2. Zolo is a pirate who's trying to avoid the marine officer Toshigi in Logtown. Zolo has to meet his captain, whereas Toshigi has to meet her superiors. There are two strategic places they can meet at: the main place and the docks. If Zolo goes to the docks and Toshigi doesn't, he gets a payoff of 12 and Toshigi a payoff of 4. If Toshigi happens to be there as well, he gets a payoff of 0 and Toshigi a payoff of 16. When Zolo sees Toshigi at the main place, he gets a payoff of 4 and Toshigi gets a payoff of 12. Finally, Zolo gets a payoff of 16 if he is at the main place and Toshigi is at the docks. In that case, Toshigi gets a payoff of 8.

(a) Write down the payoff matrix, putting Zolo is row and Toshigi in columns. Find the Nash equilibria in pure and mixed strategies of this simultaneous moves game, if any. (2 points)

(b) Assume this game is played once everyday for a week. What are the Subgame Perfect Nash Equilibria? (1 point)

(c) Assume now the game is played once, but sequentially: Zolo chooses where to go first, and Toshigi gets to learn about it from spies located throughout the town. Draw the corresponding game tree (in the payoffs, show Zolo's payoff first) and find all the Subgame Perfect Nash Equilibria. Show your work and make sure you express the SPNE correctly. (3 points)

(d) Imagine now that Toshigi goes first, and Zolo learns about it before making his move. However, spies in town can inform Toshigi of that too so she can decide to stay where she is at or switch locations. Draw the new game tree (the last moves Toshigi can make are "Stay" or "Switch". In the payoffs, show Zolo's payoff first). Find all the Subgame Perfect Nash Equilibria. Make sure you express the SPNE correctly (think about each player's strategy!). (5 points)