The St. Petersburg Paradox
The idea is that there is a lottery where someone flips a coin. If it comes up heads, you lose. If it comes up tails, you win a dollar and keep going. Every time it comes up tails, your winnings double (2 dollars, 4, 8, 16...), until you hit heads and it stops.
How much would you be willing to pay to play a lottery like that?
Most people's answers (including my own) would be "Not very much". The paradox is that, no matter what the cost of the lottery is, over time, you'd be expected to come out ahead. Over infinite plays, you'd be expected to win infinite dollars.
I don't find most of the resolutions on the Wikipedia page very satisfying.
The marginal utility explanation seems to assume a person will think, "Well, gee, I already won 20 million dollars. Is it really worth the time to keep playing this stupid coin game over and over until I hit 2 billion?" That might be how I'd think about it, but that seems contrary to human nature. How many people really strike it rich and say, "You know what? That's enough money." Maybe Bill Waterson; that's probably it.
I think the probability weighting explanation makes the most sense, but Wikipedia glosses over it.
The way I approached the problem was to code a simulation. After it played the game a few million times, it had won an average of $5 per game. So it sure as hell isn't worth paying "any" amount to play the game; about $5 would be it.
Then, instead of a payout of 2^n per tails, I tried 4^n. One time it actually hit $4 billion! But it usually seemed to hang around $20,000.
I think when it comes to playing a game for a finite number of trials, if something doesn't have a high probability of happening, it won't happen. Let's say if the probability isn't 50%, it doesn't happen. Well, in that case, if you're only gonna play one game, you should only bet $1. Cause chances are, you're just gonna lose that dollar.
If you're willing to play 8 games, you have a 50% chance of winning up to 3 games, so you could be willing to bet up to $8.
So, you should only be willing to bet up to any amount if you are willing to play infinite games. The human lifespan is probably not long enough for you to be able to play long enough to have a reasonable chance at hitting a really big score.
It is true that no one should offer this lottery for other people to play, but I think that's a totally separate issue.
The idea is that there is a lottery where someone flips a coin. If it comes up heads, you lose. If it comes up tails, you win a dollar and keep going. Every time it comes up tails, your winnings double (2 dollars, 4, 8, 16...), until you hit heads and it stops.
How much would you be willing to pay to play a lottery like that?
Most people's answers (including my own) would be "Not very much". The paradox is that, no matter what the cost of the lottery is, over time, you'd be expected to come out ahead. Over infinite plays, you'd be expected to win infinite dollars.
I don't find most of the resolutions on the Wikipedia page very satisfying.
The marginal utility explanation seems to assume a person will think, "Well, gee, I already won 20 million dollars. Is it really worth the time to keep playing this stupid coin game over and over until I hit 2 billion?" That might be how I'd think about it, but that seems contrary to human nature. How many people really strike it rich and say, "You know what? That's enough money." Maybe Bill Waterson; that's probably it.
I think the probability weighting explanation makes the most sense, but Wikipedia glosses over it.
The way I approached the problem was to code a simulation. After it played the game a few million times, it had won an average of $5 per game. So it sure as hell isn't worth paying "any" amount to play the game; about $5 would be it.
Then, instead of a payout of 2^n per tails, I tried 4^n. One time it actually hit $4 billion! But it usually seemed to hang around $20,000.
I think when it comes to playing a game for a finite number of trials, if something doesn't have a high probability of happening, it won't happen. Let's say if the probability isn't 50%, it doesn't happen. Well, in that case, if you're only gonna play one game, you should only bet $1. Cause chances are, you're just gonna lose that dollar.
If you're willing to play 8 games, you have a 50% chance of winning up to 3 games, so you could be willing to bet up to $8.
So, you should only be willing to bet up to any amount if you are willing to play infinite games. The human lifespan is probably not long enough for you to be able to play long enough to have a reasonable chance at hitting a really big score.
It is true that no one should offer this lottery for other people to play, but I think that's a totally separate issue.
