Which Ball is the Roundest? - A Suggested Tournament Stability Index

All sports have components of randomness that cause the “best” individual or team not to win every game. According to many spectators this uncertainty is part of the charm when following a competition or a match. Have different sports more or less of this unpredictability? We suggest here a general measure, a tournament stability index, together with its associated p-value which we denote the "coin-tossing-index." These indexes are aimed to quantify the randomness factor for different tournaments, and different sports. As an illustration we exemplify and discuss these measures for basketball, squash, and soccer. Some additional results will also be given on a few tournaments in ice-hockey, and handball. Furthermore, we discuss a couple of combinatorial optimization questions that turned up on the way.

[1]  Bruce A. Craig,et al.  Hybrid Paired Comparison Analysis, with Applications to the Ranking of College Football Teams , 2005 .

[2]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[3]  H. A. David,et al.  The method of paired comparisons , 1966 .

[4]  G. Fechner.  Elemente der Psychophysik , 1998 .

[5]  Ranking paired contestants , 1990 .

[6]  J. M. Roberts,et al.  Modeling hierarchy: Transitivity and the linear ordering problem , 1990 .

[7]  Teofilo F. Gonzalez,et al.  P-Complete Approximation Problems , 1976, J. ACM.

[8]  W. Neale,et al.  The Peculiar Economics of Professional Sports , 1964 .

[9]  Leszek Klukowski,et al.  The nearest adjoining order method for pairwise comparisons in the form of difference of ranks , 2000, Ann. Oper. Res..

[10]  Brad R. Humphreys,et al.  Alternative Measures of Competitive Balance in Sports Leagues , 2002 .

[11]  Rodney Fort,et al.  Pay Dirt: The Business of Professional Team Sports. , 1993 .

[12]  Peter P. Wakker,et al.  On solving intransitivities in repeated pairwise choices , 1995 .

[13]  W. A. Thompson,et al.  Maximum-likelihood paired comparison rankings. , 1966, Biometrika.

[14]  A. F. Smith,et al.  AN ALGORITHM FOR DETERMINING SLATER'S i AND ALL NEAREST ADJOINING ORDERS , 1974 .

[15]  H. Vries,et al.  Finding a dominance order most consistent with a linear hierarchy: a new procedure and review , 1998, Animal Behaviour.

[16]  M. Newman,et al.  A network-based ranking system for US college football , 2005, physics/0505169.

[17]  R. Ranyard.  AN ALGORITHM FOR MAXIMUM LIKELIHOOD RANKING AND SLATER'S i FROM PAIRED COMPARISONS , 1976 .