69 research outputs found
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
Tournaments with Gaps
A standard tournament contract specifies only tournament prizes. If agents’ performance is measured on a cardinal scale, the principal can complement the tournament contract by a gap which defines the minimum distance by which the best performing agent must beat the second best to receive the winner prize. We analyze a tournament with two risk averse agents. Under unlimited liability, the principal strictly benefits from a gap by partially insuring the agents and thereby reducing labor costs. If the agents are protected by limited liability, the principal sticks to the standard tournament
Bonus Pools and the Informativeness Principle
Previous work on moral-hazard problems has shown that, under certain conditions, bonus contracts create optimal individual incentives for risk-neutral workers. In our paper we demonstrate that, if a firm employs at least two workers, it may further bene.t from combining worker compensation via a bonus-pool contract and relative performance evaluation. Such combination leads to saved rents under a wide class of luck distributions. In addition, if the employer is wealth-constrained, complementing individual bonus contracts by the possibility of pooling bonuses can increase the set of implementable effort levels. All our results hold even though workers’ outputs are technically and stochastically independent so that, in view of Holmstrom’s informativeness principle, individual bonus contracts would be expected to dominate bonus-pool contracts
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. "
Bayesian Posteriors For Arbitrarily Rare Events
We study how much data a Bayesian observer needs to correctly infer the
relative likelihoods of two events when both events are arbitrarily rare. Each
period, either a blue die or a red die is tossed. The two dice land on side
with unknown probabilities and , which can be arbitrarily low. Given
a data-generating process where , we are interested in how much
data is required to guarantee that with high probability the observer's
Bayesian posterior mean for exceeds times that for .
If the prior densities for the two dice are positive on the interior of the
parameter space and behave like power functions at the boundary, then for every
there exists a finite so that the observer obtains such an
inference after periods with probability at least whenever
. The condition on and is the best possible. The result
can fail if one of the prior densities converges to zero exponentially fast at
the boundary
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
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Monotone Imitation Dynamics in Large Populations
We analyze a class of imitation dynamics with mutations for games with any finite number of actions, and give conditions for the selection of a unique equilibrium as the mutation rate becomes small and the population becomes large. Our results cover the multiple-action extensions of the aspiration-and-imitation process of Binmore and Samuelson [Muddling through: noisy equilibrium selection, J. Econ. Theory 74 (1997) 235–265] and the related processes proposed by Benaı¨m and Weibull [Deterministic approximation of stochastic evolution in games, Econometrica 71 (2003) 873–903] and Traulsen et al. [Coevolutionary dynamics: from finite to infinite populations, Phys. Rev. Lett. 95 (2005) 238701], as well as the frequency-dependent Moran process studied by Fudenberg et al. [Evolutionary game dynamics in finite populations with strong selection and weak mutation, Theoretical Population Biol. 70 (2006) 352–363]. We illustrate our results by considering the effect of the number of periods of repetition on the selected equilibrium in repeated play of the prisoner's dilemma when players are restricted to a small set of simple strategies.Economic
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
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