622 research outputs found
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
Dynamics of a spherical minority game
We present an exact dynamical solution of a spherical version of the batch
minority game (MG) with random external information. The control parameters in
this model are the ratio of the number of possible values for the public
information over the number of agents, and the radius of the spherical
constraint on the microscopic degrees of freedom. We find a phase diagram with
three phases: two without anomalous response (an oscillating versus a frozen
state), and a further frozen phase with divergent integrated response. In
contrast to standard MG versions, we can also calculate the volatility exactly.
Our study reveals similarities between the spherical and the conventional MG,
but also intriguing differences. Numerical simulations confirm our analytical
results.Comment: 16 pages, 3 figures; submitted to J. Phys.
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
On the transition to efficiency in Minority Games
The existence of a phase transition with diverging susceptibility in batch
Minority Games (MGs) is the mark of informationally efficient regimes and is
linked to the specifics of the agents' learning rules. Here we study how the
standard scenario is affected in a mixed population game in which agents with
the `optimal' learning rule (i.e. the one leading to efficiency) coexist with
ones whose adaptive dynamics is sub-optimal. Our generic finding is that any
non-vanishing intensive fraction of optimal agents guarantees the existence of
an efficient phase. Specifically, we calculate the dependence of the critical
point on the fraction of `optimal' agents focusing our analysis on three
cases: MGs with market impact correction, grand-canonical MGs and MGs with
heterogeneous comfort levels.Comment: 12 pages, 3 figures; contribution to the special issue "Viewing the
World through Spin Glasses" in honour of David Sherrington on the occasion of
his 65th birthda
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
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
Statistical mechanics and stability of a model eco-system
We study a model ecosystem by means of dynamical techniques from disordered
systems theory. The model describes a set of species subject to competitive
interactions through a background of resources, which they feed upon.
Additionally direct competitive or co-operative interaction between species may
occur through a random coupling matrix. We compute the order parameters of the
system in a fixed point regime, and identify the onset of instability and
compute the phase diagram. We focus on the effects of variability of resources,
direct interaction between species, co-operation pressure and dilution on the
stability and the diversity of the ecosystem. It is shown that resources can be
exploited optimally only in absence of co-operation pressure or direct
interaction between species.Comment: 23 pages, 13 figures; text of paper modified, discussion extended,
references adde
Market response to external events and interventions in spherical minority games
We solve the dynamics of large spherical Minority Games (MG) in the presence
of non-negligible time dependent external contributions to the overall market
bid. The latter represent the actions of market regulators, or other major
natural or political events that impact on the market. In contrast to
non-spherical MGs, the spherical formulation allows one to derive closed
dynamical order parameter equations in explicit form and work out the market's
response to such events fully analytically. We focus on a comparison between
the response to stationary versus oscillating market interventions, and reveal
profound and partially unexpected differences in terms of transition lines and
the volatility.Comment: 14 pages LaTeX, 5 (composite) postscript figures, submitted to
Journal of Physics
- …