game with a not too obvious solution

[doormat]

Ok, I write down a number on a piece of paper. Then I flip a coin - heads I write a larger number on the other piece of paper, tails I write a smaller one. To my way of thinking, no matter which paper you choose and no matter what number you see on it, it is a random equal chance that it will be high or low because of the coin flip. So exactly what is your function that will predict how my coin flips?

doormat

[Bozeman]

Yes, this is true, but he will switch more often with the lower # than with the higher number!

[squiffy]

Told my college roommate about the problem. He was a Phi Beta Kappa math major and studied Real Analysis in college, which dealt a lot with one to one mapping of infinite number lines to say points on the circumference of a circle, etc. His approach was to explain that you can simplify the problem by showing that the fact that the number choices stretch from negative infinity to positive infinity does not really affect the basic problem. So you can convert the problem into one where the numberline is finite, say from 0 to 100. And that problem is easier to solve intuitively.

But that really is the hardest step, realizing that the fact that the number choices are infinite does not change the problem.

You can still take a number and assign a probability to it so that you hold a higher number a larger percentage of the time than any lower number.

This makes sense to me. Though I certainly cannot follow the math.

[well]

[ QUOTE ]
Well, can you explain your solution in layman's terms. I don't understand the functions or numbers.

[/ QUOTE ]

I am not familiar with the term layman's terms. But I will try to give it a less mathematical try...

The two numbers are written, and you hold one of them.
Since you want the biggest one, you would like to switch with the smaller and keep the bigger one.
But you don't know wich one you got.
Agree that it would be good if you switched the small one strictly more of the time than you would switch the bigger one.
So all you have to do is to add a probability to switch to each number, such that for every two distinct numbers
the biggest one is least likely to be switched.

Now mathematics is unevitable for adding such probabilities to these numbers.
If you look at the above, you could easiliy see that you need a strictly decreasing function that maps {..,-2,-1,0,1,2,..} onto (0,1).

Maybe now you can look at the posts and plot one of the appropriate functions. See if you understand.

Hope I have been clear.

Next Time.

[Bozeman]

"that maps {..,-2,-1,0,1,2,..} onto (0,1)."

Actually, there is no such function, since (0,1) is a different order of infinite than {...,-1,0,1,...}. You map onto a subset of (0,1)

Craig

[aces961]

This may be being quite picky regarding the problem, but the case where player II puts the same number on both pieces of paper is not ruled out. In this case the players would split the pot every single time and the ev for player 1 would be zero which is not strictly positive.

[well]

[ QUOTE ]
"that maps {..,-2,-1,0,1,2,..} onto (0,1)."

Actually, there is no such function, since (0,1) is a different order of infinite than {...,-1,0,1,...}. You map onto a subset of (0,1)

Craig



[/ QUOTE ]

You might be right there. I don't know exactly every English term for the math,
I'm from the Netherlands, but it is a function from {..,-2,-1,0,1,2,...} to (0,1).
And indeed no surjection of such exists. Maybe the term 'onto' I used implies surjection.

Next Time.

[well]

[ QUOTE ]
This may be being quite picky regarding the problem, but the case where player II puts the same number on both pieces of paper is not ruled out. In this case the players would split the pot every single time and the ev for player 1 would be zero which is not strictly positive.

[/ QUOTE ]

You're right, I thought I already stated that, but I did not.

[ QUOTE ]
The ante for this game is $1 for both you (I) and your opponent (II). II takes two pieces of paper and writes secretly on both of them a whole number between minus and plus infinity.

[/ QUOTE ]

These numbers indeed must be distinct, otherwise it evidently results in a EV=0 strategy for II.

Next Time.

[irchans]

This "game with a not too obvious solution" is similar to the envelope problem (posted 9/29/02 by me).



[ QUOTE ]

Two envelope questions to think about:

1) A kind man tells you that he has two envelopes each with money in them. He says that you can choose either envelope, open it, and keep the money in that envelope, or trade the contents of the opened envelope for the unopened one. Is there any strategy that has a higher expectation than randomly choosing, opening, and not trading?

2) A millionaire is funding a game between you and another person. The other person is given 1000 one dollar bills and two identical envelopes. He is instructed to put any integral dollar amount up to $333 in the first envelope and put exactly twice that amount into the second envelope. He shuffles the envelopes and presents them to you. As in 1) you get to choose one envelope, open it, and then decide whether to trade for the unopened envelope. If you end up with the envelope containing less money, he gets $100. If you end up with the envelope containing more $, he gets nothing. What is a good strategy for you? What is a good strategy for him? What is his worst strategy?



[/ QUOTE ]

How do you create a hyperlink to an old post?

[well]

The similarity of the first problem is obvious, but there are some things I miss about the second.

[ QUOTE ]
A millionaire is funding a game between you and another person. The other person is given 1000 one dollar bills and two identical envelopes. He is instructed to put any integral dollar amount up to $333 in the first envelope and put exactly twice that amount into the second envelope. He shuffles the envelopes and presents them to you. As in 1) you get to choose one envelope, open it, and then decide whether to trade for the unopened envelope. If you end up with the envelope containing less money, he gets $100. If you end up with the envelope containing more $, he gets nothing. What is a good strategy for you? What is a good strategy for him? What is his worst strategy?



[/ QUOTE ]

Do I get to keep the money in the envelope I picked?
Does II return the remaining of the 1000 dollar bills to the millionaire?
I suppose II cannot choose $0 for otherwise his payoff is not completely defined?

I am solving the game under these assumptions right now.

[ QUOTE ]
How do you create a hyperlink to an old post?

[/ QUOTE ]

I do not know.

Next Time.