Comments on: Gambling with the devil

By: Elliot Reed

Elliot Reed — Mon, 01 Mar 2004 05:42:47 +0000

I think the problem involves a conceptual mistake. Stepping back from the framework of utility theory, the only sense I can make of the idea of something being infinitely good is to see it as something I would prefer any positive chance of to anything else, and similarly with infinite badness. Put this way, there can’t be an infinite good and an infinite bad for the same reason there can’t be both an immovable object and an irresistable force.

By: zombiefreak

zombiefreak — Sun, 29 Feb 2004 00:22:26 +0000

Hey I’m just your average joe no math genius etc but the first day is the only day that you can take the bet and have a chance of winning based on the wording this is the only day it makes sense to take the deal , the devil will toss a fair coin once and if it comes up heads you are free (but if tails then you face eternal torment with no possibility of reprieve). If you wait consecutive days it does not increase the odds based on the consecutive tosses all tosses after the first toss are irrelevant if its tails you suffer eternal damnation if its heads your free 1 heads your free 1 tails your screwed wait as many days as you want your fate is still determined by the first toss. Read the fine print when dealing with the devil or you may get burned Lol Thanks Just my Opinion Zombiefreak

By: Michael Hoke

Michael Hoke — Wed, 03 Dec 2003 01:01:25 +0000

Whoa, what a mess of confusion! Lots of people with conflicting assumptions arguing over whose conclusion is least wrong! Lots of people dodging the question through casual theology! Lots of confused thinking about what a utility function represents! Lets try to sort through this mess. We start by assuming that the problem was stated fairly; that is (assumption 1): the coin is fair, Satan will stick to the bargain, there is no small print to the deal, time spent in hell could be an infinite sequence of days, etc. Chrismn almost described the problem correctly as a choice over lotteries, and dsquared almost pointed out the problem with that formulation (in his comments about infinite selves). The fact is that on any given day there are only two alternatives to choose from: flip now or wait. The ranking of those two options may change as time passes (which, evidently, is measured in days even in Hell), but presumably one has no way to commit oneself today to flip the coin a week from now, so how I feel today about the coin flip I'll face next week is irrelevant a week from now (assumption 2: no commitment mechanisms are available). Then the answer is simple: a rational person will choose to flip the coin when he prefers the lottery the flip represents to not flipping on that day (question: how does he evaluate how he feels about either option? Who cares? He has to evaluate them to make a "rational" decision, and I am unwilling to stipulate a method just yet). Because the lotteries faced each day change, and because the person deciding may feel differently from day to day (hey, he spent another day in Hell - he can change his mind about how he feels about it), minimal definitions of rational choice place no restriction whatever on when the coin might be flipped. It might be the first day, it might be the seventeenth, it might be the googolth (sp?), it might be never. Oh, that's crazy, you say. Naturally, any sensible person will have preferences that may be represented by a time-separable expected utility function where the arguments are days in heaven and hell, or something similarly banal, you say. Surely, this is a paradox, you say. Fine. State the assumptions: days already spent in Hell have no bearing on the decision except insofar as they determine the number of flips (assumption 3: no information gain), the relative merits of each option (flip today or wait) may be evaluated by comparing some real-valued function of the odds of winning to some real-valued function of the number of days that have passed (assumption 4: trivial in the two-option world), and these functions are the same every day (assumption 5: time consistent preferences, a very strong assumption). A good deal more hocus-pocus and we get that the preferences may be represented by some nice time-separable (subjective?) expected-utility function (assumption 6 involves strong restrictions on the shapes of these functions, such that each option may be represented as a complex lottery over simple lotteries over states, such as being in heaven or hell for a single day, as well as a whole mess of stuff like the vN-M axioms or Savage's postulates). Place restrictions on the relative values of a day in Heaven and Hell (assumption 7) and you can solve the problem definitively, up to a discount factor, or a patience parameter, or whatever. Whew. See, the problem is that we're collectively indecisive about what we require of a rational decision maker, and we're indecisive about what we think of the torments of Hell, and we forget how restrictive the various utility "theories" are (BTW, they're supposed to be restrictive - that's how they allow us to solve ever more complicated problems: by focusing our attention on fewer options). Fact is, this group has disagreed on the merits of every assumption I made. Who's surprised that there's no single answer to which we'd all agree?

By: ads

ads — Wed, 12 Nov 2003 03:15:02 +0000

The Devil, being a known liar, has hidden something from his victim. The only thing that the unfortunate soul burning in Hell can get from that offer is—hope. So, to get the only possible benefit from the situation, the doomed soul must forever wait until tomorrow before taking the coin toss.

