Imagine that one day, a big bloke with wings taps you on the shoulder. It’s OK, he says, Brian sent me. To offer you this potential wager, on behalf of God, who has more or less given up on the human race except as a subject for philosophy conundrums.

In the envelope in my left hand, he says, I have a number, called X. At some point in the recent past, X was drawn by God from a uniform distribution over the real numbers from 0 to 1 inclusive. You can have a look at it if you like.

In my right hand, he says, I have a mobile telephone which will allow me to receive a message from God with another number, Y, which will also be drawn by God from a uniform distribution on the line 0 to 1 inclusive.

The wager is this; if you accept the wager, and X and Y are equal, then every human being currently alive on the planet earth will be horribly tortured for the next ninety million trillion years and then killed. If you accept the wager but X and Y are not equal, then a small, relatively undeserving child somewhere, will be given a lollipop.

So, do you take the wager or not?

“Go on”, says the angel. “Look at it as a problem of utility maximisation. Just look at the utility associated with each possible outcome, multiplied by the probability of that outcome.

“In fact, it’s quite easy. Define p as the probability that Y equals X. In the favourable outcome (which occurs with probability 1-p), a child gets a lollipop, which increases the sum of utilities by a small amount. In the unfavourable outcome (occuring with probability p), the sum utilities is reduced by a massive amount. So what is p?

“Well, what’s the probability that a continuous random variable Y will be equal to a particular value X?

“It’s zero.

“Therefore, if you look at the calculation, accepting the bet gives a zero chance of a very horrible outcome, and therefore a (1-0) certain small increment to utility, so you should take it”

“Hurray”, you say, aware that angels have in the past not be so forthcoming in explaining the mathematics. But you are still nagged by doubts; wouldn’t the laugh be rather terribly on you if Y turned out to equal X after all?

{ 68 comments }

dsquared 09.17.04 at 3:35 am

I think that there are two clues here as to what’s going on:

1: The probability of a continuous variable taking on any particular value is

definedas zero, but I suspect that this formal metamathematical statement is weaker than the metaphysical statement that it couldn’t happen.2: I’m bringing God in to do the drawing for a reason; I’m using Him to obfuscate the fact that I’m playing funny buggers with the axiom of choice. How do you go about picking a single point from the real number line, with no particular identifying features of that point to go on (which is what a uniform distribution means)?

Kieran Healy 09.17.04 at 3:38 am

This looks like Pascal’s wager in disguise.

Kieran Healy 09.17.04 at 3:41 am

Except for the clever bits which direct attention to the probability that a draw taken from a distribution of that sort is defined as zero.

dsquared 09.17.04 at 3:43 am

Yes, it’s got a lot of the same structure; it’s appealing to the attractive decision-theoretic heuristic that one ought not to take on even small risks of very bad outcomes, and like Pascal’s Wager, the queasy feeling it engenders is the result of someone trying to persuade you that a probability which you believe to be zero is actually a small, non-zero probability. Think of it as a Bayesian fable about dogmatic priors …

Kieran Healy 09.17.04 at 3:44 am

Ah, well you wrote that down above while my second comment was processing I see. Perhaps I’ll just be quiet now.

alkali 09.17.04 at 3:54 am

Fortunately, for the angel to jot down all the digits of that real number Y will take an infinite amount of time, so the negative payoff will never take place. (The positive payoff would be realized in finite time; as soon as one digit of Y differed from the corresponding digit of X, it’s lollipop time.)

alkali 09.17.04 at 3:55 am

Fortunately, for the angel to jot down all the digits of that real number Y will take an infinite amount of time, so the negative payoff will never take place. The positive payoff would be realized in finite time; as soon as one digit of Y differed from the corresponding digit of X, it’s lollipop time.

Brad DeLong 09.17.04 at 3:57 am

Is God going to write out both numbers so that we can see them? Then the fact that we have finite lives and light travels at a finite speed puts limits on the precision with which he can express the numbers, so the probability is non-zero.

On the other hand, rig it so that the only thing God tells you is that the first difference between the numbers occurs in the nth decimal place, and you are safe going for the lollipop…

Henry 09.17.04 at 3:58 am

Was recently reading a comment James Morrow makes on the conditions for von Neumann Morgenstern utility functions that seems relevant. Quoting from p. 31 of his _Game Theory for Political Scientists_

bq. [t]he third condition (that for each consequence there exists a lottery involving the most and least preferred outcome so that the actor is indifferent between the lottery and the outcome) also contains a very important pair of assumptions – the best outcome is not so desirable that the decider will accept any lottery with a nonzero chance of the best outcome over any other outcome for certain, and the worst outcome is not so undesirable that the decider will accept any other outcome for certain over any lottery with a nonzero chance of the worst outcome. Put another way, the third condition excludes infinite utilities for outcomes. Infinite utilities produce very bizarre behavior. For example, some argue that nuclear war has a negative infinite utility. If one accepts this position then you must prefer any outcome without nuclear war – imagine your own personal nonnuclear nightmare here – for certain to any lottery that gives you any nonzero probability of nuclear war. The nonnuclar nightmare is preferred to a lottery in which everyone on earth flips a coin simultaneously and if all five billion-plus coins come up heads, a nuclear war results, while if even one coin comes up tails, world peace (or your own personal idea of bliss) is realized. I find it hard to believe that you would choose the nonnuclear nightmare for certain over the lottery. My point here is not that infinite utilities do not “exist” but, rather, that infinite utilities have some bizarre consequences for the choices predicted.

