On the experience of confusion

I recently discovered something about myself: I have a particularly strong aversion to the experience of confusion. For example, yesterday I was looking into the relationship between common knowledge of rationality and Nash equilibrium in game theory. I had planned to spend just an hour on this, leisurely dipping into the material and perhaps coming out with a clarified understanding. Instead, something else happened. I became monomanically focused on this task. I found some theorems, but there was still this feeling that things were just slightly off, that my understanding was not quite right. I intensely desired to track down the origin of the feeling. And to destroy the feeling. I grew restless, especially because I was making some progress: I wasn’t completely stuck, it felt like I must be on the cusp of clarity. The first symptom of this restlessness was skipping my pomodoro breaks, usually a sure sign that I am losing self-control and will soon collapse into an afternoon nap. The second smyptom was to develop an unhelpful impatience, opening ever more new tabs to search for the answer elsewhere, abandoning trains of thought earlier and earlier. In the end I didn’t have time to do any of the other work I had planned that day!

This happens to me about once a week.

I don’t know if this description was at all effective at communicating my experience. It’s something far more specific than simple curiosity. I’m fine with not knowing things. I’m even happy to have big gaping holes in my knowledge, like a black rectangle on an otherwise-full map of the city. Provided the rectangle has clear boundaries and I know that, as a matter of principle, I could go explore that part of the city, and if I made no mistakes, I could draw the correct map.

Here’s another way of putting this. I’m not at all bothered if a tutor tells me: “The proof of this theorem, in appendix 12.B., relies on complicated maths. You may never understand it. But you have a good grasp of what the theorem states.” I have a picture in my head like:

\[\text{Premise} \rightarrow \text{lots of complicated maths} \rightarrow \text{conclusion}\]

I am infuriated if a tutor tells me: “When there are sticky prices, equation A looks like this.” What do we mean by sticky prices? And how does the equation follow? Tutor: “Here’s the mathematical statement of sticky prices. It involves completely different objects than equation A. Also, here’s a vague, hand-wavey intuition why the two are related.”

The problem here is not that there’s an empirical fact that I don’t know, or a proof step I don’t understand. I don’t even have a label to put on my confusion. It’s not that I don’t see how the conclusion follows, it’s that I don’t see how it could follow. It’s not that the map has dark patches. I don’t even know if I’m holding the map rightside up or upside down, and the map is written in cyrillic.

In school, I used to make myself unpopular by pursuing these lines of inquiry as far as they would let me, leading to long back-and-forths with my teachers. These conversations were often unproductive. Sometimes the implication was that I should just learn the words of the vague hand-wavey intuition as a kind of password. Naturally, I resented this. Both possibilities were enraging: either the educators themsleves believed that the words could pass for real understanding, or they just expected me to shut up and learn the password. Sometimes I was gently chided (or complimented?) for my curiosity, my apparent desire to know EVERYTHING, not to rest until the whole map was filled in. This too felt wrong: I’m not complaning about a small corner of the map left unfilled. The entire eastern part of the map is in cyrillic!

Although I hope that some people reading this might relate to my experiences, I suspect that I am out of the ordinary in the strength of my aversion to confusion. I have long thought that any of the success I’ve had in my academic pursuits was not due to intelligence, but to my refusal of explanations that felt unsatisfying in some sublte way. I say this not to humble-brag: I have good evidence that I am less intelligent than many of my peers. In school everyone used to participate in this maths competition every year. The questions required clever problem-solving, I consider them pretty close to an IQ test. They were completely different from our maths exams, which prized definitional clarity and rewarded practice. I was around the class median at the competition, but among the best at the exams. As another piece of evidence, I am seriously terrible at mental arithmetic: I routinely get simple sums wrong at the bakery, and not for lack of trying!

So I had long been aware that there was something different about how I asked questions, but only recently did I acquire the language to describe it accurately. I used to think it was “intellectual curiosity”, but as we have seen, “visceral aversion to even slight confusion” would be a more accurate label. I loathe contradiction and dissonance, not ignorance or uncertainty.

I have already talked a bit about how I think I’ve benefitted from this habit of thought. I think it may be one thing that people who get really into analytic philosophy have in common. It also comes with costs, mostly in the form of getting sucked into a productivity-wrecking hole of confusion, like with the game theory example. It would be much more rational to remain clam and composed, let the confusion go for a day or two, and then decide whether it makes sense to allocate more time to it. Part of why I’m getting sucked in so much, I suspect, is because I fear that if I stop, I will let the confusion slip by. I find that thought distressing. Perhaps it’s because I don’t want to forget I was confused, later remember the password, and adopt the confused knowledge that comes with it.

One way to help solve this is to keep a list of everything I am confused about. Then I can set a time limit on my intellectual escapades, and if I’m still confused by the end, I can write it down. Even if I never return to it, it feels much more satisfying to have a degree of meta-clarity (clarity about what I’m confused about) than to let confusion slip into a dark corner of my mind.

August 6, 2017

How much donation splitting is there, and should you split?

Table of contents

  1. Table of contents
  2. Summary
  3. Many of us split
    1. Individual examples
    2. EA funds data
      1. Naive approach: distribution of allocation percentages
      2. Less naive apprach: weighted distribution of allocation percentages
      3. Best approach: user totals
    3. Other data
  4. Arguments for splitting
    1. Empirical uncertainty combined with risk aversion
    2. Moral uncertainty
    3. Diminishing returns
    4. Achieving a community-wide split
      1. Cooperation with other donors
      2. Lack of information
    5. Remaining open-minded or avoiding confirmation bias
    6. Memetic effects
  5. Recommendation
  6. Appendix: R code


Many aspiring effective altruists report splitting their donations between two or more charities. I analyse EA funds data to estimate the extent of splitting. Expected utility reasoning suggests that for small donations, one should never split, and always donate all the money to the organisation with the highest expected cost-effectiveness. So prima facie we should not split. Are there any convincing reasons to split? I review 6 arguments in favour of splitting. I end with my recommendation.

