Say that Alicia and Badal are siblings home for the holidays and they are presented with a chocolate bar which has five sections, like this: 12345. They decide to make a game out of eating it. They take turns eating one (and only one) section at a time, and in each turn, each sibling can eat the left-most section or the right-most section. Alicia goes first, and can choose either to eat section 1 or section 5. Say Alicia chooses to eat section 5, and so the chocolate bar now looks like 1234. Then Badal eats section 1, and so the chocolate bar looks like 234. Then Alicia eats section 4, and the chocolate bar looks like 23, and so forth. What makes the game fun is that section 4 is "special" because it is the only section with peanuts in it, which both Alicia and Badal like. Whoever eats the "special" section wins the game, and the game immediately ends.

a. Model this as an extensive form game and find a subgame perfect Nash equilibrium (you don't have to write down all subgame perfect Nash equilibria, just one of them).

b. Now say that the chocolate bar has six sections, and looks like 123456. Section 4 is still the special section. Which section should Alicia eat on Alicia's first move? Who wins the game? You can figure this out by writing out the full extensive form game, like in part a. above, or you can just think about it and explain your answer.

c. Now say that the chocolate bar has nine sections, and looks like 123456789. The special section is 7. Who wins the game? Please don't write out the full extensive form game, which is large. Just think about it and explain your answer.

d. Now say that the chocolate bar has n sections and the special section is m. Who wins the game (how does this depend on m and n)? This question seems pretty complicated at first, but once you start playing around with it (trying out lots of different examples is a good way to start) it's not so bad. Please explain your answer.