By: chrismn

chrismn — Fri, 07 Nov 2003 03:45:29 +0000

I should have added— There is also nothing wrong with assuming that one is indifferent between all paths that eventually get to heaven. There is also nothing wrong with assuming that among paths which eventually get you to heaven, you prefer those which do it sooner. If heaven is infinite bliss, who’s to say it doesn’t make insignificant any finite length in hell?

By: chrismn

chrismn — Fri, 07 Nov 2003 03:38:57 +0000

Bill, I wasn’t clear. Your A, B, C preferences, where A is preferred to B and B to C, but C is so bad that the person will not accept any probability of C, no matter how small, in order to get an extra $900, are perfectly valid preferences. They are transitive and so forth. They do violate one of the expected utility axioms as you point out. In fact, they are what are called “lexicographic preferences,” as in a lexicon (or dictionary). All that matters in alphabetical ordering is the first letter, unless there is a tie, and then the second letter matters and so on. I was arguing that in my humble opinion, we shouldn’t like lexigraphic preferences. This is a well know example which violates the axiom you point out. I was simply arguing that to me, they don’t ring true. Again, my argument is that if you aren’t willing to accept any probability of C, no matter how small for an extra $900, then what you “really” care about is avoiding C. But if all you care about is avoiding C, then you are basically indifferent between A and B, and then the whole expected utility toolbox can be used again. But this is a taste issue. Nothing you said is wrong.

By: bill

bill — Fri, 07 Nov 2003 02:12:03 +0000

“In my example with no discounting above, you are indifferent between any sequences which eventually get you to heaven and prefer these to the sequence which has you in hell forever. This again can be represented by a U(eventually get to heaven) = 1, U(always stay in hell) = 0 utility function.” I think I’ve read all the comments carefully; I don’t see the argument that this utility function is correct. If the devil says “I give you two choices: Hell for a day, then Heaven forever; or Hell for two days, then Heaven forever,” I don’t see why I’m necessarily indifferent.

By: chrismn

chrismn — Thu, 06 Nov 2003 12:13:20 +0000

Bill writes: Let’s look at the following prospects: A – win $1000 B – win $100 C – “infinitely bad prospect” like eternity in Hell. One of the axioms of utility theory is something called “continuity” : it means that if A>B>C (true in my example) then there is some probability “p” between zero and 1 where you are indifferent between – getting B for sure and – getting a lottery where you have a p chance at A and a (1-p) chance at C. In my example, if p=1, then you prefer the lottery. If p is less than 1, then you prefer B for sure. You can see the discontinuous jump there is no p where you are indifferent. (End of quote). Bill is right that these preferences violate the utility theory axioms. On the other hand, it’s the utility axioms which are reasonable (IMHO) not this example. One way to put this is that if you are willing to accept no probability of C, no matter how small, in order to get an extra $900, then that is a different way of saying that all that really matters to you is not getting C. If you are indifferent between A and B and prefer both to C, then all the axioms are back and these preferences can be represented by a utility function U(A) = U(B) = 1, U( C) = 0. Then lotteries are ordered by their probability of delivering not C. In my example with no discounting above, you are indifferent between any sequences which eventually get you to heaven and prefer these to the sequence which has you in hell forever. This again can be represented by a U(eventually get to heaven) = 1, U(always stay in hell) = 0 utility function. The problem with this example of preferences is not finiteness of payoffs (or not), it’s that there are an infinite number of choices and no best choice.

By: Bill

Bill — Thu, 06 Nov 2003 08:49:56 +0000

(sorry for multiples; the symbol for “less than” messes up HTML real good :-) When you have “infinitely bad” prospects in the mix, utility theory breaks down. In other words, there is no such thing as “infinite utility”; it is (literally) a contradiction in terms. Proof: Let’s look at the following prospects: A – win $1000 B – win $100 C – “infinitely bad prospect” like eternity in Hell. One of the axioms of utility theory is something called “continuity” : it means that if A>B>C (true in my example) then there is some probability “p” between zero and 1 where you are indifferent between – getting B for sure and – getting a lottery where you have a p chance at A and a (1-p) chance at C. In my example, if p=1, then you prefer the lottery. If p is less than 1, then you prefer B for sure. You can see the discontinuous “jump”: there is no p where you are indifferent. This may seem trivial, but it is in violation of an axiom of utility theory. Therefore, you shouldn’t use the phrase “infinite utility”; to define utility, you can’t have discontinuities like that.

