A common belief within the Effective Altruism movement that you should not diversify charity donations when your donation is small compared to the size of the charity. This is counter-intuitive, and most people disagree with this. A Mathematical Explanation of Why Charity Donations Shouldn't Be Diversified has already been written, but it uses a simplistic model. Perhaps you're uncertain about which charity is best, charities are not continuous, let alone differentiable, and any donation is worthless unless it gives the charity enough money to finally afford another project, your utility function is nonlinear, and to top it all off, rather than accepting the standard idea of expected utility, you are risk-averse.

Standard Explanation:

If you are too lazy to follow the link, or you just want to see me rehash the same argument, here's a summary.

The utility of a donation is differentiable. That is to say, if donating one dollar gives you one utilon, donating another dollar will give you close to one utilon. Not exactly the same, but close. This means that, for small donations, it can be approximated as a linear function. In this case, the best way to donate is to find the charity that has the highest slope, and donate everything you can to it. Since the amount you donate is small compared to the size of the charity, a first-order approximation will be fairly accurate. The amount of good you do with that strategy is close to what you predicted it would do, which is more than you'd predict of any other strategy, which is close to what you'd predict for them, so even if this strategy is sub-optimal, it's at least very close.

Corrections to Account for Reality:

Uncertainty:

Uncertainty is simple enough. Just replace utility with expected utility. Everything will still be continuous, and the reasoning works pretty much the same.

Nonlinear Utility Function:

If your utility function is nonlinear, this is fine as long as it's differentiable. Perhaps saving a million lives isn't a million times better than saving one, but saving the millionth life is about as good as the one after that, right? Maybe each additional person counts for a little less, but it's not like the first million all matter the same, but you don't care about additional people after that.

In this case, the effect of the charity is differentiable with respect to the donation, and the utility is differentiable with respect to the effect of the charity, so the utility is differentiable with respect to the donation.

Risk-Aversion:

If you're risk-averse, it gets a little more complicated.

In this case, you don't use expected utility. You use something else, which I will call meta-utility. Perhaps it's expected utility minus the standard deviation of utility. Perhaps it's expected utility, but largely ignoring extreme tails. What it is is a function from a random variable representing all the possibilities of what could happen to the reals. Strictly speaking, you only need an ordering, but that's not good enough here, since it needs to be differentiable.

Differentiable is more confusing in this case. It depends on the metric you're using. The way we'll be using it here is that having a sufficiently small probability of a given change, or a given probability of a sufficiently small change, counts as a small change. For example, if you only care about the median utility, this isn't differentiable. If I flip a coin, and you win a million dollars if it lands on heads, then you will count that as worth a million dollars if the coin is slightly weighted towards heads, and nothing if it's slightly weighted towards tails, no matter how close it is to being fair. But that's not realistic. You can't track probabilities that precisely. You might care less about the tails, so that only things in the 40% - 60% range matter much, but you're going to pick something continuous. In fact, I think we can safely say that you're going to pick something differentiable. If I add a 0.1% chance of saving a life given some condition, it will make about the same difference as adding another 0.1% chance given the same condition. If you're risk-averse, you'd care more about a 0.1% chance of saving a life it's takes effect during the worst-case scenario than the best-case, but you'd still care about the same for a 0.1% chance of saving a life during the worst case as for upgrading it to saving two lives in that case.

Once you accept that it's continuous, the same reasoning follows as with expected utility. A continuous function of a continuous function is continuous, so the meta-utility of a donation with respect to the amount donated is continuous.

To make the reasoning more clear, here's an example:

Charity A saves one life per grand. Charity B saves 0.9 lives per grand. Charity A has ten million dollars, and Charity B has five million. One or more of these charities may be fraudulent, and not actually doing any good. You have $100, and you can decide where to donate it.

The naive view is to split the $100, since you don't want to risk spending it on something fraudulent. That makes sense if you care about how many lives you save, but not if you care about how many people die. They sound like they're the same thing, but they're not.

If you donate everything to Charity A, it has $10,000,100 and Charity B has $5,000,000. If you donate half and half, Charity A has $10,000,050 and Charity B has $5,000,050. It's a little more diversified. Not much more, but you're only donating $100. Maybe the diversification outweighs the good, maybe not. But if you decide that it is diversifying enough to matter more, why not donate everything to Charity B? That way, Charity A has $10,000,000, and Charity B has $5,000,100. If you were controlling all the money, you'd probably move a million or so from Charity A to Charity B, until it's well and truly diversified. Or maybe it's already pretty close to the ideal and you'd just move a few grand. You'd definitely move more than $100. There's no way it's that close to the optimum. But you only control the $100, so you just do as much as you can with that to make it more diversified, and send it all to Charity B. Maybe it turns out that Charity B is a fraud, but all is not lost, because other people donated ten million dollars to Charity A, and lots of lives were saved, just not by you.