Many of us split

Individual examples

For example, in CEA Staff’s Donation Decisions for 2016, out of 14 staff members who disclosed significant donations, I count 10 who report splitting. (When only small amounts are donated to secondary charities, it is sometimes ambiguous what counts as splitting.) In 2016, Peter Hurford gave about 2/3 to Rethink Charity, and 1/3 to other recipients. Jeff Kaufman and Julia Wise gave about equal amounts to AMF and the EA Giving Group Fund.

EA funds data

I wanted to study EAs’ splitting behaviour more systematically, so I looked at anonymised data from the EA funds, with permission from CEA.

In the following sections, I describe various possible analyses of the data. You can skip to “best approach: user totals” if you just want the bottom line. The R code I used is in the appendix.

I was given access to a list of every EA funds donation between 2017-03-23 and 2017-06-19. Data on allocation precentages was included. For example, if a donor went to the EA funds website and gave $1000, setting the split to 50% “global health and development” and 50% “long-term future”, there would be two entries, each for $500 and with an allocation percentage of 50%. In the following, I call these two entries EA funds donations, and the $1000 an EA funds allocation.

Naive approach: distribution of allocation percentages

The simplest analysis is to look at a histogram of the “allocation percentage” variable. The result looks like this1:


Here, most of the probability mass is on the left, because most donations are strongly split. But what we really care about is how much of the money being donated is split. For that we need to weight by donation size.

Less naive apprach: weighted distribution of allocation percentages

I compute a histogram of allocation percentages weighted by donation size. In other words, I ask: “if I pick a random dollar flowing through EA funds, what is its probability of being part of an EA funds donation which itself represents X% of an EA funds allocation?”, and then plot this for 20 buckets of Xs2.


Here, much more of the probability mass is on the right hand side. This means larger donors split less, and are much more likely to set the allocation percentage to 100%.

But this approach might still be problematic, because it is not invariant to how donors decide to spread their donations across allocations. For instance, suppose we have the following:

Allocation ID Name Fund Allocation % Donation amount
2 Alice Future 100% $1000
1 Alice Health 100% $1000
3 Bob Health 50% $1000
3 Bob Future 50% $1000

Here, Alice and Bob both split their $2000 donations equally between two funds. They merely used the website interface differently: Alice by creating two separate 100% allocations (perhaps the next month), and Bob by creating just one allocation but setting the sliders for each of the funds to 50%.

However, if we used this approach, we would count Alice as not splitting at all.

It’s an open question how much time should elapse between two donations to different charities until it is no longer considered splitting, but rather changing one’s mind. In the individual examples I gave above, I took one month, which seems like a clear case of splitting. Up to a year seems reasonable to me. Since we have less than a year of EA funds data, it’s plausible to consider any donations made to more than one fund as splitting. This is the approach I take in the next section.

Best approach: user totals

For each user, I compute:

  • Their fund sum, i.e. for each fund they donated to, the sum of their donations to that fund
  • Their user totals, i.e. the sum of all their donations to EA Funds

This allows me to create a histogram of the fraction of a user total represented by each fund sum, weighted by the fund sum3.


This is reasonably similar to the weighted distribution of allocation percentages, but with a bit more splitting.

Other data

One could also look at the Donations recorded for Vipul Naik database, or Giving What We Can’s data, and conduct similar analyses. The additional value of this over the EA funds analysis seemed limited, so I didn’t do it.

Arguments for splitting

Empirical uncertainty combined with risk aversion

Sometimes being (very) uncertain about which donation opportunity is best is presented as an argument for splitting. For example, the EA funds FAQ says that “there are a number of circumstances where choosing to allocate your donation to multiple Funds might make more sense” such as “if you are more uncertain about which ways of doing good will actually be most effective (you think that the Long-Term Future is most important, but you think that it’s going to be really difficult to make progress in that area)”.

High uncertainty is only a reason to split or diversify if one is risk averse. Is it sensible to be risk averse about one’s altruistic decisions? No. As Carl Schulman writes:

What am I going to do with my tenth vaccine? Vaccinate another kid!

While Sam’s 10th pair of shoes does him little additional good, a tenth donation can vaccinate a tenth child, or a pay for the work of a tenth scientist doing high impact research such as vaccine development. So long as Sam’s donations don’t become huge relative to the cause he is working on (using up the most efficient donation opportunities) he can often treat a charitable donation of $1,000 as just as worthwhile as a 1 in 10 chance of a $10,000 donation.

Moral uncertainty

The EA funds FAQ says that another reason for splitting could be “If you are more uncertain about your values (for example, you think that Animal Welfare and the Long-Term Future are equally important causes)”.

Does it make any difference if the uncertainty posited is about morality or our values rather than the facts? In other words, is it reasonable for a risk-neutral donor facing moral uncertainty to split?

This depends on our general theory for dealing with cases of moral uncertainty. (Will MacAskill has written his thesis on this.) We can start by distinguising moral theories which value acts cardinally (like utilitarianism) from moral theories which only value acts ordinally. The latter category would include theories which only admit of two possible ranks, permissible and impermissible (like some deontlogical theories), as well as theories with finer-grained ranking.

