Consistency of vanishing smooth fictitious play

We discuss consistency of Vanishing Smooth Fictitious Play, a strategy in the context of game theory, which can be regarded as a smooth fictitious play procedure, where the smoothing parameter is time-dependent and asymptotically vanishes. This answers a question initially raised by Drew Fudenberg and Satoru Takahashi.Comment: 17 page

Reinforcement learning with restrictions on the action set

Consider a 2-player normal-form game repeated over time. We introduce an adaptive learning procedure, where the players only observe their own realized payoff at each stage. We assume that agents do not know their own payoff function, and have no information on the other player. Furthermore, we assume that they have restrictions on their own action set such that, at each stage, their choice is limited to a subset of their action set. We prove that the empirical distributions of play converge to the set of Nash equilibria for zero-sum and potential games, and games where one player has two actions.Comment: 28 page

Convergence of generalized urn models to non-equilibrium attractors

Generalized Polya urn models have been used to model the establishment dynamics of a small founding population consisting of k different genotypes or strategies. As population sizes get large, these population processes are well-approximated by a mean limit ordinary differential equation whose state space is the k simplex. We prove that if this mean limit ODE has an attractor at which the temporal averages of the population growth rate is positive, then there is a positive probability of the population not going extinct (i.e. growing without bound) and its distribution converging to the attractor. Conversely, when the temporal averages of the population growth rate is negative along this attractor, the population distribution does not converge to the attractor. For the stochastic analog of the replicator equations which can exhibit non-equilibrium dynamics, we show that verifying the conditions for convergence and non-convergence reduces to a simple algebraic problem. We also apply these results to selection-mutation dynamics to illustrate convergence to periodic solutions of these population genetics models with positive probability.Comment: 29 pages, 2 figure

Quasi-stationary distributions for randomly perturbed dynamical systems

We analyze quasi-stationary distributions $\{\mu^{\varepsilon}\}_{\varepsilon>0}$ of a family of Markov chains $\{X^{\varepsilon}\}_{\varepsilon>0}$ that are random perturbations of a bounded, continuous map $F:M\to M$, where $M$ is a closed subset of $\mathbb{R}^k$. Consistent with many models in biology, these Markov chains have a closed absorbing set $M_0\subset M$ such that $F(M_0)=M_0$ and $F(M\setminus M_0)=M\setminus M_0$. Under some large deviations assumptions on the random perturbations, we show that, if there exists a positive attractor for $F$ (i.e., an attractor for $F$ in $M\setminus M_0$), then the weak* limit points of $\mu_{\varepsilon}$ are supported by the positive attractors of $F$. To illustrate the broad applicability of these results, we apply them to nonlinear branching process models of metapopulations, competing species, host-parasitoid interactions and evolutionary games.Comment: Published in at http://dx.doi.org/10.1214/13-AAP923 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

A Renewed Analysis of Cheating in Contests: Theory and Evidence from Recovery Doping

In rank-order tournaments, players have incentives to cheat in order to increase their probability of winning the prize. Usually, cheating is seen as a technology that allows individuals to illegally increase their best potential performances. This paper argues that cheating can alternatively be seen as a technology that ensures that the best performances are reached more often. We call this technology recovery doping and show that it yields new insights on the effects of cheating: recovery doping lowers performance uncertainty, thereby changing the outcome of the contest in favour of the best players. We develop this theory in a game with player heterogeneity and performance uncertainty and then study the results of the cross-country skiing World Cup between 1987 and 2006. In line with our theoretical predictions, race-specific rankings were remarkably stable during the 1990s, subsequently becoming more volatile. This pattern reflects the rise and fall of synthetic EPO and the emergence of blood testing and profiling

Precursors predicted by artificial neural networks for mass balance calculations: quantifying hydrothermal alteration in volcanic rocks

This study proposes an artificial neural networks-based method for predicting the unaltered (precursor) chemical compositions of hydrothermally altered volcanic rock. The method aims at predicting precursor’s major components contents (SiO2, FeOT, MgO, CaO, Na2O, and K2O). The prediction is based on ratios of elements generally immobile during alteration processes; i.e. Zr, TiO2, Al2O3, Y, Nb, Th, and Cr, which are provided as inputs to the neural networks. Multi-layer perceptron neural networks were trained on a large dataset of least-altered volcanic rock samples that document a wide range of volcanic rock types, tectonic settings and ages. The precursors thus predicted are then used to perform mass balance calculations. Various statistics were calculated to validate the predictions of precursors’ major components, which indicate that, overall, the predictions are precise and accurate. For example, rank-based correlation coefficients were calculated to compare predicted and analysed values from a least-altered test dataset that had not been used to train the networks. Coefficients over 0.87 were obtained for all components, except for Na2O (0.77), indicating that predictions for alkali might be less performant. Also, predictions are performant for most volcanic rock compositions, except for ultra-K rocks. The proposed method provides an easy and rapid solution to the often difficult task of determining appropriate volcanic precursor compositions to rocks modified by hydrothermal alteration. It is intended for large volcanic rock databases and is most useful, for example, to mineral exploration performed in complex or poorly known volcanic settings. The method is implemented as a simple C++ console progra

Residential exposure to solar ultraviolet radiation and incidence of childhood hematological malignancies in France

Few studies have investigated the relationship between solar ultraviolet radiation (UV) and childhood hematological malignancies (CHM). This study addresses the associations between residential UV exposure at diagnosis and the incidence of types and subtypes of CHM, by age and gender, in France, over a long period, on the fine scale of the 36,326 Communes that constitute mainland France. The 9,082 cases of acute leukemia and 3,563 cases of lymphoma diagnosed before the age of 15 years from 1990 to 2009 were provided by the French National Registry of Childhood Hematological Malignancies. The incidence of CHM was calculated by Commune, year, age and gender and expressed as the standardized incidence ratio (SIR). UV data from 1988 to 2007 were extracted from the EUROSUN database. The annual daily average UV exposure of the children ranged from 85.5 to 137.8 J/cm2. For each additional 25 J/cm2, there was a significant increase in precursor B-cell acute lymphoblastic leukemia (PBC-ALL) in children aged less than 5 years (SIR 1.18; 95 % CI 1.10–1.27). Further analysis of PBC-ALL in the young children suggested a better fit of models with a threshold, with the risk increasing above 100 J/cm2, for which the SIR was 1.24 (95 % CI 1.14–1.36) for a 25 J/cm2 increase. The results remained stable in analyses stratifying by deprivation index or degree of urbanization of the Communes. The study suggests that higher residential UV exposure may be positively associated with a higher incidence of PBC-ALL in early childhood