Variance Distributions from Round Robin Series of Zero-Sum Competitions
When observations involve the outcomes of many series of simple zero-sum games, with competitor i facing competitor j in some of those games, competitor i’s observed value will not be independent of competitor j’s value. Therefore, the sampling distribution of the sum of squared deviations from the mean can not be assumed to have a chi-squared distribution. In this paper we derive moments and the sampling distributions for the sum of squared deviations in these situations and show that they are asymptotically gamma distributions. For n competitors and sufficiently large number of games g in each series, the sampling distribution of sum of squared deviations will have a gamma distribution, with parameters a = (n-1)=2 and b = 2n/(n-1). This distribution approaches the chi-squared distribution as the number of competitors approaches infinity.
Non-cooperative Games, Gamma distribution, Asymptotic Distribution Theory.
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