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From: Iain Stewart <ids.domain.name.hidden>

Date: Wed, 05 May 1999 17:26:18 +0100

Stefan Rueger <smr3.domain.name.hidden> wrote:

*> Iain,
*

*>
*

*> Let box one contain m pounds and box two 2m pounds, where m is
*

*> any real positive number. You may choose a box look at the
*

*> amount of money inside and decide either to keep this amount or
*

*> the other box's amount. (For this exercise you may think of
*

*> cheques with real numbers on it and a special bank account where
*

*> you can actually bank any real number of pounds.)
*

*>
*

*> In Peter's inaugural, he suggested that if you pick a box at
*

*> random and find x pounds, you might conclude that the other box
*

*> either contains x/2 or 2x pounds, the average of which is 1.25x.
*

*> So you are better off (on average by 25%!) choosing the other
*

*> box. He left it to the audience to decide whether this conclusion
*

*> is valid or not.
*

*>
*

*> It was immediately clear to me - and I am sure, it is immediately
*

*> clear to you - that the argument is flawed.
*

Errrm... do you mean the argument exactly as given above (with the

conclusion that choosing randomly and then always switching - which is

of course just a different style of "choosing randomly"! - is 1.25 times

better than not switching) is flawed? If so, then you and I certainly

agree it's flawed, and probably Pete agrees too! (Though he'll have to

speak for himself of course.) Or do you mean the "meta-argument" (which

correctly points out the absurdity of one random 50-50 choice generation

protocol being 1.25 times better than another, but goes on to conclude

sweepingly that *no* choice protocol, random or non-random, can be

better than any other for this problem) is flawed? If this latter, then

I agree with you that the "meta-argument" is flawed - though maybe for

different reasons than yours! - and I think Pete possibly *disagrees*

and takes the meta-argument to be just fine, sweeping conclusion and

all! (Again, of course, he'll have to speak for himself... I really

shouldn't play guessing games with other people's opinions like this. I

wasn't even at his inaugural talk after all!!)

*>
*

*> We both know that
*

*> choosing a box at random (eg, using a fair coin) and keeping the
*

*> contents gives you 1.5m pounds on average. The same is true by
*

*> symmetry if you choose a box randomly and keep the contents of
*

*> the **other** box: both methods must add up to the 3m pounds.
*

*>
*

Exactly. That's the "correctly points out" clause at the beginning of

what I'm calling the meta-argument above. Fine so far. And indeed any

other *non-quantity-of-money-driven* elaboration of "50-50 random

choice" would be no better or worse too. You can think up really

glorious protocols here, like choosing a box, choosing an integer to say

how many times to "dither back and forth" (of course this integer may as

well be restricted to 0 or 1 since dithering back and forth is cyclic

mod 2), and then duly "dithering" that many times before finally

announcing the box one settles on. All these

non-quantity-of-money-driven elaborations are the same as just flipping

a coin and getting the thing over with.

*>
*

*> Peter's argument has the following flaw: The core argument is
*

*> that reading one box's cheque (say, amount x) makes the contents
*

*> of the other box's cheque amount a random variable with values 2x
*

*> and x/2 and a uniform distribution (50% probability for either).
*

*> But the contents of the other box is not random at all! It is
*

*> either 2x or x/2, and this depends on the experimentor's mood
*

*> when the boxes were set up.
*

I agree here - everything depends on the distribution the experimenter

used to generate the secret number m (the number which is used to fill

the boxes with m and 2m) in the first place. Even if the experimenter

later says something like "But I didn't use a distribution! I just shut

my eyes and thought of a number on a whim!", that statement is

scientifically meaningless - after all their "whim" has to be encoded

somehow in their brain state, and so the distribution is "whatever

probabilities brains in that whim-state have for thinking up numbers".

By hook or by crook, there's a distribution for m lurking in there!

*> This is a
*

*> classical example for
*

*> assuming a probability distribution for something that you don't
*

*> know. People who do this
*

...(including me!)...

