Attacking a target optimally: an example of war game theory from RAND 1957

The Prisoner’s Dilemma. The doctrine of mutually assured destruction (MAD). These are just two of the game theory ideas that came out of the RAND corporation during the 1950s, forever associating game theory with the non-profit think tank.

But there were other interesting game theory applications, too. I came across a 1957 document from the RAND corporation called War Gaming that was a fun read.

Today I want to discuss an example about the best way to attack an enemy base.

(This is a type of Colonel Blotto game, by the way, but it is interesting to deserve its own post)

The setup

A general is planning an attack. The target is comprised of two enemy bases, with unequal resources. One base contains 1/3 of the resources, the other contains the remaining 2/3.

At his disposal are two fighter planes. The general knows that if one of the planes successfully reaches a target, it can destroy the entire base.

But the enemy can defend itself. Intelligence indicates the enemy has 2 defense missiles that work with 100 percent accuracy. What is not known is which bases are being defended.

The general has to consider the following outcomes:

If a base is not defended, or the general sends 2 planes and the base has 1 missile, then the attack succeeds in destroying the base.

If a base has a number of missiles equal to the number of planes sent, then the base is defended and the resources are saved.

The general’s goal is to destroy as much of the enemy resources. How should the general best deploy the fighter planes?

How to model the game

Let us analyze the situation using game theory notation. First, let’s model the strategies for the general (attacker) and for the enemy (defender).

The general has three strategies: he can deploy 2 planes to either of the two sites, or he can deploy one plane to each site.

We will write the strategy of sending two planes to the more valuable site as (2, 0), one plane to each site as (1, 1), and two planes to the less valuable site as (0, 2).

Similarly, the enemy can defend itself by allocating its missile supply. The enemy also has three analogous strategies, which will similarly be written as (2, 0), (1, 1), and (0, 2).

Now we can calculate and write out the payoffs for the resulting 3 x 3 matrix.

Here are the payoffs in terms of the proportion of resources that are destroyed (from the attacker’s perspective)

Finding the solution

There are no pure strategy equilibria, so we will solve the game by finding mixed strategies.

There is a cool online solver for zero sum games of 5 strategies or less.

Here is a link: zero sum game solver.

The optimal strategy is for the attacker is 4/7 of the time to go after the less valuable site, 2/7 to split resources, and 1/7 to attack for the more valuable site.

In contrast, the enemy will defend the more valuable site 4/7 of the time, then split the missiles 2/7, and place both missiles on the less valuable site 1/7 of the time.

The attacker can destroy 2/7 of the resources on average, about 0.28 of the resources.

Why defend the less valuable base?

If you think about it, there is something odd about the solution. For 1/7 of the time, the defender is putting both of its missiles on the less valuable base that contains 1/3 of the resources.

Why would that ever be sensible?

To see why, let’s go through the following logic. Assume the enemy will never defend the less valuable site with 2 missiles. Then the game collapses into the following 3 x 2 matrix:

We can solve this game fairly easily. The defender has a weakly dominant strategy of playing (2,0): it can sustain fewer losses by just defending the more valuable site, regardless of what the attacker does.

In turn, the attacker will best respond by going after the less valuable site (0,2). The attacker ends up getting a payoff of 1/3, which is slightly larger than 2/7.

This result explains why the enemy wants to defend its less valuable base occasionally: to prevent the attacker from exclusively going after the less valuable base and gaining an edge.

The idea is to create doubt about what is being defended which causes the attacker to mix its strategy.

The importance of using a mixed strategy

This game also demonstrates the extreme value of randomization.

If the attacker had a predictable plan, the enemy could put the missiles at the correct bases, and then shoot down its planes with 100 percent accuracy: it wouldn’t be much of a game at all!

The attacker must mix its moves to create doubt in the enemy. By threatening to attack multiple sites, occasionally the attacker will find some sites unguarded and prevail.