Posts

Utility need not be bounded 2020-05-14T18:10:58.681Z · score: 31 (12 votes)
Who lacks the qualia of consciousness? 2019-10-05T19:49:52.432Z · score: 27 (17 votes)
Storytelling and the evolution of human intelligence 2019-06-13T20:13:03.547Z · score: 17 (7 votes)

Comments

Comment by richard_kennaway on Radical Probabilism [Transcript] · 2020-06-28T16:18:36.528Z · score: 2 (1 votes) · LW · GW
Jeffrey wanted to handle the case where you somehow become 90% confident of X, instead of fully confident

How does this differ from a Bayesian update? You can update on a new probability distribution over X just as you can on a point value. In fact, if you're updating the probabilities in a Bayesian network, like you described, then even if the evidence you are updating on is a point value for some initial variable in the graph, the propagation steps will in general be updates on the new probability distributions for parent variables.

Comment by richard_kennaway on Atemporal Ethical Obligations · 2020-06-27T10:17:01.426Z · score: 2 (1 votes) · LW · GW

This is saving yourself from the mob by running ahead of it.

Comment by richard_kennaway on Abstraction, Evolution and Gears · 2020-06-26T20:53:50.998Z · score: 11 (3 votes) · LW · GW

I heard it a long long time ago in a physics lecture, but I since verified it. The variation in where a ball is struck is magnified by the ratio of (distance to the next collision) / (radius of a ball), which could be a factor of 30. Seven collisions gives you a factor of about 22 billion.

I also tried the same calculation with the motion of gas molecules. If the ambient gravitational field is varied by an amount corresponding to the displacement of one electron by one Planck length at a distance equal to the radius of the observable universe, I think I got about 30 or 40 collisions before the extrapolation breaks down.

Comment by richard_kennaway on Abstraction, Evolution and Gears · 2020-06-26T09:37:32.326Z · score: 5 (3 votes) · LW · GW

To expand on the billiard ball example, the degree of sensitivity is not always realised. Suppose that the conditions around the billiard table are changed by having a player stand on one side of it rather than the other. The difference in gravitational field is sufficient that after a ball has undergone about 7 collisions, its trajectory will have deviated too far for further extrapolation to be possible — the ball will hit balls it would have missed or vice versa. Because of exponential divergence, if the change were to move just the cue chalk from one edge of the table to another, the prediction horizon would be not much increased.

Comment by richard_kennaway on The "hard" problem of consciousness is the least interesting problem of consciousness · 2020-06-23T14:40:09.120Z · score: 2 (1 votes) · LW · GW
But if we started with two problems and ended with one, then one of them is solved.

You won't escape an excess baggage charge by putting both your suitcases into one big case.

Comment by richard_kennaway on Utility need not be bounded · 2020-06-22T18:28:13.621Z · score: 2 (1 votes) · LW · GW

(I also posted this to the Open Thread—I'm not sure which is more likely to be seen.)

Since posting the OP, I've revised my paper, now called "Unbounded utility and axiomatic foundations", and eliminated all the placeholders marking work still to be done. I believe it's now ready to send off to a journal. If anyone wants to read it, and especially if anyone wants to study it and give feedback, just drop me a message. As a taster, here's the introduction.

Several axiomatisations have been given of preference among actions, which all lead to the conclusion that these preferences are equivalent to numerical comparison of a real-valued function of these actions, called a “utility function”. Among these are those of Ramsey [11], von Neumann and Morgenstern [17], Nash [8], Marschak [7], and Savage [13, 14].
These axiomatisations generally lead also to the conclusion that utilities are bounded. (An exception is the Jeffrey-Bolker system [6, 2], which we shall not consider here.) We argue that this conclusion is unnatural, and that it arises from a defect shared by all of these axiom systems in the way that they handle infinite games. Taking the axioms proposed by Savage, we present a simple modification to the system that approaches infinite games in a more principled manner. All models of Savage’s axioms are models of the revised axioms, but the revised axioms additionally have models with unbounded utility. The arguments to bounded utility based on St. Petersburg-like gambles do not apply to the revised system.
Comment by richard_kennaway on Open & Welcome Thread - June 2020 · 2020-06-22T18:26:50.798Z · score: 6 (3 votes) · LW · GW

Since posting this, I've revised my paper, now called "Unbounded utility and axiomatic foundations", and eliminated all the placeholders marking work still to be done. I believe it's now ready to send off to a journal. If anyone wants to read it, and especially if anyone wants to study it and give feedback, just drop me a message. As a taster, here's the introduction.

