1. Three players (Cornelius, Oscar, and Reid) enter a game. Cornelius uses the conservative approach to decision making, Oscar uses the optimistic approach, and Reid uses the minimax regret approach. The game is played as follows: each player chooses heads or tails, and the payoffs for each player are a function of the collective decisions of all three players. There are several possible outcomes:

(1) HHH, payoffs 2, -1, -1

(2) TTT, payoffs -4, 3, 1

(3) HHT, payoffs -2, 2, 0

(4) TTH, payoffs -2, 0, 2

(5) HTH, payoffs -1, -3, 4

(6) THT, payoffs 0, 2, -2

(7) THH, payoffs 3, -1, -2

(8) HTT, payoffs 4, -2, -2

In these outcomes the players are ordered Cornelius, Oscar, Reid, and the payoffs are given in the same order. So, for example, case (3) means that Cornelius and Oscar play heads, and Reid plays tails. The payoff for Cornelius is -2, the payoff for Oscar is 2, and the payoff for Reid is 0. Another example: in case (7) Cornelius plays tails, while the other two play heads. Cornelius receives a payoff of 3, Oscar a payoff of -1, and Reid's payoff is -2.

(a) What strategy will each of the players use?

(b) Suppose that Cornelius and Oscar both play heads and tails with equal probability. Then the probability of them both playing heads is 0.25, the probability that they both play tails is 0.25, the probability that Cornelius plays heads and Oscar plays tails is 0.25, and the probability that Cornelius plays tails and Oscar plays heads is 0.25.

If Reid states that he would accept a gamble of winning 2 with probability 0.5 and losing 1 with a probability of 0.5, and if Reid is a utility maximizer with a utility function u(x) = 1 - e^-λx, then what are the expected utilities of playing heads and playing tails?

2. Ms. French, a bank robber, robs banks by entering the bank at night when the bank is empty and then entering the bank vault. She is considering robbing three different banks. She will only rob bank 2 if she successfully robs bank 1, and she will only rob bank 3 if she successfully robs bank 2. She can, however, choose not to rob a bank even if she was successful in robbing the preceding bank. If the banks are successfully robbed, she expects to make $10 thousand, $50 thousand, and $75 thousand from banks 1, 2, and 3, respectively.

To successfully rob a bank, two things must occur. First, Ms. French must enter the bank without setting off the outside alarms and then, she must enter the vault without setting off the vault alarms. The probability of entering the first bank without setting off the outside alarms is 0.6, and the probability of entering the vault without setting off the vault alarms is 0.25. For bank 2 those probabilities are 0.4 and 0.2, and for bank 3 the probabilities are 0.35 and 0.2. If Ms. French fails in entering the bank, she is capable of escaping before the police arrive, thus she will not lose anything in such a scenario. If she sets off a vault alarm she will be arrested and incur a penalty of $30,000.

Ms. French can hire a sidekick before any of the robberies, but once the sidekick is hired, he will continue to be used for any and all subsequent robberies. The cost of the sidekick is $2500 for bank 1, $12,500 for bank 2, and $18,750 for bank 3. This amount must be paid whether or not the robbery is successful.

If the sidekick is hired, the probability of gaining entry to the vaults at banks 1, 2, and 3 improves to 0.3, 0.25, and 0.25, respectively. The effect of the sidekick on entering the bank is unclear, but we do know that the conditional probability of having a sidekick given that bank 1 was successfully entered is 0.6. For bank 2 this probability was 0.75, and for bank 3 this probability was 0.8.

Develop a decision tree for this problem and determine Ms. French's optimal strategy (which banks to rob and when, if ever, to hire a sidekick) if she wishes to maximize her expected reward.

3. A Wall Street firm is planning its strategy for next week (Monday through Friday). In particular, they are interested in day trading a particular stock. The firm plans to observe the price of the stock at 15 minute intervals each day, starting at 9:30am and ending at 4:00pm. The fluctuations in price in each of these intervals is assumed to be a normally distributed variable with a mean of 0.01 and a standard deviation of 0.25. Suppose that we have $10,000 to use for trading. The opening price each day is equal to the closing price of the previous day, and the stock began the week with a price of $100. We are interested in two possible trading schemes:

(a) Suppose the stock's price to begin the day is t. We buy the stock the first time that we observe a price less than or equal to 99.5% of the day's beginning price. In other words, if p is the price at which we buy the stock, it must be that p ≤ 0.995t. We sell our entire position when either (i) the price of the stock increases by at least 1% from where we bought it, i.e., if the price is 1.01p or more or (ii) when we reach the end of the day, whichever comes first. We never will buy twice in the same day.

(b) The same as (a), except that we buy whenever p ≥ 1.005t instead of when p ≤ 0.995t. Simulate 10 weeks of trading under each scenario and report on the average weekly profits/losses.

Note: Whenever we buy, we use all of our money. So, for example, if on a particular day we have $9000, and we buy when the stock reaches $90, we would buy $9000/$90 = 100 shares.

4. Airline Inc. overbooks its flights to account for the possibility of passengers not showing up for the flights. On its 200 seat airplanes it will take reservations for up to 210 passengers. Past data suggests that the number of reservations made for each flight is a uniformly distributed random variables taking on values between 190 and 210. The number of no-shows for each flight ranges from 0 to 10, with probabilities given in the below table. Develop a simulation of this situation and run it 500 times.

Determine the average number of seats used per flight and the probabilities that 201, 202, 203,...,210 passengers show up for a flight (in other words, the probabilities that we have to turn away 1,2,3,..,10 passengers).

5. A merchant has a contractual obligation to sell two diamonds over the next three months. He currently does not own any diamonds, but can buy them from his partners at any time. The price he must pay fluctuates from month to month as shown below. Once the merchant acquires a diamond he can sell it in the current month or save it to sell in a later month. The sales prices for each of the three months are also given below. We can buy or sell two diamonds in the same month but, because of insurance stipulations, only one diamond can be held at the end of a given month. Use dynamic programming to determine the merchant's optimal plan for acquiring and selling two diamonds.

Hint: let x_n = (# diamonds on hand at the beginning of stage n, cumulative # sold up to the beginning of stage n).

6. A politician is trying to make the most out of her $500 thousand promotional budget over the next four weeks. Promotions may be purchased in three states: Nevada, Wisconsin, and New Jersey. Each dollar spent in a state creates a certain promotional impact which is computed by multiplying the amount spent on promotion by the weight of the state in the week in which the promotion is made. The following table shows the weeks in which each state can be promoted and the weight of the state for the week.

Additionally, we have the following constraints: In weeks 1 and 2 we must spend at least as much in NV as we spend in WI and we must spend at least as much in WI as we spend in NJ. In week 3 we must spend at least as much in WI as we spend in NJ. Finally, we cannot spend more than $200 thousand in a single state during a single week. Use dynamic programming to find the optimal allocation the politician's funds.

So, for example, if $100 is spent in WI in week 2 and $200 is spent in NJ in week 3, the total promotional impact is 20*100 + 40*200 = 10,000. Additionally, we have the following constraints: In weeks 1 and 2 we must spend at least as much in NV as we spend in WI and we must spend at least as much in WI as we spend in NJ. In week 3 we must spend at least as much in WI as we spend in NJ. Finally, we cannot spend more than $200 thousand in a single state during a single week.