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 for the U.S. troops to withdraw, the Taliban forces responded to the announcement with a surge in attacks followed by a decline as the withdrawal date approached. In order to better understand these, at first, counterintuitive phenomena, this paper addresses the question of how knowing versus concealing the exact length of a strategic interaction changes the optimal equilibrium strategy by studying a two-player, zero-sum game of known and unknown duration. We show why the equilibrium dynamics is non-stationary under known duration and stationary otherwise. We show that when the duration is known the performance and effort might be increased or impaired depending on the length itself and on the nature of the interaction. We then test the model by using data available for soccer matches in the major European leagues. Most importantly, we exploit the 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 of goals during the added time.