Solutions for HW #2

1.

(1) What is the probability that a chosen number will win?

There are 1000 numbers possible for any single draw, so we get a probability of 1/1000.

(2) What is the utility (in dollars) of a single play?

(1/1000)($500) - $1 = $0.50 - $1 = -$0.50

(3) Suppose that you decide to play three times in one day, and that you choose the same number each time. (You hold three tickets at a cost of $3.) What is the expected utility (in dollars) of your triple play?

In this case, if your number is drawn, you win 3 x ($500) = $1500. Thus, we get

(1/1000)($1500) - $3 = $1.50 - $3 = -$1.50

(4) Suppose that you decide to play three times in one day, and that you choose a different number each time. What is the expected utility (in dollars) of this triple play?

In this case, your winnings will be $500 if any number is drawn, but you now have a 3 in a 1000 chance. Thus, we get

(3/1000)($500) - $3 = $1.50 - $3 = -$1.50

2.

(1) What is the probability that no merger will occur?

P(no merger) = 1- P(merger) = 1 - .4 = .6

(2) What is the expected utility of investing in the money-market fund?

With the money-market fund, we are guaranteed a return of $1200, so p=1.

EV_MM = (1)($1200) - $1000 = $200.

(3) What is the expected utility of investing in the stock?

Here we are dealing with two possible states of the world, a merger or no merger. We can set up the following matrix:

	Merger	No Merger
Stock	P = .4 $1800 - $1000	P = .6 $900 - $1000

EV_Stock = (.4)($800) + (.6)(-$100) = $320 - $60 = $260.

(4) What decision rule should the investor use?

Here we have a decision under risk and should thus use the Expected Value Strategy.

(5) What is the rational decision in this case?

Using Expected Value Strategy, we should invest in the stock because it has a higher expected value than investing in the money-market.

3.

(1) Is the decision under risk, a decision under certainty, or a decision under uncertainty?

Since we have no values for the probability of rain or of no rain on the date in question, the decision is under uncertainty.

(2) Is it reasonable in this case to correlate units of utility with dollars gained for a worthy cause?

Since the problem states that the sole purpose of the dinner party is to raise money for a worthy cause, it is reasonable to correlate units of utility with the dollars gained (rather than, say, your appearance to your boss).

(3) What decision rule would you follow in this case?

If we rank value the possible outcomes based on the profit for each situation, we would get the following decision matrix:

	No Rain	Rain
Indoor Dinner Party	3^rd $170	2^nd $440
Outdoor Picnic	1^st $500	4^th $80

We don’t have a best option in this case, and from the wording of the problem it appears there is no satisfactory option (we want to raise as much as possible). We must now choose between applying the Play-it-Safe Strategy and the Gambler’s Strategy.

(4) Apply the decision rule, and state what decision you would make. (Show your work.)

Play-it-Safe dictates an Indoor Party (the least you could make here, $170 is greater than $80); the Gambler’s strategy says hope for the biggest number, $500, by having the outdoor picnic.

4.

(1) What type of decision are you faced with now?

We now have probabilities for our possible states of the world (i.e., rain or no rain), so we have a decision under risk.

(2) What decision rule should you follow?

We would apply the Expected Value Strategy.

(3) One the basis of that rule, what is the decision? (Show your work.)

With the probabilities for rain and no rain now given, we have the following matrix:

	No Rain	Rain
Indoor Dinner Party (IDP)	P=2/3 $170	P=1/3 $440
Outdoor Picnic (OP)	P=2/3 $500	P=1/3 $80

EV_IDP = (2/3)($170) + (1/3)($440) = 340/3 + 440/3 = 780/3 = $260.

EV_OP = (2/3)($500) + (1/3)($80) = 1000/3 + 80/3 = 1080/3 = $360.

Using the Expected Value Strategy, we would choose having the outdoor picnic, because it has the higher expected value.

5.

(1) What decision rule would you use in this situation?

Since we have probabilities given to us, this is a decision under uncertainty, and we would use the Expected Value Strategy.

(2) What would your decision be? (Show your work.)

We have a probability of 1 that using the old special (OS) will give us a profit of $100. Thus,

EV_OS = (1)($100) = $100.

We can set up the following decision matrix for the new special (NS):

	Success	No Success
New Special	P=.5 $250	P=.5 -$50

EV_NS = (.5)($250) + (.5)(-$50) = $100.

Here we have the same expected values for our possible choices. What other factors could be considered in making your decision?

6. For Contest 1 we have an expected value of (.001)($5000) = $5.

For Contest 2 we have an expected value of (0.10)($100) = $10.

It seems reasonable to use the Expected Value Strategy in this case.

