The Endgame

On December 1st, 2009 President Obama announced that the U.S. troops would have started leaving Afghanistan on July 2011. Rather than simply waiting "the U.S. troops out," the Taliban forces responded with a spike in attacks followed by a decline as the withdrawal date approached. These, at first, counter-intuitive phenomena, are addressed by studying a two-player, zero-sum game where the duration of the strategic interaction is either known or unknown to players. We find that under known duration, players' equilibrium strategies depend on the time remaining in the game and their relative positions at that time of play. Under unknown duration the equilibrium strategies are independent of time and continuation probability. We test the model on data available for soccer matches in the major European leagues. Most importantly, we exploit a change in rule adopted by FIFA in 1998 requiring referees to publicly disclose the length of the added time at the end of the 90 minutes of play. We study how the change in rule has affected the probability of scoring both over time and across teams' relative performance and find that the rule's change led to a 28% increase in the probability of scoring during the added time.