Omaha
Omaha
Joined: Sep 18, 2014
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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
thecesspit
Joined: Apr 19, 2010
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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
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
Doc
Joined: Feb 27, 2010
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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
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
Omaha
Joined: Sep 18, 2014
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September 23rd, 2014 at 8:41:22 PM permalink
Quote: andysif

says who



Every mathemetician
Wizard
Administrator
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.
It's not whether you win or lose; it's whether or not you had a good bet.
andysif
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
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
Neutrino
Joined: Feb 20, 2014
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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.

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