Several axiomatisations have been given of preference among actions, which all lead to the conclusion that these preferences are equivalent to numerical comparison of a real-valued function of these actions, called a “utility function”. Among these are those of Ramsey [11], von Neumann and Morgenstern [17], Nash [8], Marschak [7], and Savage [13, 14].
These axiomatisations generally lead also to the conclusion that utilities are bounded. (An exception is the Jeffrey-Bolker system [6, 2], which we shall not consider here.) We argue that this conclusion is unnatural, and that it arises from a defect shared by all of these axiom systems in the way that they handle infinite games. Taking the axioms proposed by Savage, we present a simple modification to the system that approaches infinite games in a more principled manner. All models of Savage’s axioms are models of the revised axioms, but the revised axioms additionally have models with unbounded utility. The arguments to bounded utility based on St. Petersburg-like gambles do not apply to the revised system.
Comment by richard_kennaway on Memory is not about the past · 2020-06-20T20:37:00.794Z · score: 2 (1 votes) · LW · GW
Few activities are as quintessentially human as being on the cusp of falling asleep and suddenly be assaulted by a memory that has us relive an embarrassing episode that we thought long forgotten.

Really? *does not raise hand*

Comment by richard_kennaway on When is it Wrong to Click on a Cow? · 2020-06-20T19:57:07.964Z · score: 2 (1 votes) · LW · GW

"Only one thing is serious for all people at all times. A man may be more aware of it or less aware of it but the seriousness of things will not alter on this account.

"If a man could understand all the horror of the lives of ordinary people who are turning round in a circle of insignificant interests and insignificant aims, if he could understand what they are losing, he would understand that there can be only one thing that is serious for him—to escape from the general law, to be free. What can be serious for a man in prison who is condemned to death? Only one thing: How to save himself, how to escape: nothing else is serious."

Gurdjieff, as quoted in Ouspensky, "In Search of the Miraculous".

Comment by richard_kennaway on When is it Wrong to Click on a Cow? · 2020-06-20T19:56:02.673Z · score: 2 (1 votes) · LW · GW

Well, what do you want? What will you do to get it?

Personally, I have no inclination to read trashy novels or watch the Kardashians (or inform myself of who they might be), so the issue of whether to do that does not exist for me.

When is it wrong to click on a cow? When your better self (the one that is smarter and better informed than you, your personal coherent extrapolated volition) would not.

Comment by richard_kennaway on Tips/tricks/notes on optimizing investments · 2020-06-17T11:09:29.960Z · score: 2 (1 votes) · LW · GW

Inferential distance? Or simply knowledge distance.

You lose me at "With portfolio margin". You're talking about financial instruments that, so I understand, you have a lot of professional experience in using, but I know nothing about these things. I googled "box spread financing", and it turns out to be a complicated instrument involving four separate options that, I'm still not sure what the purpose is. No criticism of yourself intended, but if a complete stranger started talking to me about box spread financing, despite it being a real thing I'd assume they were touting a scam. I don't know what "withdrawing excess "equity" from my margin account" means, nor the quote from Goldman Sachs (which would not come to my attention anyway).

And personally, I'm in the UK and a lot of what you're talking about is US-specific, but I can't even tell which parts are and which aren't. CD? FDIC? I do not know of a UK bank offering more than derisory interest on a savings account (typically 0.01% for instant access, 0.35% if you never withdraw money), but perhaps the banks I know of (retail banks) are not the sort of banks you're talking about. The Wikipedia page for Goldman Sachs suggests it is not involved in retail banking.

Comment by richard_kennaway on The "hard" problem of consciousness is the least interesting problem of consciousness · 2020-06-13T08:19:38.796Z · score: 2 (1 votes) · LW · GW

You can't "make everything be conscious". The thing we have experience of and call consciousness works however it works. It is present wherever it is present. It takes whatever different forms it takes. How it works, where it is present, and what forms it takes cannot be affected by pointing at everything and saying "it's conscious!"

Comment by richard_kennaway on The "hard" problem of consciousness is the least interesting problem of consciousness · 2020-06-12T23:26:21.102Z · score: 2 (1 votes) · LW · GW

A piano-shaped bunch of quarks and electrons is a piano. The causal powers of the piano are exactly the same as a piano-shaped bunch of quarks and electrons. Mentioning the quarks and electrons is doing no work, because we can talk of pianos without knowing anything about quarks and electrons.

