St. Petersburg Paradox

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How much would you pay to play the St. Petersburg Lottery?

I would not play.
10
23%
$1
13
30%
$10
11
25%
$100
5
11%
$1,000
1
2%
1% of my portfolio value
1
2%
10% of my portfolio value
1
2%
100% of my portfolio value
0
No votes
100% of my portfolio value plus all of the money that I could borrow
2
5%
 
Total votes: 44

Topic Author
tpm871
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St. Petersburg Paradox

Post by tpm871 » Sat Feb 04, 2012 1:12 pm

A recent WSJ article mentions the St. Petersburg Paradox while discussing the possibility of investing in the Facebook IPO:

http://online.wsj.com/article/SB1000142 ... inance_PF4

Wikipedia describes the St. Petersburg Paradox as:
Consider the following game of chance: you pay a fixed fee to enter and then a fair coin is tossed repeatedly until a tail appears, ending the game. The pot starts at 1 dollar and is doubled every time a head appears. You win whatever is in the pot after the game ends. Thus you win 1 dollar if a tail appears on the first toss, 2 dollars if a head appears on the first toss and a tail on the second, 4 dollars if a head appears on the first two tosses and a tail on the third, 8 dollars if a head appears on the first three tosses and a tail on the fourth, etc.
How much would you pay to play this game, given that you could only ever play it once?

My knee jerk reaction was that I would pay very little, since the probability of making "at least X dollars" is fairly low, where X is a significant amount of money to me. For example, the odds are:

100% chance of getting at least $1
50% chance of getting at least $2
25% chance of getting at least $4
12.5% chance of getting at least $8
...
1.49E-06% chance of getting at least $67,108,864
...

(note that the last probability listed above is roughly the same odds as being hit by a meteorite sometime during your lifetime)

But another way of looking at this is that your "expected return" is actually infinity dollars. That's because the expected value of playing is:

0.5*1 + 0.25*2 + 0.125*4 + ... + 1.49E-06%*67,108,864 + ...
= 0.5 + 0.5 + 0.5 + ... + 0.5 + ...
= $infinity

How much would you pay to potentially have infinite money?

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wbond
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Re: St. Petersburg Paradox

Post by wbond » Sat Feb 04, 2012 1:22 pm

Since it is the selling of lottery tickets (and other gambling) that is valuable, not the buying, you might also ask yourself, what is the lowest price you would sell a ticket to such a game of chance to customers?

Random Walker
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Re: St. Petersburg Paradox

Post by Random Walker » Sat Feb 04, 2012 1:53 pm

I read the article and found that question very interesting. When I thought about it, I came up with $1.30. Guess I'm not very adventurous. So I chose $1 in the poll.

The article was fun. But I must say, if I were given a column like that to describe why IPOs are bad investments, I would try to put in a lot more historical data and spend a lot less time on this game. Explaining the poor historical results and our behavioral interest in lottery effect might be more productive.

I'm sure there is a statistical way to determine the right amount to bet, and I think it would come out a bit above a dollar (perhaps 1.25-1.50). It is an interesting behavioral experiment to see what the poll shows as the popular amounts to bet versus the statistically rational answer. Of course a poll of the population as a whole rather than just the Boglehead population would be really interesting!

Dave

Random Walker
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Re: St. Petersburg Paradox

Post by Random Walker » Sat Feb 04, 2012 1:55 pm

Looks like somebody already beat me to the punch on thinking about it in terms of expected value :)

Dave

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G-Money
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Re: St. Petersburg Paradox

Post by G-Money » Sat Feb 04, 2012 2:09 pm

I'd be a fool not to play for $1, since getting that $1 is the worst case scenario. After that, I'd need to think about it.
Don't assume I know what I'm talking about.

chaz
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Re: St. Petersburg Paradox

Post by chaz » Sat Feb 04, 2012 2:47 pm

Don't gamble.
Chaz | | “Money is better than poverty, if only for financial reasons." Woody Allen | | http://www.bogleheads.org/wiki/index.php/Main_Page

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GregLee
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Re: St. Petersburg Paradox

Post by GregLee » Sat Feb 04, 2012 3:37 pm

It's clearly worth more than $1, but I guess not much more. I would have chosen $2, if that had been an option. Being something of a daredevil, I chose $10.
Greg, retired 8/10.