Of course, his argument is about lotteries with _very small_ probabilities, rather than lotteries with zero probabilities, and the alternative is world peace rather than lollipops but (a) if your qualification (1) stands, and (b) world peace (or whatever idea of bliss) has a very high but not infinite positive utility, then Morrow’s analogy is fairly close to your own.

John Quiggin 09.17.04 at 3:59 am

Ask to look. If you can see the number it must have a finite description (e.g. as a decimal or surd), and this is an event with probability zero.

So, whatever your priors, you can conclude with probability one that the description given to you by the angel is wrong.

Steve Laniel 09.17.04 at 4:04 am

I don’t know the formal measure theory, but my understanding is that this isn’t a matter of mere definition — except to the extent that all axiomatisations are mere definition.

Let’s imagine instead that X and Y are drawn from the discrete uniform distribution on the points {1/n,2/n,…,(n-1)/n,1}. X=Y at precisely n points ({X=1/n,Y=1/n}, {X=2/n,Y=2/n},…,{X=1,Y=1}), out of the n^2 possible points that the ordered pair (X,Y) can take on. So Pr(X=Y)=n/n^2 = 1/n. Letting n go to infinity, this tends to 0.

Is that not a rigorous enough argument? Again, I’m not familiar with the measure theory, so there may be glaring problems with this. I’d love (seriously!) to hear what they are.

John Quiggin 09.17.04 at 4:11 am

Lots of simultaneous comments here!

Let me take the chance to plug my own work on this. Once you make the probabilities very small instead of zero you can use rank-dependent expected utility theory which allows for high weights on low-probability extreme outcomes. If you want, you can make the weight discontinuous at zero.

So whatever your utility function, you can justify refusing bets which have a low probability of a very bad outcome.

Dan Kervick 09.17.04 at 4:24 am

Two points:

The probability of the numbers being equal is only zero if we assume probabilities can only take values in the real interval [0,1]. But if we allow non-standard distributions, and let probabilities take infinitesimal values, then the probability that the numbers are equal is not zero, but rather 1 divided by aleph-1, assuming the truth of the continuum hypothesis. The expected utility calculation still says you should take the bet, since the values of the outcomes involved are finite. But at least this accounts for the intuition that the proposition that the two numbers are different is not an epistemic certainty (probability = 1) but instead has a probability infinitesimally less than 1.

Second point: The description of the problem specifies not only that the two numbers will be randomly selected by God, but also that they will be communicated to the participants by (i) being inscribed on something, presumably a piece of paper, contained in the envelope in the angels hand, and (ii) via a cellphone call.

These are finite physical systems, and a little thought should convince you that in each case there is a finite upper bound on the number of real numbers that can be represented by the use of these systems. So the probability that the two numbers are equal is actually some very, very small, but non-zero, real number.

That at least opens up the possibility that the expected value of taking the bet is lower than the expected value of not taking it. I’m inclined to think that taking the bet will still come out on top, but it is now a very difficult calculation.

dargie 09.17.04 at 4:40 am

*tsk* God and that angel both need lollipops of their own.

Martin Bento 09.17.04 at 5:15 am

Ah, yes, the precision of the number as represented matters quite a bit. If the precision is one, then the probability of either outcome is .5. It will round to zero or to one. Precision 2? Stay away, grasshopper. With finite precision, you could always end up screwing the pooch. What? The angel didn’t say anything about precision? The underlying question in monkey’s paw stories: Is God pulling my chain here? You gotta remember to look for the trick. For that matter: is anything random to the eye of God?

PS. I know this doesn’t matter to the question, but giving sugar to kids is a bad outcome, not a good one. This is one perverse Angel.

Fleb 09.17.04 at 5:24 am

It’s a bad idea to take the bet, because if anyone could make something impossible happen, it’d be God. So go buy a kid a lollipop yourself.

WillieStyle 09.17.04 at 6:04 am

Negative infinity multiplied by zero is

notidenticaly zero. If the infinity represented by the bad outcome (and lets face it, it is infinitely bad) is greater than the infinity of real numbers in [0,1], then the product will actually be negative infinity, not zero.Actually, the puzzle would be much more interesting if the bad outcome wasn’t quite so bad.

Andrew Boucher 09.17.04 at 6:20 am

Unless the point is to play with the probability that the two real numbers are equal is 0, then I don’t see much difference in this example than the classic of selling out-of-the-money puts on the S&P. The theory says, on average, you will earn money. But on the other hand, when there’s a crash, you lose a fortune. Basically, one accepts a limit to how much one can lose in any single event, so one doesn’t sell an extremely huge number of puts.

