TR-139: The Theory and Application of Uni-Dimensional Random Races With Probabilistic Handicaps

Abstract

We consider the problem of M racers running towards a goal, where, at each instant, racer Ri moves towards the goal with a probability of Si and stays where he is with a probability of (1-Si)- Additionally, we permit each racer to be granted a certain handicap which allows him to start closer to the goal. This handicap may be stochastically assigned. In the simplest model, the racers run on multiple tracks, and in this scenario, each racer has his own track, thus disallowing interference between the racers. However, in a more general setting, the racers may run on a single track, in which case, interferences between racers are specified in terms of overtaking rules. In this paper, which we believe is a pioneering paper in this area, we first examine random races in a one dimensional space subject to the constraint that the race has multi-tracks and that exactly one racer moves towards the goal at any given instant. A powerful result is derived for the case when the individual racers are given no handicaps at all. Subsequently, a variety of results are proven for the cases when the racers are given handicaps which are either uniformly or geometrically distributed. In each of these cases, the results proved have been obtained for the setting when the length of the track is finite and for the asymptotic condition when the race is arbitrarily long. Analogous results for the single track race are also conjectured and these conjectures are strengthened by numerous simulations.