Pythagorean Expectation and the Matching Law

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  • ピタゴラス勝率とマッチング法則
  • ピタゴラス ショウリツ ト マッチング ホウソク

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Pythagorean expectation is a simple yet experimentally robust method that utilizes a single parameter for predicting the winning percentages of sports teams based on the score ratio. The score ratio can be defined as runs scored divided by runs allowed, and the Pythagorean probability can be identified as a parameter of logistic distribution. In this study, we aim to estimate the winning probability by incorporating actual sports data and to interpret the Pythagorean probability by using the generalized matching law of behavioral economics. In addition, the probabilistic models for runs scored and runs allowed, which can be considered as two independent Gumbel distributions, are estimated with maximum likelihood parameters from the actual data of a student baseball league. We compare four types of models: the Gumbel model, Poisson model, Pythagorean model, and histogram-based resampling model. For the Gumbel model, three types of methods are applied, namely maximum likelihood (ML), ordinary least squares (OLS), and EAP (expected a posteriori). Further, a series of Monte Carlo simulations shows that the histogram is best, and the Pythagorean is quite good; however, the Gumbel models slightly outweigh the Pythagorean model in terms of the mean square prediction error.

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