4 ms·

I think the problem is with the assumption that expected wins are a good guide of behavior at the granularity of individual plays of the game. The expected win

I think the problem is with the assumption that expected wins are a good guide of behavior at the granularity of individual plays of the game. The expected win is infinite because very rare scenarios have huge payouts. However, any one play of the game has a 50% chance of paying out nothing. If I get to play the game a large enough number of times for the rare scenarios to actually occur, then I would be willing to pay a higher price than if I only got one shot at it.

jameshart 11y ago
The way I read the rules, there's no chance of a zero payout. You're guaranteed at least $2, if the first toss comes up tails. So you should be willing to pay at least $2 to enter, since if the first toss comes up heads, you're then on for a chance to win at least $4.
- oconnor663 11y ago
  Sure. The question is more like, should you be willing to pay $1 million to play? And honestly, if you can't afford to burn $1 million, that would be a pretty bad idea. However! Suppose you were allowed to modify things ever so slightly. Suppose you get to say, "Sure, I'll pay $1 million to play. But instead of playing once and taking my winnings, I want to play a million times, and only take one millionth of whatever I win each time." You haven't done anything to change the average payout of the game. But you've made it a much better idea to play, by reducing the variance. That's what it means to say the game has infinite expected value. You really should be willing to pay anything to play it, as long as you're allowed to repeat the game over and over. Of course, if you're stuck with playing the game just once, then lots of games look like a bad idea despite having a positive expected value. The St. Petersburg game is hardly alone here. For example, suppose you have a 1-in-a-million chance of winning $10 billion, but it costs $1000 to play. Should you play? The expected value is good, but I don't think most people would touch it. Not much of a paradox there.
  - grkvlt 11y ago
    Actually, for the 1/1,000,000 chance of GBP 10,000,000,000 then GBP 1,000 seems OK? My EV is GBP 10,000 or 10.0 times my stake, which is _much_ better that buying 1,000 lottery tickets at GBP 1.00 each, for a 1/14,000 chance of GBP 10,000,000 (nominal; neglecting rollovers) and an EV of GBP 714.28 (approximately) which is an under unity gain (0.71 of the stake returned). However using the Kelly criterion for placing bets, I think it turns out that even though the return is high for the ten billion wager, you should only place the GBP 1,000 bet if you have a bankroll of 1,000,000,000 already ;( Corrections to my maths welcome!
  - morgante 11y ago
    > For example, suppose you have a 1-in-a-million chance of winning $10 billion, but it costs $1000 to play. Should you play? I'd happily take those odds.
    - dsjoerg 11y ago
      It's not only the odds but the magnitude of the bet that matter. Because losing $1000 wouldn't bother you much, you're happy to do it. But scale the proposition up enough, perhaps 10^3, and you wouldn't be so relaxed about it. E.g. 1-in-a-million chance of winning $10 trillion, but it costs $1 million to play.
ikeboy 11y ago
Do you think that maximizing expected utility in general is not how a rational person should act? If yes, how do you deal with https://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Morgenstern_utility_theorem https://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Morgenster...?
- Zakharov 11y ago
  Expected utility is great, expected money is silly.
  - ikeboy 11y ago
    I don't think that's enough to solve this problem. If you replace money by utility, it seems like it would still be a paradox.
- Retric 11y ago
  On of the failings of decision theory is our 'utility functions' are not necessarily static. The most obvious case being people don't necessarily order the same thing every time they go to a restaurant. Arguably you could model this with hidden variables. So: A mathematician might go to vegas and spend 1,000$ gamboling with the expectation that they will lose that money. With novelty being the reason that becomes a reasonable trade-off. In that context maximizing expected utility becomes hard because you can't accurately model your utility function.
ericras 11y ago
>>The expected win is infinite because very rare scenarios have huge payouts. Not quite. The expected win is infinite because of the one single scenario where the games continues on an infinite basis. The average win is approx $46 if the bankroll is all the money in the world.
- jonahx 11y ago
  Thinking about the one single scenario where it goes on forever, as if it were an event unto itself, is misleading imo -- by that reasoning, in that scenario, you would win an infinite amount even if you were only getting paid $1 on each flip. But the EV is clearly not infinite for that game. The increasingly huge rare payouts are essential to the EV calc.
vilhelm_s 11y ago
Yes, I think so too. Tumblr user su3su2u1 wrote a post which argued the same thing in more detail: http://su3su2u1.tumblr.com/post/120899546078/resolving-the-st-petersburg-lottery-paradox-also http://su3su2u1.tumblr.com/post/120899546078/resolving-the-s...

