Incentives: How to Outrun an Army with Three Arrows

Can you out run an army? Economic thought provides the ability to solve problems that lie outside the areas generally covered in economics textbooks. Today’s entry is an example of this. This question and answer is taken from a wonderful book called Hidden Order: The Economics of Everyday Life by David D. Friedman.

Q: A soldier has escaped from an enemy prison and is being chased by ten enemy soldiers. If he can make it to the border, he will be safe, but if he is caught, he will be killed. The soldier’s horse is tiring, and at the current pace, it will grow too tired before the border. Therefore, he will be unable to out run the soldier that are chasing him. His only assets, besides his wits, are three arrows in a quiver (the fact that the soldier has only three arrows is visible to the pursuers). He can shoot three of the chasing soldiers, but then the other seven will catch him. How can the solder use economic thought to escape?

A: The soldier wants to give the pursuers an incentive to slow down. He must do this because his horse will tire out otherwise. If he had ten arrows, he could shoot all ten pursuers, and get away. He has only three arrows, however. What he must do then, is force each individual soldier to not want to catch him. He can do this by shooting the closest pursuer, and declaring his intention to shoot the next closest one. If he proceeds in this manner, each individual soldier has an incentive to hang back an not pursue. The end result is that the entire group has an incentive to let the soldier escape.

The power of this method lies in the fact that each soldier has an incentive to hang back individually, even though the group would certainly catch the escapee if they acted as a group.