Game Theory In Bbc's Sherlock

BBC’s Sherlock is a modern take on Sir Arthur Conan Doyle’s famous sleuth. It is one of the most well plotted shows on television today, and implicitly imbibes various strategic concepts. In this project, we have used our knowledge of game theory to analyze situations from the show. Some of the concepts we have tackled include dominant strategies, sequential games, repeated games, Mexican standoff etc. Our aim has been to derive an equilibrium using game theory and then compare it to the actual outcome on the show. The games have been selected from the entire timeline of the show, and are presented as follows, in chronological order.
A Study in Pink
The first episode in the BBC series Sherlock sees the British sleuth team up with his new flat mate and freshly returned Afghan veteran, Dr. Watson. Together, they are trying to crack a case involving a series of

They are waiting for the British forces led by Mycroft to arrive. At this point in time there is a short game played between Magnussen and Sherlock.
Magnussen has two strategies: either to reveal the information about Mary to her enemies from the past (R) or not (DR). While Sherlock has two strategies: either to shoot Magnussen (S) or not (DS).
Before deciding the pay-offs it is important to take note of the utility functions of each player. Magnussen seems to have the upper hand since all the information is stored in his mind and there’s nothing Mycroft or Sherlock have to destroy. Thus his utility is derived from the pleasure he receives by playing with Sherlock and John. He says: “I would like to punch your face John”; after flicking John’s face he says, “I could do this all day”. He also mentions that he is a businessman and he takes advantage of people by finding their pressure points and thus exploiting them.
We have the following ordinal rankings for Magnussen (Player 1) is:
U(R, DS) > U(DR, DS) > U(R, S) = U(DR,