Title:

Variants of gambling in contests

In the SeelStrack contest (Seel and Strack [2013]), n agents each privately observe an independent copy of a drifting Brownian motion which starts above 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 amongst the set of agents. The objective of each agent is to maximise her probability of winning the contest. We will give a new derivation of the results of Seel and Strack [2013] based on a Lagrangian approach. This approach facilitates our analysis of the variants of the SeelStrack problem. We will consider a generalisation of the SeelStrack contest in which the observed processes are independent copies of some timehomogeneous diffusion. We will use a change of scale to reduce this contest to a contest in which the observed processes are diffusions in natural scale. It turns out that, unlike in the SeelStrack problem, the way of breaking ties becomes important. Moreover, we will discuss an extension of the SeelStrack contest to one in which an agent is penalised when her strategy is suboptimal, in the sense that her chosen strategy does not win the contest, but there existed an alternative strategy which would have resulted in victory. We will see that different types of penalty have different effects. Seel and Strack [2013] studied the asymmetric 2player contest in which the observed processes start from different constants. We will redrive their results using the Lagrangian method and then study a general asymmetric nplayer contest. We will find that some results in the 2player contest do not hold for the general nplayer contest. In a symmetric 2player contest, the SeelStrack model assumes that the observed processes start from the same positive constant. We will extend the results to the case where the starting values of the processes are independent nonnegative random variables that have the same distribution.
