The Game Theory Hidden in the Mind of Sherlock Holmes – Scientific American

In the early 20th century, a mathematician and economist sought the optimal strategy for Holmes to escape Moriarty’s pursuit

By Manon Bischoff edited by Daisy Yuhas

Looking at my bookshelf, I’m stricken with guilt: the collected Sherlock Holmes stories by Arthur Conan Doyle have been sitting untouched for years. Unfortunately, I never got past the fantastic television adaptation starring Benedict Cumberbatch to read the source material. But happily for Holmes, the British detective has a following the world over.

In fact, stories about the ingenious sleuth and his brilliant nemesis, Professor James Moriarty, so appealed to mathematician John von Neumann and economist Oskar Morgenstern that they played a role in the creation of game theory in the early 20th century. This mathematical discipline explores the strategies for solving various decision-making problems. Take the classic “cake problem,” which posits that the fairest way for two people to split a cake such that each gets as much as possible requires one person to try to cut the cake into two equal pieces and the other to select a slice. Morgenstern and von Neumann did not devise this solution (it has been known since ancient times), but it is a good illustration of how game theorists devise optimal strategies.

The pair were particularly taken by a scenario described by Doyle in his short story “The Final Problem,” in which Moriarty pursues Holmes to a platform at Victoria station in London. There Moriarty sees Holmes jump onto a train to Dover. Moriarty can no longer board the train. He therefore hires a single motorized railroad carriage in pursuit. Holmes’s train does not go straight to Dover, however, but stops at Canterbury on the way. So Moriarty has to make a decision: Should he stop in Canterbury, in the hope that Holmes will get off the train there, or travel all the way to Dover? Holmes, too, must weigh his choices. From Dover, he can flee to the European mainland. He knows that Moriarty may expect that outcome and wait for him there, though, so maybe Holmes should get off the train at Canterbury. But what if that is exactly what Moriarty wants Holmes to think?

This scenario intrigued Morgenstern and von Neumann, who ultimately came to the conclusion in their 1944 foundational book that “Sherlock Holmes is as good as 48% dead when his train pulls out from Victoria Station.” But how could they put such a precise figure on it? And how should Holmes act to escape his adversary? All this can be answered with the help of game theory.
A Battle of Wits

The first thing to consider is that the clever Holmes and Moriarty are each likely to guess what the other is thinking. (“If he thinks that I think that he thinks….”) These considerations could easily land Holmes in an endless logic loop with no way out.

Source Links: The Hidden Game Theory of Sherlock Holmes | Scientific American