Omaha
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September 23rd, 2014 at 8:16:59 PM permalink
To 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.

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.
thecesspit
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September 23rd, 2014 at 8:20:30 PM permalink
The paradox teaches us about desirability (or not) of variance, and the incremental utility of money.
"Then you can admire the real gambler, who has neither eaten, slept, thought nor lived, he has so smarted under the scourge of his martingale, so suffered on the rack of his desire for a coup at trente-et-quarante" - Honore de Balzac, 1829
andysif
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September 23rd, 2014 at 8:22:14 PM permalink
Quote: Omaha

.... The expected value, and therefore the players' edge, is infinite. ...



says who
Doc
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September 23rd, 2014 at 8:25:32 PM permalink
This illustrates that EV is not the only factor that affects a decision to wager (or otherwise participate in an opportunity.) Nor should it be.
andysif
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September 23rd, 2014 at 8:36:52 PM permalink
Quote: andysif

says 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.
Omaha
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September 23rd, 2014 at 8:41:22 PM permalink
Quote: andysif

says who



Every mathemetician
Wizard
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September 23rd, 2014 at 8:45:15 PM permalink
The 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.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
andysif
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September 23rd, 2014 at 8:45:16 PM permalink
Quote: Omaha

Every mathemetician


please refer to the box above
andysif
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September 23rd, 2014 at 9:07:19 PM permalink
ah my wrong. its not the harmonic series and it actually converges
Neutrino
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September 23rd, 2014 at 9:10:49 PM permalink
There's no such thing as infinite money anyway. at some point you're gaining absolutely no value by increasing the numeral amount of the money. Money represents value, where there is only a finite amount of. Trying to approach infinite money is only going to cause endless inflation.
Wizard
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September 23rd, 2014 at 9:21:13 PM permalink
Quote: Neutrino

There'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.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
PapaChubby
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September 23rd, 2014 at 9:34:07 PM permalink
The reason that the expected value is infinite is due to astronomically small probabilities of winning astronomically huge amounts of money. For any single play of the game, one might set a threshold where this probability becomes insignificant. I might set this threshold somewhere around a 1 in a million possibility of winning a million dollars. This would require 20 flips of the coin. Assuming this is the best case scenario, then the expected value of the game is $20, so a fee of $20 to play is entirely reasonable.
MangoJ
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September 23rd, 2014 at 10:11:45 PM permalink
Quote: Omaha

To 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.
Omaha
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September 23rd, 2014 at 10:18:51 PM permalink
Quote: MangoJ

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).



My point exactly. There is more to any game than mathematical ev
Dieter
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September 24th, 2014 at 3:27:24 AM permalink
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?
May the cards fall in your favor.
DicePhD
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September 24th, 2014 at 11:02:11 AM permalink
I think I have convex preferences and decreasing relative risk aversion. But id probably pay about $7.75 to play
Dieter
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September 24th, 2014 at 12:21:42 PM permalink
Back 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.
May the cards fall in your favor.
beachbumbabs
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September 24th, 2014 at 5:30:47 PM permalink
Quote: Wizard

The 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! :)
If the House lost every hand, they wouldn't deal the game.
beachbumbabs
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September 24th, 2014 at 5:41:48 PM permalink
Quote: Dieter

Back 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.
If the House lost every hand, they wouldn't deal the game.
Dieter
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September 24th, 2014 at 9:50:29 PM permalink
Quote: beachbumbabs

Not 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.
May the cards fall in your favor.
Twirdman
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September 24th, 2014 at 11:42:19 PM permalink
Quote: Wizard

The 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.
Omaha
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September 25th, 2014 at 1:33:33 AM permalink
Giving someone with $100 $20 gives the same happiness as giving some with $100k $20k? Not buying it. Before it was EV now it's all about the happiness factor.
RS
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September 25th, 2014 at 12:20:13 PM permalink
What's happiness have to do with anything? Has happiness ever been used in an equation before? Why now?
thecesspit
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September 25th, 2014 at 1:05:41 PM permalink
Quote: RS

What'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.
"Then you can admire the real gambler, who has neither eaten, slept, thought nor lived, he has so smarted under the scourge of his martingale, so suffered on the rack of his desire for a coup at trente-et-quarante" - Honore de Balzac, 1829
AceTwo
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September 25th, 2014 at 1:35:59 PM permalink
Quote: RS

What'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)
Doc
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September 25th, 2014 at 2:30:48 PM permalink
When I was a draftee in the army, we had a simple equation/formula to calculate "Happiness Factor." It was the ratio of the number of days since going on active duty to the number of days remaining until separation from active duty. Happiness Factor started 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!
RS
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September 25th, 2014 at 3:12:08 PM permalink
I'm aware of variance, utility, etc. I'm not sure why the word happiness was used. Seems very odd.
Omaha
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September 25th, 2014 at 3:34:24 PM permalink
I'm the feeling of a millionaire spending a hundred grand.

-Nas
Omaha
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September 25th, 2014 at 3:45:25 PM permalink
I'm a poor man's dream

-Nas
bigfoot66
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September 25th, 2014 at 4:08:19 PM permalink
For me the EV is greater than infinity; I am a coin controller.
Vote for Nobody 2020!
chickenman
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September 26th, 2014 at 3:13:40 AM permalink
Quote: Doc

started 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!

Excellent, a new perspective on the short-timer's calendar
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