It's the quarks and electrons that are epiphenomenal to the piano, not the other way round.

Comment by richard_kennaway on The "hard" problem of consciousness is the least interesting problem of consciousness · 2020-06-12T23:02:25.773Z · score: 4 (2 votes) · LW · GW
Well, what parts of the problem are not solved by attaching the word to everything?

All of it.

1. Rocks and water have no consciousness at all.
2. You can create brain from rocks and water.
3. Brains have consciousness.
4. Only epiphenomenal things can emerge.
5. Consciousness is not epiphenomenal.

I agree with all of that except 4. (A piano "emerges" from putting together its parts. But there is nothing epiphenomenal about it, as anyone who has had a piano fall on them will know.) But it gets no farther to explaining consciousness.

If you count logic as observation: that belief leads to contradiction.

Logic as observation observes through the lens of an ontology. If the ontology is wrong, it doesn't matter how watertight the logic is.

Comment by richard_kennaway on The "hard" problem of consciousness is the least interesting problem of consciousness · 2020-06-12T18:47:12.798Z · score: 10 (3 votes) · LW · GW

If selves exist in the same way that tables exist, that's good enough for me. *kicks table* There's nothing illusory about tables. Yes, they're made of parts, so are selves, but that doesn't make them illusions.

And assuming panpsychism, there are answers to all of the usual questions about consciousness

Here are a few questions:

How can I study the consciousness of a rock?

How can I compare the consciousness of a small rock vs. a big one?

What happens to the consciousness of an iceberg when it melts and mingles with the ocean?

Am I conscious when I am unconscious? When I am dead?

What observations could you show me that would surprise me, if I believed (as I do, for want of anything to suggest otherwise) that rocks and water have no consciousness at all?

is you main objection to panpsychism is how it interacts with "self"?

My main objection to panpsychism is that it makes no observable predictions. It pretends to solve the problem of consciousness by simply attaching the word to everything.

Comment by richard_kennaway on The "hard" problem of consciousness is the least interesting problem of consciousness · 2020-06-12T15:53:24.026Z · score: 2 (1 votes) · LW · GW
Weakly illusory - meaning non-fundamental

That makes of "weakly" a weasel word, that sucks the meaning out of the concept it is attached to. What would it be, for something to be or not be "fundamental"? You could argue (and some do) that nothing is fundamental, but then it says nothing of the self in particular that it is not "fundamental".

"self is just an ethical construct"

How did ethics come into it?

you can only choose between [list]

These are just some ideas that people have thought up. I don't have to choose any of them. I can simply say: neither I nor anyone else has any idea how to solve the hard problem. No-one even knows what a solution would look like. No-one even knows how there could be a solution. Yet here we are, stubbornly conscious in a world where everything we know about how things work has no place for it. That is the hard problem. Every purported solution I have seen amounts to either grabbing onto one side of the contradiction and insisting the other side is therefore false, or saying "la la la can't hear you" to the question.

Comment by richard_kennaway on The "hard" problem of consciousness is the least interesting problem of consciousness · 2020-06-12T09:32:30.045Z · score: 2 (1 votes) · LW · GW

Well, I don't get how anyone can take panpsychism and illusory selves seriously, so there! :)

Comment by richard_kennaway on Does equanimity prevent negative utility? · 2020-06-11T09:32:01.011Z · score: 8 (6 votes) · LW · GW

Handle the pain. Don't ignore its message.

Comment by richard_kennaway on [Poll] 'Truth' vs 'Winning' · 2020-06-10T09:21:51.650Z · score: 4 (2 votes) · LW · GW

See also.

Comment by richard_kennaway on [Poll] 'Truth' vs 'Winning' · 2020-06-09T07:58:56.430Z · score: 4 (2 votes) · LW · GW

As before, I was not imputing any dishonesty to the hypothetical reader reflexively thinking up a hypothetical counterexample to a generalisation.

Comment by richard_kennaway on Utility need not be bounded · 2020-06-08T16:09:31.291Z · score: 4 (2 votes) · LW · GW

Having now read some expositions of the Jeffrey-Bolker theory, I can answer my own question.

The Jeffrey-Bolker axioms imply the finite utility of every prospect (to be technical, the Averaging axiom fails when there are infinite utilities), but the utility can be unbounded above and below. It cannot be infinite. In this it differs from Savage's system.

