883 research outputs found
Evolutionary dynamics, intrinsic noise and cycles of co-operation
We use analytical techniques based on an expansion in the inverse system size
to study the stochastic evolutionary dynamics of finite populations of players
interacting in a repeated prisoner's dilemma game. We show that a mechanism of
amplification of demographic noise can give rise to coherent oscillations in
parameter regimes where deterministic descriptions converge to fixed points
with complex eigenvalues. These quasi-cycles between co-operation and defection
have previously been observed in computer simulations; here we provide a
systematic and comprehensive analytical characterization of their properties.
We are able to predict their power spectra as a function of the mutation rate
and other model parameters, and to compare the relative magnitude of the cycles
induced by different types of underlying microscopic dynamics. We also extend
our analysis to the iterated prisoner's dilemma game with a win-stay lose-shift
strategy, appropriate in situations where players are subject to errors of the
trembling-hand type.Comment: 14 pages, 12 figures, accepted for publication by Phys. Rev.
Two-population replicator dynamics and number of Nash equilibria in random matrix games
We study the connection between the evolutionary replicator dynamics and the
number of Nash equilibria in large random bi-matrix games. Using techniques of
disordered systems theory we compute the statistical properties of both, the
fixed points of the dynamics and the Nash equilibria. Except for the special
case of zero-sum games one finds a transition as a function of the so-called
co-operation pressure between a phase in which there is a unique stable fixed
point of the dynamics coinciding with a unique Nash equilibrium, and an
unstable phase in which there are exponentially many Nash equilibria with
statistical properties different from the stationary state of the replicator
equations. Our analytical results are confirmed by numerical simulations of the
replicator dynamics, and by explicit enumeration of Nash equilibria.Comment: 9 pages, 2x2 figure
Stochastic waves in a Brusselator model with nonlocal interaction
We show that intrinsic noise can induce spatio-temporal phenomena such as
Turing patterns and travelling waves in a Brusselator model with nonlocal
interaction terms. In order to predict and to characterize these quasi-waves we
analyze the nonlocal model using a system-size expansion. The resulting theory
is used to calculate the power spectra of the quasi-waves analytically, and the
outcome is tested successfully against simulations. We discuss the possibility
that nonlocal models in other areas, such as epidemic spread or social
dynamics, may contain similar stochastically-induced patterns.Comment: 13 pages, 6 figure
Stochastic transport in the presence of spatial disorder: fluctuation-induced corrections to homogenization
Motivated by uncertainty quantification in natural transport systems, we
investigate an individual-based transport process involving particles
undergoing a random walk along a line of point sinks whose strengths are
themselves independent random variables. We assume particles are removed from
the system via first-order kinetics. We analyse the system using a hierarchy of
approaches when the sinks are sparsely distributed, including a stochastic
homogenization approximation that yields explicit predictions for the extrinsic
disorder in the stationary state due to sink strength fluctuations. The
extrinsic noise induces long-range spatial correlations in the particle
concentration, unlike fluctuations due to the intrinsic noise alone.
Additionally, the mean concentration profile, averaged over both intrinsic and
extrinsic noise, is elevated compared with the corresponding profile from a
uniform sink distribution, showing that the classical homogenization
approximation can be a biased estimator of the true mean.Comment: 16 pages, 8 figure
Minority games, evolving capitals and replicator dynamics
We discuss a simple version of the Minority Game (MG) in which agents hold
only one strategy each, but in which their capitals evolve dynamically
according to their success and in which the total trading volume varies in time
accordingly. This feature is known to be crucial for MGs to reproduce stylised
facts of real market data. The stationary states and phase diagram of the model
can be computed, and we show that the ergodicity breaking phase transition
common for MGs, and marked by a divergence of the integrated response is
present also in this simplified model. An analogous majority game turns out to
be relatively void of interesting features, and the total capital is found to
diverge in time. Introducing a restraining force leads to a model akin to
replicator dynamics of evolutionary game theory, and we demonstrate that here a
different type of phase transition is observed. Finally we briefly discuss the
relation of this model with one strategy per player to more sophisticated
Minority Games with dynamical capitals and several trading strategies per
agent.Comment: 19 pages, 7 figure
Random replicators with asymmetric couplings
Systems of interacting random replicators are studied using generating
functional techniques. While replica analyses of such models are limited to
systems with symmetric couplings, dynamical approaches as presented here allow
specifically to address cases with asymmetric interactions where there is no
Lyapunov function governing the dynamics. We here focus on replicator models
with Gaussian couplings of general symmetry between p>=2 species, and discuss
how an effective description of the dynamics can be derived in terms of a
single-species process. Upon making a fixed point ansatz persistent order
parameters in the ergodic stationary states can be extracted from this process,
and different types of phase transitions can be identified and related to each
other. We discuss the effects of asymmetry in the couplings on the order
parameters and the phase behaviour for p=2 and p=3. Numerical simulations
verify our theory. For the case of cubic interactions numerical experiments
indicate regimes in which only a finite number of species survives, even when
the thermodynamic limit is considered.Comment: revised version, removed some mathematical parts, discussion of
negatively correlated couplings added, figures adde
Dynamics of adaptive agents with asymmetric information
We apply path-integral techniques to study the dynamics of agent-based models
with asymmetric information structures. In particular, we devise a batch
version of a model proposed originally by Berg et al. [Quant. Fin. 1 (2001)
203], and convert the coupled multi-agent processes into an effective-agent
problem from which the dynamical order parameters in ergodic regimes can be
derived self-consistently together with the corresponding phase structure. Our
dynamical study complements and extends the available static theory. Results
are confirmed by numerical simulations.Comment: minor revision of text, accepted by JSTA
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
Intrinsic noise in systems with switching environments
We study individual-based dynamics in finite populations, subject to randomly
switching environmental conditions. These are inspired by models in which genes
transition between on and off states, regulating underlying protein dynamics.
Similarly switches between environmental states are relevant in bacterial
populations and in models of epidemic spread. Existing piecewise-deterministic
Markov process (PDMP) approaches focus on the deterministic limit of the
population dynamics while retaining the randomness of the switching. Here we go
beyond this approximation and explicitly include effects of intrinsic
stochasticity at the level of the linear-noise approximation. Specifically we
derive the stationary distributions of a number of model systems, in good
agreement with simulations. This improves existing approaches which are limited
to the regimes of fast and slow switching.Comment: 15 pages, 11 figure
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