2,051 research outputs found
Monotone methods for equilibrium selection under perfect foresight dynamics
This paper studies equilibrium selection in supermodular games
based on perfect foresight dynamics. A normal form game is played
repeatedly in a large society of rational agents. There are frictions:
opportunities to revise actions follow independent Poisson processes.
Each agent forms his belief about the future evolution of action distribution
in the society to take an action that maximizes his expected
discounted payo�. A perfect foresight path is de�ned to be a feasible
path of the action distribution along which every agent with a revision
opportunity takes a best response to this path itself. A Nash
equilibrium is said to be absorbing if there exists no perfect foresight
path escaping from a neighborhood of this equilibrium; a Nash equilibrium
is said to be globally accessible if for each initial distribution,
there exists a perfect foresight path converging to this equilibrium.
By exploiting the monotone structure of the dynamics, a unique Nash
equilibrium that is absorbing and globally accessible for any small degree
of friction is identi�ed for certain classes of supermodular games.
For games with monotone potentials, the selection of the monotone
potential maximizer is obtained. Complete characterizations of absorbing
equilibrium and globally accessible equilibrium are given for
binary supermodular games. An example demonstrates that unanimity
games may have multiple globally accessible equilibria for a small
friction
Stochastic approximations and differential inclusions II: applications
We apply the theoretical results on "stochastic approximations and differential inclusions" developed in Benaim, Hofbauer and Sorin (2005) to several adaptive processes used in game theory including: classical
and generalized approachability, no-regret potential procedures (Hart and Mas-Colell), smooth fictitious play (Fudenberg and Levine
Phase transitions for rock-scissors-paper game on different networks
Monte Carlo simulations and dynamical mean-field approximations are performed
to study the phase transitions in rock-scissors-paper game on different host
networks. These graphs are originated from lattices by introducing quenched and
annealed randomness simultaneously. In the resulting phase diagrams three
different stationary states are identified for all structures. The comparison
of results on different networks suggests that the value of clustering
coefficient plays an irrelevant role in the emergence of a global oscillating
phase. The critical behavior of phase transitions seems to be universal and can
be described by the same exponents.Comment: 4 pages, 4 figures, to be published in PR
Strictly Dominated Strategies in the Replicator-Mutator Dynamics
The replicator-mutator dynamics is a set of differential equations frequently used in biological and socioeconomic contexts to model evolutionary processes subject to mutation, error or experimentation. The replicator-mutator dynamics generalizes the widely used replicator dynamics, which appears in this framework as the extreme case where replication is perfectly precise. This paper studies the influence of strictly dominated strategies on the location of the rest points of the replicator-mutator dynamics, at the limit where the mutation terms become arbitrarily small. It can be proved that such limit rest points for small mutation are Nash equilibria, so strictly dominated strategies do not occur at limit stationary points. However, we show through a simple case how strictly dominated strategies can have an influence on the location of the limit rest points for small mutation. Consequently, the characterization of the limit rest points of the replicator-mutator dynamics cannot in general proceed safely by readily eliminating strictly dominated strategiesJCyL (GREX251-2009 and VA006B09), Ministry of Science and Innovation (TIN2008-06464-C03-02, DPI2010-16920 and CSD2010-00034), Ministry of Education (grant JC2009-00263
Periodicity of mass extinctions without an extraterrestrial cause
We study a lattice model of a multi-species prey-predator system. Numerical
results show that for a small mutation rate the model develops irregular
long-period oscillatory behavior with sizeable changes in a number of species.
The periodicity of extinctions on Earth was suggested by Raup and Sepkoski but
so far is lacking a satisfactory explanation. Our model indicates that this is
a natural consequence of the ecosystem dynamics, not the result of any
extraterrestrial cause.Comment: 4 pages, accepted in Phys.Rev.
Statistical mechanics of ecosystem assembly
We introduce a toy model of ecosystem assembly for which we are able to map
out all assembly pathways generated by external invasions. The model allows to
display the whole phase space in the form of an assembly graph whose nodes are
communities of species and whose directed links are transitions between them
induced by invasions. We characterize the process as a finite Markov chain and
prove that it exhibits a unique set of recurrent states (the endstate of the
process), which is therefore resistant to invasions. This also shows that the
endstate is independent on the assembly history. The model shares all features
with standard assembly models reported in the literature, with the advantage
that all observables can be computed in an exact manner.Comment: Accepted for publication in Physical Review Letter
Impact of generalized benefit functions on the evolution of cooperation in spatial public goods games with continuous strategies
Cooperation and defection may be considered as two extreme responses to a
social dilemma. Yet the reality is much less clear-cut. Between the two
extremes lies an interval of ambivalent choices, which may be captured
theoretically by means of continuous strategies defining the extent of the
contributions of each individual player to the common pool. If strategies are
chosen from the unit interval, where 0 corresponds to pure defection and 1
corresponds to the maximal contribution, the question is what is the
characteristic level of individual investments to the common pool that emerges
if the evolution is guided by different benefit functions. Here we consider the
steepness and the threshold as two parameters defining an array of generalized
benefit functions, and we show that in a structured population there exist
intermediate values of both at which the collective contributions are maximal.
However, as the cost-to-benefit ratio of cooperation increases the
characteristic threshold decreases, while the corresponding steepness
increases. Our observations remain valid if more complex sigmoid functions are
used, thus reenforcing the importance of carefully adjusted benefits for high
levels of public cooperation.Comment: 8 two-column pages, 8 figures; accepted for publication in Physical
Review
Noise and Correlations in a Spatial Population Model with Cyclic Competition
Noise and spatial degrees of freedom characterize most ecosystems. Some
aspects of their influence on the coevolution of populations with cyclic
interspecies competition have been demonstrated in recent experiments [e.g. B.
Kerr et al., Nature {\bf 418}, 171 (2002)]. To reach a better theoretical
understanding of these phenomena, we consider a paradigmatic spatial model
where three species exhibit cyclic dominance. Using an individual-based
description, as well as stochastic partial differential and deterministic
reaction-diffusion equations, we account for stochastic fluctuations and
spatial diffusion at different levels, and show how fascinating patterns of
entangled spirals emerge. We rationalize our analysis by computing the
spatio-temporal correlation functions and provide analytical expressions for
the front velocity and the wavelength of the propagating spiral waves.Comment: 4 pages of main text, 3 color figures + 2 pages of supplementary
material (EPAPS Document). Final version for Physical Review Letter
Co-evolution of strategy and structure in complex networks with dynamical linking
Here we introduce a model in which individuals differ in the rate at which
they seek new interactions with others, making rational decisions modeled as
general symmetric two-player games. Once a link between two individuals has
formed, the productivity of this link is evaluated. Links can be broken off at
different rates. We provide analytic results for the limiting cases where
linking dynamics is much faster than evolutionary dynamics and vice-versa, and
show how the individual capacity of forming new links or severing inconvenient
ones maps into the problem of strategy evolution in a well-mixed population
under a different game. For intermediate ranges, we investigate numerically the
detailed interplay determined by these two time-scales and show that the scope
of validity of the analytical results extends to a much wider ratio of time
scales than expected
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