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## #1 2016-01-21 16:53:45

Relentless
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Registered: 2015-12-15
Posts: 624

I am disturbed by what is known as the St. Petersburg paradox.

Suppose I have an infinite bankroll, and I propose to you the following game: I will put \$2 in a jar and flip a fair coin, and each time it comes up heads I will double the money in the jar until it comes up tails, at which point you will get all the money in the jar. The problem is: How much should you pay to play this game?

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## #2 2016-01-22 12:58:27

evene
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Registered: 2015-10-18
Posts: 272

How much does it cost to play the game? And other factors come in weather or not I have enough money. But that's irrelevant.

Although I'm not sure how this is a paradox. So far, it seems like a regular problem. I give you \$2.00 and we start to play the game. Am I just being a total idiot here? Please explain the paradox part.

Last edited by evene (2016-01-22 12:58:45)

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## #3 2016-01-22 15:16:59

Relentless
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Registered: 2015-12-15
Posts: 624

Ok,  I will explain

I put \$2 in a jar, and flip a coin: if it comes up tails I will give you the \$2, and if it comes up heads I will double the money to \$4, and repeat. So if you were especially lucky and I flipped four heads and then a tail, you would get 2^5 = \$32.

The paradox part comes when you work out the expectation of the game. First, there is a 50% chance you will get tails first and get \$2 - this possibility is worth \$1 (outcome times probability). Then there is a one in four chance that you get tails on the second flip and earn \$4 - again, this possibility is worth \$1. Then there is a one in eight chance that tails comes up third and you get \$8 - expectation \$1 again.

This process continues indefinitely, and so when you add the value of the possibilities you get \$1 + \$1 + \$1 + ... infinitely many times, which tends to infinity.

In a nutshell, this game has an infinite expected value. Decision theory suggests that you should be willing to give up your house, millions, billions, anything at all to play this game: but half the time you walk away with \$2 and 87.5% of the time you walk away with \$8 or less. That's the paradox.

So, yes, the ordinary problem is: How much should you pay to play this game? (Anything)

The real problem is: Now come on, what should you REALLY pay to play this game?

Last edited by Relentless (2016-01-22 15:33:38)

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## #4 2016-01-22 21:47:31

bob bundy
Registered: 2010-06-20
Posts: 8,323

HI Relentless

Let's call the bankroller A and the player B.

I agree, the expected win for B is infinite.  A doesn't ever get a chance to 'win' so eventually B must win.

Conclusion.  Don't be A.

Bob

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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## #5 2016-01-22 22:09:25

Relentless
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Registered: 2015-12-15
Posts: 624

Hi bob

But doesn't that conclusion bother you? Would you give up your house to be the player in this game, where your chance of earning your house's value or better is worse than 250,000 to 1?

Last edited by Relentless (2016-01-23 00:07:02)

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## #6 2016-01-23 08:01:17

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,850
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I think you should regularize the divergent series..yielding an expectation of 4.

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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## #7 2016-01-23 08:14:52

Relentless
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Registered: 2015-12-15
Posts: 624

Hi Agnishom,

I have heard of a solution where dollar value is replaced by utility, or risk-aversion, and so the higher rewards that are much less likely become exponentially less valuable and the expectation is a bit over \$4. Can you please explain what is involved in this resolution?

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## #8 2016-01-23 15:25:00

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
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It is a mathematical trick as opposed to any philosophical or economical idea.

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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## #9 2016-01-23 15:54:38

Relentless
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Registered: 2015-12-15
Posts: 624

You get a prize of n * 2^x where x is the number of heads. And you get an expectation of the sum of n/2 + n/2 + n/2 + ... which seems to me to tend to infinity whenever n is positive.

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## #10 2016-01-23 16:39:16

Relentless
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Registered: 2015-12-15
Posts: 624

I have just learned that there is a result in mathematics where 1 + 1 + 1 + 1 + 1 + ... = -1/2

Perhaps that is the expectation, and the banker profits after all xD

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## #11 2016-01-23 18:18:26

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,850
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I'm sorry, that is not the correct way to regularise the series.

<will write back when free>

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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## #12 2016-01-23 18:28:15

Agnishom
Real Member
From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,850
Website

I might have been wrong in those posts, I'll check my math..

The divergent expectation is making me go crazy

Last edited by Agnishom (2016-01-23 18:38:23)

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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## #13 2016-01-24 20:34:45

Relentless
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Registered: 2015-12-15
Posts: 624

Hi,

I could be wrong because I am just working with borrowed knowledge, and I do not like the conclusion, but for the case of starting with n I am getting a regularised expectation of:

Last edited by Relentless (2016-01-24 20:39:06)

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## #14 2016-02-05 22:01:44

Relentless
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Registered: 2015-12-15
Posts: 624

So can anyone give a reasonable explanation of why this game is worth four dollars, an infinite amount, minus 50 cents, minus 8.3 recurring cents, or something else? Lol

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## #15 2016-04-30 19:48:57

Relentless
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Registered: 2015-12-15
Posts: 624

Does anyone yet have more to contribute to resolving this matter? I am afraid I have never quite gotten over it

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## #16 2016-05-01 04:34:10

Member
Registered: 2016-04-16
Posts: 1,086

Since the minimum returns is \$2 it is worth \$2. Anything you get above it is your profit.

{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

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## #17 2016-05-01 04:53:56

Relentless
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Registered: 2015-12-15
Posts: 624

That's true, it is worth at least \$2. But the question is: What is the maximum entry fee (if any) that you should be willing to pay to play this game? Or: As the coin tosser, how high do I need to set the entry fee to make this game worthwhile (if it ever can be)? Or: What is the value of this game?

Last edited by Relentless (2016-05-02 01:33:01)

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## #18 2016-05-02 19:26:01

Relentless
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Registered: 2015-12-15
Posts: 624

Nutted it out. Here's my answer:

The discrepancy between intuition and mathematics stems from several unrealistic features in the mathematical description of the game. Given the formal statement of the game, the player wishing only to maximise money should indeed pay any price at all to play. This conclusion is difficult to accept primarily because of bias towards risk-aversion, and the widespread, correct judgement that money in reality does not possess a linear utility. In fact, money usually possesses diminishing utility, and is sometimes modelled as proportional to the base-10 logarithm of wealth rather than to wealth itself. Applying this principle, the amount of utility derived from the game can be calculated as that given by precisely \$4. However, the paradox reemerges for a variant of the game which has infinite utility: the pot starts at \$100 and is multiplied by 10 to the power of 2 to the power of the number of heads for each additional heads. A like intuition might possibly suggest that this second game cannot, as a matter of certainty, be worth more than \$10 septillion, and is virtually certain to be worth closer to \$100,000; in any case, here too intuition does not consider it to be of infinite value.
The reason for this further discrepancy is the widespread and, again, correct judgement that there exists some huge amount of wealth after which absolutely no more utility is possible from gaining money. Suppose, for example, this amount is \$75 trillion, approximately world GDP. For the original game, this brings the monetary value down to \$47.06 floored, and the utility-adjusted value down to \$3.99 floored. For the variant game, this brings the monetary value down to \$9,375,012,502,550 (about \$9.375 trillion), and the utility-adjusted value down to \$54,247.86 floored. Clearly, the utility-and-maximum-wealth-adjusted values can provide reasonable answers given real human preferences.

Last edited by Relentless (2016-05-02 19:35:03)

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