Random Walker
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Re: St. Petersburg Paradox

Post by Random Walker » Sat Feb 04, 2012 3:43 pm

I thought about it some more. I think the value is greater than 1.50 and less than 2

Dave

gkaplan
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Re: St. Petersburg Paradox

Post by gkaplan » Sat Feb 04, 2012 4:13 pm

St. Petersburg, Florida, or St. Petersburg, Russia?
Gordon

chaz
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Re: St. Petersburg Paradox

Post by chaz » Sat Feb 04, 2012 4:35 pm

gkaplan wrote:St. Petersburg, Florida, or St. Petersburg, Russia?
Leningrad?
Chaz | | “Money is better than poverty, if only for financial reasons." Woody Allen | | http://www.bogleheads.org/wiki/index.php/Main_Page

Zephyr120
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Re: St. Petersburg Paradox

Post by Zephyr120 » Sat Feb 04, 2012 5:20 pm

No one mentioned counterparty risk - that the 'house' would not be able to pay off $100 trillion!

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Epsilon Delta
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Re: St. Petersburg Paradox

Post by Epsilon Delta » Sat Feb 04, 2012 5:22 pm

The expected value calculation is misleading. It assumes the sponsor has infinite resources. In practice the sponsor will be bankrupt well before the 40th round so in real life the expected value is less than $40.

Ed 2
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Re: St. Petersburg Paradox

Post by Ed 2 » Sat Feb 04, 2012 5:27 pm

gkaplan wrote:St. Petersburg, Florida, or St. Petersburg, Russia?
For the first one I may pay a buck, the second not even a cent.
"The fund industry doesn't have a lot of heroes, but he (Bogle) is one of them," Russ Kinnel

Ed 2
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Re: St. Petersburg Paradox

Post by Ed 2 » Sat Feb 04, 2012 5:29 pm

chaz wrote:
gkaplan wrote:St. Petersburg, Florida, or St. Petersburg, Russia?
Leningrad?
Chaz, are you from Russia? Where you got such knowledge?
"The fund industry doesn't have a lot of heroes, but he (Bogle) is one of them," Russ Kinnel

chaz
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Re: St. Petersburg Paradox

Post by chaz » Sat Feb 04, 2012 6:57 pm

Ed 2 wrote:
chaz wrote:
gkaplan wrote:St. Petersburg, Florida, or St. Petersburg, Russia?
Leningrad?
Chaz, are you from Russia? Where you got such knowledge?
Qien sabe?
Chaz | | “Money is better than poverty, if only for financial reasons." Woody Allen | | http://www.bogleheads.org/wiki/index.php/Main_Page

Topic Author
tpm871
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Re: St. Petersburg Paradox

Post by tpm871 » Sat Feb 04, 2012 9:43 pm

Here's the wikipedia page:

http://en.wikipedia.org/wiki/St_Petersburg_Paradox

Yes, one of the things to consider is that the "house" would run out of money if you were lucky enough. But for this poll, let's assume that never happens.

I voted $100, because it seems like playing would have a good entertainment value. It's a potential black swan event, and $100 wouldn't bankrupt me. But in real life, I don't play the lottery and I have a pretty solid investment strategy that I follow consistently.

I am surprised that some people wouldn't play at all, since you're guaranteed to at least break even when paying $1.

Mudpuppy
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Re: St. Petersburg Paradox

Post by Mudpuppy » Sat Feb 04, 2012 9:53 pm

As with most games of chance, price is based on people's inability to do basic statistics and instead relying on their gut. The statistical distribution for the described game is the binomial distribution, where the probability of the flip ending up heads (p) is 0.5 (assuming an unweighted coin). One part of the statistics in the binomial distribution is the expected number of heads in n flips: E(heads) = n * p. For this game, we're toast if the expected number of heads is greater than 1, so plugging in 1 = n * 0.5 and solving for n gives us 2 coin flips. On average, the game will be over within 2 coin flips.

Mathematically, the odds of winning more than $2 are not in your favor. It's a betting game beyond that point, where the allure is the massive payoff if the rare event happens. The question then becomes "how much are you willing to gamble"?

And the answer to that often depends on who benefits from profits made off the entry money... people who might be willing to buy a St. Jude Dream Home ticket for $100 may be unwilling to play this game for $20 if the profit from the game went to something perceived as "greedy" (even though there's a greater chance of breaking even in this game at $20 than winning the house in the St. Jude raffle).

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Quasimodo
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Re: St. Petersburg Paradox

Post by Quasimodo » Sat Feb 04, 2012 10:03 pm

I selected the last option, all the money I had and could borrow, because it sounds as though the worst case is I win back my original bet.

In reality, I probably wouldn't play at all, because my experience with gambling situations is that something unexpected usually happens that is not good. (Jack of spades jumps from sealed deck, squirts cider in ear, etc.)

