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?
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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.
Holmes should therefore assume that Moriarty will foresee his decision in either case and limit the damage accordingly. In other words, the detective ought to optimize his decision with the most pessimistic assumptions in mind. This strategy was published by von Neumann as early as 1928 and was used to demonstrate that a player’s profit can be maximized if one assumes that one’s opponent intends to deal the greatest possible damage.
With no clear winning strategy—unlike in the cake problem—only chance can help. Consider games such as rock-paper-scissors: as soon as one player picks a pattern, the opponent can exploit it to win. The best strategy is therefore to select scissors, rock and paper equally, with a probability of one third each. On average, both parties should then win and lose equally often, minimizing their damage.
The case of Holmes and Moriarty is a bit more complex. To understand this point, it helps to go through the various possible scenarios individually and weight them using numbers, as von Neumann and Morgenstern did. The two mathematicians decided to use values between –100 and 100, with a high value symbolizing a particularly rewarding situation for a given person. The exact numerical values (known as payoffs) chosen for each situation are subjective, but this subjective weighting can then be used to make an optimal decision from an objective point of view.
Morgenstern and von Neumann determined that four different situations could ultimately occur. First, Moriarty and Holmes could both travel to Dover, where Moriarty would assassinate the detective. For Moriarty, this is optimal, so it corresponds to a payoff of 100. For Holmes, on the other hand, it is a disastrous –100 outcome.
Second, Moriarty could get off the train at Canterbury while Holmes travels to Dover. This is bad news for Moriarty because Holmes could flee to the European continent, making it even harder to catch him. This situation is therefore weighted at –50 for Moriarty. For Holmes, on the other hand, it is a positive outcome, so von Neumann and Morgenstern give it a value of 50.
In the third scenario, Moriarty travels to Dover, but Holmes has already disembarked at Canterbury. This is bad for Moriarty but at least better than the case described above. The situation can therefore be weighted 0 for him; the same applies to Holmes, who is still stuck in England.
In the final case, both Moriarty and Holmes disembark in Canterbury. This would be optimal for Moriarty, a clear 100, and would mean death for Holmes, whose payoff is –100.
Each person aims to maximize their payoff. With no clear optimum decision, however, Holmes and Moriarty must rely on chance. Here things get more interesting. For example, they could each flip a coin to decide whether they get off at Canterbury or Dover. If Moriarty stops at Canterbury, the expected value of Holmes’s payoff is: 0.5 × 50 – 0.5 × 100 = –25. If, on the other hand, Holmes gets off the train at Canterbury, the expected value for Holmes is –0.5 × 100 + 0.5 × 0 = –50. In total, Holmes’s expected payoff is therefore –0.5 × 25 – 0.5 × 50 = –37.5. Moriarty’s payoffs have the same magnitude but the opposite sign.
Even worse: In a scenario when their decision hinges on a coin flip, Holmes will die with a probability of 50 percent. That’s because Moriarty will murder the detective if both men get off at the same place, which has a probability of 0.5 in each scenario. This results in a probability of death of 0.5 × 0.5 + 0.5 × 0.5 = 50 percent.
Playing with Probabilities
Holmes has statistically better odds if he follows a different probability distribution—if, for instance, he flips a coin that will land on heads or tails unequally. Let’s assume that Holmes chooses Dover with a probability of p and that Moriarty does so with a probability of q (correspondingly, the two players travel to Canterbury with a probability of 1 – p and 1 – q, respectively). If Moriarty travels to Dover, Holmes’s expected payoff is: –100 × p + 0 x (1 – p) = –100p. If, on the other hand, Moriarty gets off at Canterbury, Holmes’s payoff is: 50 × p – 100 x (1 – p) = 150p – 100.
In the first case (if Moriarty travels to Dover), Holmes’s payoff decreases as p increases; in the second, it increases. To prepare for the worst situation, Holmes should therefore choose the p for which the payoffs are the same—regardless of Moriarty’s decision. To do this, both expected values must be made equal: 150p – 100 = –100p. If you solve the equation for p, you get the value 0.4. This means that Holmes should travel to Dover with a probability of 40 percent and leave the train at Canterbury with a probability of 60 percent.
Incidentally, the same reasoning applies to Moriarty, only in reverse. If you carry out the calculation in the same way, you end up with q = 0.6; this means that Moriarty should travel to Dover with a 60 percent probability. Holmes’s overall chance of survival in this scenario is therefore: (probability that Holmes is in Dover) × (probability that Moriarty is in Canterbury) + (probability that Holmes is in Canterbury) × (probability that Moriarty is in Dover) = 52 percent, slightly higher than if both had flipped a coin.
In this way, von Neumann and Morgenstern cracked the dilemma Holmes faced, at least from a mathematical point of view. But what happens in the short story?
Holmes and Moriarty have neither a rigged coin nor a random number generator with them. Nevertheless, they follow the laws of game theory. Holmes gets off the train in Canterbury and watches as Moriarty happily travels toward Dover in his single carriage, unaware that Holmes has evaded him.
The fact that Doyle opted for this version is more remarkable when you consider that game theory didn’t exist yet, and he couldn’t have known this was an optimal solution. It may have been coincidence—or he may have had good instincts. Either way, I’m reminded to take another look at his writing sometime soon.
This article originally appeared in Spektrum der Wissenschaft and was reproduced with permission.