The St. Petersburg paradox
On the average, half of all players should be paid $2 by winning on the first flip, a quarter will receive $4 with heads on the second, an eighth will get $8 on the third, and so on. Fractions such as one out of 1,024 will get $1,024 with on the 10th flip, one out of 1,048,576 will receive $1,048,576 on the 20th, and one out of 33,554,432 will pick up $33,554,432 on the 25th.
Bernoulli wanted to see how much people would pay for the gamble. The expected value, and therefore the players' edge, is infinite. On this basis alone, any solid citizen should play no matter what the fee. Infinite expected return, an unending amount of money, would always be a smart move. But, Bernoulli found and contemporary decision support experts agree that few people would buy a chance for more than the equivalent of $20 or $25.
Quote: Omaha.... The expected value, and therefore the players' edge, is infinite. ...
says who
Quote: andysifsays who
although for each term 1/2*2 + 1/4*4 + 1/8*8 .... = 1+1+1+... i think there is something fishy since the harmonic series 1/2 + 1/4 + 1/8 = divergent, and doesn't add up to 1 (100%) as sum of all probabilities required.
Quote: andysifsays who
Every mathemetician
That said, the indifference point to playing the game depends on the wealth of the person being asked to play. For example, if the person being asked to play already has a wealth of $1,000,000, then he should be indifferent to paying $20.87 to play. Either way, the expected happiness is exactly 6, using a base-10 logarithm.
Quote: OmahaEvery mathemetician
please refer to the box above
Quote: NeutrinoThere's no such thing as infinite money anyway.
I've argued before that there is no such thing as an infinite quantity of anything, and that infinity is just a mathematical concept. However, let's not revisit that debate.
What is paradoxical about the paradox is that the win must be finite, because the coin must land on the ending side eventually, but the expected win is infinite. However, when it comes to infinity, there are lots of paradoxes.
Quote: OmahaTo play, you buy-in for a specified fee. The house keeps this money regardless of what happens. You then flip a coin. On heads, the chance of which is one out of two, you're paid $2 and the game ends. If the coin shows tails, you flip again. Heads this time pays $4, given that the chance of tails followed by heads is one out of four. Tails on the second round leads to a third flip; now, heads pays $8 the chance of tails-tails-heads being one out of eight. Flips continue, paying $16, $32, $64, and so forth with chances qual to one out of 16, 32, and 64, etc, respectively, until heads finally shows and the game ends.
To make a long story short: the game pays 2^n with probability 1/2^n (for any n>1). You are asking for a fair ante.
You are right, the EV is infinite. But EV is not utility, i.e. EV is not the quantity you should seek to maximize (if it were, you would sell your whole fortune for a single entry into that game for which anyone would laugh at you).
So utility is something different. Some people claim log(wealth) is a good utiltiy. It is probably better than EV, but even with a log-utility you can construct games with infinite log-utility you would never play for any larger ante.
One possible solution is simple: If you have a bazillion of USD, i.e. you are a effectively an unlimited source of dollars: Congratulations - you can do almost nothing with it. Economy will simply switch to a different currency because stability of a currency is essential.
Having some billions of dollars is nice (i.e. scale of the the largest companies in the US), having a trillion (the scale of the gross domestic product of the US) is not.
Hence at somewhere distinctive size of wealth any reasonable utility does not increase anymore, and at best stays constant. Then any expected utility stays finite, and you can calculate your best ante for such games.
Quote: MangoJYou are right, the EV is infinite. But EV is not utility, i.e. EV is not the quantity you should seek to maximize (if it were, you would sell your whole fortune for a single entry into that game for which anyone would laugh at you).
My point exactly. There is more to any game than mathematical ev
Quote:To play, you buy-in for a specified fee. The house keeps this money regardless of what happens. You then flip a coin.
This is such an intriguing game.
Do I get to use my own coin, in addition to tossing it myself?
2 or more players would gather, usually summoned by the challenge of "get your quarter out**".
Everyone flipped a coin.
If you flipped a tail, you won*.
If you flipped a head, you were still "in", flipping again.
You "lost" if you were the only person to flip a head on a certain round, and you got the honor of buying a soda for all the other players, plus enduring the ribbing.
Some people bought far more often than others. Some of us practiced our coin tosses.
A $20 stake would mean that you just need 5 in a row to profit - not terribly uncommon in my experience. A streaks of 8-10 is pretty readily doable.
If you're going to try it, work on your toss first. Flipping a coin with your thumb is difficult to control, and there are other methods. (I use an index/social finger toss.)