John
Many wealthy people are little more than janitors of their possessions. | | Frank Lloyd Wright, architect (1867-1959)

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Imperabo
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Re: St. Petersburg Paradox

Post by Imperabo » Sat Feb 04, 2012 10:18 pm

tpm871 wrote:
How much would you pay to potentially have infinite money?

An infinitely small chance at infinite money? About 2 bucks.

555
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Re: St. Petersburg Paradox

Post by 555 » Sun Feb 05, 2012 12:02 am

Random Walker wrote:Looks like somebody already beat me to the punch on thinking about it in terms of expected value :)
Dave
Umm, yes that would be Nicolas Bernoulli in 1713.

555
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Re: St. Petersburg Paradox

Post by 555 » Sun Feb 05, 2012 12:05 am

Quasimodo wrote:I selected the last option, all the money I had and could borrow, because it sounds as though the worst case is I win back my original bet.
No, the worst case is you get $1.

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docneil88
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Re: St. Petersburg Paradox

Post by docneil88 » Sun Feb 05, 2012 5:17 am

At age 21 I went to a casino and I kept doubling down on each prior loss of blackjack until I won, then repeated the process about 10 times. I thought, "Wow, there's a very low chance I'll lose at this game. I have outfoxed the casino." What I didn't sufficiently factor in was how much I could lose, namely every cent to my name. Luckily it never came close to that.

M B
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Re: St. Petersburg Paradox

Post by M B » Sun Feb 05, 2012 7:28 am

Regarding counter-party risk:
- If you trust the counter-party up to a million dollars, then the expected value is $11.
- If you trust the counter-party up to a billion dollars, then the expected value is $16.
- If you trust the counter-party up to a million billion dollars, then the expected value is $26.

And then of course there is the difference between the average gain and the average utility. If to you making ten billions instead of one billion makes little difference, then the game is worth at most $16.

At $1 you cannot lose (at worst you break even), so it is silly not to play.
At $10, you must trust your counter-party to a quarter million (or care little about making more than a quarter million).
At $100, the odds cannot realistically be in your favor.

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magician
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Re: St. Petersburg Paradox

Post by magician » Sun Feb 05, 2012 12:24 pm

555 wrote:
Random Walker wrote:Looks like somebody already beat me to the punch on thinking about it in terms of expected value :)
Dave
Umm, yes that would be Nicolas Bernoulli in 1713.
So, in all fairness, Bernoulli appears to have had a head start over Random Walker.
Simplify the complicated side; don't complify the simplicated side.

TheEternalVortex
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Re: St. Petersburg Paradox

Post by TheEternalVortex » Sun Feb 05, 2012 12:38 pm

M B wrote:Regarding counter-party risk:
- If you trust the counter-party up to a million dollars, then the expected value is $11.
- If you trust the counter-party up to a billion dollars, then the expected value is $16.
- If you trust the counter-party up to a million billion dollars, then the expected value is $26.

And then of course there is the difference between the average gain and the average utility. If to you making ten billions instead of one billion makes little difference, then the game is worth at most $16.

At $1 you cannot lose (at worst you break even), so it is silly not to play.
At $10, you must trust your counter-party to a quarter million (or care little about making more than a quarter million).
At $100, the odds cannot realistically be in your favor.
Yeah, $10 seems reasonable. e.g., casinos regularly pay out $1 million. But $1 billion is a stretch.

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baw703916
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Re: St. Petersburg Paradox

Post by baw703916 » Sun Feb 05, 2012 12:52 pm

TheEternalVortex wrote: Yeah, $10 seems reasonable. e.g., casinos regularly pay out $1 million. But $1 billion is a stretch.
There is, in fact a historical precedent:

http://en.wikipedia.org/wiki/Pepsi_Bill ... weepstakes

I actually made money on that one (as a BRK shareholder).

Brad
Most of my posts assume no behavioral errors.

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Imperabo
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Re: St. Petersburg Paradox

Post by Imperabo » Sun Feb 05, 2012 1:00 pm

baw703916 wrote:
TheEternalVortex wrote: Yeah, $10 seems reasonable. e.g., casinos regularly pay out $1 million. But $1 billion is a stretch.
There is, in fact a historical precedent:

http://en.wikipedia.org/wiki/Pepsi_Bill ... weepstakes

I actually made money on that one (as a BRK shareholder).

Brad

So most of that billion was to paid out in a balloon payment in 40 years. It bugs me that games and lotteries are able to claim a payout of $xxx when it won't be paid for decades. Hey guess what, I'm a billionaire! If you take the future value of my current assets in a few hundred years.

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