One equation allocates playing time in competitive team sports: the rulebook sets the arithmetic and the league sets the habit
Abstract
Every team sport that lets a team replace players rations the replacements, and the ration settles who plays and for how long. Across eight codes, three in both sexes, we measure how playing time is allocated and find one equation behind it, complete in the one-way codes, its terms separating what the rulebook fixes from what a competition chooses. Under scarcity every league behaves alike: loaded starters are kept on at the same rate on four continents. Under freedom each expresses a preference, and the size of a league's departure is priced near one-for-one by what its own withdrawal minute prices (lambda = 0.87, 95% interval 0.54 to 1.21, twelve leagues), a law confirmed in four out-of-sample predictions and refused in one. The ordering of leagues by appetite persists across a rule change, and club wealth does not price it. Rules are the geometry of a competition; culture is its material.
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BibTeX
@misc{ricou2026equation,
title = {{One equation allocates playing time in competitive team sports: the rulebook sets the arithmetic and the league sets the habit}},
author = {Gustavo Pedro Ricou},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.05584},
eprint = {2609.05584},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.05584},
}
RIS
TY - UNPB
TI - One equation allocates playing time in competitive team sports: the rulebook sets the arithmetic and the league sets the habit
AU - Gustavo Pedro Ricou
PY - 2026
DA - 2026/09/04
PB - arXiv
DO - 10.48550/arxiv.2609.05584
UR - https://arxiv.org/abs/2609.05584
AB - Every team sport that lets a team replace players rations the replacements, and the ration settles who plays and for how long. Across eight codes, three in both sexes, we measure how playing time is allocated and find one equation behind it, complete in the one-way codes, its terms separating what the rulebook fixes from what a competition chooses. Under scarcity every league behaves alike: loaded starters are kept on at the same rate on four continents. Under freedom each expresses a preference, and the size of a league's departure is priced near one-for-one by what its own withdrawal minute prices (lambda = 0.87, 95% interval 0.54 to 1.21, twelve leagues), a law confirmed in four out-of-sample predictions and refused in one. The ordering of leagues by appetite persists across a rule change, and club wealth does not price it. Rules are the geometry of a competition; culture is its material.
ER -