*If the last two people flipped tails, everyone was back in, to start it all over.
**It wasn't uncommon to suffix this challenge with a rude word or two.
Quote: WizardThe reason why you shouldn't pay an infinite amount of money for the bet is because happiness is not proportional to your wealth. I like to estimate it as proportional to log(wealth). This suggests that if you multiply anybody's wealth by x, then happiness is increased by log(x). For example, the same increase in happiness is achieved by giving somebody with $100 another $20 as giving somebody with $100,000,000 another $20,000,000.
That said, the indifference point to playing the game depends on the wealth of the person being asked to play. For example, if the person being asked to play already has a wealth of $1,000,000, then he should be indifferent to paying $20.87 to play. Either way, the expected happiness is exactly 6, using a base-10 logarithm.
Since it's all the same, may I volunteer to be the person with $100,000 receiving another $20K? Thanks in advance! :)
Quote: DieterBack when I worked in the factory, we used to do a variant of this.
2 or more players would gather, usually summoned by the challenge of "get your quarter out**".
Everyone flipped a coin.
If you flipped a tail, you won*.
If you flipped a head, you were still "in", flipping again.
You "lost" if you were the only person to flip a head on a certain round, and you got the honor of buying a soda for all the other players, plus enduring the ribbing.
Some people bought far more often than others. Some of us practiced our coin tosses.
A $20 stake would mean that you just need 5 in a row to profit - not terribly uncommon in my experience. A streaks of 8-10 is pretty readily doable.
If you're going to try it, work on your toss first. Flipping a coin with your thumb is difficult to control, and there are other methods. (I use an index/social finger toss.)
*If the last two people flipped tails, everyone was back in, to start it all over.
**It wasn't uncommon to suffix this challenge with a rude word or two.
Dieter,
Not sure if you ever served in the US military, but this is long-standing tradition among many; each unit has a "challenge coin" they earn somehow after assigned, and always have to have them. Their insignia is embossed on the coin, and it's usually about the size of a dollar slug. It's a matter of unit pride. They use the coins to determine who pays when guys are out together, and this is one of the ways. I always thought it was pretty cool.
Quote: beachbumbabsNot sure if you ever served in the US military, but this is long-standing tradition among many
Nope, didn't serve, but it's interesting to hear the origin of this particular game.
I just know it's what got me interested in advantage play, and it was nice to drink a lot of free sodas.
Quote: WizardThe reason why you shouldn't pay an infinite amount of money for the bet is because happiness is not proportional to your wealth. I like to estimate it as proportional to log(wealth). This suggests that if you multiply anybody's wealth by x, then happiness is increased by log(x). For example, the same increase in happiness is achieved by giving somebody with $100 another $20 as giving somebody with $100,000,000 another $20,000,000.
That said, the indifference point to playing the game depends on the wealth of the person being asked to play. For example, if the person being asked to play already has a wealth of $1,000,000, then he should be indifferent to paying $20.87 to play. Either way, the expected happiness is exactly 6, using a base-10 logarithm.
While that is one way around this specific version of the St. Petersburg paradox you can build ramp ups in wins such that any unbounded happiness function you use would still lead to infinite happiness. For log you would just need to have the game return 10^(2n) if you flip n heads in a row.
Quote: RSWhat's happiness have to do with anything? Has happiness ever been used in an equation before? Why now?
Happiness is a way of measuring the utility of money. The idea is that $1 of money is not equally useful depending on the $ you have right now.
The other way to look at it is simply that EV is not the only factor in a 'good' bet. Variance is as well. A positive EV bet that only comes of one time in 10 million is possibly just not worth it.
Quote: RSWhat's happiness have to do with anything? Has happiness ever been used in an equation before? Why now?
Of course it has. It's just not called happiness, but utility. Call it 'money hapiness'.
And the principle is that the same absolute amount of money mean different things for people with different wealth and they will accordingly take different decisions.
A bet of $1 million for a billionaire with 60% prob of winning $1 million and 40% of losing $1 million. ie EV 20% is a very good bet for him.
But for someone with Wealth of $1 million is a very bad idea. Despite the 20% EV he has 40% probability of becoming destitude.
Economic theories are full of Utility and Marginal Utility theories and all sort of equations with them (from what I remember from my economics courses)
-Nas
-Nas
Excellent, a new perspective on the short-timer's calendarQuote: Docstarted out with a value of zero and a slope of zero. For us two-year draftees, it took a full year for Happiness to attain a value of 1.0, but on the final day of active duty, our Happiness was infinite and increasing at an infinite rate!
