The Urinal Problem: Optimal Strategies Under Uncertainty

This paper presents a game-theoretic analysis of urinal selection behavior, modeling the scenario as a spatial optimization problem with social distance constraints. The "urinal problem" has been discussed informally in popular culture and treated mathematically in recreational contexts, but this paper represents the first rigorous empirical test of theoretical predictions against actual human behavior.

We formalize the problem as follows: given an array of n urinals, some subset of which may be occupied, an entering individual seeks to maximize minimum distance from all occupied urinals while respecting the boundary constraints that the first and last urinals offer only one-sided buffering. When no solution preserves a buffer of at least one empty urinal, additional factors (proximity to exit, wall presence) become decisive.

Theoretical analysis yields optimal selection algorithms for all configurations up to n=20. We tested these predictions through concealed observation of 2,400 urinal selection events across 15 men's restrooms varying in array length (3-12 urinals), finding that 89% of selections matched optimal predictions.

The 11% deviation rate reveals systematic biases: subjects showed unexpected preference for urinals near walls (even when this reduced buffer distance), and aversion to urinals near trash cans or hand dryers. These "comfort modifiers" can be incorporated into an expanded model that achieves 94% prediction accuracy.

We also document the phenomenon of "strategic timing delay," wherein subjects approaching a restroom with suboptimal configurations available will pause at sinks or hand dryers until a configuration improvement occurs. This delay behavior was observed in 23% of subjects when entering during high-occupancy periods.

The paper concludes with design recommendations for architects, including optimal array lengths and the strategic placement of privacy barriers to maximize user comfort while maintaining space efficiency.