For Savage's axioms, unbounded utility implies the existence of gambles like St. Peterburg, of infinite utility, and all the rest of the menagerie of infinite games listed in this SEP article. From these a contradiction with Savage's axioms can be found. Hence all models of Savage's axioms have bounded utility.

In the Jeffrey-Bolker system, gambles cannot be constructed at will. The set of available gambles is built into the world that the agent faces. The agent is an observer: it cannot act upon the world, only have preferences about how the world is. None of the paradoxical games exist in a model of the Jeffrey-Bolker axioms. They do allow the existence of non-paradoxical infinite games, games such as Convergent St. Petersburg, which is St. Petersburg modified to have arithmetically instead of geometrically growing payouts. However, I note that one of Jeffrey's verbal arguments against St. Petersburg — that no-one can offer the game because it requires them to be able to cover arbitrarily large payouts — applies equally to Convergent St. Petersburg.

Comment by richard_kennaway on [Poll] 'Truth' vs 'Winning' · 2020-06-08T15:28:25.439Z · score: 4 (2 votes) · LW · GW

You made up six stories here. I was not imputing any dishonesty, only pointing out that they are fiction.

OTOH, you just said of the other stories presented here that "we don't have any evidence". The stories I was referring to are jimmy's story of preventing swelling in an injured joint, and his account of Conor McGregor. These stories purport to be of real things that happened. To say that his account is no evidence of that looks very like what you took me to be doing.

Comment by richard_kennaway on Everyday Lessons from High-Dimensional Optimization · 2020-06-08T14:08:13.808Z · score: 2 (1 votes) · LW · GW

You can't take logarithms of non-positive numbers. Even if you know some parameter is non-positive, you still have a free choice of scale, so the problem of comparing scales of different parameters does not go away.

Comment by richard_kennaway on [Poll] 'Truth' vs 'Winning' · 2020-06-08T13:58:31.328Z · score: 4 (2 votes) · LW · GW

The examples that people have given are real ones. Yours are fictional. It's easy to make up stories of how the world would look, conditional upon any proposition whatever being true. (V cerqvpg gung ng yrnfg bar ernqre jvyy vafgnagyl erfcbaq gb guvf pynvz ol znxvat hc n fgbel va juvpu vg vf snyfr.) In this light, the "least convenient possible world" for one's interlocutors is the most convenient possible for oneself, the one in which the point at issue is imagined to be true.

Comment by richard_kennaway on Should we stop using the term 'Rationalist'? · 2020-06-02T16:55:14.843Z · score: 10 (6 votes) · LW · GW

"Rationalism" has the baggage of having meant the idea of finding truth by pure reason, without needing to look at the world.

"Empiricism" has the baggage of having meant the idea of finding truth just by looking at the world, without applying reason to discern its inner structures.

"Bayesianism" is far too narrow.

"Baconianism" might be close enough, but too obscure.

There does not appear to be any word that means "finding the truth by reason and observation, not separate from each other, but different aspects of a single method, as described in the Sequences", however many of the individual ideas there can be found in sources predating them.

Comment by richard_kennaway on TruetoThis's Shortform · 2020-06-02T14:50:35.866Z · score: 3 (2 votes) · LW · GW

I am not sure what that is.

It still takes effort to travel along a path. And there are many paths to choose from.

Comment by richard_kennaway on TruetoThis's Shortform · 2020-06-01T07:29:03.207Z · score: 3 (2 votes) · LW · GW

Yes. Therefore you should not do that. "Least resistance" and "going with the flow" are for those who want to remain asleep, to do nothing, to be nothing.

Comment by richard_kennaway on Should we stop using the term 'Rationalist'? · 2020-05-31T18:14:12.832Z · score: 3 (2 votes) · LW · GW

In English it means a particular kind of amateur: one without commitment, a dabbler, whose knowledge is merely superficial. "Amateur" is also used in the same sense, although it has not entirely lost the meaning of one who engages in something for the love of it, and may be (and occasionally is) the equal of a professional.

Comment by richard_kennaway on Are "superforecasters" a real phenomenon? · 2020-05-30T12:34:53.175Z · score: 2 (1 votes) · LW · GW

By definition, the top 2% are always better than the other 98%.

Comment by richard_kennaway on Why aren’t we testing general intelligence distribution? · 2020-05-26T17:34:52.629Z · score: 5 (3 votes) · LW · GW

To what extent has that been empirically tested?