I think you can probably phrase the problem as follows (again, unless the whole idea is the zero probability factor). Suppose there is 999 999 999/1 000 000 000 probability of killing 10 people and 1/1 000 000 000 of killing 8 000 000 000 people. If this is only one event, which do you choose?

Chan 09.17.04 at 6:30 am

The problem isn’t really one of anylzing the utility function. It is a question of who to trust. Anyone willing to torture all humanity for 10 E15 years will cheat. Never let dishonorable opponents define the terms.

Jacques Distler 09.17.04 at 6:43 am

Two points:

1) The idea that we should overweight low-probability strongly-negative outcomes is a commonplace. The insurance industry is based on that premise. After all, the insurance company is only willing to insure you if the present-value of the expected payout is less than the premium you pay them. Nevertheless, you happily pay the premium, knowing that you are covered, should disaster befall you.

2) As others have noted, the statement of the problem is bogus. Anything that can be written down on a piece of paper and stuck in an envelope or spoken over a finite-time cell-phone call is, at best, a finite-precision decimal number. The probability that two such numbers are equal is certainly nonzero.

On the other hand, God, with his impecable penmanship, can easily fit 100 decimal digits onto a slip of paper and, speaking quickly, can recite a similar number of digits over the phone.

The probability that two such hundred-digit numbers coincide is slightly larger than 10^{-200}.

A very small number indeed (the number of protons in the universe being a mere 10^{80}).

Nick Simmonds 09.17.04 at 6:55 am

The numbers that God draws are never

explicitlystated to be random. Thus, bad bargain–the probability is not dependent any longer on the infinity of real numbers but rather the presumption of God’s goodwill, on which it is not reasonable to bet 4.7 x 10^30 human-hours of torture against a lollipop.For that matter, I would argue that any time you are offered such a deal, no matter how explicitly the terms are stated, you should decline; even if the probability of a bad outcome is zero, the probability of your understanding the problem incorrectly is non-zero.

Then there’s the argument that giving a lollipop to a random undeserving child is not necessarily desireable at all. If both outcomes have negative utility, there’s no reason at all to accept.

Also, the 90 million billion years of torure may be finite, but I would argue that the death of all humanity actually has infinite negative utility, in that it is an unrecoverably bad state. Thus, the real negative utility of the situation is not just infinite, but a negative infinity plus a very large finite number. Even multiplied by zero, an infinity is an undeterminable but

finitenumber. That finite number has a very large probability of being higher than the very small utility of giving a child a lollipop, even if the child deserves one to begin with. As williestyle says, “the puzzle would be much more interesting if the bad outcome wasnâ€™t quite so bad.” Even then, though, I’d decline the arrangement based on any of the above points.John Quiggin 09.17.04 at 8:21 am

“The idea that we should overweight low-probability strongly-negative outcomes is a commonplace. The insurance industry is based on that premise.”

The explanation of insurance offered in expected-utility theory is unrelated to probability weights – it depends on marginal utility of income. Consistent models of choice under uncertainty nonlinear in the probabilities weren’t developed until the 1980s. Mark Machina has some good survey articles which you could find with Google if you’re interested.

Chris 09.17.04 at 8:26 am

Three words (well, an abbreviation and two words): St. Petersburg Paradox. This is why expected value theory is such a problem, even for things like Pascal’s Wager, in which even the remotest probabilities yield an infinite expected reward or cost, and therefore the rational thing to do is to “wager” the maximum possible amount.

Of course, things get even weirder when you step outside of the world of theoretical decisions and expected value calculations, and consider actual decision making behavior.

Jor 09.17.04 at 8:36 am

I really need to take a class on measure theory to figure this stuff out.

Paul Orwin 09.17.04 at 8:52 am

Aha, I’ve hit upon the answer (or, I’m really tired). You must take the bet. There are two possibilities; One, you really are dealing with God and thus he would never let X=Y (that’s why he’s God), or Two, you are actually dealing with the Devil, in which case you are screwed anyway, so you might as well take your chances.

As a side note, I can’t understand Dan Kervick’s comment above. How can the probability value p lie outside the real interval [0,1]. Are there complex p values? I’d love an explanation, although it may be beyond my ken.

bad Jim 09.17.04 at 9:03 am

I’m with

fleb. Who in his right mind would trust a messenger of god? For that matter, giving a kid a lollipop might not actually be a good thing (granting that the prospective recipient might disagree).Arbitrarily many real numbers may be precisely specified by their properties. Who is unacquainted with e or pi?

Anyone familiar with Fortran IV immediately understands that GOD is real, unless declared integer. Programmers of that time were nonetheless prone to making equality tests on reals. One of the questions that comes to mind, were one to take this seriously, is what language is god using?

John Quiggin 09.17.04 at 9:33 am

Chris, the rank-dependent model, which is one of the precursors of cumulative prospect theory was designed largely to deal with the Allais “paradox”.

Chris 09.17.04 at 10:36 am

John Q.,

Denzel played you pretty well in the movie ;).

