Power Laws Can Forecast Team Managers' Tenure Length

November 18, 2008

Managers of British soccer teams—for that matter, managers of any sports teams—would be wise to bone up on their mathematics. It could help them predict their future.

Researchers have found that a strong mathematical trend underlies how long managers keep their jobs. Modeled by a "power law," their longevity has less to do with ability or talent and more to do with what the British term "sacking and poaching" thresholds and randomness in teams' success.

Toke S. Aidt, Bernard Leong, William C. Saslaw, and Daniel Sgroi developed a formula relating the length of a soccer manager's career, t, to the number of managers, n, dismissed at that time in their career. The formula had the form of a power law: n = a(t + 1)^b, where a and b are constants that vary from country to country and league to league.

To find the rules underlying this result, the authors constructed a mathematical model that focused on how a manager's reputation is either enhanced or diminished by the results of each match played by the team. The model's core was a round-robin tournament in which 20 teams play each other once at home and once away (just like in Britain's Premier League). Based on historical records, teams had a 37% chance of winning, 26% chance of losing, and 37% chance of playing to a draw.

In the model, managers started with a given (numerical) reputation, then gained or lost points as teams played. Termination of tenure occurred when a manager's reputation fell below a "firing threshold," another team offered a better deal, the manager got too old, or the team was demoted out of the top league.

With these rules in place, the researchers ran many simulations, varying the random parameters in each run. They found that a power-law distribution for tenure lengths emerged for a broad range of starting parameters, suggesting that random game results and firing/poaching thresholds play an important role in deciding how long a manager stays with a team.

The researchers did concede, however, that their model does not entirely prove the existence of this phenomenon in all sporting worldwide. Nonetheless, don't feel sorry for British "Premier League" managers when they're abruptly let go. The risk is worth the rewards because their compensation averages more than £2 million a year.

The results appear in the paper "A Power-Law Distribution for Tenure Lengths of Sports Managers," published in the journal Physica A.

Source: Plus, Oct. 28, 2008.