The page you linked only gives a weak argument (3 genes to give a normal distribution of colour in maize?) and no references to empirical observations of the distribution. The video on the page, talking about skin colour, does not claim anything about the distribution, beyond the fact that there is a continuous range. Even with all of the mixing that has taken place in the last few centuries, the world does not look to me like skin colour is normally distributed.

Even Fisher's original paper on the subject says only "The great body of available statistics show us that the deviations of a human measurement from its mean follow very closely the Normal Law of Errors" near the beginning, then proceeds with pure mathematics.

I can think of several ways in which a polygenic trait might not be normally distributed. I do not know whether these ever, rarely, or frequently happen. Only a small number of genes involved. Large differences in the effects of these genes. Multiplicative rather than additive affect. The central limit theorem doesn't work so well in those situations.

And the graph of raw scores in the OP is clearly not a normal distribution. Would you justify transforming it into a normal distribution because that is how the "real" thing "must" be distributed? That would render the belief in normal distributions untestable.

Comment by richard_kennaway on How to convince Y that X has committed a murder with >0.999999 probability? · 2020-05-23T12:12:05.148Z · score: 2 (1 votes) · LW · GW

And yet, despite saying "Inconceivable!" they did collect their winnings and buy the mansion.

Comment by richard_kennaway on Utility need not be bounded · 2020-05-22T10:37:21.414Z · score: 3 (2 votes) · LW · GW
If you add +1 up from 0 and do -1 from w you never cross because those are part of separate archimedean fields.

Ah, you are using "field" in a different sense (than "something with addition and multiplication obeying the usual laws").

Comment by richard_kennaway on How to convince Y that X has committed a murder with >0.999999 probability? · 2020-05-21T21:51:33.786Z · score: 2 (1 votes) · LW · GW
I don't know why the last comment is relevant. I agree that 1 in a million odds happen 1 in a million times. I also agree that people win the lottery. My interpretation is that it means "sometimes people say impossible when they really mean extremely unlikely", which I agree is true.

The point was not that people win the lottery. It's that when they do, they are able to update against the over 100 million-to-one odds that this has happened. "No, no," say the clever people who think the human mind is incapable of such a shift in log-odds, "far more likely that you've made a mistake, or the lottery doesn't even exist, or you've had a hallucination." The clever people are wrong.

Comment by richard_kennaway on How to convince Y that X has committed a murder with >0.999999 probability? · 2020-05-21T17:54:33.067Z · score: 2 (1 votes) · LW · GW

In the log-odds space, both directions look the same. You can wander up as easily as down.

I don't know what probability space you have in mind for the set of all possible phenomena leading to an error, that would give a basis for saying that most errors will lie in one direction.

When I calculated the odds for the Euromillions lottery, my first calculation omitted to divide by a factor to account for there being no ordering on the chosen numbers, giving a probability for winning that was too small by a factor of 240. The true value is about 140 million to 1.

I have noted before that ordinary people, too ignorant to know that clever people think it impossible, manage to collect huge jackpots. It is literally news when they do not.

Comment by richard_kennaway on Utility need not be bounded · 2020-05-21T17:29:49.797Z · score: 2 (1 votes) · LW · GW
With reals you can easily compare finite vs finite and infinite vs infinite but doing a comparison across multiple archimedean fields gets tricky.

Did you mean non-archimedean fields? I regard those as not the real real numbers. For practical purposes in the present context, I don't think you can beat the Dedekind-complete ordered field (i.e. the real numbers), with nominal ∞ and –∞ symbols added as a shorthand for more verbose statements about infinite integrals and sums.

Comment by richard_kennaway on How does publishing a paper work? · 2020-05-21T14:35:17.350Z · score: 27 (12 votes) · LW · GW

1. Format. This depends on the journal. Any journal should provide LaTeX and Word templates for their preferred style. It may not be necessary to use them for the initial submission — check their author guidelines and do exactly what they say. It doesn't matter what you think of their style: their journal, their rules. When you want something from someone, do not give them reasons to say no.

2. Yes: you have to write the paper. Get other people to read it and tell you how bad it is. Give presentations of it and field hostile questions. It builds character.

3. Different institutions may have their own rules about author order. A frequent convention is that major contributors go first, the head of the research group goes last, and everyone else is in the middle. Alphabetical order for ties.

4. You can only submit a paper to one place at a time. Multiple submission is pretty much never acceptable. If detected it may be grounds for instant rejection. You can also only publish it once. A further paper must be a sufficient advance on previous work.