Discontinuity:

The final problem to look at is that the effects of donations aren't continuous. The time I've seen this come up the most is when discussing vegetarianism. If you don't it meat, it's not going to make enough difference to keep the stores from ordering another crate of meat, which means exactly the same number of animals are slaughtered.

Unless, of course, you were the straw that broke the camel's back, and you did keep a store from ordering a crate of meat, and you made a huge difference.

There are times where you might be able to figure that out before-hand. If you're deciding whether or not to vote, and you're not in a battleground state, you know you're not going to cast the deciding vote, because you have a fair idea of who will win and by how much. But you have no idea at what point a store will order another crate of meat, or when a charity will be able send another crate of mosquito nets to Africa, or something like that. If you make a graph of the number of crates a charity sends by percentile, you'll get a step function, where there's a certain chance of sending 500 crates, a certain chance of sending 501, etc. You're just shifting the whole thing to the left by epsilon, so it's a little more likely each shipment will be made. What actually happens isn't continuous with respect to your donation, but you're uncertain, and taking what happens as a random variable, it is continuous.

A few other notes:

Small Charities:

In the case of a sufficiently small charity or large donation, the argument is invalid. It's not that it takes more finesse like those other things I listed. The conclusion is false. If you're paying a good portion of the budget, and the marginal effects change significantly due to your donations, you should probably donate to more than one charity even if you're not risk-averse and your utility function is linear.

I would expect that the next best charity you manage to find would be worse by more than a few percent, so I really doubt it would be worth diversifying unless you personally are responsible for more than a third of the donations.

An example of this is keeping money for yourself. The hundredth dollar you spend on yourself has about a tenth of the effect the thousandth does, and the entire budget is donated by you. The only time you shouldn't diversify is if the marginal benefit of the last dollar is still higher than what you could get donating to charity.

Another example is avoiding animal products. Avoiding steak is much more cost-effective than avoiding milk, but once you've stopped eating meat, you're stuck with things like avoiding milk.

Timeless Decision Theory:

If other people are going to make similar decisions to you, your effective donation is larger, so the caveats about small charities applies. That being said, I don't think this is really much of an issue.

If everyone is choosing independently, even if most of them correlate, the end result will be that the charities get just enough funding that some people donate to some and others donate to others. If this happens, chances are that it would be worth while for a few people to actually split their investments, but it won't make a big difference. They might as well just donate it all to one.

I think this will only become a problem if you're just donating to the top charity on GiveWell, regardless of how closely they rated second place, or you're just donating based purely on theory, and you have no idea if that charity is capable of using more money.

I have a master's project I'm having trouble working on. It's something I've wanted to do, and I even started working on, long before I started my master's degree. If I can't even enjoy that, then I'm doomed to spend eight hours a day doing something I hate for the rest of my life. Even if I manage to improve my willpower, I doubt I'll be very productive doing something I don't want to do.

Does anyone have any idea how I can enjoy working more?

I made Trust in God, or, The Riddle of Kyon, a Haruhi fanfic by Eliezer Yudkowsky, into a visual novel. At least, I started it. It still needs quite a bit of work. If anyone wants to edit it, message me.

I was reading a review of Trust in God, or, The Riddle of Kyon. The reviewer went through step by step and listed problems with the fic. Some of them I agreed with, and others did not. What really stood out was in a comment agreeing with it: "reading that fic usually just evoked vague anger and other unpleasantness." Not unlike the vague feeling of anger and other unpleasantness I felt upon reading the review.

I don't consider myself someone whose opinion can be trusted on the quality of the original fic. In addition to being every bit as biased as any hater, although in the opposite direction, I have Asperger's syndrome, and I don't trust myself to notice such things as people acting out of character. However, because of this, I know my revulsion cannot be due to the quality of the review. I looked for it in hopes of weakening any bias I have. I think I can safely say that my revulsion will prevent that from happening.

So, any idea on what I can do? I've always thought haters should just stay away from what they hate. That would work fine if I just hated ponies or something, but I don't think it's such a good idea in cases where ideology is involved. And if nothing else, I don't want a vague feeling of anger and unpleasantness to ruin a perfectly good fanfic, like The Death of Haruhi Suzumia.

Timeless physics is what you end up with if you take MWI, assume the universe is a standing wave, and remove the extraneous variables. From what I understand, for the most part you can take a standing wave and add a time-reversed version, you end up with a standing wave that only uses real numbers. The problem with this is that the universe isn't quite time symmetric.

