Would an EA world with limited money fund costly treatments?

During EA Global London 2017, Emilia Wilson gave an interesting five minute talk, which I've transcribed below:

I'll just start by saying that this is kind of an ethics project, but it's also about movement building. And if either of those are your interest, please do come talk to me. I'll be at the office hours but I'll also be at the poster session. The starting point of this project is the fact that in my experience, when you introduce the idea of effective altruism to groups of people, you almost always hear "but if everyone gave to AMF, what's going to happen to cancer research?" And this is a really bad critique. Many people have pointed out that this is a really bad critique for a few different reasons. And I'm not going to dwell on that. What I'm interested in is when you point out that this is a really bad critique to the people raising it, they don't seem terribly convinced even if they understand why. And so I was looking at whether there's kind of a better concern hiding in there. Is there something more that they're getting at?

And I argue that there is at least a coherent concern that they're perhaps getting at here. And so it centers around how if we think of an EA world, what is the world we're thinking about? Now I don't think we can describe that world in any particular detail, which is kind of it's own interesting issue. But I think we can say that there would be causes which would go unfunded because they are too ineffective, too expensive. Jonathan touched on this as kind of an inescapable principle that if you have two interventions and one of them is going to save more lives, you go for that intervention. And I actually think this might be the thing that is causing people some concern. Essentially what I'm talking about is people who if you give them the example of Sightsavers vs. guide dogs they agree we should donate to Sightsavers, but they don't want to live in a world without guide dogs.

And I think this concern breaks down into two parts. So I think we firstly have a concern about the value of hope - the possibly quite practical value of hope. If people know they're going to receive no help, they lose hope - this is a loss of welfare, this is a negative thing. And I think that's a more practical concern that effective altruism can deal with quite easily. We can say well we can just put that into our model. That's just another welfare concern.

But I think the second part of this concern is less practical and is an idea about a world without chances, a world that lacks fairness. And the concern here is that when you say that a particular group of people are just going to receive no help whatsoever and have no chance of savior, that is unfair in some way. And so there seems to be an underpinning idea here that if you were to die having had no chance of savior that is somehow worse and somehow unfair compared to you dying having had some chance that you missed out on. So I think if you hold this view of a world without chances, you think it's unfair to distribute resources in this way, what you probably want is something like a weighted lottery. So you distribute resources according to impact - less effective interventions get less funding, but they won't get none, so everyone has some chance of help.

So we then need to ask: is that idea at all reconcilable with effective altruism? And I've argued that it really doesn't seem to be. It seems like if effective altruism tries to accommodate this idea, what it's really doing is abandoning maximization or at least a typical understanding of maximization. Now I confess since being at the conference and discussing the poster with a few people, some individuals seem to have broader conceptions of effective altruism than I'm familiar with. But it certainly seems like softening to this kind of weighted lottery principle would really change what we understand effective altruism to be by massively redefining how we approach maximization. And to me this seems like a significant concern. Because we don't actually know how common this view is and I think that's in itself a bit worrying.

But if even a fraction of the people who raised the "Ah but if everyone gave to AMF" concern is actually holding this view, then that's going to pose a massive issue to our movement growth because it isn't something we can adopt. And so essentially what I would like or what I think it'd be good to have is if any of you happen to be in marketing or psychology or statistics, I think it'd be fantastic to have quantitative data on how many people hold these kinds of views, what are the typical responses to EA. Because currently we have a data driven movement that seems to have a severe lack of data on how we actually grow this movement. And I think if these kinds of views - which seem to clash with principles that we hold to be very obvious - if these kinds of views are common, that is going to to be a huge barrier to the growth of the movement and this is information we desperately need. 

To explore the idea of a weighted lottery system, let's consider a hypothetical:

There are 100,000 people with disease A; it costs $5,000 to cure one person of disease A.

There are 2,400,000 people with disease B; it costs $30,000 to cure one person of disease B.

The two diseases cause an equal amount of suffering per person.

We have $100,000,000 to spend. How do we spend it?

If our goal is to maximize the number of people cured, then we would spend all of the money on curing people of disease A (resulting in 20,000 people being cured of disease A).

If our goal is to maximize utility (including hope), then we would presumably choose to spend some amount of money on curing people of disease B.

But what if we want to use a weighted lottery system? This system could be implemented in several different ways. One way is to allocate money such that the probability of an individual being cured is inversely proportional to the cost of the cure (meaning that the probability of being cured of a given disease goes down as the cost of curing a person of that disease goes up). I'll refer to this as the proportional chance principle. To figure out how much to allocate to curing people of each disease under this principle, we need to solve a system of equations.

Here are our variables:

N_A = number of people who have disease A [100,000]

N_B = number of people who have disease B [2,400,000]

C_A = cost of curing a person of disease A [$5,000]

C_B = cost of curing a person of disease B [$30,000]

S_T = total amount of money to be spent [$100,000,000]

S_A = amount of money to be spent curing people of disease A [to be determined]

S_B = amount of money to be spent curing people of disease B [to be determined]

Here are our equations:

1: S_A + S_B = S_T

2: C_B / C_A = ((S_A / C_A) / N_A) / ((S_B / C_B) / N_B) [On the left side, we have the ratio of the cost of curing a person of disease B to the cost of curing a person of disease A. On the right side, we have the ratio of the fraction of people with disease A who are cured to the fraction of people with disease B who are cured. To calculate the fraction of people with a given disease who are cured, we divide the amount allocated to curing people of that disease by the cost of curing a person of that disease to obtain the number of people with that disease who will be cured and then divide the number of people with that disease who will be cured by the total number of people with that disease.]

