My friend asked me this many years ago and we discussed it for hours but didn't reach a satisfactory conclusion.
Suppose you entered a casino and a dealer suggested this game to you.
He flips a coin. If it comes up tales then you lose and the game ends. If it comes up heads you win $1 and he flips the coin again. On the second flip if it comes up tales you lose and the game ends, if it comes up head you win an additional $2 and he flips another coin. This continues until you eventually get a tales with the reward for each consecutive head increasing exponentially i.e. $1, $2, $4, $8...
What would you pay to play this game? That is, what is it worth?
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If you solve for the infinite series you will find that the game is worth infinite dollars. As the award increases exponentially as the probabilities decrease exponentially.
1/2 * 1 + 1/4 * 2 + 1/8 * 4 ...
Or generalised:
Sigma n=1 to n=infinity
1/(2^n) * 2^(n-1)
Each of the infinite terms = 1/2 so the series adds up to infinity.
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So the mathematics tells us that the game is worth infinity but clearly you wouldn't bet infinite dollars to play this game. But why not? The expected payout is infinity and if you only play the game a few times you'd probably win less than $100 each time in practice.
What if another game paid out $1000 with the first head and $2000 for the second head etc. You'd clearly pay more to play the second game but you can't distinguish between the two games mathematically.
What is the game worth?
Suppose you entered a casino and a dealer suggested this game to you.
He flips a coin. If it comes up tales then you lose and the game ends. If it comes up heads you win $1 and he flips the coin again. On the second flip if it comes up tales you lose and the game ends, if it comes up head you win an additional $2 and he flips another coin. This continues until you eventually get a tales with the reward for each consecutive head increasing exponentially i.e. $1, $2, $4, $8...
What would you pay to play this game? That is, what is it worth?
---
If you solve for the infinite series you will find that the game is worth infinite dollars. As the award increases exponentially as the probabilities decrease exponentially.
1/2 * 1 + 1/4 * 2 + 1/8 * 4 ...
Or generalised:
Sigma n=1 to n=infinity
1/(2^n) * 2^(n-1)
Each of the infinite terms = 1/2 so the series adds up to infinity.
---
So the mathematics tells us that the game is worth infinity but clearly you wouldn't bet infinite dollars to play this game. But why not? The expected payout is infinity and if you only play the game a few times you'd probably win less than $100 each time in practice.
What if another game paid out $1000 with the first head and $2000 for the second head etc. You'd clearly pay more to play the second game but you can't distinguish between the two games mathematically.
What is the game worth?
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I think I've found your answer and all it took was a little clever Googling... how I love Google.
http://en.wikipedia.org/wiki/St._Petersb…
http://en.wikipedia.org/wiki/St._Petersb…
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Simple, bet one dollar. If you lose, bet 2 dollars the second time. If you lose, bet 3 dollars, if you lose bet 6 dollars. Continue to always bet your losses, and you will eventually win, putting you back to square one and having lost no money