7. We can set up the following decision matrix using the given probabilities, setting utilities equal to ticket cost, and considering the three possible states of the world:

	Go to exactly 3 operas	Go to exactly 4 operas	Go to exactly 5 operas
Buy Single Tickets	P=1/3 -$75	P=1/3 -$100	P=1/3 -$125
Buy a Season Ticket	P=1/3 -$100	P=1/3 -$100	P=1/3 -$100

EV_Single = (1/3)(-$75) + (1/3)(-$100) + (1/3)(-$125) = -75/3 –100/3 – 125/3

= -300/3 = -$100.

EV _Season = (1/3)(-$100) + (1/3)(-$100) + (1/3)(-$100) = -$100.

Since the expected values for buying single tickets and a season ticket are equal, there is no added utility in buying the season ticket. When we think about this problem more carefully, we see that the option of buying single tickets can be thought of in steps: buy a single ticket the first night if you're able to go, etc. Since there's no expected advantage in going for a season ticket, it makes sense to collect more information each night as to whether you can go or not.

8.

(1) What is the probability that you will get the flu if you are not vaccinated?

We assume that if there is no epidemic, you will not get the flu. So the probability you will get the flu depends (i) on how likely there is to be an epidemic and (ii) how likely you are to get the flu if there is an epidemic. So the probability asked for in the question is 0.6 (the prob of an epidemic) x 0.4 (the prob of getting flu if there is an epidemic) = 0.24

Since we’ll need it below, we can also calculate the probability of not getting the flu if one is not vaccinated. It is 0.6 (the prob of an epidemic) x (1 – 0.4) (the prob of not getting the flu if there is an epidemic) = 0.36

(2) What is the expected utility of not being vaccinated?

Before answering this question, we need to set up a decision matrix. There are three relevant mutually exclusive and exhaustive outcomes.

	Epidemic but don’t get flu	Epidemic but do get flu	No epidemic (hence, no flu)
Not vaccinated	P = 0.36 V = 0	P = 0.24 V = - 9	P = 0.4 V = 0

E_NV = (0.24)(- 9) = - 2.16

(3) What is the expected utility of being vaccinated?

It appears from the way the problem is stated that we are to assume that if you decide to get vaccinated there is zero chance of getting the flu. (This does not seem like a very realistic assumption but since we aren’t given any probability covering this case, we assume that the value is at least approximately zero.)

So all we have to worry about is the allergic reaction. So the expected value of this outcome does not depend on whether there is an epidemic or not.

	Allergic reaction	No allergic reaction
Vaccinated	P = 0.1 V = -2	P = 0.9 V = 0

(4)

(5)

E_V = - 0.2

(4) If you follow the rule of maximizing expected utilities, will you get the flu shot or not?

Getting vaccinated will give us the least negative expected value, so we should get the shot.

9.

(1) Is it rational for the person to choose the gamble rather than the gift?

The expected value of the gamble is (5/6) ($100,000) or about $83,333! This is a lot bigger than the sure $1000 so it is the choice dictated by rational decision theory

(2) Why or why not?

However, intuitively it seems like the 100% chance of saving the business is the best choice.

(3) What does this problem suggest about the practice of equating units of utility with units of money?

Maybe one could argue that the actual value of the $100,000 is not that much greater than the value of the $1000 because $1000 is all that the person desperately needs.

(4) What does this problem suggest to you about considering risks as well as possible gains when contemplating an action?

Although our flow chart doesn’t have a play-it-safe option in the case where probabilities are known, maybe it should have!

10.

It looks like we should include in our matrix a factor for how much less we like the $110 jacket in addition to the price! But we aren’t told how strong that sentiment is so let’s just use dollars as utilities and only invoke the style factor it there is a tie.

	Get Jacket You Like	Have to Buy Jacket You Don’t Like
Buy Now	P=1 -$100	P=0 $0
Wait 3 Days	P=.5 -$50	P=.5 -$110

EV_BuyNow = (1)(-$100) + (0)($0) = -$100.

EV_Wait = (.5)(-$50) + (.5)(-$110) = -$80.

Applying the Expected Value Strategy, we should wait 3 days to buy a jacket.

11. We can set up the following decision matrix using the given values and probabilities:

	Buyers for 50 Prints	Buyers for 60 Prints	Buyers for 70 Prints
Order 50 Prints	P=.9 $2500-$1000 = $1500	P=.07 $2500-$1000 = $1500	P=.03 $2500-$1000 = $1500
Order 100 Prints	P=.9 $2500-$1500 = $1000	P=.07 $3000-$1500 = $1500	P=.03 $3500-$1500 = $2000

EV₅₀ = (.9)($1500) + (.07)($1500) + (.03)($1500) = $1500

EV₁₀₀ = (.9)($1000) + (.07)($1500) + (.03)($2000) = 900 + 105 + 60 = $1065

Using the Expected Value Strategy, we should choose to only have 50 prints made.