Assignment 1: Decision Theory
Q1. Monty Hall
You are on a game show where you select one of three doors, A, B, or C. Behind one door is a great prize of payoff $10,000. Behind the other two doors are goats, with payoff 0. Your prior belief is that the door with the prize has been chosen completely at random such that the initial probability of any door being the right one is ⅓. You get an opportunity to select, but not necessarily open, a door, A, B, or 3
C. The game show host will then select a door you have not chosen that contains a goat. (If there are two available doors with goats, assume he chooses one or the other with 50% probability. If there is only one available door with a goat, obviously he opens that one.) You then have a choice to remain at the door you initially chose (R), or move to the other available door (M).
a. Draw the complete game tree, with the two players being Nature, who initially selects the right door and then reveals a door with a goat, and You/Player 1/[insert your name], who selects a door and then decides whether to remain or move.
b. In words, explain why your initial choice of door (A, B, or C) doesn't matter.
c. Suppose you initially selected door A. As will always happen, the host reveals another door with a goat. Calculate the expected payoff of choosing R and the expected payoff of choosing M.
What should you do, or does it matter?

Q2. Value of Information
You have $100 to invest and are wondering where to put it. You are considering whether to buy your favorite stock (B) or put it in your couch (C), either decision occurring today. You know the stock price today, conveniently enough $100 per share, but you don't know the stock price tomorrow. For simplicity, it could either be double ($200) or nothing ($0), with the probability of double being p, and the probability of nothing being 1 - p. If you invest in your couch you get your $100 back tomorrow.
Your discount factor is 8, and you are risk neutral (i.e. your payoffs only depend on time discounting and money.)
a. Draw the complete game tree, with nature and player 1.
b. At what level of p should you buy your favorite stock?
c. Why doesn't 8 affect part (b)?
d. Martha Stewart knows what the stock price is going to be tomorrow, and offers to sell this information to you. You trust that she knows and is being honest because her written offer is finely hand-crafted using everyday household items. Assuming your only concern is making money and discounting for time (instead of moral qualms, or going to prison), how much would you be willing to pay for this information?

Assignment 1: Decision Theory Q1. Monty Hall You are on a game sh...