5. The journal editor sometimes makes the decision himself, without sending it out. When he turns it down, this is called a "desk rejection", because it never gets past the editor's desk. That aside, anonymous peer review is the way it's done almost everywhere, although it is lately a subject of some contention. Sometimes it's doubly anonymous: the reviewers don't know who the authors are, although realistically, it is often not difficult to guess.

6. I haven't encountered conference-journals, but there does exist the opposite: the authors of the best papers at a conference (as judged by the conference committee) may be invited to brush them up for publication (usually with peer review) in a special issue of a journal.

7. Mm, not getting into that one, beyond mentioning public-access journals like the PLoS family. Beware, however, of predatory journals and conferences that accept everything for a fee, even randomly generated nonsense, but which nobody reads. Subscription journals aren't the only rent-seekers around.

8. Arxiv: Yes. Also biorXiv for biological sciences. There is some small hurdle to pass, I forget exactly what. In physics, I've heard, everyone reads arXiv to keep up, and eventual journal publication is just a formal stamp of approval that the experts don't need to see because they're the ones who gave it that stamp. For junk and crackpottery that can't even get into arXiv, there is viXra, which is only technically publishing, i.e. anyone can read it but nobody will.

9. Sci-hub: Yes. Well, exactly who is committing an offence and in what jurisdiction I'll leave to the lawyers. BTW, it's worth just googling the title of a paper, because you can often find a copy put online by the author, maybe a pre-publication version. You can also ask the author directly for a copy. They will usually be happy to do so. Also, not all journals are paywalled, e.g. PLoS, and some journals publish a mixture of open and paywalled articles.

10. Value of publishing: In the general sense of making your work public, yes, or why are you doing it? (Well, maybe to make money from your discoveries, which you don't want to reveal until you've made a huge pile, but that's not the usual way things go.) Putting it in a journal or on arXiv gives it permanence and a standard way to refer to it which you won't get by putting it on your blog. As for prestige, except for academic promotion committees who just add up the impact factors of the journals you've published in, the prestige comes from the work, not the venue. The most that the venue can do for you is bring the work to more people's attention. After that, the venue no longer matters.

Comment by richard_kennaway on How to convince Y that X has committed a murder with >0.999999 probability? · 2020-05-20T22:08:48.347Z · score: 0 (2 votes) · LW · GW
Yes, see response to Dagon. But, 0.99999999 seems overconfident to me. You have to account not only for "I might be insane" (what are the base rates on that?), but simpler things like "I misread the question or had a brain fart." 

Those could go either way.

Comment by richard_kennaway on How to convince Y that X has committed a murder with >0.999999 probability? · 2020-05-20T21:57:33.609Z · score: 4 (2 votes) · LW · GW

Having sat on a jury (for a rather dull case of a failed burglary), I concur with this.

Jury confidentiality is taken seriously in the UK, so I can't comment on our deliberations, but the consensus was that it was him wot dunnit. He looked resigned rather than indignant when the verdict was read out, so with that and the evidence I'm as sure as I need to be that we got it right. I couldn't put a number on it, but 0.000001 is way smaller than a reasonable doubt.

Comment by richard_kennaway on How to convince Y that X has committed a murder with >0.999999 probability? · 2020-05-20T21:50:01.433Z · score: 0 (2 votes) · LW · GW

If X has confessed, how can he be on trial?

Comment by richard_kennaway on How to convince Y that X has committed a murder with >0.999999 probability? · 2020-05-20T21:14:10.251Z · score: 5 (3 votes) · LW · GW

If I buy a ticket in the Euromillions lottery, I am over 0.99999999 sure I will lose. (There are more than 100 million possible draws.)

Comment by richard_kennaway on Utility need not be bounded · 2020-05-18T11:41:18.598Z · score: 2 (1 votes) · LW · GW

Yes, St. Peterburg and Petrograd (= St. Petersburg with all payouts increased by one) are given the same infinite utility. Neither is preferable to the other, despite the intuition saying that Petrograd is better. While intuition can be a guide, it is an untrustworthy one, a castle in the air that requires a foundation to be built underneath it.

The problem with comparing infinities is that if you impose conditions on the preference relation that seem reasonable for finite games, then before you know it — literally so in Savage's case — you end up excluding all the infinities, and neither St. Peterburg nor Petrograd exist. To avoid doing that, you have to give up some of those conditions. Savage's P2, for example, sounds perfectly reasonable if you don't think about infinite games, but as soon as you do, you can see that it must fail. Not that there's anything special about P2, it's really the basic ontology of the system that is at fault.

