Tuesday, March 16, 2010

Two armies, one island; one fights, one retreats....or does it?

Here's a classic problem in the theory of extensive form games.

Two armies are in conflict over an island that has magical properties. One army is headed by a man named V (he must not be named), and another is headed by a man named D. Both armies want the island, but Army D is in currently possession of it.

The island is situated between the land of D and the land of V, as in the following diagram.
As the diagram illustrates, a bridge connects each army's homeland with the island. Here's how the game proceeds. First, Army V decides whether to attack Army D. Second, Army D decides whether to fight back or retreat.

So, what should happen? It all depends on the payoffs to the armies. Assume that a battle over the island is the worst possible outcome for both armies, but the island is really great, so it might be worth risking a fight. That is, we can think of three distinct outcomes:

(1) Army D gets the island with no fight,
(2) Army V gets the island with no fight, and
(3) The armies fight

Army D prefers (1) over (2), and (2) over (3). Army V prefers (2) over (1), and (1) over (3). One way to represent this is through the following game tree.**

In this game tree, I depict that Army V chooses whether to fight. After seeing this decision, Army D decides on whether to retreat. We solve games like this by starting at the end and working backward. One way to think about this is that Army V's general puts himself in Army D's shoes. V asks, "What would D do if I attacked?"

Neither army likes a fight, so if D were attacked, he would order his army to retreat. Army V knows this, so V assumes this will happen. Based on this assumption, V knows that it is best to attack. And, the outcome of the game is that V attacks, D retreats, and Army V captures the island.

It is no good to be Army D in that version of the story.

Fortunately for D, there's an extension of this story. Put yourself in Army D's shoes. You're on the island, and you know that an attack is imminent from V. How can you prevent V from attacking? After all, if you prevent an attack, you get to keep the island, which is the best scenario for you.

Army V hates fighting just as much as Army D does. So, you might think that sending a note to Army V that says "If you attack, I will attack you back. You don't want that. Do you?" would do the trick. Unfortunately, such a note is no good. It is not a credible commitment. Economists call such signals cheap talk because although their goal is to induce the other army to act differently, the other army knows your best response once he attacks.

With this in mind, let's try a different tactic. Suppose that Army D's general burns his own bridge, effectively taking away the option to retreat (the army has no boats, and the soldiers cannot swim). In this case, D will have no choice but to fight back if Army V attacks. V knows this, and hence, Army V will not attack.

This is better for D, so the general will burn his own bridge. The map will look like this.

There's some irony in the story. Army D wins the island by burning the bridge, but D's soldiers can never go home, and their island is connected to the wrong homeland.

Viewed in this way, the solution seems silly, but this strategy of "burning bridges" is useful beyond just armies, islands, and bridges. Burning the bridge represents making a credible commitment to an action that would be costly if you were not serious about keeping the island. Making credible commitments explains how come men buy engagement rings, why couples who want children get married, how someone can more effectively get up in the morning, and even how someone could commit to a diet.

** If you want to learn more about how to solve extensive form games, I recently posted a video to my YouTube channel that explains how. Here's the embedded video:

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