If the only theories in which you have non-zero credence are cardinal theories, we can simply treat our normative uncertainty like empirical uncertainty, by computing the expected value. (MacAskill argues persuasively against competing proposals like ‘my favourite theory’, see Chapter 1).

What if you also hold some credence in merely ordinal theories? In that case, according to MacAskill, you should treat the situation as a voting problem. Each theory can “vote” by ranking your possible actions, and gets a number of votes that is proportional to your credence in that theory.

The question is which voting rule to use. Different voting rules have different properties. A simple property might be:

Unanimity: if all voters agree that X>Y, then the ouput of the voting rule must have X>Y.

Let’s say we are comparing the following acts:

  1. Donate $1000 to charity A
  2. Donate $500 to charity A and $500 to charity B.

Unanimity implies that if all the first-order theories in which you have credence favour (1), then your decision after accounting for moral uncertainty will also favour (1). So provided our voting rule satisfies unanimity, moral uncertainty provides no additional reason to split. (In fact, a much weaker version of unanimity will usually do, if you have sufficiently low credence in pro-splitting moral theories.)

Diminishing returns

A good reason to split would be if you face diminishing returns. At what margins do we begin to see returns diminish sufficiently to justify splitting? This depends on how much donation opportunities differ in cost-effectiveness.

Suppose there are two charities, whose impact \(f\)$ and \(g\) are monotone increasing with monotone decreasing first derivatives. Then you should start splitting at \(D\) such that \(g'(0)>f'(D)\).

If you have \(X\), you should donate \(X-A\) to charity f and \(A\) to charity g such that \(g'(A)=f'(X-A)\).

It’s generally thought that for small (sub-six-figure) donors, \(A=0\), that is, returns aren’t diminishing noticeably compared to the difference in cost-effectiveness between charities.

However, many people believe that at the level of the EA community, there should be splitting. What does this imply in the above model?

Let’s assume that the EA community moves \(X = $100 million\) per year (including Good Ventures). Some people take the view that the best focus area is more than an order of magnitude more cost-effective than others (although it’s not always clear which margin this claim applies to). Under some such view, marginal returns would need to diminish by more than 10 times over the 0-100M range in order to get a significant amount of splitting. To me, this seems intuitively unlikely. (Of course, some areas may have much faster diminishing returns than others4.) Michael Dickens writes:

The US government spends about $6 billion annually on biosecurity5. According to a Future of Humanity Institute survey, the median respondent believed that superintelligent AI was more than twice as likely to cause complete extinction as pandemics, which suggests that, assuming AI safety isn’t a much simpler problem than biosecurity, it would be appropriate for both fields to receive a similar amount of funding. (Sam Altman, head of Y Combinator, said in a Business Insider interview, “If I were Barack Obama, I would commit maybe $100 billion to R&D of AI safety initiatives.”) Currently, less than $10 million a year goes into AI safety research.

Open Phil can afford to spend something like $200 million/year. Biosecurity and AI safety, Open Phil’s top two cause areas within global catastrophic risk, could likely absorb this much funding without experiencing much diminishing marginal utility of money. (AI safety might see diminishing marginal utility since it’s such a small field right now, but if it were receiving something like $1 billion/year, that would presumably make marginal dollars in AI safety “only” as useful as marginal dollars in biosecurity.)

To take another approach, let’s look at animal advocacy. Extrapolating from Open Phil’s estimates, its grants on cage-free campaigns are probably about ten thousand times more cost-effective than GiveDirectly (if you don’t heavily discount non-human animals, which you shouldn’t) (more on this later), and perhaps a hundred times better after adjusting for robustness. Since grants on criminal justice reform are not significantly more robust than grants on cage-free campaigns, the robustness adjustments look similar for each, so it’s fair to compare their cost-effectiveness estimates rather than their posteriors.

Open Phil’s estimate for PSPP suggests that cage-free campaigns are a thousand times more effective. If we poured way more money into animal advocacy, we’d see diminishing returns as the top interventions became more crowded, and then less strong interventions became more crowded. But for animal advocacy grants to look worse than grants in criminal justice, marginal utility would have to diminish by a factor of 1000. I don’t know what the marginal utility curve looks like, but it’s implausible that we would hit that level of diminished returns before increasing funding in the entire field of farm animal advocacy by a factor of 10 at least. If I’m right about that, that means we should be putting $100 million a year into animal advocacy before we start making grants on criminal justice reform.

I find this line of argument moderately convincing. Therefore, my guess is that people who believe that their preferred focus area is orders of magnitude better than others, should generally also believe that the whole EA community should donate only to that focus area.

Achieving a community-wide split

Suppose you do think, for reasons like those described in the previous section, that because of diminishing returns, the community’s split should be \(A=30%X\). (There may be other reasons to believe this, for instance if the impact of different causes is multiplicative rather than additive.)

There are two ways that this could lead to you prefer splitting your individual donation: cooperation with other donors, and lack of information.

Cooperation with other donors

Suppose that at time t, before you donate, the communty splits \(A_t = 50%X\). You are trying to move the final allocation to \(A= 30%X\), so you should donate everything to \(f\) (assuming your donation is small relative to the community). If the community’s allocation was \(A_t = 90%X\), however, you should donate everything to \(g\). We can call this view the single-player perspective.

From this perspective, it’s very important to find out what the community’s current allocation is, since this completely changes how you should act.

