Another sadistic angel problem

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 defined as 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)?

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.

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

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.

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.

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.

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.

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.

Negative infinity multiplied by zero is not identicaly 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.

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.

The numbers that God draws are never explicitly stated 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 finite number. 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 wasn’t quite so bad.” Even then, though, I’d decline the arrangement based on any of the above points.

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.

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.

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

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.

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 impossible that you end up tortured?

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 “It’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?

>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”???

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 independent—and assuming this is the sort of puzzle where the actors tell the truth—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_!

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

What flavour is the lollipop?

Jack wrote:

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.

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

Perhaps this is a loophole: if God actually has to demonstrate to me that X and Y are equal, then the unpleasant fate of mankind can be infinitely postponed.

Quiggin wrote:

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.

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 not to 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.)

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.

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.

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.

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!

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 fixed finite 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) each have 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…

(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.”

Given that you don’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.

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.

A few small notes:

* Non-standard analysis has nothing to 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.

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.

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.

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

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

Because this is merely a gedankenexperiment“?

‘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,