If I ignore that complex numbers ever were used in quantum physics, it seems unlikely that complex numbers is the correct solution. Is there another one? Should I be reversing charge and parity as well as time when I make the standing real-only wave?

leeping Beauty is put to sleep on Sunday. If the coin lands on heads, she is awakened only on Monday. If it lands on tails, she is awaken on Monday and Tuesday, and has her memory erased between them. Each time she is awoken, she is asked how likely it is the coin landed on tails.

According to the one theory, she would figure it's twice as likely to be her if the coin landed on tails, so it's now twice as likely to be tales. According to another, she would figure that the world she's in isn't eliminated by heads or tails, so it's equally likely. I'd like to use the second possibility, and add a simple modification:

The coin is tossed a second time. She's shown the result of this toss on Monday, and the opposite on Tuesday (if she's awake for it). She wakes up, and believes that there are four equally probable results: HH, HT, TH, and TT. She then is shown heads. This will happen at some point unless the coin has the result HT. In that case, she is only woken once, and is shown tails. She now spreads the probability between the remaining three outcomes: HH, TH, and TT. She is asked how likely it is that the coin landed on heads. She gives 1/3. Thanks to this modification, she got the same answer as if she had used SIA.

Now suppose that, instead of being told the result of second coin toss, she had some other observation. Perhaps she observed how tired she was when she woke up, or how long it took to open her eyes, or something else. In any case, if it's an unlikely observation, it probably won't happen twice, so she's about twice as likely to make it if she wakes up twice.

Edit: SIA and SSA don't seem to be what I thought they were. In both cases, you get approximately 1/3. As far as I can figure, the reason Wikipedia states that you get 1/2 with SIA is that it uses sleeping beauty during the course of this experiment as the entire reference class (rather than all existent observers). I've seen someone use this logic before (they only updated on the existence of such an observer). Does anyone know what it's called?

Utilitarianism seems to be a common theme on this site. I suggest checking out felicifia.org, a Utilitarianism forum. That is all.

1. God said, Let there be bliss: and there was bliss.

From what I understand, the Kolmogorov axioms make no mention of conditional probability. That is simply defined. If I really want to show how probability actually works, I'm not going to argue "by definition". Does anyone know a modified form that uses simpler axioms than P(A|B) = P(A∩B)/P(B)?

An improper prior is essentially a prior probability distribution that's infinitesimal over an infinite range, in order to add to one. For example, the uniform prior over all real numbers is an improper prior, as there would be an infinitesimal probability of getting a result in any finite range. It's common to use improper priors for when you have no prior information.

The mark of a good prior is that it gives a high probability to the correct answer. If I bet 1,000,000 to one that a coin will land on heads, and it lands on tails, it could be a coincidence, but I probably had a bad prior. A good prior is one that results in me not being very surprised.

With a proper prior, probability is conserved, and more probability mass in one place means less in another. If I'm less surprised when a coin lands on tails, I'm more surprised when it lands on heads. This isn't true with an improper prior. If I wanted to predict the value of a random real number, and used a normal distribution with a mean of zero and a standard deviation of one, I'd be pretty darn surprised if it doesn't end up being pretty close to zero, but I'd be infinitely surprised if I used a uniform distribution. No matter what the number is, it will be more surprising with the improper prior. Essentially, a proper prior is better in every way. (You could find exceptions for this, such as averaging a proper and improper prior to get an improper prior that still has finite probabilities and they just add up to 1/2, or by using a proper prior that has zero in some places, but you can always make a proper prior that's better in every way to a given improper prior).

Dutch books also seems to be a popular way of showing what works and what doesn't, so here's a simple Dutch argument against improper priors: I have two real numbers: x and y. Suppose they have a uniform distribution. I offer you a bet at 1:2 odds that x has a higher magnitude. They're equally likely to be higher, so you take it. I then show you the value of x. I offer you a new bet at 100:1 odds that y has a higher magnitude. You know y almost definitely has a higher magnitude than that, so you take it again. No matter what happens, I win.

You could try to get out of it by using a different prior, but I can just perform a transformation on it to get what I want. For example, if you choose a logarithmic prior for the magnitude, I can just take the magnitude of the log of the magnitude, and have a uniform distribution.

There are certainly uses for an improper prior. You can use it if the evidence is so great compared to the difference between it and the correct value that it isn't worth worrying about. You can also use it if you're not sure what another person's prior is, and you want to give a result that is at least as high as they'd get no matter how much there prior is spread out. That said, an improper prior is never actually correct, even in things that you have literally no evidence for.