But now suppose that there other donors, who also use the single-player perspective. For the sake of simplicity we can assume they also believe the correct community-wide split is \(A= 30%X\)5. The following problem occurs:

Everyone is encouraged to spend a lot of time looking into current margins, to work out what the best cause is. Worse, if the community as a whole is being close to efficient in allocation, in fact what is best at the margin changes a whole lot as things scale up, and probably isn’t much better than second- or third-best thing. This means that it’s potentially lots of work to form a stable view on where to give, and it doesn’t even matter that much.6

Imagine the donors could all agree to donate in the same proportional split as the optimal community allocation (call this the cooperative perspective). They would obtain the same end result of a 70%/30% split, while saving a lot of effort. When everyone uses the single-player perspective, the group is burning a lot of resources on a zero-sum game.

From a rule-consequentialist perspective, you should cooperate in prisonner’s dilemmas, that is, you should use the cooperative perspective, even if, to the best of you knowledge, this will lead to less impact.

Even if we find rule-consequentialism unconvincing, act-consequentialism would still recommend investing resources to make it more likely that the community as a whole cooperates. This could include publicly advocating for the cooperative perspective, or getting a group of high-profile EA donors to promise to cooperate amongst themselves.

Lack of information

Suppose information about the community’s split was impossible or prohibitively expensive to come by. Then someone using the single-player perspective would have to rely on their priors. One reasonable-sounding prior would be one that is symmetrical on either side of \(A_t=30%X\), or otherwise has the expected value \(A_t=30%X\). This prior assumes that given no information about where others are donating, they are equally likely to collectively undershoot as to overshoot their preferred community-wide split.

On this prior, the best thing you can do is to donate 70% to \(f\) and 30% to \(g\). So given some priors, and when there is no information about others’ donations, the single-player perspective converges with the cooperative perspective.

Remaining open-minded or avoiding confirmation bias

Because of confirmation bias and consistency effects, donating 100% to one charity may bias us in the direction of believing that this charity is more cost-effective. For example, one GiveWell staff member writes7:

I believe that it is important to keep an open mind about how to give to help others as much as possible. Since I spend a huge portion of my time thinking within the GiveWell framework, I want to set aside some of my time and money for exploring opportunities that I might be missing. I am not yet sure where I’ll give these funds, but I’m currently leaning toward giving to a charity focused on improving farm animal welfare.

I tend to find this type of argument from bias less convincing than other members of the EA community8. I suspect that the biases involved are insensitive to the scope of the donations, that is, it’s sufficient to donate a nomial amount to other causes in order to reduce or eliminate the bias. Then such considerations would offer no reason for significant splitting. It’s also questionable whether such self-deception is even likely to work. Claire Zabel’s post “How we can make it easier to change your mind about cause areas” also offers five techniques for reducing this bias. Applying these techniques seems like a less costly approach than sacrificing significant expected impact to splitting.

Memetic effects

Sometimes people justify splitting like so: “splitting will reduce my direct impact, but it will allow me to have more indirect impact by affecting how others view me”.

For example:

In the past we’ve split our donations approximately 50% to GiveWell-recommended charities and 50% to other promising opportunities, mostly in EA movement building. […] GiveWell charities are easier to talk about and arguably allow us to send a less ambiguous signal to outsiders.

And also:

I’ll probably also give a nominal amount to a range of different causes within EA (likely AMF, GFI and MIRI), in order to keep up to date with the research across the established cause areas, and signal that I think that other cause areas are worthwhile.

The soundness of these reasons depends very much on each donor’s personal social circumstances, so it’s hard to evaluate any specific instance. A few general points to keep in mind are:

  • There may be memetic costs as well as benefits to splitting. For example, donating to only one charity reinforces the important message that EAs try to maximise expected value.
  • From a rule-consequentialist perspective, it may be better to always be fully transparent, and not to make donations decisions based on how they will affect what others think of us.
  • There could be cheaper ways of achieving the same benefits. For example, saying “This year I donated 100% to X, but in the past I’ve donated to Z and Y” or “Many of my friends in the community donate to Z and Y” could send some of the intended signals without requiring any actual splitting.


I’m not convinced by most of the reasons people give for splitting. Cooperation with donors appears to me to be the best proposed reason for splitting.

To some degree, we may be using splitting to satisfy our urge to purchase “fuzzies”. I say this without negative judgement, I agree with Claire Zabel that we should “de-stigmatize talking about emotional attachment to causes”. I think we should satisfy our various desires, like emotional satisfaction or positive impact, in the most efficient way possible. It may not be psychologically realistic to plan to stop splitting altogether. Instead, one could give as much as possible to the recipient with the highest expected value, while satisfying the desire to split with the small remaining part. Personally, I donate 90% to the Far Future EA fund and 10% to the Animal Welfare fund for this reason.

