I posted earlier about my Diplomacy game (read the full post through the link to the left, otherwise there is a short recap below) and how I was determined to view the game through a Bayesian lens in an attempt to avoid biases.
Here is a quick recap from the original post:
- I started the game as Italy and was mostly concerned with my neighbors, (France and Austria).
- Austria appeared willing to work with me, so I decided to attack France.
- I spoke to England about working together against France but only sensed mild interest.
- After the first turn, I immediately suspected a Franco-English alliance and assigned an 80% probability.
- After the second turn, France, and England both moved to Belgium indicating they may not have an alliance. Using Baye’s Theorem, the probability of a Franco-English alliance was reduced to 66.67%
- After England built a fleet in the south instead of the north, the probability of a Franco-English alliance was further reduced to 57.14%
I ended the post at the end of 1901 with a 57.14% probability of a Franco-English alliance. Normally when I play without considering Bayes Theorem, I would have already put the probability of the alliance at zero. The fact that Baye’s formula kept the probability so high after two strong pieces of evidence, surprised me. I will now continue with the Spring 1902 turn and see if Baye’s Theorem knows something I do not.
In the screenshot above there are a few things going on. First of all, the new English fleet in London went to the English Channel and the army in Yorkshire went to London posing a direct threat to the French supply center of Brest. Additionally, France has moved from Portugal to the Mid-Atlantic Ocean instead of the southern coast of Spain. This is possibly defensive against England and most helpful for me, the Italian player. I have a difficult time imagining a Franco-English alliance given the Spring 1902 moves by both countries. Now, I will see what Baye’s Theorem has to say. I will focus on the English move to the English Channel as it is the strongest piece of evidence. I will assign a minuscule 5% probability that England would move to the English Channel instead of the North Sea, given that there is a Franco-English alliance. I will assign a 40% chance of England moving to the English Channel instead of the North Sean if there is not a Franco-English alliance. The North Sea is just a better move for England, in my opinion regardless of an alliance with France. Given my 57.14% prior probability of an alliance, I now get the following:
- Prior probability of a Franco-English alliance P(A) = 57.14%
- Probability of England moving into the English Channel given a Franco-English alliance P(EC given A) = 5%
- Probability of England moving into the English Chanel with no Franco-English alliance = 40%
- Probability of England moving into the English Chanel P(EC) = (.5714)(.05) + (.4286)(.40) = 20%
The math: the probability of a Franco-English alliance given that England moved into the English Channel = P(A) 57.14% * P(EC given A) 5% / P(EC) 20% = 14.29%
So, the prospect of an alliance appears to be fading. There is still a 14.29% probability remaining. After three turns of strong evidence against an alliance, it seems that Baye’s Theorem is reluctant to forget prior probabilities completely.
Now for Fall 1902.
Now it appears that Baye’s Theorem knows something that I do not. Working things out in my own head, I would have put the probability of a Franco-English alliance at zero. Baye’s Theorem was still clinging to a 14.29% probability and for good reason. In Fall 1902, England and France appear to have worked things out. England left the English Channel and France sent their fleet to the southern coast of Spain. While it may not be an outright alliance, there is clearly some type of arrangement between the two. Baye’s Theorem never lost sight of the possibility while I was always eager to throw it out at the slightest hint of a fracture. The game is not quite over yet but France just surrendered after Germany overran their north and I overtook their south. As long as my alliance with Austria (and Germany who was added later) holds, I should emerge semi-victorious with a three-way draw in the near future.
My final verdict, a Bayesian framework is a valuable tool for Diplomacy and likely to be equally useful in financial analysis.