*> call themselves Bayesians.
*

*> The problem
*

*> with the Bayesian approach is that in some cases, when you are
*

*> not careful, it can produce misleading results.
*

Ah, but all good Bayesians are careful. :-)

I think what you mean by being "not careful", in the context of this

example, is taking the distribution for the question "Is the box I've

just opened the m box or the 2m box?" to always be 50-50, *even after

you've read the quantity of money it contains*. That would indeed be not

just careless but wrong - it is, after all, 50-50 *before* you've read

the quantity of money, and *therefore*, for that very reason, is not in

general 50-50 *after* you've read the quantity of money. If you had

access to the experimenter's secret distribution for generating m in the

first place (which is not the same thing as access to the value of m

itself on this occasion, I hasten to add!), you could perform the update

in a fully proper and knowledgeable way. If you are a Bayesian, then

even in the absence of access to the experimenter's distribution for m,

you take it upon yourself to do the next best thing to this: you make

your own guess as to "what kind of distributions experimenters like when

playing games with people", you weight *these* distributions according

to a guessed set of weightings ("meta-distribution"?) of your own, you

collapse the meta-distribution of "ordinary" distributions into one

consolidated distribution by the standard

"multiply-it-all-out-and-add-it-all-up" method, and you use *that*

next-best-thing consolidated distribution (and your just-obtained

knowledge of the quantity of money in the box you opened) to update your

weightings - previously 50-50 - for that dangling question this long and

rambling paragraph started off with: "Is the box I've just opened the m

box or the 2m box?". Then, of course, you switch iff your updated

weightings favour the unopened box. Phew! :-)

*>
*

*> Thinking deeper about Peter's box problem after the inauguration,
*

*> I thought that there might be a similar experiment that actually
*

*> increases your expected return of 1.5m.
*

*>
*

*> You may find this astonishing, but I actually came up with a
*

*> strategy that has an expected return of **more than** 1.5m -
*

*> no matter what m is (as long as m is bigger than zero).
*

*>
*

*> In order to increase suspension a little and give you some time
*

*> to think about this, I'll reveal my method to you - after you
*

*> reply to this mail saying something to the effect of "It would be
*

*> astonishing and completely unbelievable that such a method should
*

*> exist".
*

*>
*

*> Regards!
*

*>
*

*> Stefan
*

Well, as should now be clear, I wouldn't use *those* words - and in fact

the algorithm I said a good Bayesian would follow (the business about

the consolidated distribution, in my long and rambling paragraph above)

*would*, I believe, increase the expected return above 1.5m in most

real-life cases. Even using a very crude caricature of what I called the

meta-distribution in that paragraph would probably raise the expected

return a wee bit above 1.5m. So, I'm not saying such an algorithm would

be "astonishing and completely unbelievable". Not at all! But of course

I'd still like you to tell me your method. I'm too tired to work out a

particular instantiation of my own "algorithm in words only" in my long

and rambling paragraph above! :-)

Bye for now... (and apologies to those on the Theories of Everything

mailing list for whom this might all be a bit out of context - but it is

a fairly well-known "toy probability paradox" in the literature, and

Stefan's explanation of it at the beginning is quite self-contained.)

Iain.

Date: Wed, 05 May 1999 17:26:18 +0100

Stefan Rueger <smr3.domain.name.hidden> wrote:

Errrm... do you mean the argument exactly as given above (with the

conclusion that choosing randomly and then always switching - which is

of course just a different style of "choosing randomly"! - is 1.25 times

better than not switching) is flawed? If so, then you and I certainly

agree it's flawed, and probably Pete agrees too! (Though he'll have to

speak for himself of course.) Or do you mean the "meta-argument" (which

correctly points out the absurdity of one random 50-50 choice generation

protocol being 1.25 times better than another, but goes on to conclude

sweepingly that *no* choice protocol, random or non-random, can be

better than any other for this problem) is flawed? If this latter, then

I agree with you that the "meta-argument" is flawed - though maybe for

different reasons than yours! - and I think Pete possibly *disagrees*

and takes the meta-argument to be just fine, sweeping conclusion and

all! (Again, of course, he'll have to speak for himself... I really

shouldn't play guessing games with other people's opinions like this. I

wasn't even at his inaugural talk after all!!)