Appendix: R code


f <- data.frame(read_csv("~/split/Anonomized EA Funds donation spreadsheet - Amount given by fund.csv"))

exchr <- 1.27

# convert everything to usd
f$dusd <- ifelse(f$Currency=="GBP",exchr*f$Donation.amount,f$Donation.amount)

#naive histogram of allocation percentages
n <- hist(f$Allocation.percentage, breaks=bseq, freq=FALSE, xlab="Allocation Percentage", main="")
n_n <- data.frame(bucket=w$breaks[2:length(n$breaks)],prob=(n$counts/sum(n$counts)))

# weighted histogram
w <- weighted.hist(f$Allocation.percentage,f$dusd,breaks=bseq,freq = FALSE, xlab="Allocation Percentage", ylab = "Density, weighted by donation size")
w_n <- data.frame(bucket=w$breaks[2:length(w$breaks)],prob=(w$counts/sum(w$counts)))

#user totals
f <- ddply(f,.(UserID),transform,usersum=sum(dusd))
u <- ddply(u,.(UserID),transform,usersum=sum(dusd))

#user totals by fund
f <- ddply(f,.(UserID, Fund.Name),transform,usersum_fund=sum(dusd))

# fundfrac
f$fundfrac <- f$usersum_fund/f$usersum

# remove appropriate duplicates
f$isdupl <- duplicated(f[,c(2,5)])
f2 <- subset(f, isdupl == FALSE)

# weighted histogram of fundfrac
z <- weighted.hist(f2$fundfrac,f2$usersum_fund,breaks=bseq/100,freq = FALSE, xlab="Fund fraction (per user)", ylab = "Density, weighted by fund sum (per user)")
z_n <- data.frame(bucket=z$breaks[2:length(z$breaks)],prob=(z$counts/sum(z$counts)))
  1. The underlying data are:

    allocation percentage bucket probability
    5% 0.088
    10% 0.137
    15% 0.105
    20% 0.166
    25% 0.098
    30% 0.050
    35% 0.037
    40% 0.039
    45% 0.067
    50% 0.061
    55% 0.011
    60% 0.016
    65% 0.005
    70% 0.011
    75% 0.006
    80% 0.019
    85% 0.004
    90% 0.007
    95% 0.002
    100% 0.069

  2. The data:

    allocation percentage bucket probability
    5% 0.009
    10% 0.011
    15% 0.025
    20% 0.020
    25% 0.043
    30% 0.060
    35% 0.032
    40% 0.066
    45% 0.015
    50% 0.037
    55% 0.083
    60% 0.004
    65% 0.015
    70% 0.002
    75% 0.004
    80% 0.007
    85% 0.022
    90% 0.003
    95% 0.075
    100% 0.467

  3. The data are:

    fraction of user total bucket probability
    5% 0.013
    10% 0.019
    15% 0.021
    20% 0.024
    25% 0.044
    30% 0.052
    35% 0.074
    40% 0.065
    45% 0.025
    50% 0.033
    55% 0.097
    60% 0.074
    65% 0.017
    70% 0.003
    75% 0.004
    80% 0.008
    85% 0.025
    90% 0.003
    95% 0.002
    100% 0.395

  4. One extreme example would be a disease eradication programme, where returns stay high until they go to zero after eradicaiton has been successful, vs. cash transfers where returns diminish very slowly. 

  5. The extension to the general case would go like this: everyone truthfully states their preferred split and donation amount, and a weighted average is used to compute the resulting community-preferred spit. See also “Donor coordination under simplifying assumptions”

  6. Adapted from Owen Cotton-Barratt, personal communication. 

  7. In addition, this OpenPhil post on worldview diversification, and this comment give reasons a large funder may want to make diversified donations in order to retain the ability to pivot to a better area. Some of them may transfer to the individual donor case. 

  8. The points in this paragraph apply similarly to oher “arguments from bias”, such as donating for learning value or to motivate oneself to do reasearch in the future (both of which I have seen made). 

August 3, 2017

The logical empiricist picture of the world

If we take in our hand any volume; of divinity or school metaphysics, for instance; let us ask, Does it contain any abstract reasoning concerning quantity or number? No. Does it contain any experimental reasoning concerning matter of fact and existence? No. Commit it then to the flames: for it can contain nothing but sophistry and illusion.

— David Hume, An Enquiry Concerning Human Understanding, 1777, Section XII, Part 3

I made this diagram as an attempted summary of logical empiricism.1

We can then think of various counters to logical empiricism within this framework. For example, Kripke denies the Kantian thesis. In “Two dogmas of empiricism”, Quine attacks all three vertices of the golden triangle. I may write a post about these in the future.

  1. The above construction, in javascript, might break one day. Backups are in .png and .xml

August 1, 2017

Why ain't causal decision theorists rich? Some speculations

Note added on 24 June 2018: This is an old post which no longer reflects my views. It likely contains mistakes.


CDT is underspecified

Standard causal decision theory is underspecified. It needs a theory of counterfactual reasoning. Usually we don’t realise that there is more than one possible way to reason counterfactually in a situation. I illustrate this fact using the simple case Game, described below, where a CDT agent using bad counterfactuals loses money.

But before that I need to set up some formalisms. Suppose we are given the set of possible actions an agent could take in a situation. The agent will in fact choose only one of those actions. What would have happened under each of the other possible actions? We can think of the answer to this question as a list of counterfactuals.

Let’s call such a list K of counterfactuals a “causal situation” (Arntzenius 2008). The list will have n elements when there are n possible actions. Start by figuring out what all the possible lists of counterfactuals K are. They form a set P which we can call the “causal situation partition”. Once you have determined P, then for each possible K, figure out what the expected utility:

\[E[U_K(A)]= \sum_{j}Cr(O_j | A \land K)*U(O_j)\]

of each of your n acts \(\{A_1,A_2,...,A_n\}\) is. (Where \(O_j\) are the outcomes, \(U(.)\) is the utility function and \(Cr(.)\) is the credence function.) Then, take the average of these expected utilities, weighted by your credence in each causal situation:

\[E[U(A)]=\sum_{K} Cr(K) * E[U_K(A)]\]

Perform the act with the highest \(E[U(A)]\).

