The long-run behavior of the stochastic replicator dynamics

Fudenberg and Harris' stochastic version of the classical replicator dynamics is considered. The behavior of this diffusion process in the presence of an evolutionarily stable strategy is investigated. Moreover, extinction of dominated strategies and stochastic stability of strict Nash equilibria are studied. The general results are illustrated in connection with a discrete war of attrition. A persistence result for the maximum effort strategy is obtained and an explicit expression for the evolutionarily stable strategy is derived.Comment: Published at http://dx.doi.org/10.1214/105051604000000837 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Time averages, recurrence and transience in the stochastic replicator dynamics

We investigate the long-run behavior of a stochastic replicator process, which describes game dynamics for a symmetric two-player game under aggregate shocks. We establish an averaging principle that relates time averages of the process and Nash equilibria of a suitably modified game. Furthermore, a sufficient condition for transience is given in terms of mixed equilibria and definiteness of the payoff matrix. We also present necessary and sufficient conditions for stochastic stability of pure equilibria.Comment: Published in at http://dx.doi.org/10.1214/08-AAP577 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Imitation Processes with Small Mutations

This note characterizes the impact of adding rare stochastic muta- tions to an "imitation dynamic," meaning a process with the properties that any state where all agents use the same strategy is absorbing, and all other states are transient. The work of Freidlin and Wentzell [10] and its extensions implies that the resulting system will spend almost all of its time at the absorbing states of the no-mutation process, and provides a general algorithm for calculating the limit distribution, but this algorithm can be complicated to apply. This note provides a sim- pler and more intuitive algorithm. Loosely speaking, in a process with K strategies, it is sufficient to find the invariant distribution of a K x K Markov matrix on the K homogeneous states, where the probability of a transit from "all play i" to "all play j" is the probability of a transition from the state "all agents but 1 play i, 1 plays j" to the state "all play j. "

Uniform approximation of eigenvalues in Laguerre and Hermite beta-ensembles by roots of orthogonal polynomials

We derive strong uniform approximations for the eigenvalues in general Laguerre and Hermite beta-ensembles by showing that the maximal discrepancy between the suitably scaled eigenvalues and roots of orthogonal polynomials converges almost surely to zero when the dimension converges to infinity. We also provide estimates of the rate of convergence. --Gaussian ensemble,random matrix,rate of convergence,Weyl?s inequality,Wishart matrix

Promoting nutrition sensitive and climate smart agriculture through increased use of traditional underutilised species in the Pacific

Poster presented at Tropentag 2014. International Conference on Research on Food Security, Natural Resource Management and Rural Development. "Bridging the Gap between Increasing Knowledge and Decreasing Resources" Prague (Czech Republic) Sep 17-19 2014

Bayesian and maximin optimal designs for heteroscedastic regression models

The problem of constructing standardized maximin D-optimal designs for weighted polynomial regression models is addressed. In particular it is shown that, by following the broad approach to the construction of maximin designs introduced recently by Dette, Haines and Imhof (2003), such designs can be obtained as weak limits of the corresponding Bayesian Φq-optimal designs. The approach is illustrated for two specific weighted polynomial models and also for a particular growth model. --

Maximin and Bayesian optimal designs for regression models

For many problems of statistical inference in regression modelling, the Fisher information matrix depends on certain nuisance parameters which are unknown and which enter the model nonlinearly. A common strategy to deal with this problem within the context of design is to construct maximin optimal designs as those designs which maximize the minimum value of a real valued (standardized) function of the Fisher information matrix, where the minimum is taken over a specified range of the unknown parameters. The maximin criterion is not differentiable and the construction of the associated optimal designs is therefore difficult to achieve in practice. In the present paper the relationship between maximin optimal designs and a class of Bayesian optimal designs for which the associated criteria are differentiable is explored. In particular, a general methodology for determining maximin optimal designs is introduced based on the fact that in many cases these designs can be obtained as weak limits of appropriate Bayesian optimal designs. --maximin optimal designs,Bayesian optimal designs,nonlinear regression models,parameter estimation,least favourable prior

Uniform approximation of eigenvalues in Laguerre and Hermite â-ensembles by roots of orthogonal polynomials

We derive strong uniform approximations for the eigenvalues in general Laguerre and Hermite beta-ensembles by showing that the maximal discrepancy between the suitably scaled eigenvalues and roots of orthogonal polynomials converges almost surely to zero when the dimension converges to infinity. We also provide estimates of the rate of convergence

Port Cities within Port Regions: Shaping Complex Urban Environments in Gdańsk Bay, Poland

Port cities located within various metropolitan or functional regions face very different development scenarios. This applies not only to entire municipalities but also to particular areas that play important roles in urban development - including ports as well as their specialized parts. This refers also to the various types of maritime industries, including the processing of goods, logistics operations, shipbuilding, or ship repairing, to name just a few. Since each of these activities is associated with a different location, any transformation process that creates changes in geographic borders or flows will dynamically affect the port cityscape. Municipalities may evolve in different directions, becoming 'major maritime hubs,' 'secondary service centers,' 'specialized waterfront cities,' or just distressed urban areas. Within each metropolitan area, one can find several cities evolving in one of the above-mentioned directions, which results in the creation of a specific regional mosaic of various types of port cities. These create specific ‘port regions’ with specific roles assigned to each of these and shape the new (regional) dimension of the geography of borders and flows. As a result, these port regions are created as porous structures where space is discontinuous. To further develop the issue of the creation and evolution of port regions, the authors present the case study of the Gdańsk Bay port region. This study in particular allowed for the development of both the theoretical background of this phenomenon and the presentation of a real-life example

Bayesian and Maximum Optimal Designs for Heteroscedastic Regression Models

The problem of constructing standardized maximin D-optimal designs for weighted polynomial regression models is addressed. In particular it is shown that, by following the broad approach to the construction of maximin designs introduced recently by Dette, Haines and Imhof (2003), such designs can be obtained as weak limits of the corresponding Bayesian Φ_q-optimal designs. The approach is illustrated for two specific weighted polynomial models and also for a particular growth model