Yeah, I know the rank-dependent model, and prospect theory. I still think our behavior is weird, even if explainable. The more I learn about decision making (with all its sub-optimality, flaws in intertemporal reasoning, valuation and devaluation effects, etc.), the less I ever want to make a decision again.

Jack 09.17.04 at 11:00 am

The infinite length of the real number is a red herring. The probability of it taking very long at all to show a difference in decimal expansions is low.

I think if the angel was cheating or god is beyond randomness then it isn’t clear to me that the preson making the “wrong” choice is responsible in any case.

My solution is to turn down the wager and buy a kid a lolipop myself but as with others I’m suffering from vertigo.

You could make the question weirder by offering the choice based on whether there is a three in the decimal expansion.

Jack 09.17.04 at 11:10 am

Paul O,

Dan is, I think, talking about a Non-Standard Analysis approach to the issue. Non Standard Analysis is roughly speaking the result of treating infinitesimals as real objects rather than just convenient fictions.

Much of the time this doesn’t make much difference but sometimes it allows finite ways of achieving things normally done with infinite parts of calculus and other times, as here, it allows a convenient way of keeping track of auxillary facts.

dsquared 09.17.04 at 11:31 am

Just one or two technical adustments:

First, although the angel has a mobile phone in his hand, it’s playing a purely symbolic role. When God makes his random draw, he communicates the number Y to you by direct divine revelation, which is of course instantaneous.

Second, I don’t think John’s first solution works. Say you take a look at X and it’s 0.25. You’re still faced with fundamentally the same problem; are you confident enough abut the validity of measure theory that you’re going to say that it’s

impossiblethat you end up tortured?Jack 09.17.04 at 12:20 pm

If he presents an argument in measure theoretic friendly terms and they turn out not to be applicable then the god is cheating.

If god is cheating then what I decide in the game is of no importance, he could just penalise me for fun anyway and trying to second guess god has to be a waste of time.

Even if god didn’t present the number to me by divine revelation it would only take seconds to know within a one in a google chance whether it was right. In fact there is a reason for accepting the bet — what would, with probability one, take an infinite amount of time to demonstrate is that the two numbers were the same. Maybe the entire human race would be put to work verifying the identity of the two numbers. Or it might be X=0, Y=1 if there is a zero of the zeta function with real part not equal to a half and 0 otherwise. There had to be a reason the angel was so specific about the endpoints.

Steve 09.17.04 at 1:30 pm

Hey, I just forge a memo that says what I want ….

Really, though, I give a kid a lollipop paid for out of my own pocket and skip the risk, or ….

I take the representation that the probablility is zero as the defining conditional (thus

â€œItâ€™s zero.”being a condition precedent to acceptance) and if the angel goes “hey, I was wrong” or “I lied” then what else would he be wrong about.After all, who dealing with an angel is truly rational or stays on topic?

John Quiggin 09.17.04 at 1:39 pm

DD, I think you misinterpret my solution.

Obviously my faith in angels who know Brian is just about absolute, but say I have a 10^-10 subjective probability that the angel is lying to me.

When I see 0.25, which has probability zero if the angel is telling the truth, but some finite probability if the angel is lying, I update my priors and conclude with probability 1 that the angel is lying.

I don’t have a solution for how to deal with dishonest sadistic angels, but as soon as I finish the details of my new plan to bring peace and democracy to Iraq I’ll get to work on it.

a different chris 09.17.04 at 3:33 pm

>Then the fact that we have finite lives and light travels at a finite speed puts limits on the precision with which he can express the numbers

Naw. I’m not even God, and yet how many digits do I express just by writing “3*PI”???

Matt Weiner 09.17.04 at 4:05 pm

I decline the bet, because I don’t think it’s rational to maximize expected utility in situations like this. An Allais-paradox type thing.

Jeff Erickson 09.17.04 at 4:35 pm

The angel never explicitly stated that X and Y would be drawn from *independent* uniform distributions. (Perhaps drawing X in the past and drawing Y in the future is _supposed_ to imply that the distributions are independent, but it doesn’t.) So I would not take the wager.

If the distributions are independentâ€”and assuming this is the sort of puzzle where the actors tell the truthâ€”then you might as well take the bet and give the kid her lolly with probability 1. Torturing every human being for ninety million trillion years and then killing them is not _infinitely_ bad, merely mind-bogglingly unbelieveably hugely (but finitely) bad. Torturing everyone for ninety _billion_ trillion years would be worse, but still only finitely bad. Torturing everyone for Graham’s number of years would be much _much *much*_ worse, but still only finitely bad. Most finite numbers are incomprehensibly huge, but they’re all infinitely smaller than infinity, and their product with zero is still zero.

John Quiggen comments “When I see 0.25, which has probability zero if the angel is telling the truth, but some finite probability if the angel is lying, I update my priors and conclude with probability 1 that the angel is lying.” But you could make precisely the same argument no matter what number is written on the paper; *every* number has probability zero of being X. So why should you open the envelope?

