional game theory has been how to
modify it to accommodate the bounded rationality of all real-world players. A
recurring issue in statistical physics is how best to approximate joint
probability distributions with decoupled (and therefore far more tractable)
distributions. This paper shows that the same information theoretic
mathematical structure, known as Product Distribution (PD) theory, addresses
both issues. In this, PD theory not only provides a principled formulation of
bounded rationality and a set of new types of mean field theory in statistical
physics. It also shows that those topics are fundamentally one and the same.Comment: 17 pages, no figures, accepted for publicatio
Oscillatory Dynamics in Rock-Paper-Scissors Games with Mutations
We study the oscillatory dynamics in the generic three-species
rock-paper-scissors games with mutations. In the mean-field limit, different
behaviors are found: (a) for high mutation rate, there is a stable interior
fixed point with coexistence of all species; (b) for low mutation rates, there
is a region of the parameter space characterized by a limit cycle resulting
from a Hopf bifurcation; (c) in the absence of mutations, there is a region
where heteroclinic cycles yield oscillations of large amplitude (not robust
against noise). After a discussion on the main properties of the mean-field
dynamics, we investigate the stochastic version of the model within an
individual-based formulation. Demographic fluctuations are therefore naturally
accounted and their effects are studied using a diffusion theory complemented
by numerical simulations. It is thus shown that persistent erratic oscillations
(quasi-cycles) of large amplitude emerge from a noise-induced resonance
phenomenon. We also analytically and numerically compute the average escape
time necessary to reach a (quasi-)cycle on which the system oscillates at a
given amplitude.Comment: 25 pages, 9 figures. To appear in the Journal of Theoretical Biolog
Solving Stochastic B\"uchi Games on Infinite Arenas with a Finite Attractor
We consider games played on an infinite probabilistic arena where the first
player aims at satisfying generalized B\"uchi objectives almost surely, i.e.,
with probability one. We provide a fixpoint characterization of the winning
sets and associated winning strategies in the case where the arena satisfies
the finite-attractor property. From this we directly deduce the decidability of
these games on probabilistic lossy channel systems.Comment: In Proceedings QAPL 2013, arXiv:1306.241
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