What will turn out to be crucial for our purposes is that there is more than one causal situation partition P one can consistently use. So it’s not just a matter of figuring out “the” possible Ks that form “the” P. We also need to choose a P among a variety of possible Ps.

In other words, there is the following hierarchy:

  • Choose a causal situation partition P out of the set of possible partitions \(\{P_1,P_2,...,P_k\}\) (the causal situation superpartition?).
  • This partition defines a list of possible causal situations: \(P = \{K_1,K_2,...,K_j\}\).
  • Each causal situation K defines a list of counterfactuals of length n: \(K = \{C_1,C_2,...,C_n\}\). Where each counterfactual \(C_i\) is of the form “\(A_i \mathbin{\square\!\mathord\to} X\)”. You have a credence distribution over Ks.


Now let’s consider the following case, Game, also from Arntzenius (2008):

Harry is going to bet on the outcome of a Yankees versus Red Sox game. Harry’s credence that the Yankees will win is 0.9. He is offered the following two bets, of which he must pick one:

(1) A bet on the Yankees: Harry wins $1 if the Yankees win, loses $2 if the Red Sox win

(2) A bet on the Red Rox: Harry wins $2 if the Red Sox win, loses $1 if the Yankees win.

What are the possible Ps? According to Arntzenius, they are:

P1: Yankees Win, Red Sox Win

P2: I win my bet, I lose my bet

To make this very explicit using the language I describe above, we can write that the set of causal situations (the “superpartition”) is \(\{ P_b , P_w\}\). (I use \(P_b\) for “baseball” and \(P_w\) for “win/lose”.)

Let’s first deal with the baseball partition: \(P_b= \{K_y,K_s\}\). (I use \(K_y\) for “Yankees win” and \(K_s\) for “Sox win”.)

\(K_y = \{C_1,C_2 \}\)
\(C_1 = \text{Harry bets on the Yankees} \mathbin{\square\!\mathord\to} \text{Harry +1\$}\)
\(C_2 = \text{Harry bets on the Sox} \mathbin{\square\!\mathord\to} \text{Harry -1\$}\)

\(K_s = \{C_3,C_4 \}\)
\(C_3 = \text{Harry bets on the Yankees} \mathbin{\square\!\mathord\to} \text{Harry -2\$}\)
\(C_4 = \text{Harry bets on the Sox} \mathbin{\square\!\mathord\to} \text{Harry +2\$}\)

And Harry has the following credences1:


When using this partition and the prodecure described above, Harry finds that the expected value of betting on the Yankees is 70c, whereas the expected value of betting on the Sox is -70c, so he bets on the Yankees. This is the desired result.

And now for the win/lose partition: \(P_w= \{K_w,K_l\}\). (I use \(K_w\) for “Harry wins his bet” and \(K_l\) for “Harry loses his bet”.)

\(K_w = \{C_5,C_6 \}\)
\(C_5 = \text{Harry bets on the Yankees} \mathbin{\square\!\mathord\to} \text{Harry +1\$}\)
\(C_6 = \text{Harry bets on the Sox} \mathbin{\square\!\mathord\to} \text{Harry +2\$}\)

\(K_l = \{C_7,C_8 \}\)
\(C_7 = \text{Harry bets on the Yankees} \mathbin{\square\!\mathord\to} \text{Harry -2\$}\)
\(C_8 = \text{Harry bets on the Sox} \mathbin{\square\!\mathord\to} \text{Harry -1\$}\)

What are Harry’s credences in \(K_w\) and \(K_l\)? It turns out that it doesn’t matter. Arntzenius writes: “no matter what \(Cr(K_w)\) and \(Cr(K_l)\) are, the expected utility of betting on the Sox is always higher”.

So Harry should bet on the Sox regardless of his credences. But the Yankees win 90% of the time, so once Harry has placed his bet, he will correctly infer that \(Cr(K_l)=0.9\). Harry will lose 70c in expectation, and he can foresee that this will be so! It’s because he is using a bad partition.


Now consider the case Predictor, which is identical to Game except for the fact that:

[…] on each occasion before Harry chooses which bet to place, a perfect predictor of his choices and of the outcomes of the game announces to him whether he will win his bet or lose it.

Arntzenius crafts this thought experiment as a case where, purportedly:

  • An evidential decision theories predictably loses money.2
  • A causal decision theorist using the baseball partition predictably wins money.

I’ll leave both of these claims undefended for now, taking them for granted.

I’ll also skip over the crucial question of how one is supposed to systematically determine which partition is the “correct” one, since Arntzenius provides an answer3 that is long and technical, and I believe correct.

What is the point of proposing Predictor? We know that EDT does predictably better than CDT in Newcomb. Predictor is a case where CDT does predictably better than EDT, provided that it uses the appropriate partition. But we already knew this from more mundane cases like Smoking lesion (Egan 2007).


The value of WAYR arguments

Arntzenius’ view appears to be that “Why ain’cha rich?”-style arguments (henceforth WAYRs) give us no reason to choose any decision theory over another. There is one sense in which I agree, but I think it has nothing to do with Predictor, and, more importantly, that this is not an argument for being poor, but instead a problem for decision theory as currently conducted.

One way to think of decision theory is as a conceptual analysis of the word “rational”, i.e. a theory of rationality. Some causal decision theorists say that in Newcomb, rational people predictably lose money. But this, they say, is not an argument against CDT, for in Newcomb, the riches were reserved for the irrational: “if someone is very good at predicting behavior and rewards predicted irrationality richly, then irrationality will be richly rewarded” (Gibbard and Harper 1978).