The finite-representation thing seems like a red herring. I’m willing to assume that God can correctly (and will honestly) compare any two real numbers, even though that’s a formally undecidable problem. After all, he’s _God_!

Jason 09.17.04 at 4:46 pm

Another Chris

“Naw. Iâ€™m not even God, and yet how many digits do I express just by writing â€œ3*PIâ€???”

Yes, you represent a lot of digits, but not so many in base pi. The problem is that suppose the number were to be communicated to you in some misture of English, standard Mathematical expressions, and a few other languages thrown in. There are still only a finite number of symbols to use (in an extreme, only a finite number of atoms in the universe etc.), and so the number of values expressible in a finite amount of time is finite.

Sebastian Holsclaw 09.17.04 at 5:33 pm

Clearly you should take the bet. “allow me to receive a message from God with

anothernumber, Y,”.Is it cheating to abuse lexical ambiguity to take the problem out of the problem?

I think the objection that you have a non-zero chance of misinterpreting the problem is an excellent non-mathematical answer.

dsquared 09.17.04 at 5:36 pm

John; what if I offered you a choice between the first bet I offered, and a second bet under which you lose if Y is a rational number?

Tim Lambert 09.17.04 at 6:04 pm

What flavour is the lollipop?

Brian Weatherson 09.17.04 at 6:07 pm

Depending on how exactly our physics works, it may be possible to do this without worries about communication. Let X be the _exact_ temperature in one room, and Y the _exact_ temperature in an adjacent room that by human standards seems to be at the same temperature. Now there are worries with this method. Some of them have to do with vagueness, though I think that could be avoided by a precise definition of the room. Some of them have to do with worries about whether temperatures can take any real value, or whether there are weird quantum effects that mean temperatures are discrete. (This is sloppy because I know next to no relevant physics, but hopefully the idea is clear enough.) But if the temperatures can be any real value, and if God can measure them exactly, and if you believe both of the above, then there’s a reason to think seriously about Daniel’s bet.

Me, I take the torturing to be infinitely worse than the kid getting a lollipop, so I don’t take the bet. That doesn’t mean it has infinite disutility (whatever that means) it just means it is infinitely worse than the alternative.

In general I think there’s a fascinating question here about how expected utility theory handles zero probabilities. Take out the torture from the case, so we just have the following bet.

If the two rooms’ temperatures are identical, a small undeserving child gets a lollipop. Otherwise nothing happens.

Provided you put any positive value on small undeserving children getting lollipops, the bet looks like it should have strictly positive value for you. But according to orthodoxy, it has zero value. I take that to be a reason to not like numerical measurement of the values of bets, but some others may disagree.

Jacques Distler 09.17.04 at 6:33 pm

Jack wrote:

It take a finite amount of time to show that two real numbers X and Y are unequal. But it takes an

infiniteamount of time to show that they areequal.Perhaps this is a loophole: if God actually has to

demonstrateto me that X and Y are equal, then the unpleasant fate of mankind can be infinitely postponed.Quiggin wrote:

Thanks for the references, but why not apply similar reasoning here? Why is a straightforward expected-utility calculation supposed to yield the “right” answer here, whereas a more sophisticated analysis is required in the insurance “game”?

For the hell of it, let me rephrase D^2’s game. To avoid the red-herring of measure theory, X and Y will be 100-digit decimal numbers, still chosen with independent uniform distributions on the interval [0,1]. But now, rather than merely giving you the choice of accepting the wager, the angel demands $100

notto take the cell-phone call.If you refuse to cough up the $100, there is a finite (but extraordinarily tiny) probability that mankind will suffer a ghastly fate.

I can make the dilemma even more pointed by noting that there are a panoply of vastly more probable (but still unlikely) calamities that you would be able to forestall if you hold onto the $100. (So much for the marginal utility of income.)

Jacques Distler 09.17.04 at 6:34 pm

Jack wrote:

It take a finite amount of time to show that two real numbers X and Y are unequal. But it takes an

infiniteamount of time to show that they areequal.Perhaps this is a loophole: if God actually has to

demonstrateto me that X and Y are equal, then the unpleasant fate of mankind can be infinitely postponed.Quiggin wrote:

Thanks for the references, but why not apply similar reasoning here? Why is a straightforward expected-utility calculation supposed to yield the “right” answer here, whereas a more sophisticated analysis is required in the insurance “game”?

For the hell of it, let me rephrase D^2’s game. To avoid the red-herring of measure theory, X and Y will be 100-digit decimal numbers, still chosen with independent uniform distributions on the interval [0,1]. But now, rather than merely giving you the choice of accepting the wager, the angel demands $100

notto take the cell-phone call.If you refuse to cough up the $100, there is a finite (but extraordinarily tiny) probability that mankind will suffer a ghastly fate.

I can make the dilemma even more pointed by noting that there are a panoply of vastly more probable (but still unlikely) calamities that you would be able to forestall if you hold onto the $100. (So much for the marginal utility of income.)

Brett Bellmore 09.17.04 at 7:12 pm

Obviously, you decline the bet. It’s not a philosophical problem, it’s a test of your moral values, and if you accept the bet, you go to Hell.

