If everybody reasons as you describe then everyone will guess 1/∞ and everyone will tie. You can't get closer to 2/3 of an infinitesimal than an infinitesimal, so it's stable.
Disclaimer: I'm not mathy. Maybe you actually can get closer to 2/3 of an infinitesimal than an infinitesimal.
The question required us to provide real numbers, and infinitesimals are not real numbers. Even if you allowed infinitesimals, though, 0 would still be the Nash equilibrium. After all, if 1/∞ is a valid guess, so is (1/∞)*(2/3), etc., so the exact same logic applies: any number larger than 0 is too large. The only value where everyone could know everyone else's choice and still not want to change is 0.
I use Firefox with NoScript and StumbleUpon. If I stumble a site that does not show any content because it needs javascript, I do not white list it (I do not need to see the dancing pigs), I just go on to the next stumble.
An implementation that does not need client side scripting could reach a larger audience.
Doing this with server side scripting is crazy. You'd have to submit a zillion forms and take a second to get the answer for each try. This is precisely the sort of thing client-side scripting is meant for.
Of course, the page would explain that it needed JavaScript, if you had JavaScript disabled, not just show a blank page.
I got the wrong rule, but it said I was right because I made only one mistake. I thought the rule was that a sequence was awesome if it was an increasing arithmetic progression. The only one of your examples at the end that contradicted this was 2, 9, 15. All the other awesome ones were, in fact, increasing arithmetic progressions: five out of the six awesome sequences you gave at the end. You should probably cut that down to two or three, so I'd have lost.
there's no term for "how surprised I was" in Bayes' Theorem.
Not quite. The intuitive notion of "how surprised you were" maps closely to bayesian likelihood ratios.
Regarding your die/beads scenarios:
In your die scenario, you have one highly favored model that assigns equal probability to each possible number. In the beads scenario you have many possible models, all with low probability; averaging their predictions gives equal probability to each possible color.
To simplify things, let's say our only models are M, which predicts the outcomes are random and equally likely (i.e. a fair die or jar filled with an even ratio of 12 colors of beads), and not-M (i.e. a weighted die or jar filled with all the same color beads). In the beads scenario we might guess that P(M)=.1; in the die scenario P(M)=.99. In both cases, our probability of red/one is 1/12, because neither of our models tell us which color/number to expect. But our probability of winning the bet is different -- we only win if M is correct.
That clears things up a lot. I hadn't really thought about the multiple-models take on it (despite having read the "prior probabilities as mathematical objects" post). Thanks.
Do you have empirical evidence that Mormons are more likely to cause harm than atheists? (Let's say in the clear-cut sense of stabbing people instead of in the sense of spreading irrationality.)
I'll claim that, yes, I do have such evidence. The Mormon Church funded many advertisements in favor of California Proposition 8 which denies civil rights to homosexuals.
Even accepting the premise that voting for the proposition was clearly wrong, that's a single anecdote. It does nothing to demonstrate that Mormons are overall worse people than atheists. It is only a single point in the atheists' favor. I could respond with examples of atheists doing terrible things, e.g., the amount of suffering caused by communists.
Anecdotes are not reliable evidence; you need a careful, thorough, and systematic analysis to be able to make confident statements. It's really surprised me how commonly people supply purely anecdotal evidence here and expect it to be accepted (and how often it is accepted!). This is a site all about promoting rationalism, and part of that is reserving judgment unless you have good evidence.
I really don't think a systematic analysis of the morality of Mormons vs. atheists exists, for any given utility function. That kind of analysis is probably close to impossible, in fact, even if you can precisely specify a utility function that a lot of people will agree on. To begin with, it would absolutely have to be controlled to be meaningful ― the cultural, etc. backgrounds of atheists are surely not comparable on average to those of Mormons.
I think this is an issue that rationalists just need to admit uncertainty about. That's life, when you're rational. Only religious people get to be certain most of the time about moral issues. A Mormon asked the same question would be able to say with confidence that the atheists caused more evil, since not following Mormonism is so evil that it would clearly outweigh any minor statistical differences between the two groups in terms of things like violent crime. If you believe in utility functions that depend on all sorts of complex empirical questions, you really can't answer most moral questions very confidently.
I think this post could have been more formally worded. It draws a distinction between two types of probability assignment, but the only practical difference given is that you'd be surprised if you're wrong in one case but not the other. My initial thought was just that surprise is an irrational thing that should be disregarded ― there's no term for "how surprised I was" in Bayes' Theorem.
But let's rephrase the problem a bit. You've made your probability assignments based on Omega's question: say 1/12 for each color. Now consider another situation where you'd give an identical probability assignment. Say I'm going to roll a demonstrated-fair twelve-sided die, and ask you the probability that it lands on one. Again, you assign 1/12 probability to each possibility.
