Passing before possession loss: Optimal initiation of a developing receiving opportunity

Abstract

Receiver movement can increase the value of a pass during the interval in which a ball carrier remains exposed to dispossession. This study determines the initiation time of one specified pass under a fixed waiting policy. Expected utility includes the outcome after initiation, possession loss before initiation and an additional charge for time spent waiting. Differentiation gives the condition r′(t) > λ(t)[r(t) + L(t)] + c(t) for a locally beneficial delay, where r is conditional pass value, λ is the loss hazard, L is the magnitude of the loss payoff and c is the waiting-cost rate. Exponential development of pass value and constant hazard yield an exact optimum within a finite initiation interval. With initial value 0.30, development amplitude 0.60, development rate 0.90 s⁻¹, loss payoff −0.75, time charge 0.04 utility units per second and a 3-s limit, hazards of 0.08, 0.35 and 0.70 s⁻¹ produce optimal delays of 1.366, 0.216 and 0 s. The largest positive delay occurs at a unique development rate: increasing that rate first lengthens and then shortens optimal waiting. The three maximum delays are 1.440, 0.413 and 0.215 s. Numerical optimisation and integration agree with the closed-form calculations to within 3.34 × 10⁻¹⁶ utility units across 10,062 parameter combinations. These are calculated properties of the specified functions, without athlete measurements. Their application to passing depends on measuring movement initiation, retaining premature losses and valuing every timing strategy over the same outcome horizon.