#dvd commentary

So!

Holmes and Watson, trying to escape Moriarty, get on a train going to Dover. Soon after, Moriarty commissions his own train to Dover, which will eventually catch Holmes' train. At the moment, though, Moriarty is well behind and can't see Holmes' train, so Holmes can try to pull a fast one if he wants.

Holmes has two options: continue to Dover or get off in Canterbury, the one and only intermediate stop. Once he chooses, though, he's committed, he can't change his mind.

Moriarty also has the same two options. If he chooses the same thing as Holmes, he can catch Holmes and win. If he misses his guess, Holmes escapes and Moriarty loses.

In this situation, what should each of them do?

Doyle/Holmes argued that since Moriarty is going to Dover, Holmes should get off in Canterbury. THIS IS A WRONG ANSWER.

Because see, Moriarty knows that Holmes could get off in Canterbury! Therefore, he should stop in Canterbury, where Holmes is. THIS IS ALSO A WRONG ANSWER.

Because see, Holmes knows that Moriarty knows that Holmes could get off in Canterbury! So he should get off in Dover, where Moriarty won't be. THIS IS ALSO A WRONG ANSWER.

Because see, Moriarty knows that Holmes knows that Moriarty knows that…

…and you're seeing The Princess Bride connection, right? This argument can go on to infinity, and you get nothing but wrong answers, because THEY BOTH KNOW EVERYTHING THE OTHER KNOWS AND NEITHER OF THEM CAN FORCE A WIN.

Now, at this point a certain theorem kicks in and a game theorist applies a bunch of arithmetic to it all, calculating the odds of each player catching the other at each station and the values to each player of winning and losing in each scenario and you don't need to know all that. What you need to know is. That the correct answer is.

Holmes should flip a coin.

Because you can't fucking outguess a coin.

And Moriarty, mathematically, should do the same thing.

And sure, he still has a fifty-fifty chance of losing. But mathematically, there's no way to do better than fifty-fifty on this game. That's the best odds, the best pay-off, that either of them can get.

(Technically, this is called a "mixed solution", which is math-speak for "flip the coin, roll the dice, they can't out-guess you if you're leaving it up to chance.")

THAT'S MATH BABY

So this is something that the math side of Holmes fandom has known more-or-less ever since game theory was invented. (Fun fact: Holmes fandom is older than game theory! We probably shouldn't laugh at Doyle about getting it wrong -- after all, the solution hadn't been mathematically proven yet -- but I will confess: we totally do.)

Once upon a time I considered writing Final Problem fic about this, before giving it up as a bunch of uncompelling didactic wankery. Time passed. I moved on. I changed fandoms. I wrote other stories…

…and no shit, there I was, long time mathematician and Holmes fan, writing Hornblower and Bush fleeing Caillard across the French countryside, trying to pick a destination port -- and I said, Fuck me. This is isomorphic to The Final Problem.

(Isomorphic is a fun math word for "the same as".)

(Mathematicians, don't at me for that lay definition.)

At this point I knew one true thing in the whole entire world: I WAS NOT GOING TO FUCK UP THE MATH LIKE ARTHUR CONAN DOYLE DID.

Now, as many on this website know, I am also a William Bush fangirl. He often gets the short end of the stick in canon, Hornblower going on about how Bush is not very smart, he can't play whist, he can barely work navigation equations, he's not a mathematician like our super special smart boy Hornblower here.

So I thought it would be good clean fun to let Hornblower fall into the Vizzini/Holmes trap of believing this is a game of wits. Of believing it's possible to out-think your opponent in a game like this. Of being trapped on the "I know that you know that I know that you know" merry-go-round.

And in the process, letting Bush be the one to say: So let's not play whist. let's play dice. And letting him be RIGHT.

Because remember, according to the math, the perfect play is to flip a fucking coin.

(While I'm on the topic, I maintain that Bush being right is in character here. We know he doesn't play whist; it makes sense that dice would be more his game. And canonically, Hornblower's math is not impeccable: he believes in the Gambler's Fallacy. In the TV show, Bush even says to him that the Gambler's Fallacy doesn't make sense! It gets brushed off in a Bush-is-stupid-Hornblower-is-smart kind of moment, BUT BUSH IS RIGHT. THAT'S BAD MATH MR HORNBLOWER, AND YOU SHOULD FEEL BAD.)

(Note: most people should not feel bad about messing up the math! Mr Hornblower, who prides himself on his math and thinks belittling thoughts about how other people are not as smart as he is, SHOULD FEEL BAD.)

Anyway. Back to your ask! This passage is my earnest attempt to hand Bush a moment of glory by explaining this nugget of game theory in an in-character, non-technical, and period-appropriate way. I've always been self-conscious about this passage, because I'm never quite sure that I pulled it off? Whenever I look at it, it looks like this wart of a math lecture sitting there in the middle of an otherwise decent narrative. But damn it, I tried. Oh, how I tried.

At least I can sleep well at night, knowing that when I was handed The Final Problem, I didn't let my hero fuck up the math. ;-)

Thank you for the ask! Do let me know if you have any follow-up questions!