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" as expected by several commentators, the Taliban forces responded to the announcement with a spike in attacks followed by a decline as the withdrawal date approached. In order to better understand these, at rst, counter-intuitive phenomena, this paper addresses the question of how knowing versus concealing the exact length of a strategic interaction changes the path of the optimal equilibrium strategy. We study a two-player, zero-sum game of known and unknown duration. Under known duration, whether the performance and eort are increased or impaired depends on the time remaining in the game and the players' relative positions. Under unknown duration the equilibrium strategies are stationary. We then 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 aected the probability of scoring both over time and across teams' relative performance and nd that the rule's change led to a 28% increase in the probability of scoring during the added time.