I have to wonder how strong a mathematical background some of the people who have published on the subject had. Attempting to construct a total ordering on all functions from a probability space to the real numbers, or even just on the measurable functions, seems doomed to failure.

Comment by richard_kennaway on Utility need not be bounded · 2020-05-18T09:38:06.848Z · score: 2 (1 votes) · LW · GW
What matters is whether the agent has some nonzero credence that they are being offered such a game. I for one am such an agent, and you would be too if you bought into the "0 is not a probability" thing, or if you bought into solomonoff induction or something like that.

In fact I don't buy into those things. One has to distinguish probability at the object level from probability at the metalevel. At the metalevel it does not exist, only true and false exist, 0 and 1. So when I propose a set of axioms whereby measures of probability and utility are constructed, the probability exists within that framework. The question of whether the framework is a good one matters, but it cannot be discussed in terms of the probability that it is right. I have set out the construction, which I think improves on Savage's, but people can study it themselves and agree or not. It rules out the Pasadena game. To ask what the probability is of being faced with the Pasadena game is outside the scope of my axioms, Savage's, and every set of axioms that imply bounded utility. Everyone excludes the Pasadena game.

No, actually they don't. I've just come across a few more papers dealing with Pasadena, Altadena, and St. Petersburg games, beginning with Terrence Fine's "Evaluating the Pasadena, Altadena, and St Petersburg Gambles", and tracing back the references from there. From a brief flick through, all of these papers are attempting what seems to me to be a futile activity: assigning utilities to these pathological games. Always, something has to be given up, and here, what is given up is any systematic way of assigning these games utilities; nevertheless they go ahead and do so, even while noticing the non-uniqueness of the assignments.

So there is the situation. Savage's axioms, and all systems that begin with a total preference relation on arbitrary games, require utility to be bounded in order to exclude not only these games, but also infinite games that converge perfectly well to intuitively natural limits. I start from finite games and then extend to well-behaved limits. Others try to assign utility to pathological games, but fail to do so uniquely.

I'm happy to end the conversation here, because at this point there is probably little for us to say that would not be repetition of what has already been said.

Comment by richard_kennaway on Utility need not be bounded · 2020-05-16T17:49:46.224Z · score: 2 (1 votes) · LW · GW

With my axioms, utility can be unbounded, and the St. Petersburg game is admitted and has infinite utility. I don't regard this as paradoxical. The game cannot be offered, because the offerer must have infinite resources to be able to cover every possible outcome, and on average loses an infinite amount.

St. Petersburg-like games with finite expected utility also exist, such as one where the successive payouts grow linearly instead of exponentially. These also cannot be offered in practice, for the same reason. But their successive approximations converge to the infinite game, so it is reasonable to include them as limits.

Both types are excluded by Savage's axioms, because those axioms require bounded utility. This exclusion appears to me unnatural and resulting from an improper handling of infinities, hence my proposal of a revised set of axioms.

Comment by richard_kennaway on Utility need not be bounded · 2020-05-16T10:25:45.687Z · score: 2 (1 votes) · LW · GW

The St. Petersburg game variant in which the payoffs are dollars can only exist in Savage's system if there is a limit on the number of utilons that any amount of dollars could buy. No more utility than that exists. But that game is not paradoxical. It has a finite expected value in utilons, and that is an upper bound on the fee it is worth paying to play the game.

Comment by richard_kennaway on Utility need not be bounded · 2020-05-16T10:18:45.319Z · score: 2 (1 votes) · LW · GW

Is there such a thing as surreal non-standard analysis? It's not a subject I follow, but I understood that there still wasn't a good concept of integration for the surreal numbers.

My general attitude to other systems of non-standard numbers is the same as my attitude to ultrafinitism. Non-standard real numbers are really a way of thinking about the Platonic reals, in which what look like infinite real numbers from within the system are finite real numbers so enormous that in a limited system of reasoning they cannot be reached by any of the available constructions. But I don't have a formalisation of this idea.

Comment by richard_kennaway on Utility need not be bounded · 2020-05-15T17:12:36.864Z · score: 2 (1 votes) · LW · GW

Re Jeffrey-Bolker, the only system I studied in detail was Savage's, but my impression is that the fix I applied to that system can be applied all the others that paint themselves into the corner of bounded utility, and with the same effect of removing that restriction. Do the Jeffrey-Bolker axioms either assume or imply bounded utility?

