This Week's 538 Riddle

[silverscreenselect]

It's somewhat similar to the one a couple of weeks ago that sparked a lot of interest here:

A man in a trench coat approaches you and pulls an envelope from his pocket. He tells you that it contains a sum of money in bills, anywhere from $1 up to $1,000. He says that if you can guess the exact amount, you can keep the money. After each of your guesses he will tell you if your guess is too high, or too low. But! You only get nine tries. What should your first guess be to maximize your expected winnings?

As with the one two weeks ago, the obvious answer would appear to be $500, but I'm almost positive that's wrong. There's no prize other than getting a shoutout on the site next week if you get it right and they pick your entry. Here's the link:

http://fivethirtyeight.com/features/can ... ench-coat/

[BackInTex]

The number would be higher than 500 because you are trying to maximize your winnings, not maximize your chance of winning. So there is higher value in narrowing down a higher number rather than narrowing down a lower number.

In other words, each of the 9 guesses are better spent working towards a number above 500 rather than below.


This is for smarter people than me to figure out, but I think my initial logic is correct.

[SpacemanSpiff]

The number would be higher than 500 because you are trying to maximize your winnings, not maximize your chance of winning. So there is higher value in narrowing down a higher number rather than narrowing down a lower number.

In other words, each of the 9 guesses are better spent working towards a number above 500 rather than below.


This is for smarter people than me to figure out, but I think my initial logic is correct.

My guess --

$500? (lower)
$250? (lower)
$100? (lower)
Why bother any more? (lower...)

[Pastor Fireball]

For no scientific reason whatsoever, I would start at $667 in order to split the field into "one-third of the amounts are higher than $667" and "two-thirds of the amounts are lower than $667". I would keep splitting the fields in thirds until I (hopefully) got lucky with the correct number.

[Bob78164]

It's somewhat similar to the one a couple of weeks ago that sparked a lot of interest here:

A man in a trench coat approaches you and pulls an envelope from his pocket. He tells you that it contains a sum of money in bills, anywhere from $1 up to $1,000. He says that if you can guess the exact amount, you can keep the money. After each of your guesses he will tell you if your guess is too high, or too low. But! You only get nine tries. What should your first guess be to maximize your expected winnings?

As with the one two weeks ago, the obvious answer would appear to be $500, but I'm almost positive that's wrong. There's no prize other than getting a shoutout on the site next week if you get it right and they pick your entry. Here's the link:

http://fivethirtyeight.com/features/can ... ench-coat/

If your first guess is 744, then whenever the actual number is that or more, you're guaranteed to win via the obvious strategy. You've still got a chance when the number is lower. That would be my first thought. --Bob

[Bob78164]

It's somewhat similar to the one a couple of weeks ago that sparked a lot of interest here:

A man in a trench coat approaches you and pulls an envelope from his pocket. He tells you that it contains a sum of money in bills, anywhere from $1 up to $1,000. He says that if you can guess the exact amount, you can keep the money. After each of your guesses he will tell you if your guess is too high, or too low. But! You only get nine tries. What should your first guess be to maximize your expected winnings?

As with the one two weeks ago, the obvious answer would appear to be $500, but I'm almost positive that's wrong. There's no prize other than getting a shoutout on the site next week if you get it right and they pick your entry. Here's the link:

http://fivethirtyeight.com/features/can ... ench-coat/

If your first guess is 744, then whenever the actual number is that or more, you're guaranteed to win via the obvious strategy. You've still got a chance when the number is lower. That would be my first thought. --Bob

Repeating this strategy (decreasing the power of two by one at each step) means that you'll win whenever the number is 489 or higher, and never otherwise. This gives you an expectation value of $381.184. --Bob

[Bob78164]

It's somewhat similar to the one a couple of weeks ago that sparked a lot of interest here:

Aman in a trench coat approaches you and pulls an envelope from his pocket. He tells you that it contains a sum of money in bills, anywhere from $1 up to $1,000. He says that if you can guess the exact amount, you can keep the money. After each of your guesses he will tell you if your guess is too high, or too low. But! You only get nine tries. What should your first guess be to maximize your expected winnings

As with the one two weeks ago, the obvious answer would appear to be $500, but I'm almost positive that's wrong. There's no prize other than getting a shoutout on the site next week if you get it right and they pick your entry. Here's the link:

http://fivethirtyeight.com/features/can ... ench-coat/

If your first guess is 744, then whenever the actual number is that or more, you're guaranteed to win via the obvious strategy. You've still got a chance when the number is lower. That would be my first thought. --Bob

Repeating this strategy (decreasing the power of two by one at each step) means that you'll win whenever the number is 489 or higher, and never otherwise. This gives you an expectation value of $381.184. --Bob

I think I got the arithmetic wrong. I think this strategy guarantees a win whenever the number is 481 or higher. --Bob

[Bob78164]

Fencepost error. Your first guess should be 745, not 744. That does guarantee a win when the number is 490 or higher. --Bob

[TheCalvinator24]

When I ran a simulation, starting at 745 didn't get me a guarantee win in 9 tries. It took 10 tries.

Of course, I did it manually, so I may have screwed it up somewhere.

[Bob78164]

When I ran a simulation, starting at 745 didn't get me a guarantee win in 9 tries. It took 10 tries.

Of course, I did it manually, so I may have screwed it up somewhere.

You're never guaranteed a win with 9 tries. You can only cover 511 of the 1,000 possibilities. Choosing 745 lets you arrange things so that the 511 possibilities you can cover are the most valuable ones.

Look at it this way. If you only had one guess, you'd maximize your expectation value by choosing 1,000. That's just as likely as any other number and it's more valuable than any of the others. Choosing 745 when you have 9 guesses generalizes that insight. --Bob

[TheCalvinator24]

"That does guarantee a win when the number is 490 or higher."

Maybe I'm misreading this line, or like I said, maybe I screwed up my simulation, but this is the result I can't replicate.

[Bob78164]

"That does guarantee a win when the number is 490 or higher."

Maybe I'm misreading this line, or like I said, maybe I screwed up my simulation, but this is the result I can't replicate.

When the number is lower than 745, your next choice will be 617. If it continues to be lower, you choose, in order, 553, 521, 505, 497, 493, 491, and 490. Any time the number is higher than your previous choice, you choose the midpoint of the two endpoints that you know about. --Bob

[TheCalvinator24]

"That does guarantee a win when the number is 490 or higher."

Maybe I'm misreading this line, or like I said, maybe I screwed up my simulation, but this is the result I can't replicate.

When the number is lower than 745, your next choice will be 617. If it continues to be lower, you choose, in order, 553, 521, 505, 497, 493, 491, and 490. Any time the number is higher than your previous choice, you choose the midpoint of the two endpoints that you know about. --Bob

I rounded up at some point when I had XX(odd).5

That was where I went wrong.