5,042 research outputs found
Log-Distributional Approach for Learning Covariate Shift Ratios
Distributional Reinforcement Learning theory suggests that distributional fixed points could play a fundamental role to learning non additive value functions. In particular, we propose a distributional approach for learning Covariate Shift Ratios, whose update rule is originally multiplicative
Statistical equilibrium in simple exchange games I
Simple stochastic exchange games are based on random allocation of finite
resources. These games are Markov chains that can be studied either
analytically or by Monte Carlo simulations. In particular, the equilibrium
distribution can be derived either by direct diagonalization of the transition
matrix, or using the detailed balance equation, or by Monte Carlo estimates. In
this paper, these methods are introduced and applied to the
Bennati-Dragulescu-Yakovenko (BDY) game. The exact analysis shows that the
statistical-mechanical analogies used in the previous literature have to be
revised.Comment: 11 pages, 3 figures, submitted to EPJ
Winning quick and dirty: the greedy random walk
As a strategy to complete games quickly, we investigate one-dimensional
random walks where the step length increases deterministically upon each return
to the origin. When the step length after the kth return equals k, the
displacement of the walk x grows linearly in time. Asymptotically, the
probability distribution of displacements is a purely exponentially decaying
function of |x|/t. The probability E(t,L) for the walk to escape a bounded
domain of size L at time t decays algebraically in the long time limit, E(t,L)
~ L/t^2. Consequently, the mean escape time ~ L ln L, while ~
L^{2n-1} for n>1. Corresponding results are derived when the step length after
the kth return scales as k^alpha$ for alpha>0.Comment: 7 pages, 6 figures, 2-column revtext4 forma
Cooperation in a resource extraction game
An exhaustible stock of resources may be exploited by N players. An arbitrarily long duration of the game is only possible, if the utility function satisfies certain restrictions at small values R of extraction. We find that stability against unilateral defection occurs if the elasticity of the marginal utility turns out to be larger than (N - 1 )/N, however independent of the value of the discount factor. Hence we find that cooperation does not depend on the discount factor for a certain range of elasticities. Analogy to phase transitions in statistical physics is discussed.
Explicit lower and upper bounds on the entangled value of multiplayer XOR games
XOR games are the simplest model in which the nonlocal properties of
entanglement manifest themselves. When there are two players, it is well known
that the bias --- the maximum advantage over random play --- of entangled
players can be at most a constant times greater than that of classical players.
Recently, P\'{e}rez-Garc\'{i}a et al. [Comm. Math. Phys. 279 (2), 2008] showed
that no such bound holds when there are three or more players: the advantage of
entangled players over classical players can become unbounded, and scale with
the number of questions in the game. Their proof relies on non-trivial results
from operator space theory, and gives a non-explicit existence proof, leading
to a game with a very large number of questions and only a loose control over
the local dimension of the players' shared entanglement.
We give a new, simple and explicit (though still probabilistic) construction
of a family of three-player XOR games which achieve a large quantum-classical
gap (QC-gap). This QC-gap is exponentially larger than the one given by
P\'{e}rez-Garc\'{i}a et. al. in terms of the size of the game, achieving a
QC-gap of order with questions per player. In terms of the
dimension of the entangled state required, we achieve the same (optimal) QC-gap
of for a state of local dimension per player. Moreover, the
optimal entangled strategy is very simple, involving observables defined by
tensor products of the Pauli matrices.
Additionally, we give the first upper bound on the maximal QC-gap in terms of
the number of questions per player, showing that our construction is only
quadratically off in that respect. Our results rely on probabilistic estimates
on the norm of random matrices and higher-order tensors which may be of
independent interest.Comment: Major improvements in presentation; results identica
A general methodology to price and hedge derivatives in incomplete markets
We introduce and discuss a general criterion for the derivative pricing in
the general situation of incomplete markets, we refer to it as the No Almost
Sure Arbitrage Principle. This approach is based on the theory of optimal
strategy in repeated multiplicative games originally introduced by Kelly. As
particular cases we obtain the Cox-Ross-Rubinstein and Black-Scholes in the
complete markets case and the Schweizer and Bouchaud-Sornette as a quadratic
approximation of our prescription. Technical and numerical aspects for the
practical option pricing, as large deviation theory approximation and Monte
Carlo computation are discussed in detail.Comment: 24 pages, LaTeX, epsfig.sty, 5 eps figures, changes in the
presentation of the method, submitted to International J. of Theoretical and
Applied Financ
Game theory, maximum entropy, minimum discrepancy and robust Bayesian decision theory
We describe and develop a close relationship between two problems that have
customarily been regarded as distinct: that of maximizing entropy, and that of
minimizing worst-case expected loss. Using a formulation grounded in the
equilibrium theory of zero-sum games between Decision Maker and
Nature, these two problems are shown to be dual to each other, the solution
to each providing that to the other. Although Tops\oe described this connection
for the Shannon entropy over 20 years ago, it does not appear to be widely
known even in that important special case. We here generalize this theory to
apply to arbitrary decision problems and loss functions. We indicate how an
appropriate generalized definition of entropy can be associated with such a
problem, and we show that, subject to certain regularity conditions, the
above-mentioned duality continues to apply in this extended context.
This simultaneously provides a possible rationale for maximizing entropy and
a tool for finding robust Bayes acts. We also describe the essential identity
between the problem of maximizing entropy and that of minimizing a related
discrepancy or divergence between distributions. This leads to an extension, to
arbitrary discrepancies, of a well-known minimax theorem for the case of
Kullback-Leibler divergence (the ``redundancy-capacity theorem'' of information
theory). For the important case of families of distributions having certain
mean values specified, we develop simple sufficient conditions and methods for
identifying the desired solutions.Comment: Published by the Institute of Mathematical Statistics
(http://www.imstat.org) in the Annals of Statistics
(http://www.imstat.org/aos/) at http://dx.doi.org/10.1214/00905360400000055
Growth Optimal Investment and Pricing of Derivatives
We introduce a criterion how to price derivatives in incomplete markets,
based on the theory of growth optimal strategy in repeated multiplicative
games. We present reasons why these growth-optimal strategies should be
particularly relevant to the problem of pricing derivatives. We compare our
result with other alternative pricing procedures in the literature, and discuss
the limits of validity of the lognormal approximation. We also generalize the
pricing method to a market with correlated stocks. The expected estimation
error of the optimal investment fraction is derived in a closed form, and its
validity is checked with a small-scale empirical test.Comment: 21 pages, 5 figure
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