Exactly. That's the "correctly points out" clause at the beginning of

what I'm calling the meta-argument above. Fine so far. And indeed any

other *non-quantity-of-money-driven* elaboration of "50-50 random

choice" would be no better or worse too. You can think up really

glorious protocols here, like choosing a box, choosing an integer to say

how many times to "dither back and forth" (of course this integer may as

well be restricted to 0 or 1 since dithering back and forth is cyclic

mod 2), and then duly "dithering" that many times before finally

announcing the box one settles on. All these

non-quantity-of-money-driven elaborations are the same as just flipping

a coin and getting the thing over with.

I agree here - everything depends on the distribution the experimenter

used to generate the secret number m (the number which is used to fill

the boxes with m and 2m) in the first place. Even if the experimenter

later says something like "But I didn't use a distribution! I just shut

my eyes and thought of a number on a whim!", that statement is

scientifically meaningless - after all their "whim" has to be encoded

somehow in their brain state, and so the distribution is "whatever

probabilities brains in that whim-state have for thinking up numbers".

By hook or by crook, there's a distribution for m lurking in there!

...(including me!)...

Ah, but all good Bayesians are careful. :-)

I think what you mean by being "not careful", in the context of this

example, is taking the distribution for the question "Is the box I've

just opened the m box or the 2m box?" to always be 50-50, *even after

you've read the quantity of money it contains*. That would indeed be not

just careless but wrong - it is, after all, 50-50 *before* you've read

the quantity of money, and *therefore*, for that very reason, is not in

general 50-50 *after* you've read the quantity of money. If you had

access to the experimenter's secret distribution for generating m in the

first place (which is not the same thing as access to the value of m

itself on this occasion, I hasten to add!), you could perform the update

in a fully proper and knowledgeable way. If you are a Bayesian, then

even in the absence of access to the experimenter's distribution for m,

you take it upon yourself to do the next best thing to this: you make

your own guess as to "what kind of distributions experimenters like when

playing games with people", you weight *these* distributions according

to a guessed set of weightings ("meta-distribution"?) of your own, you

collapse the meta-distribution of "ordinary" distributions into one

consolidated distribution by the standard

"multiply-it-all-out-and-add-it-all-up" method, and you use *that*

next-best-thing consolidated distribution (and your just-obtained

knowledge of the quantity of money in the box you opened) to update your

weightings - previously 50-50 - for that dangling question this long and

rambling paragraph started off with: "Is the box I've just opened the m

box or the 2m box?". Then, of course, you switch iff your updated

weightings favour the unopened box. Phew! :-)

Well, as should now be clear, I wouldn't use *those* words - and in fact

the algorithm I said a good Bayesian would follow (the business about

the consolidated distribution, in my long and rambling paragraph above)

*would*, I believe, increase the expected return above 1.5m in most

real-life cases. Even using a very crude caricature of what I called the

meta-distribution in that paragraph would probably raise the expected

return a wee bit above 1.5m. So, I'm not saying such an algorithm would

be "astonishing and completely unbelievable". Not at all! But of course

I'd still like you to tell me your method. I'm too tired to work out a

particular instantiation of my own "algorithm in words only" in my long

and rambling paragraph above! :-)

Bye for now... (and apologies to those on the Theories of Everything

mailing list for whom this might all be a bit out of context - but it is

a fairly well-known "toy probability paradox" in the literature, and

Stefan's explanation of it at the beginning is quite self-contained.)

Iain.

-- I have discovered a truly marvellous Theory of Everything. Unfortunately this T-shirt is too small to contain it. (Or: yeah, sure, the Theory of Everything will fit on a T-shirt. But in what size of font?)Received on Wed May 05 1999 - 09:33:37 PDT

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