The Hidden Game Theory of Sherlock Holmes
By looking at my library, I am struck with guilt: the collected stories of Sherlock Holmes by Arthur Conan Doyle have been seated for years. Unfortunately, I have never exceeded the fantastic television adaptation with Benedict Cumberbatch to read the source material. Fortunately for Holmes, the British detective followed the whole world.
In fact, stories about the ingenious Sleuth and his brilliant Nemesis, Professor James Moriarty, therefore called on mathematician John von Neumann and the economist Oskar Morgenstern that they played a role in the creation of game theory at the beginning of the 20th century. This mathematical discipline explores strategies for resolving various decision -making problems. Take the classic “cake problem”, which postulates that the fairest way for two people to divide a cake so that everyone gets as much as possible requires that a person is trying to cut the cake into two equal pieces and the other to select a slice. Morgenstern and Von Neumann have not designed this solution (it has been known since ancient times), but it is a good illustration of the way in which the theorists of the game develop optimal strategies.
The couple was particularly followed by a scenario described by Doyle in their new “the final problem”, in which Moriarty continues Holmes to a platform of the Victoria station in London. There, Moriarty sees Holmes jump on a train for Dover. Moriarty can no longer board the train. He therefore hires a single motorized railway car in pursuit. The Holmes train does not go directly to Dover, but stops in Canterbury on the way. Moriarty must therefore make a decision: should he stop in Canterbury, in the hope that Holmes will leave the train there or will go to Dover? Holmes, too, must weigh his choices. Deames, he can flee to the European continent. He knows that Moriarty can expect this result and wait for him there, so maybe Holmes should get from the train to Canterbury. But what happens if this is exactly what Moriarty wants Holmes to think?
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This scenario intrigued Morgenstern and Von Neumann, who ended up being at the conclusion of their fundamental book of 1944 as “Sherlock Holmes is as good as 48% of death when his train withdraws from Victoria Station”. But how could they put such a precise figure there? And how should Holmes act to escape his opponent? All this can be answered using game theory.
A battle of mind
The first thing to consider is that the intelligent Holmes and Moriarty are each likely to guess what the other thinks. (“If he thinks I think he thinks …”) These considerations could easily land Holmes in an endless logical loop without any outcome.
Holmes should therefore assume that Moriarty provides for her decision in both cases and limit damage accordingly. In other words, the detective should optimize his decision with the most pessimistic hypotheses in mind. This strategy was published by Von Neumann in 1928 and was used to demonstrate that the benefit of a player can be maximized if we assume that his opponent intends to do the greatest possible damage.
Without a clear winning strategy – not as in the problem of the cake – a chance can help. Consider games such as rock paper scrushes: as soon as a player chooses a model, the opponent can use it to win. The best strategy is therefore to select scissors, rock and paper equally, with a probability of a third party each. On average, both parties should then win and also lose, minimizing their damage.
The case of Holmes and Moriarty is a little more complex. To understand this point, it is useful to browse the different possible scenarios and the weights using numbers, as did von Neumann and Morgenstern. The two mathematicians decided to use values between –100 and 100, a high value symbolizing a particularly enriching situation for a given person. The exact digital values (called gains) 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 have determined only four different situations could ultimately occur. First, Moriarty and Holmes could both go to Douvres, where Moriarty would murder the detective. For Moriarty, this is optimal, so it corresponds to a gain of 100. For Holmes, on the other hand, it is a disastrous result –100.
Secondly, Moriarty could leave the train in Canterbury while Holmes went to Douvres. This is bad news for Moriarty because Holmes could flee to the European continent, which makes it even more difficult to catch it. This situation is therefore weighted at –50 for Moriarty. For Holmes, on the other hand, it is a positive result, so von Neumann and Morgenstern give him a value of 50.
In the third scenario, Moriarty went to Dover, but Holmes has already landed in Canterbury. It’s bad for Moriarty but at least better than the case described above. The situation can therefore be weighted 0 for him; The same goes for Holmes, who is still stuck in England.
In the final case, Moriarty and Holmes land in Canterbury. It would be optimal for Moriarty, a clear 100, and would mean death for Holmes, whose gain is –100.
Each person aims to maximize their gain. Without a clear optimal decision, however, Holmes and Moriarty must count on chance. Here, things become more interesting. For example, they could each return a room to decide whether they go down to Canterbury or Dover. If Moriarty stops in Canterbury, the expected value of the Holmes gain is: 0.5 × 50 – 0.5 × 100 = –25. If, on the other hand, Holmes leaves the train in Canterbury, the expected value of Holmes is –0.5 × 100 + 0.5 × 0 = –50. In total, the expected gain of holms is therefore –0.5 × 25 – 0.5 × 50 = –37.5. Moriarty’s gains have the same magnitude but the opposite sign.
Worse still: in a scenario when their decision depends on a reversal of documents, Holmes dies with a probability of 50%. Indeed, Moriarty will assassinate the detective if the two men descend to the same place, which has a probability of 0.5 in each scenario. This results in a death probability of 0.5 × 0.5 + 0.5 × 0.5 = 50%.
Play with probabilities
Holmes has statistically better ratings if he follows a different probability distribution – if, for example, he returns a piece that will unevenly land heads or tails. Suppose Holmes chooses Dover with a probability of p And that Moriarty does it with a probability of q (Consequently, the two players go to Canterbury with a probability of 1 – p and 1 – Q, respectively). If Moriarty goes to Douvres, the expected gain of Holmes is: –100 × p + 0 x (1 – p) = –100p. If, on the other hand, Moriarty descends to Canterbury, the gain from Holmes is: 50 × p – 100 x (1 – p) = 150p – 100.
In the first case (if Moriarty Voyages in Dover), the gain of Holmes decreases as p increase; In the second, it increases. To prepare for the worst situation, Holmes should therefore choose the p For which the gains are the same – without what Moriarty’s decision. To do this, the two expected values must be billed: 150p – 100 = –100p. If you resolve the equation to P, You get the value 0.4. This means that Holmes should go to Dover with a probability of 40% and leave the train in Canterbury with a probability of 60%.
By the way, the same reasoning applies to Moriarty, only upside down. If you do the calculation in the same way, you end up with q = 0.6; This means that Moriarty should go to Dover with a probability of 60%. The overall risk of Holmes survival in this scenario is therefore: (probability that Holmes is in Dover) × (probability that Moriarty be in Canterbury) + (probability that Holmes be at Canterbury) × (probability that Moriarty be in Dover) = 52%, slightly higher than if the two had cut a piece.
In this way, Von Neumann and Morgenstern fell for the dilemma with which Holmes was confronted, at least from a mathematical point of view. But what’s going on in the news?
Holmes and Moriarty have neither a greeting piece nor a generator of random numbers with them. Nevertheless, they follow the laws of game theory. Holmes leaves the train to Canterbury and watches Moriarty to go to Dover happily in his unique car, ignoring that Holmes escaped him.
The fact that Doyle opted for this version is more remarkable if we consider that the theory of games did not yet exist, and he could not know that it was an optimal solution. It may have been a coincidence-where he may have had a good instinct. Anyway, I remember taking another look at his writing very soon.
This article originally appeared in Spektrum der Wissenschaft and was reproduced with permission.
