Where Should You Park Your Car? The $\frac{1}{2}$ Rule

We investigate parking in a one-dimensional lot, where cars enter at a rate $\lambda$ and each attempts to park close to a target at the origin. Parked cars also depart at rate 1. An entering driver cannot see beyond the parked cars for more desirable open spots. We analyze a class of strategies in which a driver ignores open spots beyond $\tau L$, where $\tau$ is a risk threshold and $L$ is the location of the most distant parked car, and attempts to park at the first available spot encountered closer than $\tau L$. When all drivers use this strategy, the probability to park at the best available spot is maximal when $\tau=\frac{1}{2}$, and parking at the best available spot occurs with probability $\frac{1}{4}$.