The famous detective Sherlock Holmes is energetically trying to catch the super villain Professor Moriarty. Professor Moriarty has to meet another villain, Dr Corona, at either the busy and crowded Piccadilly Circus or at the tranquil and beautiful Kew gardens. Sherlock has of course deduced this and is now thinking about where to go. Moriarty is also thinking about where to meet Dr Corona. If Sherlock and Moriarty end up in the same location, Sherlock will arrest Moriarty (and also catch Corona) which yields a payoff of 8 to Sherlock and -6 to Moriarty. If they end up in different locations Sherlock gets a payoff of 0 and Moriarty a payoff of 5. Since Sherlock loves the beauty of Kew gardens, he will earn an additional payoff of 2 if he ends up there. The payoffs are illustrated in the game matrix below.

a) Assume that Sherlock and Moriarty decide simultaneously (without knowing their opponent's choice) where to go. Find the pure-strategy Nash equilibria (if any). Explain your result intuitively

see the attached file

b) Now, let Sherlock and Moriarty use mixed strategies. Sherlock goes to Piccadilly with probability p and Moriarty goes to Piccadilly with probability q. Derive the best- response functions/correspondences of the players and illustrate them in a diagram.

c) Find the Nash Equilibrium in mixed strategies and illustrate it in the diagram under b).

d) Now let us assume that Moriarty's payoff in the case where he is caught in Kew gardens increases by 2 (no other payoff changes). Will Moriarty's Nash equilibrium choice of q change? If so, by how much? Explain your answer carefully

Question 2 The famous detective Sherlock Holmes is energetically trying to catch the super villain Professor Moriarty. Professor Moriarty has to meet another villain, Dr Corona, at either the busy and crowded Piccadilly Circus or at the tranquil and beautiful Kew gardens. Sherlock has of course deduced this and is now thinking about where to go. Moriarty is also thinking about where to meet Dr Corona. If Sherlock and Moriarty end up in the same location, Sherlock will arrest Moriarty (and also catch Corona) which yields a payoff of 8 to Sherlock and -6 to Moriarty. If they end up in different locations Sherlock gets a payoff of 0 and Moriarty a payoff of 5. Since Sherlock loves the beauty of Kew gardens, he will earn an additional payoff of 2 if he ends up there. The payoffs are illustrated in the game matrix below. a) Assume that Sherlock and Moriarty decide simultaneously (without knowing their opponent's choice) where to go. Find the pure-strategy Nash equilibria (if any). Explain your result intuitively. (7 points) Moriarty Piccadilly Kew gardens Sherlock Piccadilly 8, -6 0, 5 Kew gardens 2, 5 10, -6 b) Now, let Sherlock and Moriarty use mixed strategies. Sherlock goes to Piccadilly with probability p and Moriarty goes to Piccadilly with probability q. Derive the best- response functions/correspondences of the players and illustrate them in a diagram. (8 points) c) Find the Nash Equilibrium in mixed strategies and illustrate it in the diagram under b). (5 points) d) Now let us assume that Moriarty's payoff in the case where he is caught in Kew gardens increases by 2 (no other payoff changes). Will Moriarty's Nash equilibrium choice of q change? If so, by how much? Explain your answer carefully. (5 points)