John Quiggin 09.17.04 at 9:47 pm

Following Jacques and Dan, I think you can rationalise rejecting the bet with maximin or discontinuous-weight rank-dependent preferences, taking the view that infinitesimals are nonzero.

The field of angelic-sadistic decision theory is in its infancy, but clearly looks set to grow rapidly, perhaps unboundedly.

Jacques Distler 09.17.04 at 10:30 pm

Actually, my reformulation of Dan’s game put it in the standard form of an “insurance” game.

Risk is not cost-free. A certain loss of $100 is preferable to a possible loss of $1 million with probability 10^{-4}.

It is a perfectly reasonable question to ask whether $100 is a reasonable premium to pay to insure against the aforementioned ghastly fate befalling humanity (with some small, but finite probability).

But it’s not reasonable to ask questions which require an infinite amount of computation to decide.

Abiola Lapite 09.17.04 at 10:55 pm

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Hash: SHA1

“Anything that can be written down on a piece of paper and stuck in an envelope or spoken over a finite-time cell-phone call is, at best, a finite-precision decimal number.”

Who says the description has to be in digit form? As long as we don’t restrict ourselves in that manner, we have all of the algebraic numbers to choose from.

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iD4DBQFBS11EOgWD1ZKzuwkRAqPTAJj4GO3/PiLE8A6hCnCBgmT9gfQBAJ9DajXV

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HP 09.17.04 at 11:22 pm

Is God a moral actor? God chooses to offer the wager, so clearly God is capable of choice. Therefore, God can also freely choose to torture humanity or to not torture humanity. If I accept the bet, and lose, who bears the responsibility for the torture of humanity? Me, or God? God has the option of choosing not to torture humanity even after I lose the wager. Am I going to then insist that he not back out on the deal?

If a kidnapper kidnaps my child, and says he will kill the child unless I pay a ransom, and I refuse to pay, and he kills the child, who is responsible for the child’s death? The kidnapper, says both law and common sense.

I would argue that the bet was made and lost long ago, and that the torture of humanity followed by ultimate destruction is a pretty good description of “life.” How does God torture humans? With red-hot irons and buckets of flaming pitch? Or with disappointment, heartbreak, frustration, grief, anxiety….

I’d take the bet. Hey, a lollipop!

Jacques Distler 09.17.04 at 11:36 pm

1) Are the algebraic numbers are uniformly-distributed in the interval [0,1]? (I would guess that they are, but that’s the sort of number theory question that I’m not too strong on.)

2) While it’s certainly true that describing an algebraic number is a finite operation (hence removing one of my objections), it’s not clear to me that proving the equality of an arbitrary pair of algebraic numbers is a finite operation.

I imagine a proof that it

isa finite operation would involve arguing that anypairof algebraic numbers lie in a finite extension of the rationals.But, heck, why do we need to go to algebraic numbers? There are an infinite number of rational numbers in the interval [0,1]. If God’s so smart, he presumably has an algorithm for choosing rational numbers in [0,1] with a uniform distribution. [This, as D^2 alluded to, is the really tricky bit, which is why he had God do the picking.]

That’ll suffice because

a) any rational number has a finite presentation

b) comparing any pair of rational numbers for equality is a finite operation.

c) there are still a (countably) infinite number of them — so the probability that two “randomly-chosen” ones are equal vanishes.

Jacques Distler 09.17.04 at 11:56 pm

Alas, I think I spoke too soon. While there are, indeed, an infinite number of rationals (or algebraic numbers) in the interval [0,1], we are interested in those which can be described by a

fixedfinite amount of data.For instance, for the rationals, we might describe X as the solution to

a X – b = 0

for two coprime integers, a and b, with a>b. But if we demand that a and b (say)

eachhave a maximum of 50 digits, then there are only a finite number of rationals that can be so-described.And similarly for other algebraic numbers.

I don’t think you can get around the limitations of “a piece of paper” that way…

g 09.18.04 at 12:34 am

A few answers to Jacques’s technical questions:

1. There’s no answer to the question “are the algebraic numbers uniformly distributed in [0,1]?”. (Ditto for the rationals.) But the following is true: There is no probability distribution on the algebraic (or the rational) numbers in [0,1] that assigns probability p to their intersection with any interval of length p. Which kinda-sorta says that there’s no way of choosing one uniformly at random.

(Proof of that fact: there are only countably many of them. So they can’t all have zero probability, by countable additivity. But if the number

qhas non-zero probability, then the probability of any small enough interval aroundqmust be bigger than some constant, hence bigger than the length of that interval.)2. Given any two (finite!) descriptions of algebraic numbers (by which I mean, as I think everyone else here does, descriptions that make it obvious that the numbers are algebraic) it’s straightforward to determine whether they are equal.

(Proof: the restriction on the descriptions is basically equivalent to specifying both numbers as “the unique root of such-and-such an integer polynomial within such-and-such an interval”. There’s an easy algorithm for finding the greatest common divisor of two such polynomials, and another for checking whether it has a root in the specified interval. That’s all we need.)