This line of reasoning appearts particularly compelling in Arntzenius’ Insane Newcomb:

Consider again a Newcomb situation. Now suppose that the situation is that one makes a choice after one has seen the contents of the boxes, but that the predictor still rewards people who, insanely, choose only box A even after they have seen the contents of the boxes. What will happen? Evidential decision theorists and causal decision theorists will always see nothing in box A and will always pick both boxes. Insane people will see $10 in box A and $1 in box B and pick box A only. So insane people will end up richer than causal decision theorists and evidential decision theorists, and all hands can foresee that insanity will be rewarded. This hardly seems an argument that insane people are more rational than either of them are.

But, others will reply: “The reason I am interested in decision theory is so that I can get rich, not so that I can satisfy some platonic notion of rationality. If I were actually facing that case, I’d rather be insane than rational.”

What is happening? The disputants are using the word “rational” in different ways. When language goes on holiday to the strange land of Newcomb, the word “rational” loses its everyday usefulness. This shows the limits of conceptual analysis.

Instead, we should use different words depending on what we are interested in. For instance, I am interested in getting rich, so I could say that act-decision theory is the theory that tells me how to get rich if I find myself in a particular situation and am not bound by any decision rule. Rule-decision theory would be the theory that tells you which rules are best for getting rich. Inspired by Ord (2009), we could even define global decision theory as the theory which, for any X, tells you which X will make you the most money.

Which X to use will depend on the context. Specifically, you should use the X which you can choose, or causally intervene on. If you are choosing a decision rule, for example by programming an AI, you should use rule-decision theory. (If you want to think of “choosing a rule for the AI” as an act, act-decision theory will tell you to choose the rule that rule decision theory identifies. That’s a mere verbal issue.) If you are choosing an act, such as deciding whether to smoke, ou should use act-decision theory.

Kenny Easwaran has similar thoughts:

Perhaps there just is a notion of rational action, and a notion of rational character, and they disagree with each other. That the rational character is being the sort of person that would one-box, but the rational action is two-boxing, and it’s just a shame that the rational, virtuous character doesn’t give rise to the rational action. I think that this is a thought that we might be led to by thinking about rationality in terms of what are the effects of these various types of intervention that we can have. […]

I think one way to think about this is […] trying to understand causation through what they call these causal graphs. They say if you consider all the possible things that might have effects on each other, then we can draw an arrow from anything to the things that it directly affects. Then they say, well, we can fill in these arrows by doing enough controlled experiments on the world, we can fill in the probabilities behind all these arrows. And we can understand how one of these variables, as we might call it, contributes causally to another, by changing the probabilities of these outcomes.

The only way, they say, that we can understand these probabilities, is when we can do controlled experiments. When we can sort of break the causal structure and intervene on some things. This is what scientists are trying to do when they do controlled experiments. They say, “If you want to know if smoking causes cancer, well, the first thing you can do is look at smokers and look at whether they have cancer and look at non-smokers and look at whether they have cancer.” But then you’re still susceptible to the issues that Fisher was worrying about. What you should actually do if you wanted to figure out whether smoking causes cancer, is not observe smokers and observe non-smokers, but take a bunch of people, break whatever causes would have made them smoke or made them not smoke, and you either force some people to smoke or force some people not to smoke.

Obviously this experiment would never get ethical approval, but if you can do that – if you can break the causal arrows coming in, and just intervene on this variable and force some people to be smokers and force others to not be smokers, and then look at the probabilities – then we can understand what are the downstream effects of smoking.

In some sense, these causal graphs only make sense to the extent that we can break certain arrows, intervene on certain variables and observe downstream effects. Then, I think, in all these Newcomb type problems, it looks like there’s several different levels at which one might imagine intervening. You can intervene on your act. You can say, imagine a person who’s just like you, who had the same character as you, going into the Newcomb puzzle. Now imagine that we’re able to, from the outside, break the effect of that psychology and just force this person to take the one box or take the two boxes. In this case, forcing them to take the two boxes, regardless of what sort of person they were like, will make them better off. So that’s a sense in which two-boxing is the rational action.

Whereas if we’re intervening at the level of choosing what the character of this person is before they even go into the tent, then at that level the thing that leaves them better off is breaking any effects of their history, and making them the sort of person who’s a one-boxer at this point. If we can imagine having this sort of radical intervention, then we can see, at different levels, different things are rational.

To what extent we human beings can intervene at the level our acts, or at the level of our rules, is, I suspect, an empirically and philosophically deep issue. But I would be delighted to be proven wrong about that.

A problem for any decision theory?

I think using these distinctions can solve much of the confusion about WAYRs in Newcomb and analogous cases. But Insane Newcomb hints at a more fundamental problem. Both EDT and CDT can be made vulerable to an WAYR, for example in Insane Newcomb.

Moreover, any decision theory can be made vulnerable to WAYRs. Imagine the following generalised Newcomb problem.

The predictor has a thousand boxes, some transparent and some opaque, and the opaque boxes have arbitrary amounts of money in them. Suppose you use decision theory X, which, conditional on your credences, determines a certain pattern of box-taking (e.g. take box 1, leave boxes 2 and 4, take boxes 3 and 5, etc). The predictor announces that if he has predicted that you will take boxes in this pattern, he has put $0 in all opaque boxes, while otherwise he has put $1000 in each opaque box.