By: Bill

Bill — Thu, 06 Nov 2003 08:48:26 +0000

When you have “infinitely bad” prospects in the mix, utility theory breaks down. In other words, there is no such thing as “infinite utility”; it is (literally) a contradiction in terms. Proof: Let’s look at the following prospects: A – win $1000 B – win $100 C – “infinitely bad prospect” like eternity in Hell. One of the axioms of utility theory is something called “continuity” : it means that if A>B>C (true in my example) then there is some probability “p” between zero and 1 where you are indifferent between – getting B for sure and – getting a lottery where you have a p chance at A and a (1-p) chance at C. In my example, if p=1, then you prefer the lottery. If p>1, then you prefer B for sure. You can see the discontinuous “jump”: there is no p where you are indifferent. This may seem trivial, but it is in violation of an axiom of utility theory. Therefore, you shouldn’t use the phrase “infinite utility”; to define utility, you can’t have discontinuities like that.

By: Bill

Bill — Thu, 06 Nov 2003 04:16:12 +0000

By: Paul L

Paul L — Wed, 05 Nov 2003 16:32:57 +0000

This might have been alluded to in comments of ‘disutility’). If the torture/torment is infinitely experienced before the devil comes back the second day- the added ‘benefit’ of waiting another day is moot- you have already experienced eternal torment. It seems then that the devil’s trick is to negate an increasing finite probability (waiting days) with infinity (eternal torment).

By: chrismn

chrismn — Wed, 05 Nov 2003 04:03:13 +0000

The choice here is over a menu of lotteries over infinite lives. The first item on the menu is a 50/50 chance of receiving (Hell,Hell,Hell,…) and (Heaven, Heaven, Heaven, …). This is the lottery you get if you flip the first day. The next item is a 25/75 lottery over the sequences (Hell,Hell,…) and (Hell,Heaven,Heaven,…). This is the lottery if you flip the second day. The third is a 12.5/87.5 lotter over (Hell,Hell,…) and (Hell,Hell,Heaven,…) and so on. There are a countably infinite number of such choices. I wrote earlier that if people have a constant rate of discount beta and heaven and hell give finite per day payoffs, then there exists a probability p = 2(1-beta)/(2-beta) such that you are indifferent between flipping or not. For probabilities greater than p you wait, less than you flip. What does it mean to have a constant rate of discount and finite per-period payoffs of heaven and hell? Basically, rationality only imposes that people can order the lotteries above in a transitive way. If I prefer lottery A to lottery B and lottery B to lottery C, then I prefer lottery A to C. One way to do this is to find a way to map every certain sequence into a number and then have preferences over lotteries determined by the expected value of this lottery. This isn’t the only way to do it, but you can run into real problems otherwise. Suppose for instance that if lottery A has a lower probability than lottery B of staying in Hell forever, then I prefer lottery A. These preferences are transitive. Then which lottery should I choose. Why the best one of course! That is the basis of the theory of rational choice. If faced with a menu of choices, choose the best one. If there are a finite number of choices and you can rank them, (you don’t need to put numbers on them), then there is always at least one best one. (There could be ties, in which case there could be more than one best choice). But if there are an infinite number of items on the menu, it is possible there is no best choice. Suppose I am facing a choice of choosing whatever counting number I want and I like higher numbers better than lower numbers. Then there is no best choice. That is the same problem here. If one wants to minimize the probability of spending forever in hell, there is no best choice. But if our model of choice is we choose the best thing on the menu, and there is no best choice, then the problem is simply ill defined.

By: pw

pw — Tue, 04 Nov 2003 21:34:14 +0000

For those arguing about things like discount rates for future suffering, this problem is pretty much the same as the Supernova Problem for currency values. In this neck of the woods it should be called the Red Giant Problem, but what the heck. If you knew with certainty that the dollars you were about to receive for doing some job would be worth half their value tomorrow, you’d want twice as many of them as you would if you knew they would be worth the same tomorrow as today. Simple stuff: current value of monetary instruments depends on the expected future value of the money. Well, some day a few billion years from now, all the dollars on earth will be worth exactly nothing, because they (along with mountains and oceans and everything else but a small refractory core) will be boiled away by the aging sun. The day before that happens, dollars will also be worthless because they’re about to be vaporized, so who wants them. The day before that… and so on, with the result that no possible discount rate can explain the fact that dollars and yen and ecu are worth something today. Of course they are, and for good reason, but you won’t find it in the simplified math.

By: meursalt

meursalt — Tue, 04 Nov 2003 21:13:41 +0000

A revision to my earlier comment. Everyday the “devil” is going to come and entice you with a better offer the “next day” and next thing you know, you’re in hell forever. Do it now. Do it today. There’s always a reason to do it tomorrow.