Game Theory of Military Battle Strategy

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The US Military Needs Your Intellect ⚓

Okay Math 450, today is the day. It is the 1940's and you all have been drafted into the United States military. Your commanding officer knows that you're a smart cookie and wants you to apply game theory to help the Allies become victorious. I am here today to help expedite your understanding of how this can be done. Before I jump into the material, can you brainstorm a couple of theoretical applications in which different war type scenarios can be modeled using game theory? Let me know in the comments section below! 

What's the connection between Game Theory and battle decisions? 🔫

I'm glad you asked! The overarching goal in our decision making process is to select a course of action that yields a successful outcome relative to any decisions made by the Axis powers during an operation. This is a doctrine based on enemy capabilities, according to the doctrine of decision of the armed forces of the United States. This doctrine is conservative in nature, molding a course of action based off of all of the possible choices the enemy is "capable" of making.  In the particular example we are going to use, both “players” are blind to each other’s choices, making this a two-person zero-sum game.  Using this framework, we can create and solve a decision matrix that allows the commander to make the best decision. Does anybody want to take a shot at what I mean when I say two-person zero-sum? Let me know below!... So, what I mean by two-person zero-sum is that, two groups are involved (the Allied 😀 and Axis 😷  powers) and are coupled into a pursuit-evasion game. And in this game, anything that is lost or gained only pertains to the groups playing. For example, if I am playing poker with Mac and he wins, then he gains my poker chips and I lose all of mine. 

Maximizing success and minimize casualties 💣

Earlier in the year we looked at combinatorial games and if you knew the rules of the game you could put yourself in a P-posistion to eventually win. However, real world applications of game theory are probabilistic in nature. So, you can't always guarantee victory, but you can maximize your chances of success. Let's take a look at a real World War II battle and the decisions that were made. We can then use game theory to evaluate the actual decision that was made by applying a matrix. 

The Rabaul-Lae Convoy Situation 🚢

In February 1943, General Kenney of the Allied Air Force struggled to regain the island of New Guinea. Luckily, some information was leaked about a Japanese convoy assembling at Rabaul and then sailing over to Lau (See Fig. 1). 

General Kenney and his troops estimated that this Japanese convoy would take three days to sail from Rabaul to Lau. The Allied forces wanted to maximize the amount of days they could bomb the Japanese. However, the issue here is that the "evaders" in this case have two possible routes shown above (See Fig. 1). They could sail along the southern route with clear visibility or along the northern route with poor visibility. Being a well educated General who was taught many battle strategies, General Kenney proceeded with his decision using the five-step Estimate of the Situation. This technique is taught at all military institutions for situations exactly like this. We are going to look at Kenney's thought process and then apply game theory to see if he made the proper decision to maximize his chances for success. The five-step Estimate of the Situation is as follows: 1. Mission 2. Situation and course of action 3. Analysis of the opposing courses of action 4. Comparison of available courses of action 5. The decision. Since General Kenney was working with limited time and resources, he had to decide which side of the island he would have the majority of his pilots patrol, while sending one additional aircraft to the opposite side of the island to alert his forces if the Axis fleet had indeed went the other way.

The commanders for the axis fleet believed that it was safest to keep their forces together and had to choose whether to go north through poor visibility or south through the open waters without knowing where Kenney’s forces would be patrolling.

Based on the choices the generals had to make, four possible outcomes exist: Both go North, both go south, the fleet goes north and the bombing force goes south or the fleet goes south, while the bombing forces head north (Fig. 2).

The Allies want to guarantee the maximal amount of bombing, whereas the Axis powers want to minimize the days that they are under fire.

If the Allies concentrated their power on the northern route, the outcomes could be at most two days of bombing. On there other side, if the Allies patrolled the south, the two outcomes could be either one day of bombing or three days of bombing. To make a long story short, General Kenney chose to patrol the Northern route because it would guarantee at least two days of bombing. According the the U.S. doctrine of decision, General Kenney took action to achieve the greatest amount of success while taking in all consideration of the enemies capabilities. 

As for the Japanese, they also chose the northern route because they did not want to take the risk of getting bombed for three full days. The independent decisions of these two commanders lead to the Battle of the Bismarck Sea. Japanese ships were spotted after a day of reconnaissance and were bombed for two days as predicted by the Allies. 

Game theory analysis and matrix 📈

There exists a way in game theory to analyze situations like the Battle of the Bismarck Sea. The doctrine of the Estimate of the Situation is synonymous with that of von Neumann's solutions of the two-person zero-sum game. Lets apply some numbers to this situation and create a matrix of options that Kenney had to select from and then use game theory to analyze it. 

To solve this "game" we attach a "Maximum of columns" and a "Minimum of row". We can do this because the Japanese select strategies from the columns and Kenney can independently select his strategies from the rows. 

By looking at the minimum value in each row, General Kenney could see the number of days that he could bomb and can make his decision accordingly.  The US military’s conservative approach is highlighted in Kenney’s employing of the “maximin” method of choice; If he went with option two then he could get three days of bombing, however there is also the possibility of only getting one, so he took the safer bet and went with a guaranteed two days of bombing.  

For the Japanese, they wanted to achieve the opposite. Since their decisions come from the columns, they take the maximum of those columns and then select the minimal possible amount of days to potentially be bombed or the "minimax". A military general might not be so quick as to see this fast and easy way to solve his pursuit-evasion "game"! 

Conclusion 🔒

Now as a class, we are all well versed in some battle strategy! Something cool to note in this example is that if there was some sort of secret spy on either side, who could find out the opponents strategy, then his opponent would have not really gained or lost anything from that. This is called a matched strategy or a Nash equilibrium; No player can gain by changing his strategy. In summary, we can now see that there exists a relationship between game theory and the U.S. doctrine of military decision. And we can use this relationship to analyze situations where a commander might attack what he believes is the enemies next move.  


Bibliography

1. Williams, Prof. Brian, and Prof. Emilio Frazzoli. "Principles of Autonomy and Decision Making." MIT OpenCourseWare. Accessed March 13, 2017. https://ocw.mit.edu/courses/aeronautics-and-astronautics/16-410-principles-of-autonomy-and-decision-making-fall-2010/lecture-notes/MIT16_410F10_lec25.pdf.

2. Haywood , O.G., Jr. " MILITARY DECISION AND GAME THEORY." Jstor.org. Accessed March 13, 2017. https://www.jstor.org/stable/pdf/166693.pdf.