First-price auction with a stochastic head start

We consider a two-player first-price sealed-bid auction for an item with a common value, where one player has a head start with a stochastic value. We show that under mild conditions, there exists a unique Bayesian Nash equilibrium in pure strategies in which the player with the head start buys the item at a price equal to its value minus the minimal value of the head start rather than its expected value. Thus, the seller extracts an information rent. Applications of the results in professional sports are considered.