Gambling in contests modelled with diffusions

摘要

In the Seel-Strack contest (J Econ Theory 148(5):2033-2048, 2013), n agents each privately observe an independent copy of a drifting Brownian motion which starts above zero and is absorbed at zero. Each agent chooses when to stop the process she observes, and the winner of the contest is the agent who stops her Brownian motion at the highest value. The objective of each agent is to maximise her probability of winning the contest. Seel and Strack derived a Nash equilibrium under a joint feasibility condition on the drift and the number of players. We consider a generalisation of the Seel-Strack contest in which the observed processes are independent copies of some time-homogeneous diffusion. This naturally leads us to consider the problem in cases where the analogue of the feasibility condition is violated. We solve the problem via a change of scale and a Lagrangian method. Unlike in the Seel-Strack problem, it turns out that the optimal strategy may involve a target distribution which has an atom, and the rule used for breaking ties becomes important.