(Actually, these assignments are spectacularly wrong, since they give a zero probability to all other colors/numbers. Nothing deserves a zero probability. But let's assume you gave a negligible but nonzero probability to everything else, and 1/12 is just shorthand for "slightly less than 1/12, but not enough to bother specifying".)
So as far as everything goes, your probability assignments for the two cases look identical up to this point. Now let's say I offer you a bet: we'll go through both events (drawing a bead and putting it back, or rolling the die) a million times. If your estimate of the probability of red/one was within 1% of correct in that sample, I give you $1000. Otherwise, you give me $1000.
In the case of the die, we would all take the bet in a heartbeat. We're very sure that our figures are correct, since the die is demonstrated to be fair, and 1% is a lot of wiggle room for the law of large numbers. But you'd have to be crazy to take the same bet on the jar, despite having assigned a precisely identical chance of winning.
So what's the difference? Isn't all the information you care about supposed to be encapsulated in your probability distribution? What is the mathematical distinction between these two cases that causes such a clear difference in whether a given bet is rational? Are we supposed to not only assign probabilities to which events will occur, but also to our probabilities themselves, ad infinitum?
Alicorn, I think it'd be appropriate to add the following link at the beginning of the article:
Related to: Priors as Mathematical Objects.
It also kinda answers your questions.
Even if Omega had asked about the bead being lilac, and you'd dutifully given a tiny probability, it would not have surprised you to see a lilac bead emerge from the jar.
I see this conclusion as a mistake: being surprised is a way of translating between intuition and explicit probability estimates. If you are not surprised, you should assign high enough probability, and otherwise if you assign tiny probability, you should be surprised (modulo known mistakes in either representation).
Predicting the second bead given the color of the first one can also be expressed as probability estimates for joint observations, made before you observe the color of the first bead. What is the probability that you'll see two reds? That you'll see a red followed by a non-red? Non-red following by a red? Two non-reds? Then crunch the numbers through the definition of conditional probability/Bayes' theorem.
I see this conclusion as a mistake: being surprised is a way of translating between intuition and explicit probability estimates. If you are not surprised, you should assign high enough probability, and otherwise if you assign tiny probability, you should be surprised (modulo known mistakes in either representation).
That's not true at all. Before I'm dealt a bridge hand, my probability assignment for getting the hand J♠, 8♣, 6♠, Q♡, 5♣, Q♢, Q♣, 5♡, 3♡, J♣, J♡, 2♡, 7♢ in that order would be one in 3,954,242,643,911,239,680,000. But I wouldn't be the least bit surprised to get it.
In the terminology of statistical mechanics, I guess surprise isn't caused by low-probability microstates ― it's caused by low-probability macrostates. (I'd have been very surprised if that were a full suit in order, despite the fact that a priori that has the same probability.) What you define as a macrostate is to some extent arbitrary. In the case of bridge, you'd probably divide up hands into classes based on their utility in bridge, and be surprised only if you get an unlikely type of hand.
In this case, I'd probably divide the outcomes up into macrostates like "red", "some other bright color like green or blue", "some other common color like brown", "a weird color like grayish-pink", and "something other than a solid-colored ball, or something I failed to even think of". Each macrostate would have a pretty high probability (including the last: who knows what Omega's up to?), so I wouldn't be surprised at any outcome.
This is an off-the-cuff analysis, and maybe I'm missing something, but the idea that any low-probability event should be surprising certainly can't be correct.
It would make her right. And that would be all it would do - if she were lucky.
Huh. Do you need me to post a few dozen links to articles detailing incidents where Mormons did evil acts because of their religious beliefs? I mean, Mormonism isn't as inherently destructive as Islam, but it's not Buddhism either.
Anyway, even if Wednesday ended up living her life without once doing harm to others or to herself because of her beliefs, deconverting would still be a good idea: At the very least, theism will distort the rest of her priorities, because they will be in competition with delusion-based priorities like "I want to please God", and "I want my friends and family to go to the highest level of Heaven". Becoming an atheist would therefore allow her to put the right importance on her real priorities.
Huh. Do you need me to post a few dozen links to articles detailing incidents where Mormons did evil acts because of their religious beliefs? I mean, Mormonism isn't as inherently destructive as Islam, but it's not Buddhism either.
Do you have empirical evidence that Mormons are more likely to cause harm than atheists? (Let's say in the clear-cut sense of stabbing people instead of in the sense of spreading irrationality.) Mormons might do more bad things because their god requires it, but atheists might do more bad things because they don't have a god to require otherwise. They might be more likely to become nihilists or solipsists and not care about other people, say, acting purely selfishly. A priori, I have no idea which one is correct.
It seems that as a rationalist, you should be wary of assigning high probabilities here without direct empirical evidence. Especially since you presumably suffer from in-group bias. But perhaps you're aware of studies that support your view that religion is harmful in a simple sense?
