BBC’S SHERLOCK — GAME THEORY
The Reichenbach Fall Finale
Analyzing one of the best episodes of television history ever written
SPOILER ALERT: This article contains spoilers for BBC’s Sherlock.
BBC’s adaptation of Sir Arthur Conan Doyle’s fan-favorite detective Sherlock Holmes is a true masterpiece. Sherlock, a consulting detective, solves crimes and mysteries with his devoted friend, Dr. John Watson, through deductions and meticulous observations to solve murders and mysteries.
While Season 1 had people hooked, Season 2 delivered beyond the expectations. The story writing and cinematography provokes such breathless speculations over a story of a fictional case, that is the genius of this show. The suspenseful yet enthralling soundtrack (Blood On The Pavement) and the captivating background songs (Sinnerman) engross the viewers. The show had just the right mix of thrill, suspense, mystery and most importantly climax.
Andrew Scott, who plays Moriarty, is a distinct 21st-century villain, sort of a twitchy, (more) flamboyant yet intellectual. His performance as Moriarty is nothing less than memorable. The characters of Sherlock and Moriarty exude class, the spellbinding conversations and the twists make them a joy to watch. In a way, Moriarty is the perfect villain for Sherlock, one who would leave clues by himself to keep the game afoot.
In the season finale, Moriarty commits a series of impossible crimes, stealing crown jewels, opening vaults and unlocking cells of Prison, but he allows himself to be arrested giving viewers this iconic photo:
“James Moriarty is for hire. A tradesman? Yes. But not the sort who’d fix your heating? No, the sort who’d plant a bomb or stage an assassination. But I’m sure he’d make a decent job of your boiler. “ — SH to Prosecutor
He demands Sherlock at his trial, only to threaten the jury and be acquitted.
Moriarty very cleverly gives Sherlock a visit afterwards and implies that it was Sherlock who had the anti-security program which helped him pulling off the heists, to the criminal underworld, thus endangering Sherlock’s life.
Under the surveillance of criminals, Sherlock, who had earned love of the ‘media’, continued to solve cases. While investigating kidnapping of the Ambassador’s children, with a skillful scheme by Moriarty, Sherlock is suspected of being a fraud (yes, he’s suspected to be too good after all) and staging all his cases himself, as no one is that skilled to crack every case, even the ones that Scotland Yard’s entire squad is unable to solve. Sherlock gets arrested, but escapes with the help of who else but John.
While on the run, Sherlock breaks into the house of an investigative journalist, Kitty Riley who approached him for a scoop before the trial, to lo and behold meet Moriarty who pretends to be Richard Brooks, an actor (with proof) who was supposedly paid by Sherlock to become the remarkable criminal mastermind Moriarty. In terms of Game Theory, this move by Moriarty is called an irrational move since it could not have been predicted by Sherlock with rational thinking. This forces Sherlock to make one last move, leading up to the final scene on the infamous roof of the hospital.
The Final Showdown:
Sherlock calls Moriarty, on the rooftop of the St. Bartholomew’s Hospital, to finally put an end to this ruse or ‘twisted’ game. Moriarty, with a lot of emotions, displays sorrow about the current state of affairs and his supposed victory. To complete the game, Sherlock has to commit suicide, but the anti-security program presents a solution, erasing Richard Brooks and revealing Moriarty’s truth saving his repute as well as dignity. Alas, Sherlock finds out that the heist program was just a hoax and not some clever hack, thus overthinking this one. Moriarty, then threatens to kill John, Mrs. Hudson and Lestrade, three of the most important people in Sherlock’s life, to force Sherlock to commit suicide.
Moriarty makes a strategic move by attacking Sherlock’s weak point — his friends — hoping to gain the upper hand. Everybody thought of Sherlock as a rational thinker, a scientist, a man who uses remarkable powers of inference and who is bereft of emotions and care. However, this move quells Sherlock’s supposed indifference to his friends, and he has to give up his life and career for the few people he loved.
“You want me to shake hands with you in hell, I shall not disappoint you.” — SH
For analyzing this move, we consider a situation known as the Prisoner’s Dilemma. In the cases of Prisoner’s Dilemma, we are presented with a similar scenario, we can map out the choices for Sherlock and Moriarty till this point to construct a payoff matrix as follows:
The matrix is a diagrammatic representation of the rooftop situation. Each player loses a point (-1) if he commits suicide or gets killed. However, if a player manages to survive, only Moriarty gets awarded (+1) whereas Sherlock’s score still remains at zero since he has to lose his friends at the cost of his existence. In the Prisoner’s Dilemma, a strictly dominant strategy is for both players to ‘confess’, regardless of the decision of the other player. But, if Moriarty chooses to ‘commit suicide’, then Sherlock can choose to either ‘commit suicide’ or ‘stay alive’ but it would always be a pre-defined choice calculated by Moriarty’s move; hence making it an example of *mixed strategy Nash Equilibrium (confused? read further to find out) unlike Prisoner’s Dilemma with a pure strategy Nash Equilibrium. In the optimal condition when both the players stay alive Sherlock can benefit from the situation.
Nash Equilibrium and Strategies:
A mixed strategy Nash equilibrium is one where one of the players executes a randomized strategy due to which the other player is not able to improve his payoff even by implementing another strategy. On the other hand, a Nash equilibrium, where no player plays a randomized strategy is called a pure strategy Nash equilibrium.
With his intelligent play, Sherlock figures out that there must be a sign to recall the assassins, and thus compels Moriarty to counteract. With a shocking twist like no other, after an awkward conversation and a handshake, Moriarty blows off his temple.
With no plan to fall back on, Sherlock throws himself off the roof.
With Moriarty succumbing to death, Sherlock was left with two choices, either to commit suicide in order to save his friends and reputation or by staying alive himself; or stay alive despite his friends’ deaths and the widespread allegations claiming that he was a phoney. From Prisoner’s Dilemma, we know that the ideal case would have been for both players to confess, or stay alive in this case, but with Moriarty at death’s door, Sherlock had to act quickly. If we can call this a game, it was one that included mixed strategy equilibrium, using which Moriarty committed suicide (an irrational move), thus preventing Sherlock from manipulating him into calling off the snipers, presenting no pure strategy to win. In other words, neither of them had an incentive to change their moves to guarantee a better outcome. The optimal strategy would have been for both the players to ‘stay alive’, but the worst outcomes for each player transpired.
The roof sequence, which culminates in an intellectual and emotional rollercoaster, is indisputably acclaimed as a masterpiece of fiction. The bland deaths and intense conversations, the slow reveal of Moriarty’s plan, Sherlock’s despair, the agony of John over losing his best friend, and Sherlock saving his friends over his own life marks the ending of the season with both the players committing suicide (-1,-1).
Had our players applied Game Theory, they wouldn’t have had to deploy such extreme moves and could have maneuvered a better result.
Authors — Aditya Choubey and Samriddhi Kejriwal,
The Indian Game Theory Society — Delhi Technological University
References:
- The Reichenbach Fall. Fandom. Available at:
https://bakerstreet.fandom.com/wiki/The_Reichenbach_Fall - Investopedia Team. “Prisoner’s Dilemma.” Investopedia. Available at: https://www.investopedia.com/terms/p/prisoners-dilemma.asp
- Matching Pennies. Wikipedia. Available at:
https://en.wikipedia.org/wiki/Matching_pennies - Dixit, Avinash K., and Barry J. Nalebuff. The Art of Strategy. W.W. NORTON & CO., 2008.
