Heads you win

wolfgang59 View Profile

Posers and Puzzles

01 Mar '11 00:04

mtthw
Joined
07 Sep '05
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35068
04 Mar '11 11:031 edit
Originally posted by wolfgang59
mtthw's interpretation of the game is correct.
Good, I'll carry on then 🙂.

Here's another step towards an answer.

(I'm ignoring the possibility of mixed strategies at the moment - will try and look at how much that makes a difference later).

Let's say there N players (including me). Let's also assume we all have the same strategy: stop at n. Which value of n maximises my (or any other individual's) chance of winning?

P(I win) = (1/2)^n [1 - (1/2)^n]^(N - 1)

This is maximised by n = ln(N)/ln(2)

So, 4 player, n = 2. 8 player, n = 3 etc.

For other values of N, I cheated and drew up a table in Excel:

N=2 => n=1
N=3 => n=2
N=4 => n=2
N=5 => n=2
N=6 => n=3
N=7 => n=3
N=8 => n=3
N=9 => n=3
N=10 => n=3
N=11 => n=3
N=11 => n=4
etc
Palynka
Upward Spiral
Halfway
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02 Aug '04
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8702
04 Mar '11 11:273 edits
Originally posted by wolfgang59
I dont think this is a "mixed strategy" solution since I allow draws.
Isn't the goal to beat your opponent rather than maximize winnings? That wasn't clear at all from the first post.

If you allow for draws (equal to winning) then N players stopping at 1 is a Nash equilibrium. You draw heads, you win and no point in going further.
Palynka
Upward Spiral
Halfway
Joined
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Moves
8702
04 Mar '11 11:34
Originally posted by JS357
Could you be less specific?
Do you want me to post a google link for you or do you think you can do it by yourself?
JS357
Joined
29 Dec '08
Moves
6788
04 Mar '11 15:18
Originally posted by wolfgang59
mtthw's interpretation of the game is correct. You cannot win if you toss a tail .. its all about how far you go.

My thinking was stick after a single head (but I'm not 100% sure)

I dont think this is a"mixed strategy" solution since I allow draws.

I do not have a clue what the solution is for 3 players. Thats the real challenge.
Do you mean you allow that a draw is a win for the drawing players?
JS357
Joined
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Moves
6788
04 Mar '11 15:36
Originally posted by mtthw
Good, I'll carry on then 🙂.

Here's another step towards an answer.

(I'm ignoring the possibility of mixed strategies at the moment - will try and look at how much that makes a difference later).

Let's say there N players (including me). Let's also assume we all have the same strategy: stop at n. Which value of n maximises my (or any other individua ...[text shortened]... N=6 => n=3
N=7 => n=3
N=8 => n=3
N=9 => n=3
N=10 => n=3
N=11 => n=3
N=11 => n=4
etc
2 questions:

So do you count it as a win if another player or other players and you survive and then all of you stop at n? e.g, 2 players; you both stop ASAP; or 8 players, say 3 including you, survive, and all 3 stop at turn 3. IOW is a draw to be counted as a win? I wouldn't think so.

Is your chance of surviving until turn n, better than anyone else's who plays until turn n? I wouldn't think so.

I'll not argue with the math but I don't see the strategy affecting the odds of winning, if all the players follow this as their guide for maximizing their odds, i.e., if all are rational optimizers.
mtthw
Joined
07 Sep '05
Moves
35068
04 Mar '11 16:47
Originally posted by JS357
2 questions:

So do you count it as a win if another player or other players and you survive and then all of you stop at n?

Is your chance of surviving until turn n, better than anyone else's who plays until turn n? I wouldn't think so.

I'll not argue with the math but I don't see the strategy affecting the odds of winning, if all the players follow this as their guide for maximizing their odds, i.e., if all are rational optimizers.
Q1: No. A win is only if I have more money than anyone else at the end of the game.

Q2: No.

The idea is this. If I stop at n, I end up with either 0 or n. If we all have the same strategy, then the only way I win is if I get n and everybody else goes bust.

As n increases - we all have the same expected winnings, but the variance increases, which is why there's a different chance of there being a single winner.

If we start considering mixed strategies, I win if I don't go bust and everybody either sticks before I do or they go bust. It's not that hard to formulate the problem, but solving it is another matter!
JS357
Joined
29 Dec '08
Moves
6788
04 Mar '11 19:00
Originally posted by mtthw
Q1: No. A win is only if I have more money than anyone else at the end of the game.

Q2: No.

The idea is this. If I stop at n, I end up with either 0 or n. If we all have the same strategy, then the only way I win is if I get n and everybody else goes bust.

As n increases - we all have the same expected winnings, but the variance increases, which is ...[text shortened]... they go bust. It's not that hard to formulate the problem, but solving it is another matter!
JS357
Joined
29 Dec '08
Moves
6788
04 Mar '11 21:04
Originally posted by mtthw
Q1: No. A win is only if I have more money than anyone else at the end of the game.

Q2: No.

The idea is this. If I stop at n, I end up with either 0 or n. If we all have the same strategy, then the only way I win is if I get n and everybody else goes bust.

As n increases - we all have the same expected winnings, but the variance increases, which is ...[text shortened]... they go bust. It's not that hard to formulate the problem, but solving it is another matter!
I believe I see my mistake. I have neglected maximizing the frequency of games that SOMEONE wins.

Sorry to have taxed anyone's patience.