In the story The Final Problem (FINA) Holmes and Watson attempted to flee England to get away from Professor Moriarty, who was pursuing Holmes for the purpose of killing him before Holmes’ investigation could bring down the Professor and his criminal empire. Holmes had a crucial decision to make about what to do to elude their pursuer. Here we will examine the thought processes and logic that were behind the decision.
(Credits: I claim no original thought for this one. Years ago while a student at the University of Wisconsin, I heard a professor lecturing on statistics and probability present his take on the familiar story. I have forgotten his name, but will give him full credit for the example of Holmes and Moriarty engaging in what is sometimes called “Gaming Theory” to determine the outcome of that chase by rail from London to Dover. At the time, I had read and knew The Canon fairly well, but was unaware of the larger umbrella of organizations’ analyses of the stories, and the extensive membership and scholarship of societies devoted to the subject. It is obvious now that my professor was a true Sherlockian.)
And now, my long-suffering readers, don’t be afraid as we digress into a relatively short introduction to that branch of mathematics mentioned. It is not a tiresome subject, but merely a formal expression of thought processes used by most everyone in day-to-day living and planning. To set the stage, let us imagine how to make monetary bets on a simple proposition.
Let us say that I am a gambling casino, and you are a prospective bettor. I, the “house”, have a game that is so simple and straightforward it would never be popular in a real casino. The game is called “The Coin Flip”. You flip a coin and you make your bet on the outcome. We both know there are two and only two possible outcomes of the coin flip, Heads or Tails, and each is equally likely to occur. Both you and the house put a dollar on the outcome. If you guess correctly you keep your dollar and the house pays you a dollar. If you guess wrong, the house keeps your dollar and you get nothing. This is a perfectly fair bet; if you guess right you win, and if you guess wrong you lose. Speaking in in the jargon of mathematics, your expectation is 1.00, that is to say, in the long run, you will win as much as you will lose, and neither the house nor the bettor has an edge. Even money, as they say.
In real life, a casino would not do this. They need their “edge” to stay in business and make a profit. If “The Coin Flip” was a real casino game, the house would keep your dollar if you lost, and pay you a smaller amount, say 90 cents, if you won. In that case your expectation would be 0.90. That is not a “good bet”, viewed as a practical matter, or viewed by a person who is mathematically astute.
All bets in real casinos are like this. The bets are more complicated, and the games are structured to be more exciting, but the reality is just as simple. You may beat the odds if you are lucky, but in the long run, the house always retains its edge. In the casino, all the bets give you an expectation lower than 1.00, some much lower than others. There are no bets with an expectation greater than 1.00. For every lucky bettor, there will be one or more unlucky bettors, and that is how casinos pay the bills and manage to make a profit.
Now let us imagine in our example, that I, as the house, decided to run our Coin Flip game with a different payoff rule. Let’s say the house goes a little nuts and says they keep your dollar if you lose, but if you guess right, the payoff will not be $1.00, but instead $1.20. Now that is a good bet! If you find this game in the casino, play it, play it, and play it again. You can’t lose! The longer you play, the more money you will make.
Or will you? This is where gaming theory comes in. In real life there is more to it than your simple statistical edge, and your expectation greater than 1.00. There is the consequence of defeat to consider. In our example, you won’t win every flip; you will guess wrong about half the time. Now let’s say the house establishes a minimum bet rule. The minimum bet becomes $10.00, with the payoff being $12.00. Same expectation as before (1.20), but now you can make money even faster! Still a very good proposition.
But what if the minimum bet is $100,000 and the payoff $120,000? Mathematically it is exactly the same, and still an attractive bet, but what if you lost? What if you called it wrong three flips in a row? Certainly possible, and for many people an intolerable result. It would ruin them. They would have to sell their house, rent a dump in a downscale neighborhood, drive an old junker, eat less meat, drink rotgut wine, and not be able to send their kids to college. For less prosperous people, the consequences would be even worse. For the wealthier, the loss might be tolerable and the game still lucrative, but for even the extremely wealthy there would be some minimum bet that could result in the threat of an intolerable loss and keep them from playing the game.
This example demonstrates there is more to it than simple probability and statistical expectations. The consequences of defeat have to be considered. In gambling, this gives an “edge” that transcends statistics and probability and provides an advantage to the player (or casino) with the biggest bankroll. Think of bluffing in a poker game, which is the obvious manifestation of the principle. Or the “gamble” of starting a new business. Your chance of success is far greater if you can ride out a few losses, and that takes capital. In gambling, application of Gaming Theory can affect the behavior of the bettor as much is the probability of winning or losing, which is an entirely different question.
Now let’s take a look at Holmes and Moriarty in The Final Problem. Professor Moriarty was a brilliant mathematician. Holmes was a very smart fellow, whether he was trained in formal mathematics or not, a bold but purely logical thinker. Holmes’ and Moriarty’s academic or instinctive knowledge of Gaming Theory affected the way the story within The Final Problem turned out. The train chase was crucial. Holmes and Watson were in London, and Holmes’ lengthy investigation was about to ruin Moriarty and bring down his criminal empire. A few more days were needed to complete the job, and Moriarty knew it. He also knew if Holmes were to be eliminated, the threat would go away. Holmes knew this too, and he knew that Moriarty knew it.
Holmes elected to flee to the European continent while the London police did their work to wrap up the case he had provided for them. Holmes and Watson took a train and headed to Dover, where they could board a steamer to Calais and leave Moriarty behind and vulnerable. Holmes knew the train schedule, as did Moriarty. There was an intermediate stop in Canterbury. By this time, Moriarty and his confederates were close upon the trail, and engaged a “Special” to chase Holmes. Holmes and Watson got off the train in Canterbury, and watched from the platform as Moriarty’s special pursuit train sped through hoping to catch Holmes in Dover and kill him there.
Now we need to examine the three possible outcomes of the pursuit, and the consequences of each:
Both Holmes and Moriarty get off at the same station, either Canterbury or Dover. The result is a high likelihood that Moriarty and his men will catch Holmes and kill him. A big win for Moriarty. An intolerable loss for Holmes.
Holmes stays on the train to Dover, and then goes by boat to Calais. Moriarty gets off at Canterbury to hunt Holmes down there, and forever loses his chance to catch him. Moriarty’s empire is doomed, he goes to prison or is hung, and Holmes is safe to oversee the downfall. An intolerable loss for Moriarty.
Moriarty stays on the train to Dover, and Holmes gets off at Canterbury. This is what happened in the story. Holmes does not get to the continent right away, and Moriarty buys some time to track him down. Kind of a “draw” for both.
Here is an excerpt, beginning with Holmes telling Watson his plan in London, as the train is leaving the station:
It is possible Holmes was wrong on this analysis, but he was probably just pulling Watson’s leg a bit, which he was known to do from time to time. We at McMurdo’s Camp believe both Holmes and Moriarty knew exactly what they had to do, and that the logic explained above relating to Gaming Theory governed their plans. Both Holmes’ and Moriarty’s primary strategy had to be to avoid the intolerable loss. Neither could afford any other course of action. They both knew and understood this prior to leaving London. Holmes even told us he did, and Moriarty, being the more refined purely mathematical reasoner, had it all figured out as well. The train chase was necessary, but its outcome was predetermined to be the “null solution” before either train departed from London.
A little lesson in applied mathematics, played for big stakes!
If you would like to read the story, click on the Sherlockian Link “Doyle’s Works” and go to Memoirs of Sherlock Holmes. Or for a synopsis, click FINA in “Story Info Sheets”.