571 research outputs found
Generating functional analysis of Minority Games with real market histories
It is shown how the generating functional method of De Dominicis can be used
to solve the dynamics of the original version of the minority game (MG), in
which agents observe real as opposed to fake market histories. Here one again
finds exact closed equations for correlation and response functions, but now
these are defined in terms of two connected effective non-Markovian stochastic
processes: a single effective agent equation similar to that of the `fake'
history models, and a second effective equation for the overall market bid
itself (the latter is absent in `fake' history models). The result is an exact
theory, from which one can calculate from first principles both the persistent
observables in the MG and the distribution of history frequencies.Comment: 39 pages, 5 postscript figures, iop styl
Order-Parameter Flow in the SK Spin-Glass II: Inclusion of Microscopic Memory Effects
We develop further a recent dynamical replica theory to describe the dynamics
of the Sherrington-Kirkpatrick spin-glass in terms of closed evolution
equations for macroscopic order parameters. We show how microscopic memory
effects can be included in the formalism through the introduction of a dynamic
order parameter function: the joint spin-field distribution. The resulting
formalism describes very accurately the relaxation phenomena observed in
numerical simulations, including the typical overall slowing down of the flow
that was missed by the previous simple two-parameter theory. The advanced
dynamical replica theory is either exact or a very good approximation.Comment: same as original, but this one is TeXabl
Dynamical Replica Theory for Disordered Spin Systems
We present a new method to solve the dynamics of disordered spin systems on
finite time-scales. It involves a closed driven diffusion equation for the
joint spin-field distribution, with time-dependent coefficients described by a
dynamical replica theory which, in the case of detailed balance, incorporates
equilibrium replica theory as a stationary state. The theory is exact in
various limits. We apply our theory to both the symmetric- and the
non-symmetric Sherrington-Kirkpatrick spin-glass, and show that it describes
the (numerical) experiments very well.Comment: 7 pages RevTex, 4 figures, for PR
Finite Size Effects in Separable Recurrent Neural Networks
We perform a systematic analytical study of finite size effects in separable
recurrent neural network models with sequential dynamics, away from saturation.
We find two types of finite size effects: thermal fluctuations, and
disorder-induced `frozen' corrections to the mean-field laws. The finite size
effects are described by equations that correspond to a time-dependent
Ornstein-Uhlenbeck process. We show how the theory can be used to understand
and quantify various finite size phenomena in recurrent neural networks, with
and without detailed balance.Comment: 24 pages LaTex, with 4 postscript figures include
Multiplpe Choice Minority Game With Different Publicly Known Histories
In the standard Minority Game, players use historical minority choices as the
sole public information to pick one out of the two alternatives. However,
publishing historical minority choices is not the only way to present global
system information to players when more than two alternatives are available.
Thus, it is instructive to study the dynamics and cooperative behaviors of this
extended game as a function of the global information provided. We numerically
find that although the system dynamics depends on the kind of public
information given to the players, the degree of cooperation follows the same
trend as that of the standard Minority Game. We also explain most of our
findings by the crowd-anticrowd theory.Comment: Extensively revised, to appear in New J Phys, 7 pages with 4 figure
Stationary states of a spherical Minority Game with ergodicity breaking
Using generating functional and replica techniques, respectively, we study
the dynamics and statics of a spherical Minority Game (MG), which in contrast
with a spherical MG previously presented in J.Phys A: Math. Gen. 36 11159
(2003) displays a phase with broken ergodicity and dependence of the
macroscopic stationary state on initial conditions. The model thus bears more
similarity with the original MG. Still, all order parameters including the
volatility can computed in the ergodic phases without making any
approximations. We also study the effects of market impact correction on the
phase diagram. Finally we discuss a continuous-time version of the model as
well as the differences between on-line and batch update rules. Our analytical
results are confirmed convincingly by comparison with numerical simulations. In
an appendix we extend the analysis of the earlier spherical MG to a model with
general time-step, and compare the dynamics and statics of the two spherical
models.Comment: 26 pages, 8 figures; typo correcte
Generating functional analysis of minority games with inner product strategy definitions
We use generating functional methods to solve the so-called inner product
versions of the minority game (MG), with fake and/or real market histories, by
generalizing the theory developed recently for look-up table MGs with real
histories. The phase diagrams of the lookup table and inner product MG versions
are generally found to be identical, with the exception of inner product MGs
where histories are sampled linearly, which are found to be structurally
critical. However, we encounter interesting differences both in the theory
(where the role of the history frequency distribution in lookup table MGs is
taken over by the eigenvalue spectrum of a history covariance matrix in inner
product MGs) and in the static and dynamic phenomenology of the models. Our
theoretical predictions are supported by numerical simulations.Comment: 30 pages, 12 figures (some lower resolution to enable submission,
originals available upon request), submitted to Journal of Physics
Statistical Mechanics of Dilute Batch Minority Games with Random External Information
We study the dynamics and statics of a dilute batch minority game with random
external information. We focus on the case in which the number of connections
per agent is infinite in the thermodynamic limit. The dynamical scenario of
ergodicity breaking in this model is different from the phase transition in the
standard minority game and is characterised by the onset of long-term memory at
finite integrated response. We demonstrate that finite memory appears at the
AT-line obtained from the corresponding replica calculation, and compare the
behaviour of the dilute model with the minority game with market impact
correction, which is known to exhibit similar features.Comment: 22 pages, 6 figures, text modified, references updated and added,
figure added, typos correcte
Incorporating Inertia Into Multi-Agent Systems
We consider a model that demonstrates the crucial role of inertia and
stickiness in multi-agent systems, based on the Minority Game (MG). The inertia
of an agent is introduced into the game model by allowing agents to apply
hypothesis testing when choosing their best strategies, thereby reducing their
reactivity towards changes in the environment. We find by extensive numerical
simulations that our game shows a remarkable improvement of global cooperation
throughout the whole phase space. In other words, the maladaptation behavior
due to over-reaction of agents is removed. These agents are also shown to be
advantageous over the standard ones, which are sometimes too sensitive to
attain a fair success rate. We also calculate analytically the minimum amount
of inertia needed to achieve the above improvement. Our calculation is
consistent with the numerical simulation results. Finally, we review some
related works in the field that show similar behaviors and compare them to our
work.Comment: extensively revised, 8 pages, 10 figures in revtex
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