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Journal of Quantitative Analysis in Sports

An official journal of the American Statistical Association

Editor-in-Chief: Rigdon, Steve

Editorial Board Member: Glickman, PhD, Mark

4 Issues per year

CiteScore 2016: 0.44

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Measures of tactical efficiency in water polo

James Graham / John Mayberry
Published Online: 2014-02-21 | DOI: https://doi.org/10.1515/jqas-2013-0127


We present a notational analysis of offensive tactics commonly employed in elite men’s water polo and address three questions related to this objective: which tactics are most effective?, which tactical performance indicators best classify the winning team?, and how accurate are predictive models based on these performance indicators? We define a new statistic, Efficiency Rating, which quantifies the importance of a tactic via a weighted average of direct and indirect goals generated by its use. By this measure, direct shot is the most efficient even strategy despite being employed far less frequently than centre or perimeter tactics. We address our second question by measuring the effect size of winning over losing teams for 25 tactical variables and find that exclusion conversion rate is the most effective discriminatory statistic in both close and unbalanced games, correctly classifying almost 90% of all contests. To address our third question, we develop and apply a simple Binomial model based on goals generated per play which correctly predicts all eight games in the medal round of the 2012 Men’s Olympics from preliminary rounds. Success probabilities are computed based on a weighted average of offensive and defensive efficiency with an optimal weight that favors defense.

Keywords: classification; notational analysis; prediction; water polo


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About the article

Corresponding author: John Mayberry, University of the Pacific – Mathematics, 3601 Pacific Ave., Stockton, California 95211, USA, Tel.: +209.946.3166, e-mail:

Published Online: 2014-02-21

Published in Print: 2014-01-01

In basketball, it is common to distinguish between plays and possessions (Kubatko et al. 2007), the latter referring to the period of game play between which a team gains control of the ball until the time at which control passes to the opposing team. In this paper, we look only at plays because (i) we are more interested in the outcome of specific tactical choices and (ii) the proportion of plays ending in non-possession ending outcomes such as corner or rebound is relatively small anyways.

We exclude exclusions resulting from exclusions in this calculation so that ε is technically the conditional probability that a power-play situation results in a goal given that the power-play resulted in a return to an even situation or counterattack.

Exceptions included penalty shots and shooting percentage for centre and direct shots, which all received values of 0 for both teams in about 40% of contests in our sample.

There was also one game in which both teams had the same ECR.

The correlation is similar if one looks just at perimeter shooting percentage.

We excluded all games against last place finishers Kazakhstan and Great Britain as well as the Serbia vs. Romania game because it occurred after Romania was eliminated from playoff contention.

Note that we round nijα to the nearest integer.

The two exceptions being USA vs Hungary and USA vs Montenegro.

In fact, one can compute the probability that team i beats j by k goals for any k≥0 in a similar manner.

Incorrect: Serbia vs Hungary and Montenegro vs Hungary. With α=0.03, we also incorrectly predicted Hungary vs USA so it appears that the Exclusion Model with low offensive weight does especially poorly in predicting matches involving Hungary.

... and USA vs Montenegro in the preliminary round.

Citation Information: Journal of Quantitative Analysis in Sports, ISSN (Online) 1559-0410, ISSN (Print) 2194-6388, DOI: https://doi.org/10.1515/jqas-2013-0127.

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