Chapter 9. Surviving with Limited Information


18.

Let’s Make a Deal

You are on a game show and are offered a choice of three closed boxes. Two boxes are empty, but the third contains an expensive prize. You don’t know which box has the prize, but the host does. You are first told to choose one of the boxes. Next, the game show host will pick an empty box that you didn’t choose and open it, showing that it’s empty. Finally, the host gives you the option to change boxes. You can switch from the box you picked to the other unopened box. Is there any advantage in switching boxes?

answer: let s make a deal you should definitely change boxes. let s label the three boxes as a, b, and c and assume that you picked box a. obviously there is a 1/3 probability that the prize is in box a and a 2/3 probability that the prize is in either box b or c. either or both boxes b or c are empty. thus, after you pick box a, it will always be possible for the host to open either box b and c to reveal an empty box. consequently, the host s actions do not affect the probability of the prize being in box a. before the host opened one of the two other boxes, there was a 1/3 chance of the prize being in box a, and after he opened one of the other boxes, there is still a 1/3 chance of the prize being in box a. let s say that the host opened box b and showed you that it was empty. now, since there is a 1/3 chance that box a has the prize, and a zero chance that box b has the prize, there must be a 2/3 chance that box c has the prize. consequently, you should switch to box c because it will double your chance of winning.

19.

The 99 Percent Accurate AIDS Test

You have just taken an AIDS test. You know a lot about how AIDS is transmitted, and, thankfully, you are almost certain that you don’t have the disease. Anything is possible, however, and you estimate that there is a 1/100,000 chance that you have AIDS.

The AIDS test you took is good but not perfect. The test is 99 percent accurate. This means that if you do have AIDS, the test is 99 percent likely to give you a positive result, while if you don’t have AIDS, there will be a 99 percent chance that the test results will come back negative.

You get back your test results, and they are positive. How concerned should you be?

answer: the 99 percent accurate aids test you actually shouldn t be that worried since you almost certainly don t have aids. to see this, imagine that 1,000,000 people, who are just like you, take the test. each of these people has a 1/100,000 chance of having aids. thus, of these 1,000,000 people, 10 have the disease and 999,990 are free of it. when these people get tested, probably nine or ten of the people with aids will get back positive results. of the 999,990 who don t have the disease 1 percent will get false positives. this means that there will be about 10,000 false positives. consequently the vast majority of people who get positive results got false positives. your chance of actually having aids after getting the positive test results are only about 1/1,000. this result seems very paradoxical since the test is 99 percent accurate. after getting your tests results, however, you have two pieces of information: the tests results and your initial belief that you almost certainly didn t have aids. you don t lose the second piece of information just because of the positive test results; rather the test results should be used to update your beliefs. these two pieces of information need to be combined. when you (sort of) average the 1/100,000 chance of having aids with the 99 percent chance of not having aids, you get an approximate 1/1,000 chance of having the disease.

20.

Newcomb’s Problem

A super-intelligent being places two closed boxes in front of you. Box A always contains $1,000. Box B contains either $1 million or nothing. You are given a choice of either taking only box A, or taking both boxes A and B.

Your choice might seem simple, but this super-intelligent being is very good at making predictions. If he thinks you are going to pick only box B, he will put $1 million in that box. If he thinks you are going to be greedy and pick both boxes, he will leave box B empty. Which box(es) should you pick? Remember that the being makes his choice before you make yours. This question is known as Newcomb’s Problem.

answer: newcomb s problem there is no generally accepted answer to newcomb s problem. since the super-intelligent being has already made his decision before you decided what box to pick, it would seem reasonable always to take both boxes. after all, regardless of whether he put any money in box b, you re always better off taking both boxes. of course, if the being really is able to predict your move, and you pick both boxes, you will get only $1,000; whereas if you took only box b, you would get $1 million.

21.

John and Kim negotiate over the sale of Kim’s car. The car is worth $5,000 to John and $3,000 to Kim. Both people know everything about the car, so there is no concern about adverse selection. John makes an offer to buy the car. If the offer is rejected, the game ends. What offer should John make?

 john should offer to buy the car for just over $3,000. since kim must either accept the offer or allow negotiations to end, if she is rational, she will accept any offer over $3,000.

