Randomness in Competitions

We study the effects of randomness on competitions based on an elementary random process in which there is a finite probability that a weaker team upsets a stronger team. We apply this model to sports leagues and sports tournaments, and compare the theoretical results with empirical data. Our model shows that single-elimination tournaments are efficient but unfair: the number of games is proportional to the number of teams N, but the probability that the weakest team wins decays only algebraically with N. In contrast, leagues, where every team plays every other team, are fair but inefficient: the top $\sqrt{N}$ of teams remain in contention for the championship, while the probability that the weakest team becomes champion is exponentially small. We also propose a gradual elimination schedule that consists of a preliminary round and a championship round. Initially, teams play a small number of preliminary games, and subsequently, a few teams qualify for the championship round. This algorithm is fair and efficient: the best team wins with a high probability and the number of games scales as $N^{9/5}$, whereas traditional leagues require N^3 games to fairly determine a champion.Comment: 10 pages, 8 figures, reviews arXiv:physics/0512144, arXiv:physics/0608007, arXiv:cond-mat/0607694, arXiv:physics/061221

Asymptotic unbiased density estimator

International audienceThis paper introduces a computationally tractable density estimator that has the same asymptotic variance as the classical Nadaraya-Watson density estimator but whose asymptotic bias is zero. We achieve this result using a two stage estimator that applies a multiplicative bias correction to an oversmooth pilot estimator. Simulations show that our asymptotic results are available for samples as low as n = 50, where we see an improvement of as much as 20% over the traditionnal estimator. Mathematics Subject Classification. 62G07, 62G20

Stochastic Event Reconstruction of Atmospheric Contaminant Dispersion Using Bayesian Inference

Environmental sensors have been deployed in various cities for early detection of contaminant releases into the atmosphere. Event reconstruction and improved dispersion modeling capabilities are needed to estimate the extent of contamination, which is required to implement effective strategies in emergency management. To this end, a stochastic event reconstruction capability that can process information from an environmental sensor network is developed. A probability model is proposed to take into account both zero and non-zero concentration measurements that can be available from a sensor network because of a sensor’s specified limit of detection. The inference is based on the Bayesian paradigm with Markov chain Monte Carlo (MCMC) sampling. Fast-running Gaussian plume dispersion models are adopted as the forward model in the Bayesian inference approach to achieve rapid-response event reconstructions. The Gaussian plume model is substantially enhanced by introducing stochastic parameters in its turbulent diffusion parameterizations and estimating them within the Bayesian inference framework. Additionally, parameters of the likelihood function are estimated in a principled way using data and prior probabilities to avoid tuning in the overall method, The event reconstruction method is successfully validated for both real and synthetic dispersion problems, and posterior distributions of the model parameters are used to generate probabilistic plume envelopes with specified confidence levels to aid emergency decisions

Inequity Measures for Evaluations of Environmental Justice: A Case Study of Close Proximity to Highways in NYC

Assessments of environmental and territorial justice are similar in that both assess whether empirical relations between the spatial arrangement of undesirable hazards (or desirable public goods and services) and socio-demographic groups are consistent with notions of social justice, evaluating the spatial distribution of benefits and burdens (outcome equity) and the process that produces observed differences (process equity. Using proximity to major highways in NYC as a case study, we review methodological issues pertinent to both fields and discuss choice and computation of exposure measures, but focus primarily on measures of inequity. We present inequity measures computed from the empirically estimated joint distribution of exposure and demographics and compare them to traditional measures such as linear regression, logistic regression and Theil’s entropy index. We find that measures computed from the full joint distribution provide more unified, transparent and intuitive operational definitions of inequity and show how the approach can be used to structure siting and decommissioning decisions

Subclinical psychosis syndromes in the general population: results from a large-scale epidemiological survey among residents of the canton of Zurich, Switzerland

Aims. Prevalence and covariates of subclinical psychosis have gained increased interest in the context of early identification and treatment of persons at risk for psychosis. Methods. We analysed 9829 adults representative of the general population within the canton of Zurich, Switzerland. Two psychosis syndromes, derived from the SCL-90-R, were applied: 'schizotypal signs' and 'schizophrenia nuclear symptoms'. Results. Only a few subjects (13.2%) reported no schizotypal signs. While 33.2% of subjects indicated mild signs, only a small proportion (3.7%) reported severe signs. A very common outcome was no 'schizophrenia nuclear symptoms' (70.6%). Although 13.5% of the participants reported mild symptoms, severe nuclear symptoms were very rare (0.5%). Because these two syndromes were only moderately correlated (r = 0.43), we were able to establish sufficiently distinct symptom clusters. Schizotypal signs were more closely connected to distress than was schizophrenia nuclear symptoms, even though their distribution types were similar. Both syndromes were associated with several covariates, such as alcohol and tobacco use, being unmarried, low education level, psychopathological distress and low subjective well-being. Conclusions. Subclinical psychosis symptoms are quite frequent in the general population but, for the most part, are not very pronounced. In particular, our data support the notion of a continuous Wald distribution of psychotic symptoms in the general population. Our findings have enabled us to confirm the usefulness of these syndromes as previously assessed in other independent community samples. Both can appropriately be associated with well-known risk factors of schizophrenia

Dynamics of Three Agent Games

We study the dynamics and resulting score distribution of three-agent games where after each competition a single agent wins and scores a point. A single competition is described by a triplet of numbers $p$, $t$ and $q$ denoting the probabilities that the team with the highest, middle or lowest accumulated score wins. We study the full family of solutions in the regime, where the number of agents and competitions is large, which can be regarded as a hydrodynamic limit. Depending on the parameter values $(p,q,t)$, we find six qualitatively different asymptotic score distributions and we also provide a qualitative understanding of these results. We checked our analytical results against numerical simulations of the microscopic model and find these to be in excellent agreement. The three agent game can be regarded as a social model where a player can be favored or disfavored for advancement, based on his/her accumulated score. It is also possible to decide the outcome of a three agent game through a mini tournament of two-a gent competitions among the participating players and it turns out that the resulting possible score distributions are a subset of those obtained for the general three agent-games. We discuss how one can add a steady and democratic decline rate to the model and present a simple geometric construction that allows one to write down the corresponding score evolution equations for $n$-agent games