Consider the following scenario:

There are two pirates operating among three islands A, B, and C. On each island, two treasures are buried: a large one worth 2 and another smaller one worth 1. The prevailing winds in the area are such that from island A you can only reach island B, from island B only island C, and from island C only island A. Once on an island, the pirates only have time to excavate one treasure, before heading to the next island. The pirates are not on good terms, and if they are at the same island at the same time they will fight, and as a result, they will find no treasures (but will not suffer any additional damage). If they do not meet, both pirates will visit all three islands and each pirate will find three treasures. Each of the pirates has to decide where to start their treasure hunt.

a. Model this setting as a normal-form game.

b. Compute a mixed Nash equilibrium of this game and argue that it is unique.

c. Compute players’ expected utilities in this equilibrium.