5,511 research outputs found
Entropy Games and Matrix Multiplication Games
Two intimately related new classes of games are introduced and studied:
entropy games (EGs) and matrix multiplication games (MMGs). An EG is played on
a finite arena by two-and-a-half players: Despot, Tribune and the
non-deterministic People. Despot wants to make the set of possible People's
behaviors as small as possible, while Tribune wants to make it as large as
possible.An MMG is played by two players that alternately write matrices from
some predefined finite sets. One wants to maximize the growth rate of the
product, and the other to minimize it. We show that in general MMGs are
undecidable in quite a strong sense.On the positive side, EGs correspond to a
subclass of MMGs, and we prove that such MMGs and EGs are determined, and that
the optimal strategies are simple. The complexity of solving such games is in
NP\&coNP.Comment: Accepted to STACS 201
The Operator Approach to Entropy Games
Entropy games and matrix multiplication games have been recently introduced by Asarin et al. They model the situation in which one player (Despot) wishes to minimize the growth rate of a matrix product, whereas the other player (Tribune) wishes to maximize it. We develop an operator approach to entropy games. This allows us to show that entropy games can be cast as stochastic mean payoff games in which some action spaces are simplices and payments are given by a relative entropy (Kullback-Leibler divergence). In this way, we show that entropy games with a fixed number of states belonging to Despot can be solved in polynomial time. This approach also allows us to solve these games by a policy iteration algorithm, which we compare with the spectral simplex algorithm developed by Protasov
Solving Irreducible Stochastic Mean-Payoff Games and Entropy Games by Relative Krasnoselskii-Mann Iteration
International audienceWe analyse an algorithm solving stochastic mean-payoff games, combining the ideas of relative value iteration and of Krasnoselskii-Mann damping. We derive parameterized complexity bounds for several classes of games satisfying irreducibility conditions. We show in particular that an ε-approximation of the value of an irreducible concurrent stochastic game can be computed in a number of iterations in O(|log(ε)|) where the constant in the O(⋅) is explicit, depending on the smallest non-zero transition probabilities. This should be compared with a bound in O(ε^{-1}|log(ε)|) obtained by Chatterjee and Ibsen-Jensen (ICALP 2014) for the same class of games, and to a O(ε^{-1}) bound by Allamigeon, Gaubert, Katz and Skomra (ICALP 2022) for turn-based games. We also establish parameterized complexity bounds for entropy games, a class of matrix multiplication games introduced by Asarin, Cervelle, Degorre, Dima, Horn and Kozyakin. We derive these results by methods of variational analysis, establishing contraction properties of the relative Krasnoselskii-Mann iteration with respect to Hilbert’s semi-norm
Distributed stochastic optimization via matrix exponential learning
In this paper, we investigate a distributed learning scheme for a broad class
of stochastic optimization problems and games that arise in signal processing
and wireless communications. The proposed algorithm relies on the method of
matrix exponential learning (MXL) and only requires locally computable gradient
observations that are possibly imperfect and/or obsolete. To analyze it, we
introduce the notion of a stable Nash equilibrium and we show that the
algorithm is globally convergent to such equilibria - or locally convergent
when an equilibrium is only locally stable. We also derive an explicit linear
bound for the algorithm's convergence speed, which remains valid under
measurement errors and uncertainty of arbitrarily high variance. To validate
our theoretical analysis, we test the algorithm in realistic
multi-carrier/multiple-antenna wireless scenarios where several users seek to
maximize their energy efficiency. Our results show that learning allows users
to attain a net increase between 100% and 500% in energy efficiency, even under
very high uncertainty.Comment: 31 pages, 3 figure
Simultaneous dense and nondense orbits for commuting maps
We show that, for two commuting automorphisms of the torus and for two
elements of the Cartan action on compact higher rank homogeneous spaces, many
points have drastically different orbit structures for the two maps.
Specifically, using measure rigidity, we show that the set of points that have
dense orbit under one map and nondense orbit under the second has full
Hausdorff dimension.Comment: 17 pages. Very minor changes to the exposition. Three additional
papers cite
Dynamic Power Allocation Games in Parallel Multiple Access Channels
We analyze the distributed power allocation problem in parallel multiple
access channels (MAC) by studying an associated non-cooperative game which
admits an exact potential. Even though games of this type have been the subject
of considerable study in the literature, we find that the sufficient conditions
which ensure uniqueness of Nash equilibrium points typically do not hold in
this context. Nonetheless, we show that the parallel MAC game admits a unique
equilibrium almost surely, thus establishing an important class of
counterexamples where these sufficient conditions are not necessary.
Furthermore, if the network's users employ a distributed learning scheme based
on the replicator dynamics, we show that they converge to equilibrium from
almost any initial condition, even though users only have local information at
their disposal.Comment: 18 pages, 4 figures, submitted to Valuetools '1
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