From Custer to Thermopylae: Last stand behavior in multi-stage contests

In a multi-stage contest known as a two-player race, players display two fundamental behaviors: (1) the laggard will make a last stand in order to avoid the cost of losing; and (2) the player who is ahead will defend his lead if it is threatened. Last stand behavior, in particular, contrasts with previous research where the underdog simply gives up. The distinctive results are achieved by introducing losing penalties and discounting into the racing environment. This framework permits the momentum effect, typically ascribed to the winner of early stages, to be more thoroughly examined. I study the likelihood that the underdog will catch up. I find that neck-and-neck races are common when the losing penalty is large relative to the winning prize, while landslide victories occur when the prize is relatively large. Closed-form solutions are given for the case where players have a common winning prize and losing penalty.