This case has the consequence that X-decision theorists will end up poor. Since X can be anything, a sufficiently powerful predictor can punish the user of any decision theory. Newcomb is a special case where the predictor punishes causal decision theorists.

So I’m inclined to say that there exists no decision theory which will make you rich in all cases. So we need to be pragmatic and choose the decision theory that works best given the cases we expect to face. But this just means using the meta-decision theory that tells you to do that.

  1. This isn’t fully rigorous, since Ks are lists of (counterfactual) propositions, so you can’t have a credence in a K. What I mean by \(Cr(K_y)=0.9\) is that Harry has credence 0.9 in every C in K, and (importantly) he also has credence 0.9 in in their conjunction \(C_1 \land C_2\). But I drop this formalism in the body of the post, which I feel already suffers from an excess of pedantry as it stands! 

  2. This is denied by Ahmed and Price (2012), but I ultimately don’t find their objection convincing. 

  3. See section 6, “Good and Bad Partitions”. Importantly, this account fails to identify any adequate partition in Newcomb, so the established conclusion that causal decision theorists tend to lose money in Newcomb still holds. 

June 28, 2017

QALYs/$ are more intuitive than $/QALYs

Cross-posted to the effective altruism forum.


Cost-effectiveness estimates are often expressed in $/QALYs instead of QALYs/$. But QALYs/$ are preferable because they are more intuitive. To avoid small numbers, we can renormalise to QALYs/$10,000, or something similar.

Cost-effectiveness estimates are often expressed in $/QALYs.

Four examples:

GiveWell, “Errors in DCP2 cost-effectiveness estimate for deworming”:1

Eventually, we were able to obtain the spreadsheet that was used to generate the $3.41/DALY estimate. That spreadsheet contains five separate errors that, when corrected, shift the estimated cost effectiveness of deworming from $3.41 to $326.43. We came to this conclusion a year after learning that the DCP2’s published cost-effectiveness estimate for schistosomiasis treatment – another kind of deworming – contained a crucial typo: the published figure was $3.36-$6.92 per DALY, but the correct figure is $336-$692 per DALY. (This figure appears, correctly, on page 46 of the DCP2.)

DCP3, “Cost-Effectiveness of Interventions for Reproductive, Maternal, Newborn, and Child Health”:


Michael Dickens, “Charities I would like to see”:

This would cost about $5 per rat per month plus an opportunity cost of maybe $500 per month for the time spent, which works out to another $5 per rat per month. Thus creating 1 rat QALYs costs $120 per year, which is $240 per human QALYs per year.

Deworming treatments cost about $30 per DALY. Thus a rat farm looks like a fairly expensive way of producing utility.

GiveWell, “Mass Distribution of Long-Lasting Insecticide-Treated Nets (LLINs)” uses cost per life saved:

LLIN distribution is one of the most cost-effective ways to save lives that we’ve seen. Our best guess estimate comes out to about $3,000 per equivalent under-5 year old life saved (or, excluding developmental impacts, $7,500 per life saved) using the total cost per net in the countries we expect AMF to work over the next few years.

QALYs/$ are preferable to $/QALYs

As long as we compare opportunities to do good by looking at the ratio of their cost-effectiveness, $/QALYs is equivelent to QALYs/$.

However, even if we know that we ought to be using ratios of cost-effectiveness, our System 1 may sometimes implicitly be using differences (subtractions) of cost-effetiveness. This can lead to problems when using $/QALYs which are entirely avoided if we use QALYs/$.

Suppose we have 20 charities \(a\)-\(t\) whose cost-effectiveness follows a log-normal distribution. I have plotted bar graphs of these values expressed in $/QALYs and in QALYs/$.

d_per_q Looking at this graph, we are immediately attracted to the right-hand side. That’s where the big, visible differences in bar height are. So we feel that the high-hand side is where most of the action is. We may have the intuition that most of the gains are to be had by switching away from from charities like \(o\), \(b\), and \(p\), in favour of charities like \(g\), \(l\) and \(m\). This is because we would implicitly be using differneces instead of ratios.

In reality, of course, what’s crucial is the left-hand side of the graph. Charity \(q\) produces about 9 times more value than charity \(a\), while charity \(b\) is only 1.5 times better than charity \(p\).

q_per_d If we had used QALYs/$, this would have been easier to see. Here, the importance of picking the best charity (rather than a merely good one) stands out visually.

When we use QALYs/$, both products and subtractions give us the correct result. That is why QALYs/$ are preferable.

Small numbers

One potential problem with using QALYs/$ is that we end up with very small numbers. Small numbers can be unintuitive. It’s hard to picture 0.05 and 0.1 of something, and easy to picture 20 and 10 of something.

But this problem can easily be solved by multiplying the small numbers by a large constant. This is what we did with the Oxford Prioritisation Project, and it’s also what Toby Ord does in “The moral imperative towards cost-effectiveness”.

Further reading

By the way, this exact phenomenon is well documented in the domain of car fuel efficiency. See “The MPG Illusion”, Science Vol. 320, Issue 5883, pp. 1593-1594, DOI: 10.1126/science.1154983.

Bastian Stern also has posts explaining how $/QALYs create problems when we use arithmetic means, and when we look at proportional improvements between charities. This is not suprising, since arithmetic means and proportions are essentially based on subtraction.


Wherever possible, we should stop using $/QALYs and use QALYs/$10,000, or something similar.

  1. Of course, there are also many examples of people correctly using QALYs/$. See for instance “The moral imperative towards cost-effectiveness”, or chapter 3 of “Doing Good Better”. 

June 15, 2017