Decision-Making Paradox

Relentless · 2016-01-21 16:53:45

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?

evene · 2016-01-22 12:58:27

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)

Relentless · 2016-01-22 15:16:59

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)

Bob · 2016-01-22 21:47:31

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

Relentless · 2016-01-22 22:09:25

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)

Agnishom · 2016-01-23 08:01:17

I think you should regularize the divergent series..yielding an expectation of 4.

Relentless · 2016-01-23 08:14:52

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?

Agnishom · 2016-01-23 15:25:00

It is a mathematical trick as opposed to any philosophical or economical idea.

What happens if you start with $n instead of $2?

Relentless · 2016-01-23 15:54:38

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.

Relentless · 2016-01-23 16:39:16

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

Agnishom · 2016-01-23 18:18:26

I'm sorry, that is not the correct way to regularise the series.

Agnishom · 2016-01-23 18:28:15

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)

Relentless · 2016-01-24 20:34:45

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)

Relentless · 2016-02-05 22:01:44

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

Relentless · 2016-04-30 19:48:57

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

thickhead · 2016-05-01 04:34:10

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

Relentless · 2016-05-01 04:53:56

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)

Relentless · 2016-05-02 19:26:01

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)

Math Is Fun Forum

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

Decision-Making Paradox

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Re: Decision-Making Paradox

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