2a. However, instead of “rational number” or “algebraic number” you could have substituted “computable number”. There is

notan algorithm for determining, given two (explicit) computable numbers, whether they are equal.(Proof: If you could do that, you could solve the halting problem. I’m too lazy to fill in the details.)

3. But, of course, if the numbers are required to be specified in a

fixedamount of space then there exists an algorithm for determining whether they are equal. Though you might not be able to carry it out :-).(Proof: there are only finitely many cases for the algorithm to test. So an algorithm that gets them all right must exist.)

-*- -*- -*- -*- -*-

There’s a lot of technical quibbling here, which is of course half the fun. But I think most of the key issues remain if we bypass all the technicalia and propose instead that God has chosen two million-digit integers, uniformly (and independently: good catch, Jeff) at random.

So, what then? (I think I take the bet, assuming I decide that having a lollipop actually benefits the child.)

“hp”: Suppose what the kidnapper says is: If you buy your son a lollipop today then I’ll torture your daughter. Otherwise I’ll release her unharmed. And suppose you buy that lollipop and your daughter gets tortured. Do you really think you don’t bear *any* of the responsibility? It seems to me that the present proposal is more like that than like a typical kidnapping-with-ransom; all you have to do to avoid the torturing of the human race is to decline the bet.

dsquared 09.18.04 at 1:09 am

(reposting material from the thread attached to Brian’s post)

How about if the angel instead says:

“Some time in the last few minutes, God set a bicycle wheel spinning, and it is currently rotating at 200rpm in a horizontal plane. Consider the point P, which is the centre of gravity of the valve on that bicycle wheel, and the circle C described by P as the wheel rotates.

Some time in the next ten seconds, step forward. If P is at the most northerly point of C at the instant when the centre of gravity of your body is exactly 1 metre from the centre of gravity of my body, everyone dies, otherwise the kid gets the lollipop.â€

Given that you donâ€™t know when and with what orientation God set the wheel spinning, then you only need the fairly reasonable Bayesian assumption that zero information should be reflected by a uniform prior distribution, and you’ve got a decision problem more or less equivalent to the one described above.

I think that this sorts out all the “piece of paper” issues related to decimal expansions; I’ve used centres of gravity because they are proper and well-defined geometric points, in order to avoid any references to possibly vague or quantised physical events.

Jack 09.18.04 at 2:26 am

Surely it is even harder to determine the location of your centre of gravity, the moment it was one metre from the wheel, theprecise position of the wheel, its fluctuation due to temperature changes and air currents caused by your movement, corrosion air resistance giving rise to leading number issues. It still assumes infinite accuracy in the measurements which will be at least as hard to determine as the random number.

dsquared 09.18.04 at 3:18 am

yes exactly; I think a lot of people on this thread are getting hung up on X and Y as “Numbers” in the sense of decimal expansions. I’m trying to move thinking toward a physical example to make people focus on an *event* of probability zero.

Dan Kervick 09.18.04 at 4:38 am

To Paul Orwin,

Jack is right; I was thinking about non-standard analysis. Intuitively, if x is the cardinality of some set, then the probability that any given real member of that set will be chosen on a random selection is 1/x. Assuming the cardinality of the reals is aleph-1, then the corresponding probability is 1 over aleph-1. Since aleph-1 is greater than any real number, 1 over aleph-1 is an infinitesimal number – it is less than any non-zero real, but greater than 0.

I’m not sure what I think about infinitesimals, but appealing to them does provide a way of accounting for the intuition that the proposition that the two numbers in God’s drawing will be the same is not an epistemic impossibility.

To dsquared,

While I see where you are going, saying that God communicates the number by direct divine revelation doesn’t avoid the problem. Since my brain is a finite system, there are only finitely many real numbers God can communicate to me.

Perhaps, though, we can capture the version of the problem you want to think about by just specifying that God will simply communicate to you whether the two randomly selected numbers were the same or different, and that you are certain – for whatever reason – that God won’t lie.

Anarch 09.18.04 at 6:07 am

A few small notes:

* Non-standard analysis has

nothingto do with cardinality considerations. The easiest way to create a non-standard model of the reals (and, in some sense, the only way to do so) is via an ultraproduct, typically with an ultrafilter over N. Infinitesimals then become [equivalence-classes of] decreasing sequences; again, nothing whatsoever to do with cardinality.Insofar as invoking reciprocated cardinality is possible, you’d have to use Conway’s universal ordered embedding field. I know very little about this, save that weirdness can, and will, ensue.

* I believe, though haven’t checked recently, that the set of computable reals — the real objects under consideration, if we’re being somewhat pedantic here — is uniformly distributed through [0,1] in some sense. [I’m fairly sure they’re equidistributed under certain “nice” orderings, in fact, but like I said I haven’t checked this.] In that sense you can actually pick one “uniformly at random”, though I’m not sure exactly what that means in this context; perhaps picking the next line of your Turing machine at random would suffice?

