Game Theory: Sherlock Holmes & Moriarty “The Final Problem”
This paper, written by Professor David K. Levine when he was a faculty at University of California Los Angeles, briefly explained a few basic concepts in the Game Theory. It described a few examples such as the well-known Prisoner’s dilemma to elaborate on the topic. However, what drew my attention in this paper is the example about Sherlock Holmes’ escape from Moriarty in “The Final Problem.”
Why is it a game? In this example, we have two players: Sherlock Holmes, and Moriarty. Although Dr. Watson was traveling as a companion of Sherlock Holmes and followed the same track as Holmes, he was not considered as a player since he was not participating in this intellectual game. Both Moriarty and Sherlock had two options: getting off at Canterbury, or continuing to Paris. If both of them get off at the same bus top, then it would mean severe consequences on Sherlock Holmes; otherwise, Sherlock Holmes would escape and get a chance to fight back. Both Sherlock Holmes and Moriarty knew very well each other’s strategies and what results would lead for each option. Sherlock Holmes made a meaningful remark after getting off at Canterbury, “There are limits, you see, to our friend’s intelligence. It would have been a coup-de-maître had he deduced what I would deduce and acted accordingly.” This comment showed that he was very well acknowledged of the potential actions that would be taken by Moriarty and what consequences would these actions lead to, and he fully appreciated the fact that Moriarty knew his options as well. Finally, the payoffs Sherlock Holmes and Moriarty receive completely depended on each other’s choice, that is, if they would get off at the same or different bus stop, and we should be able to assume that both of them value their lives and would prefer to destroy the other’s life and receive larger payoffs.
From the information above, we can construct a table representing the payoff for each player given each strategy as shown on the article. If Sherlock Holmes and Moriarty get off at the same bus stop, the payoff for Sherlock Holmes will be -1 and payoff for Moriarty will be 1 since Sherlock would very likely be dead in this case while Moriarty will be completing his mission. However, if they get off at different stops, payoff for Sherlock Holmes will be 1 and payoff for Moriarty will be -1 since Sherlock Holmes gains more time to fight back and Moriarty has to continue on tracking down Sherlock Holmes. By analyzing the table, we can derive a few pieces of information:
- If Sherlock Holmes gets off at Canterbury, Moriarty should get off at Canterbury.
- If Sherlock Holmes gets off at Paris, Moriarty should get off at Paris.
- If Moriarty gets off at Canterbury, Sherlock Holmes should get off at Paris.
- If Moriarty gets off at Paris, Sherlock Holmes should get off at Canterbury.
We can see that in this case we couldn’t form any pair of strategies that each player should perform that satisfies the Nash equilibrium. So there does not exist a pure strategy Nash Equilibrium in this case. In this case we will be considering mixed strategies. If the probability of Moriarty getting off at Canterbury is p, then the expectation of Sherlock getting off at Canterbury is –p + (1 – p), and the expectation of him getting off at Paris will be p – (1 – p). –p + (1 – p) = p – (1 – p) implies that p = 0.5. If the probability of Sherlock getting off at Canterbury is q, then the expectation of Moriarty getting off at Canterbury is q – (1 – q), and the expectation of him getting off at Paris will be – q + (1 – q). q – (1 – q) = – q + (1 – q) implies that q = 0.5. So in this case there is actually a 50-50 chance for each of them to get off at Canterbury or Paris.
I find it rather interesting that the escape of Sherlock Holmes eventually might be just pure luck and that each of them independently decided by chance the bus stop they should get off assuming that their opponent will do the same.
Source: http://levine.sscnet.ucla.edu/general/cogsci.htm
In the story Holmes and Watson take a train to Dover where a ship to France would be leaving in 15 minutes after the trains arrival. This detail changes the payoff table – if Moriarty gets off the train at Canterbury and Holmes gets off at Dover, then Holmes can safely aboard a ship to France and save his life (at least for immediate future), thus the payoff is 1 for Holmes and -1 for Moriarty. However if Holmes gets off at Canterbury and Moriarty gets off at Dover, then Holmes does not loose his life, but he does not save it as of yet either, so the payoff in that case should be 0 for both players. With this game set up the story’s characters logic makes more sense.