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We consider a random knockout tournament among players 1,... , n, in which each match involves two players. The match format is specified by the number of matches played in each round, where the constitution of the matches in a round is random. Supposing that there are numbers [v.sub.1],...,[v.sub.n] such that a match between i and j will be won by i with probability [v.sub.i]/([v.sub.i] + [v.sub.j]), we obtain a lower bound on the tournament win probability for the best player, as well as upper and lower bounds for all of the players. We also obtain additional bounds by considering the best and worst formats for player 1 in the special case [v.sub.1] > [v.sub.2] = [v.sub.3] = ... = [v.sub.n]. Funding: The fifth author was supported by, or in part by, the U.S. Army Research Laboratory and the U.S. Army Research Office [Contract/Grant W911NF-14-1-0166] and by the National Science Foundation [Contract/Grant CMMI1233337]. Keywords: games/group decisions * stochastic model applications * probability * sequential decision analysis, 1. Introduction We consider a tournament among players 1,..., n, in which each match involves two players. The tournament is assumed to be of knockout type in that the losers [...]

The number of indicators developed to measure quality of patient care has expanded rapidly as pressures to improve quality have increased. Some of these indicators are different measures of the [...], Objective. To demonstrate the value of shrinkage estimators when calculating a composite quality measure as the weighted average of a set of individual quality indicators. Data Sources. Rates of 28 quality indicators (QIs) calculated from the minimum dataset from residents of 112 Veterans Health Administration nursing homes in fiscal years 2005-2008. Study Design. We compared composite scores calculated from the 28 QIs using both observed rates and shrunken rates derived from a Bayesian multivariate normal-binomial model. Principal Findings. Shrunken-rate composite scores, because they take into account unreliability of estimates from small samples and the correlation among QIs, have more intuitive appeal than observed-rate composite scores. Facilities can be profiled based on more policy-relevant measures than point estimates of composite scores, and interval estimates can be calculated without assuming the QIs are independent. Usually, shrunken-rate composite scores in 1 year are better able to predict the observed total number of QI events or the observed-rate composite scores in the following year than the initial year observed-rate composite scores. Conclusion. Shrinkage estimators can be useful when a composite measure is conceptualized as a formative construct. Key Words. Composite measures, Bayesian models, quality indicators

ABSTRACT Accepting a sequence of independent positive mean bets that are individually unacceptable is what Samuelson called a fallacy of large numbers. Recently, utility functions were characterized where this occurs [...]