(If you consider spreading religion inherently evil, then you have more reason to presume that Mormonism is harmful. You would still have to argue that the harm outweighs any possible benefit, but you'd have a stronger case for assuming that. However, by your comparisons to Islam and Buddhism you seem to mean plain old violence and so forth.)
This post raises a whole constellation of connected questions, so here are my thoughts on all of them:
If the question is "Can Wednesday be religious and still be a smart person who's good at using rationality?", the answer is empirically yes (eg Robert Aumann).
If the question is "Can we still call Wednesday rational if she's religious?" the answer is to taboo "rational" and let the problem take care of itself.
If the question is "Is it okay for Wednesday to be religious?" the question is confused in the first place and any answer would be equally confused.
If the question is "Should Wednesday choose to believe religion?" the answer is that you don't voluntarily choose your beliefs so it doesn't matter.
If the question is "Should Wednesday, while not exactly choosing to believe religion, avoid thinking about it too hard because she thinks doing so will make her an atheist?," then she's already an atheist on some level because she thinks knowing more will make her more atheist, which implies atheism is true. This reduces to the case of deception, which you seem to be against unconditionally.
If the question is "Should I, as an outside observer, do my best to convince Wednesday religion is wrong?" the answer depends on your moral system. I'm a utilitarian, so I would say no - I think it's a background assumption here that she's happier being deceived. I know you're not a utilitarian, so you'd have to work it out in whatever system you use.
If the question is "Should we at Less Wrong exclude all theists?", my answer is of course not. If they want to come here and talk about prisoners dilemmas or the Singularity or something, then of course we should welcome their opinions.
If the question is "Should we at Less Wrong tell all theists they can't talk about how great religion is?" my answer is a qualified "yes". Not because we specifically hate religion, but for the same reason we don't allow posts explicitly about politics. There are places for those debates, this isn't one of those places, and having them completely changes the feel of a community and saps its energy.
If the question is "Should we at Less Wrong stop acting like atheism is an open-and-shut case?," my answer is "no". Sometimes in order to move on, we've got to accept certain assumptions. For example, even though there are a few hard-core steady state theorists out there, most astronomers have accepted the Big Bang as a default assumption because they can get more done by building on Big Bang theory and working out its exact implications then they can debating the last few steady-staters ad nauseum or refusing to even mention the beginning of the universe because it might exclude someone. Christians work in exactly the same way; when they want to discuss obscure points of theology, they start from the assumption that God exists and work from there, although they'll discard that assumption when they're debating an atheist. I don't hold it against these Christians - they'd hardly be able to do theology without it - and I hope they don't hold it against us.
If the question is "Should we at Less Wrong stop saying mean things about religion?" then my answer is that we should never deliberately say mean things just for the sake of saying mean things, but that if it's absolutely necessary to condemn religion to make some greater point (like to use it as an example of a bias towards anthropomorphism) then it's not worth refraining from it to prevent potentially some hypothetical theist from feeling excluded. However, writers should make sure to phrase it as neutrally and non-insultingly as possible, something atheists are generally bad at.
If the question is "What kind of person would name their daughter Wednesday?", I have no good answer. Maybe someone who really, really liked the Thursday Next books?
Also, this wins my prize for most intriguing title on LW so far.
If the question is "Should Wednesday, while not exactly choosing to believe religion, avoid thinking about it too hard because she thinks doing so will make her an atheist?," then she's already an atheist on some level because she thinks knowing more will make her more atheist, which implies atheism is true. This reduces to the case of deception, which you seem to be against unconditionally.
That's not necessarily true. Perhaps she believes Mormonism is almost certainly right, but acknowledges that she's not fully rational and might be misled if she read too many arguments against it. Most Christians believe in the idea that God (or Satan) tempts people to sin, and that avoiding temptation is a useful tactic to avoid sin. Kind of like avoiding stores where candy is on display if you're trying to lose weight, say. You know what's right in advance, but you're afraid of losing resolve.
Certainly whatever your beliefs, some people who disagree with you are sufficiently charismatic and good at rhetoric that they might persuade you if you give them the chance. (Well, for most of us, anyway.) How many atheist Less Wrongers would be able to withstand lengthy debate with very talented missionaries? Some, certainly. Most, probably. All? I doubt it.
Overall, though, an excellent response, and I agree with almost all the rest of it.
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I assume "no funny business" means "entries must be of the form 'A' or 'B/C' or 'D.E' for some numeral strings A,B,C,D,E with C nonzero".
In that case Warrigal would have said "rational" rather than "real". Numbers such as 17π would presumably be fine too, not just fractions. "No funny business" presumably means "I'd better be able to figure out whether it's the closest easily". For instance, the number "S(12)/2^n, where S is the max shifts function and n is the smallest integer such that my number is less than 100" is technically well-defined, in a mathematical sense. But if you can actually figure out what it is, you could publish a paper about it in any journal of computer science you liked.