jameshart 11y ago

The way I read the rules, there's no chance of a zero payout. You're guaranteed at least $2, if the first toss comes up tails. So you should be willing to pay at least $2 to enter, since if the first toss comes up heads, you're then on for a chance to win at least $4.

oconnor663 11y ago
Sure. The question is more like, should you be willing to pay $1 million to play? And honestly, if you can't afford to burn $1 million, that would be a pretty bad idea. However! Suppose you were allowed to modify things ever so slightly. Suppose you get to say, "Sure, I'll pay $1 million to play. But instead of playing once and taking my winnings, I want to play a million times, and only take one millionth of whatever I win each time." You haven't done anything to change the average payout of the game. But you've made it a much better idea to play, by reducing the variance. That's what it means to say the game has infinite expected value. You really should be willing to pay anything to play it, as long as you're allowed to repeat the game over and over. Of course, if you're stuck with playing the game just once, then lots of games look like a bad idea despite having a positive expected value. The St. Petersburg game is hardly alone here. For example, suppose you have a 1-in-a-million chance of winning $10 billion, but it costs $1000 to play. Should you play? The expected value is good, but I don't think most people would touch it. Not much of a paradox there.
- grkvlt 11y ago
  Actually, for the 1/1,000,000 chance of GBP 10,000,000,000 then GBP 1,000 seems OK? My EV is GBP 10,000 or 10.0 times my stake, which is _much_ better that buying 1,000 lottery tickets at GBP 1.00 each, for a 1/14,000 chance of GBP 10,000,000 (nominal; neglecting rollovers) and an EV of GBP 714.28 (approximately) which is an under unity gain (0.71 of the stake returned). However using the Kelly criterion for placing bets, I think it turns out that even though the return is high for the ten billion wager, you should only place the GBP 1,000 bet if you have a bankroll of 1,000,000,000 already ;( Corrections to my maths welcome!
- morgante 11y ago
  > For example, suppose you have a 1-in-a-million chance of winning $10 billion, but it costs $1000 to play. Should you play? I'd happily take those odds.
  - dsjoerg 11y ago
    It's not only the odds but the magnitude of the bet that matter. Because losing $1000 wouldn't bother you much, you're happy to do it. But scale the proposition up enough, perhaps 10^3, and you wouldn't be so relaxed about it. E.g. 1-in-a-million chance of winning $10 trillion, but it costs $1 million to play.

ikeboy 11y ago

Do you think that maximizing expected utility in general is not how a rational person should act? If yes, how do you deal with https://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Morgenstern_utility_theorem https://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Morgenster...?

Zakharov 11y ago
Expected utility is great, expected money is silly.
- ikeboy 11y ago
  I don't think that's enough to solve this problem. If you replace money by utility, it seems like it would still be a paradox.
Retric 11y ago
On of the failings of decision theory is our 'utility functions' are not necessarily static. The most obvious case being people don't necessarily order the same thing every time they go to a restaurant. Arguably you could model this with hidden variables. So: A mathematician might go to vegas and spend 1,000$ gamboling with the expectation that they will lose that money. With novelty being the reason that becomes a reasonable trade-off. In that context maximizing expected utility becomes hard because you can't accurately model your utility function.

ericras 11y ago

>>The expected win is infinite because very rare scenarios have huge payouts. Not quite. The expected win is infinite because of the one single scenario where the games continues on an infinite basis. The average win is approx $46 if the bankroll is all the money in the world.

jonahx 11y ago
Thinking about the one single scenario where it goes on forever, as if it were an event unto itself, is misleading imo -- by that reasoning, in that scenario, you would win an infinite amount even if you were only getting paid $1 on each flip. But the EV is clearly not infinite for that game. The increasingly huge rare payouts are essential to the EV calc.

vilhelm_s 11y ago

Yes, I think so too. Tumblr user su3su2u1 wrote a post which argued the same thing in more detail: http://su3su2u1.tumblr.com/post/120899546078/resolving-the-st-petersburg-lottery-paradox-also http://su3su2u1.tumblr.com/post/120899546078/resolving-the-s...