Comments on: Another sadistic angel problem

By: j

Tue, 21 Sep 2004 23:09:40 +0000

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 foremost—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.”

By: Andrew T

Andrew T — Sun, 19 Sep 2004 20:58:15 +0000

‘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,

By: J Thomas

J Thomas — Sun, 19 Sep 2004 07:29:08 +0000

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.

By: Abiola Lapite

Abiola Lapite — Sat, 18 Sep 2004 21:30:07 +0000

“If so, why even waste time even considering such a trivial benefit?” Because this is merely a gedankenexperiment“?

By: j

Sat, 18 Sep 2004 15:33:00 +0000

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?

By: bluemeanie

bluemeanie — Sat, 18 Sep 2004 15:22:28 +0000

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

By: Mats

Mats — Sat, 18 Sep 2004 10:38:12 +0000

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.

By: Jack

Jack — Sat, 18 Sep 2004 10:29:35 +0000

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.

By: digamma

digamma — Sat, 18 Sep 2004 08:03:50 +0000

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

By: Alan Bostick

Alan Bostick — Sat, 18 Sep 2004 07:13:22 +0000

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.

By: Jacques Distler

Jacques Distler — Sat, 18 Sep 2004 06:13:15 +0000

I’m trying to move thinking toward a physical example to make people focus on an event of probability zero.

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 precise distance 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 sequence of problems in which there is a finite probability of a ghastly outcome (of finite disutility) 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?

By: Anarch

Anarch — Sat, 18 Sep 2004 06:07:58 +0000

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.

By: Dan Kervick

Dan Kervick — Sat, 18 Sep 2004 04:38:50 +0000

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.

By: dsquared

dsquared — Sat, 18 Sep 2004 03:18:23 +0000

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.

By: Jack

Jack — Sat, 18 Sep 2004 02:26:22 +0000

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.