22.

Abe owns an antique that Bill wants. Abe values the antique at $3,000, and Bill values it at $5,000. Without knowing anything about how the negotiations will be conducted, what can we guess might happen?

abe will sell the antique for some price between $3,000 and $5,000.

23.

Use the same facts as in question 22, but now consider that Cindy also wants to buy the antique. Cindy is willing to pay $5,000 for it. Without knowing anything about how the negotiations will be conducted except that Bill and Cindy won’t collude, what can we guess might happen?

abe will sell the antique to either bill or cindy for $5,000. there is no other reasonable outcome. it would be unreasonable, for example, to assume that abe would sell the good to bill for $4,800, because cindy would be willing to offer bill more than this. as long as the price is below $5,000, we don t have a stable outcome, because the buyer who didn t get the good would be willing to pay more.

Answers

18.

Answer: Let’s Make a Deal

You should definitely change boxes. Let’s label the three boxes as A, B, and C and assume that you picked box A. Obviously there is a 1/3 probability that the prize is in box A and a 2/3 probability that the prize is in either box B or C. Either or both boxes B or C are empty. Thus, after you pick box A, it will always be possible for the host to open either box B and C to reveal an empty box. Consequently, the host’s actions do not affect the probability of the prize being in box A. Before the host opened one of the two other boxes, there was a 1/3 chance of the prize being in box A, and after he opened one of the other boxes, there is still a 1/3 chance of the prize being in box A. Let’s say that the host opened box B and showed you that it was empty. Now, since there is a 1/3 chance that box A has the prize, and a zero chance that box B has the prize, there must be a 2/3 chance that box C has the prize. Consequently, you should switch to box C because it will double your chance of winning.

19.

Answer: The 99 percent Accurate AIDS Test

You actually shouldn’t be that worried since you almost certainly don’t have AIDS. To see this, imagine that 1,000,000 people, who are just like you, take the test. Each of these people has a 1/100,000 chance of having AIDS. Thus, of these 1,000,000 people, 10 have the disease and 999,990 are free of it. When these people get tested, probably nine or ten of the people with AIDS will get back positive results. Of the 999,990 who don’t have the disease 1 percent will get false positives. This means that there will be about 10,000 false positives. Consequently the vast majority of people who get positive results got false positives. Your chance of actually having AIDS after getting the positive test results are only about 1/1,000.

This result seems very paradoxical since the test is 99 percent accurate. After getting your tests results, however, you have two pieces of information: the tests results and your initial belief that you almost certainly didn’t have AIDS. You don’t lose the second piece of information just because of the positive test results; rather the test results should be used to update your beliefs. These two pieces of information need to be combined. When you (sort of) average the 1/100,000 chance of having AIDS with the 99 percent chance of not having AIDS, you get an approximate 1/1,000 chance of having the disease.

20.

Answer: Newcomb’s Problem

There is no generally accepted answer to Newcomb’s problem. Since the super-intelligent being has already made his decision before you decided what box to pick, it would seem reasonable always to take both boxes. After all, regardless of whether he put any money in box B, you’re always better off taking both boxes. Of course, if the being really is able to predict your move, and you pick both boxes, you will get only $1,000; whereas if you took only box B, you would get $1 million.

21.

John should offer to buy the car for just over $3,000. Since Kim must either accept the offer or allow negotiations to end, if she is rational, she will accept any offer over $3,000.

22.

Abe will sell the antique for some price between $3,000 and $5,000.

23.

Abe will sell the antique to either Bill or Cindy for $5,000. There is no other reasonable outcome. It would be unreasonable, for example, to assume that Abe would sell the good to Bill for $4,800, because Cindy would be willing to offer Bill more than this. As long as the price is below $5,000, we don’t have a stable outcome, because the buyer who didn’t get the good would be willing to pay more.




Game Theory at Work(c) How to Use Game Theory to Outthink and Outmaneuver Your Competition
Game Theory at Work(c) How to Use Game Theory to Outthink and Outmaneuver Your Competition
ISBN: N/A
EAN: N/A
Year: 2005
Pages: 260

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