The Battle of the Bismarck Sea was a battle fought in February 1943 in Southeast Asia during World War II, between the Japanese Navy and the US Air Force. In game theory, its modeling was done by O. G. Haywood, Jr. in his article “Military Decision and Game Theory”, 1954. It’s a game used in game theory to analyze zero-sum games with two players.
The game, based on the actual military operation, is based on the decision General Kenney had to make. General Kenney, as Commander of the Allied Forces in the South-west Pacific Area, received intelligence reports indicating part of the Japanese Navy was about to sail from Rabaul, in the island of New Britain, to Lae, in New Guinea. Knowing this, General Kenney decided to make his five-step “Estimate of the Situation”, a technique used in US military operations.
Step 1: The Mission
General Kenney’s mission was to intercept the convoy and inflict the maximum possible amount of damage.
Step 2: Situation and Courses of Action
In addition to the intelligence report on Japanese troop movements, Kenney received weather reports indicating that rain and poor visibility was predicted for the area north of New Britain, while south of the island visibility would be good.
The Japanese commander had two possible courses of action: he could sail his convoy north of the island, or go south of the island. Any of these routes would take three days to sail.
General Kenney had therefore two possible courses of action: concentrate most of his reconnaissance aircrafts (but not all) along the northern route or along the southern route.
Step 3: Analysis of the Opposing Courses of Action
Since both commanders have two possible strategies, there are four possible outcomes.
Step 4: Comparison of Available Courses of Action
Obviously, General Kenney had to look for the best possible outcome. These are the four possible outcomes:
The first scenario (or set of opposing courses of action) shows the US Air Force’s bulk of airplanes north of New Britain, and the Japanese Navy taking the northern route. Because of poor visibility, the convoy wouldn’t be discovered until the second day, allowing for two days of bombing.
The second scenario shows again the US Air Force’s bulk of airplanes north of New Britain, but in this case the Japanese take the southern route. Because of limited reconnaissance south of the island, the convoy could be missed during the first day, allowing once again for two days of bombing.
The third scenario shows the main part of the US Air Force south of the island, and the Japanese Navy taking the northern route. Considering the poor visibility north of the island, plus limited reconnaissance, the convoy would be missed for two days, allowing for just one day of bombing.
The fourth and last scenario shows the US Air Force’s bulk of airplanes south of the island, and the Japanese taking the southern route. In this case, having the majority of airplanes in the area and having good visibility, General Kenney could hope for three days of bombing.
Step 5: The Decision
General Kenney decided to concentrate his reconnaissance airplanes north of New Britain.
In game theory, the Battle of the Bismarck Sea would be considered a simultaneous game, since both players have to make a decision at the same time, without knowing his opponent’s decision. Therefore, it can be depicted using the strategic form, by means of a matrix such as the following one.
We can solve this game analyzing the dominant and dominated strategies. The way to proceed is to eliminate for each player every strategy that seems ‘unreasonable’, which will greatly reduce the number of equilibria. This method is quite easy to use when only strictly dominated strategies are in place.
In this game, Kenney has no dominant strategy (the sum of the payoffs of the first strategy equals the sum of the second strategy), but the Japanese do have a weakly dominating strategy, which is to go North (the payoffs are equal for one strategy but strictly better for the other). Since only one of them has a dominant strategy, there is no dominant strategy equilibrium. We must then proceed by eliminating dominated strategies. As we’ve already mentioned, for the Japanese the strategy ‘go North’ weakly dominates strategy ‘go South’. Therefore, we eliminate the strategy ‘go South’ for the Japanese, who will go North. Now that we only consider the Japanese going North, Kenney’s strategy ‘go North’ is strictly dominant over strategy ‘go South’, which will be eliminated. Therefore, North-North is the weak-dominance equilibrium.
General Kenney made the right decision: he won the battle.