4. [commitment and menus] Consider a DM who, at time 1, chooses a menu of alternatives and, at time 2, chooses an item to consume from that menu. Suppose first that the DM is a standard rational agent. Let the set of all alternatives be X = {a,b,c,d,...,x,y,z}. Consider menus M1 = {w,x,y} and M2 = {w,x,y,z}. The goal here is to understand her preference over menus that derive from her underlying preferences over X.

a. Characterize, in terms of the underlying preferences over consumption items, when the DM prefers M1 over M2, M2 over M1, or is indifferent between M1 and M2. That is, for each possible preference relation over X, how does she feel about the two menus? [Hint: For example, is w is her favorite item in X, then she would choose w from M1 and w from M2 and would get the same outcome in either case, so you would be indifferent between M1 and M2. But if z is her favorite item, then she would choose z from M2 and something she likes less from M1 and in this case she would prefer M2 over M1.]

b. Generalize the above example. Suppose M3 is an arbitrary menu and that M4 is a subset of M3. What preference between M3 and M4 is impossible for the standard DM?

c. Suppose now that z represents a cheeseburger (the least healthy option of all), and that the DM would like to eat healthy food, but is concerned that if a cheeseburger is on the menu then she may be tempted to eat it anyway. If this temptation concern is strong, what will her menu preference between M1 and M2 be at the first date when she chooses the menu?

d. Provide a one sentence description of the difference between the temptation-prone DM of part (c) with the standard agent of parts (a-b).