Consider the following scenario. Three people A, B, C each decide simultaneously on a cafe to study at. There are two choices of cafes, Cafe X and Cafe Y. The problem with this situation is that none of these people mutually like each other's company. A dislikes B, B dislikes C, and C dislikes A. On the other hand, A likes C, B likes A, and C likes B. After deciding on a cafe, each individual i obtains a utility consistent with these preferences. Everyone likes Cafe X better than Cafe Y, so choosing Cafe X yields each player an additional 1 unit of utility, while going to Cafe Y yields zero additional utility. Each individual i also obtains an additional 2 units of utility if the person that they like decides to study at the same cafe and obtains an additional k units of (dis)utility if the person they dislike chose to study at the same cafe. Being alone in a cafe yields no additional utility. For example, if A, B, C all decided to study at Cafe X, then A obtains 1 + 2 k = 3 k utils.

Part a: What are all of the rationalizable strategies of each player of this game if k = 4? What are all of the rationalizable strategies of each player of this game if k = 2?

Part b: If k = 4, what are all of the (pure strategy) Nash equilibria of this game? If k = 2, what are all of the (pure strategy) Nash equilibria of this game?