Comment by richard_kennaway on Utility need not be bounded · 2020-05-15T16:56:33.109Z · score: 2 (1 votes) · LW · GW
are you sure Savage's axioms rule out the sorts of preferences I'm talking about? They don't rule out bounded utility functions, after all.

Savage's axioms imply that utility is bounded. This is what Savage did not know when he formulated them, but Peter Fishburn proved it, and Savage included the result in the second edition of his book. So Savage accidentally brute-forced the pathological games out of existence. All acts, in Savage's system, have a defined, finite expected value, and the St. Petersburg game and its variants do not exist. God himself cannot offer you these games. The utilities of the successive St. Peterburg payoffs are bounded, and cannot even increase linearly, although intuitively that version should have a well-defined, finite expected value.

In my approach, I proceed more cautiously by only considering "finite" acts at the outset: acts with only finitely many different consequences. Then I introduce acts with infinitely many consequences as limits of these, some of which have finite expected values and some infinite.

Comment by richard_kennaway on Utility need not be bounded · 2020-05-15T11:53:06.552Z · score: 2 (1 votes) · LW · GW

To give an idea of the motivation, consider the following analogy (which is actually quite close to what happens in my paper). Take the set of all functions from the reals to the reals. (In Savage's system, these are the "acts" when the set of "world states" and the set of "consequences" are both copies of the reals.)

Suppose you want some measure of the "size" of a function, intuitively visualising this as the area under the function's curve. So any probability density function would have a "size" of 1, its negative would have a "size" of -1, the constant function f(x) = 1 would have infinite "size", and so on.

So you write down a set of axioms that this "size" seems like it ought to obey, and end up defining the "size" of a function to be its integral over the whole real line.

But then you have a problem. For some functions, there is simply no such thing. The sine function, for example, can be integrated over every finite interval, but not over the whole real line. f(x) = exp(x) could be assigned infinite value, but that doesn't work for exp(x)sin(x). You might want f(x) = 1/x to have an integral of zero by symmetry, but you can get any value you like depending on how you approach the singularity at 0. And there are far wilder functions than these (the non-measurable functions, but I don't want to start explaining measure theory).

It would be silly to solve this problem by insisting that the real numbers must be bounded. The real numbers are not up for grabs. They simply are what they are, and if that is not consistent with your intuitions about the "size" of a function, so much the worse for your intuitions. (I'm aware of ultrafinitists who try to do exactly that. As a staunch Platonist, I regard their efforts as interesting, and worthwhile in the spirit of "let 10,000 views contend", but I read them as describing what the Platonic reals look like from within certain more limited axiom systems.)

What one actually does when defining integration is to start with some simple class of functions that intuitively do have obvious values for their integrals, extend this by limiting constructions as far as possible, and then accept that not every function has an integral. I do something similar in revising Savage's axioms, and as a result I can consistently admit unbounded utilities and the more well-behaved sort of infinite games.

Comment by richard_kennaway on Utility need not be bounded · 2020-05-15T10:15:39.427Z · score: 3 (2 votes) · LW · GW
Imagine if I said "I've come up with a voting system which satisfies all of Arrow's axioms, thus getting around his famous theorem" and then you qualified with "To make this work, I had to exclude certain scenarios from the purview of preference aggregation theory, namely, the ones that would make my system violate one of the axioms..."

I am actually excluding less than Savage does, not more: models of my axioms include all models of his, and more. And since Savage at first did not know that his axioms implied bounded utility, that cannot have been a consideration in his design of them.

People may give preferences involving pathological scenarios, but clearly those preferences cannot satisfy Savage's axioms (since his axioms rule them out, and even more strongly than mine do).

There is no free lunch here. You can have preferences about everything in the Tegmark level 7 universe (or however high the hierarchy goes -- somewhere I saw it extended several levels beyond Tegmark himself), but at the cost of them failing to obey reasonable sounding properties of rational preference.

Comment by richard_kennaway on Premature death paradox · 2020-05-15T08:59:06.619Z · score: 2 (1 votes) · LW · GW

Consider an atom of uranium 238. It has a constant probability of per unit time of emitting an alpha particle. Unlike people, it does not get tired and frail. The probability that it goes ping in the next second remains constant, however long it has survived already. That probability is extremely small. The half-life is more than 4 billion years, similar to the age of the earth. Whenever it happens, it was extraordinarily unlikely to happen just then. It has decayed 4 billion years ahead of its expected remaining lifetime.

But in a quarter kilogram of U238, it will happen 4 million times a second.