* D2, your most recent post doesn’t really get around the problem of specifying an inherently infinite quantity (barring quantization of space, which opens up a whole new kettle o’ fish) in a finite amount of time. Either you have to give us poor humans a way to verify equality in a finite amount of time or simply declare that God knows whether the numbers are the same or not and proceed from there.

Jacques Distler 09.18.04 at 6:13 am

I don’t see how that materially changes the issue.

Whether we are trying to determine whether two randomly-chosen real numbers are precisely equal, or trying to determine the

precisedistance between our centers-of-gravity at some particular instant, neither is achievable in finite time by mere mortals.What I find puzzling, though, is that you seem unwilling to study this problem as the limit of a

sequenceof problems in which there is afiniteprobability of a ghastly outcome (offinitedisutility) as I and several others have suggested.For each problem in this sequence, the calculation of what to do is mundanely straightforward. Why should the limit be exotic?

Alan Bostick 09.18.04 at 7:13 am

I know what I think my time is worth, and I know the value of a lollipop. The possibility of the eternal torture of all humanity is big and scary enough that I’m going to want to think for a bit and make sure everything is on the up-and-up before accepting the angel’s wager.

I value my time such that the amount of time I would need to satisfy that the bet is a good one is worth more to me than the prospect of a lollipop being given to someone else. Thus, even if I were to conclude that the probility of the numbers matching was in fact infinitely small, it’s still a negative-EV bet, because it costs me more to accept it than it does to show the angel the door without considering it.

digamma 09.18.04 at 8:03 am

I see a lot of philosophers, game theorists, and economists posting here. I suspect some

dentalexpertise would supply an easy answer to the problem.But here’s my reply to the angel: “I am not an expert on game theory, and I don’t trust you. Those two factors make the risk of torture too great for me, zero probability be damned. I decline your wager.”

Jack 09.18.04 at 10:29 am

I think the finitness issues are intrinsic in the difficculty of satisfactorily understanding the behaviour of a possible event of probability actually zero. Nobody has actually experienced such a thing.

For dan K, there is a disconnect between cardinality and measure. The example I gave above of numbers without a three (seven would work just as well etc.) in their decimal expansion has the same cardinality as the interval and has measure zero.

Mats 09.18.04 at 10:38 am

Now, if the ratio of the harm done by “every human being currently alive on the planet earth will be horribly tortured for the next ninety million trillion years and then killed” to the harm done by the child in question not having the lollipop, is *known and finite* – then we’re home safe: take the bet.

Otherwise we need more information, and I guess at least I would need more some mathematical skills beyond the concept of convergence and other stuff from basic theoretical calculus.

bluemeanie 09.18.04 at 3:22 pm

As a misanthrop, I would take the bet and hopefully lose. Then I would beg the angel to go double or nothing.

j 09.18.04 at 3:33 pm

Given the small potential positive payout, isn’t the opportunity cost of talking to this angel and considering this wager, greater than any potential benefit of winning this wager. If so, why even waste time even considering such a trivial benefit?

Abiola Lapite 09.18.04 at 9:30 pm

“If so, why even waste time even considering such a trivial benefit?”

Because this is merely a

gedankenexperiment“?J Thomas 09.19.04 at 7:29 am

Say you look at the paper and it unfolds to exactly one meter long. There is a line drawn on the paper that’s somewhere between 0 and 1 meter.

You accept the bet and the angel places the call. God sings into the phon for a time that’s somewhere between 0 and 1 minute.

Both God and the angel can tell whether the two lengths are exactly the same. These lengths can serve as numbers for God and the angel.

Something like that might work to get past the objections about infinite numbers of digits.

My question is how many human beings are there? I’ve heard of about 6 billion. But is this the only universe that has humans? What if there are an infinite number of universes with humans in them? What if the angel gives the choice an infinite number of times? And what if a loss means losing all the humans of all the earths, not just one?

He called it “earth” but that might be just how he thinks.

If he can give an uncountable number of humans the choice, then it may not be probability zero at all. Then our only chance is for a whole lot of them to say no.

Andrew T 09.19.04 at 8:58 pm

‘When I see 0.25, which has probability zero if the angel is telling the truth, but some finite probability if the angel is lying, I update my priors and conclude with probability 1 that the angel is lying.’

is an odd comment. The probability of any set number being on the paper is zero. It matters not at all if it is 0.25 or 0.3234528974523… This sort of reasoning causes people to bet on numbers that are ‘due’ in the lottery.

Regards,

j 09.21.04 at 11:09 pm

Abiola, Your answer is not a response. As Nietzsche commented,

“There came a time when people rubbed their foreheads. People are still rubbing them today. They had dreamed: first and foremostâ€”the old Kant. “By means of a faculty,” he had said, or at least meant. But is that an answer? An explanation? Or is it not rather a repetition of the question? How does opium make people sleep? “By means of a faculty,” namely, the virtus dormitiva, answered that doctor in Moliere.

Quia est in eo virtus dormitiva

Cujus est natura sensus assoupire*

But answers like that belong in comedy.”

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