# Fantastic Opportunity

wolfgang59
Posers and Puzzles 04 Jul '09 07:41
1. wolfgang59
Mr. Wolf
04 Jul '09 07:41
A man offers you this bet;
He will flip a coin and you call. The coin is fair.
If you win he gives you £2.
If you lose you give him £1.

If you win you can play on. Same odds.
(ie If you win he gives you £6 if you lose you give him the £3.)

You can only continue playing after a win.
You must invest all your money if you continue.

What is your strategy to maximise your winnings?
(ie after how many wins do you stop playing)

What are your average winnings with this strategy?
(ie if you played the game each day what would you expect to be up after a year?)
2. TheMaster37
Kupikupopo!
05 Jul '09 12:10
Originally posted by wolfgang59
A man offers you this bet;
He will flip a coin and you call. The coin is fair.
If you win he gives you £2.
If you lose you give him £1.

If you win you can play on. Same odds.
(ie If you win he gives you £6 if you lose you give him the £3.)

You can only continue playing after a win.
You must invest all your money if you continue.

What is you ...[text shortened]... his strategy?
(ie if you played the game each day what would you expect to be up after a year?)
If you keep winning, no problems.

Once you lose you can't continue and you will have a total loss of 1 pound.

In one game you'd expect to win 0.5 * 2 + 0.5 * -1 = 1/2 pounds.

After two games you'd expect to win
0.5 * -1 + 0.25 * -3 +0.25 * (6+2) = 3/4

After three games
0.5*-1 + 0.25*-3 + 0.125*-9 + 0.125*(18+6+2) = 7/8

After N games: (2^N - 1)/2^N.

Still working on optimal strategy.
3. TheMaster37
Kupikupopo!
05 Jul '09 12:41
Say p is the chance you'll continue after winning.

For 4 games:

profit = 1/2 + 1/4 p - 1/8 pp + 1/16 ppp - 5pppp

I'm not going to try to find the maximum for n games, I'll make too many mistakes ðŸ™‚
4. wolfgang59
Mr. Wolf
05 Jul '09 12:58
The question is WHEN DO YOU QUIT?

Simple game but what is optimum strategy?
5. Palynka
Upward Spiral
05 Jul '09 12:59
Originally posted by TheMaster37
Say p is the chance you'll continue after winning.

For 4 games:

profit = 1/2 + 1/4 p - 1/8 pp + 1/16 ppp - 5pppp

I'm not going to try to find the maximum for n games, I'll make too many mistakes ðŸ™‚
Well, it's easy to see that the payoff is monotonically increasing in N. This is a variant of the St. Petersburg paradox.
6. Palynka
Upward Spiral
05 Jul '09 12:591 edit
Originally posted by wolfgang59
The question is WHEN DO YOU QUIT?

Simple game but what is optimum strategy?
Never. ðŸ™‚

Edit - Obviously, the St. Petersburg paradox shows the flaws of using expected winnings maximization as the measure for optimal strategy. This is especially true when expected winnings are infinite, but require infinite time to reach.
7. wolfgang59
Mr. Wolf
05 Jul '09 13:08
Originally posted by Palynka
Never. ðŸ™‚

Edit - Obviously, the St. Petersburg paradox shows the flaws of using expected winnings maximization as the measure for optimal strategy. This is especially true when expected winnings are infinite, but require infinite time to reach.
Thanks for that Palynnka never heard of that paradox before. My question came from me trying to simplfy the best strategy for a game called Pass the Pigs.
8. wolfgang59
Mr. Wolf
05 Jul '09 20:09
As usual the internet has an answer ..

http://cs.gettysburg.edu/~tneller/nsf/pig/

not sure I understand it ... ðŸ˜•
9. Palynka
Upward Spiral
05 Jul '09 21:59
Originally posted by wolfgang59
As usual the internet has an answer ..

http://cs.gettysburg.edu/~tneller/nsf/pig/

not sure I understand it ... ðŸ˜•
Did you check the pdf in the linked zip file? It seems to explain it quite well.

http://cs.gettysburg.edu/~tneller/papers/pig.zip