Now let's solve for C_B.

C_B / C_A = ((S_A / C_A) / N_A) / ((S_B / C_B) / N_B) [Equation 2 from above]

C_B / C_A = (S_A / (C_A * N_A)) / (S_B / (C_B * N_B))

C_B * (S_B / (C_B * N_B)) = C_A * (S_A / (C_A * N_A))

S_B / N_B = S_A / N_A

S_A + S_B = S_T [Equation 1 from above]

S_A = S_T - S_B

S_B / N_B = (S_T - S_B) / N_A

S_B * N_A = (S_T - S_B) * N_B

S_B * N_A = S_T * N_B - S_B * N_B

S_B * N_A + S_B * N_B = S_T * N_B

S_B * (N_A + N_B) = S_T * N_B

S_B = S_T * N_B / (N_A + N_B)

Plugging in numbers, we get:

S_B = 100,000,000 * 2,400,000 / (100,000 + 2,400,000)

S_B = 96,000,000

S_A = 100,000,000 - 96,000,000

S_A = 4,000,000

Thus, we have the following allocation:

We will spend $4,000,000 on helping the 100,000 people with disease A (with each cure costing $5,000).

We will spend $96,000,000 on helping the 2,400,000 people with disease B (with each cure costing $30,000).

Now let's confirm that this allocation complies with the proportional chance principle. 

We cure 800 people of disease A ($4,000,000 / $5,000), which is 1/125 of people with disease A (800 / 100,000).

We cure 3,200 people of disease B ($96,000,000 / $30,000), which is 1/750 of people with disease B (3,200 / 2,400,000).

This means that the chance of a person with disease A being cured is six times higher (1/125 / 1/750).

Since it cost six times more to cure a person of disease B ($30,000 / $5,000), our allocation complies with the proportional chance principle.

There are three things worth noting about this allocation.

1. It results in significantly fewer people being cured (4,000 under this allocation (800 + 3,200) vs. 20,000 when maximizing the number of people cured).

2. Look at the equation we got for S_B: S_B = S_T * N_B / (N_A + N_B). It only contains N_A, N_B, and S_T. This means that when determining the amount of money to spend curing people of disease B (denoted by S_B), the only relevant considerations are the total amount of money (denoted by S_T), the number of people suffering from disease A (denoted by N_A), and the number of people suffering from disease B (denoted by N_B); the cost of curing a person of disease A (denoted by C_A) and the cost of curing a person of disease B (denoted by C_B) are not relevant. Thus, the same allocation would be used even if the cost of curing a person of disease B doubled.* Because the relative cost-effectiveness of the cures is irrelevant to the allocation of money under the proportional chance principle, the proportional chance principle is literally incompatible with effective altruism. 

*Let's verify that the same allocation would comply with the proportional chance principle even if the cost of curing a person of disease B doubled from $30,000 to $60,000.

We still spend $4,000,000 on helping the 100,000 people with disease A (with each cure still costing $5,000).

We still spend $96,000,000 on helping the 2,400,000 people with disease B (though each cure now costs $60,000).

We still cure 800 people of disease A ($4,000,000 / $5,000), which is 1/125 of people with disease A (800 / 100,000).

We now cure only 1,600 people of disease B ($96,000,000 / $60,000), which is 1/1,500 of people with disease B (1,600 / 2,400,000).

This means that the chance of a person with disease A being cured is 12 times higher (1/125 / 1/1,500).

Since it costs 12 times more to cure a person of disease B ($60,000 / $5,000), our new allocation complies with the proportional chance principle.

To see why doubling the cost of curing a person of disease B has no effect on the allocation, recall our equation for the proportional chance principle: C_B / C_A = ((S_A / C_A) / N_A) / ((S_B / C_B) / N_B). As the cost of curing a person of disease B increases (increasing the ratio of the cost of curing a person of disease B to the cost of curing a person of disease A), the share of people with disease B who are cured decreases (proportionally increasing the ratio of the share of people with disease A who are cured to the share of people with disease B who are cured). 

3. Under our allocation, 24 times as much money is spent on curing people of disease B ($96,000,000 / $4,000,000). There are also 24 times as many people with disease B (2,400,000 / 100,000). Is this a coincidence? Let's check by deriving the ratio of spending on curing people of disease A to spending on curing people of disease B under the proportional chance principle.

We start by deriving the equation for S_A, the amount spent on curing people of disease A.

S_B = S_T * N_B / (N_A + N_B) [This is the equation for S_B that we derived earlier.]

S_A + S_B = S_T [Equation 1 from above]

S_A = S_T - S_B

S_A = S_T - S_T * N_B / (N_A + N_B)

S_A = S_T * (1 - N_B / (N_A + N_B))

S_A = S_T * ((N_A + N_B) / (N_A + N_B) - N_B / (N_A + N_B))

S_A = S_T * (N_A + N_B - N_B) / (N_A + N_B)

S_A = S_T * N_A / (N_A + N_B)

Now let's calculate the ratio of S_A to S_B.

S_A / S_B = (S_T * N_A / (N_A + N_B)) / (S_T * N_B / (N_A + N_B))

S_A / S_B = (N_A * (S_T / (N_A + N_B))) / (N_B * (S_T / (N_A + N_B)))

S_A / S_B = N_A / N_B

The ratio of funding for curing people of disease A to funding for curing people of disease B is the same as ratio of the number of people with disease A to the number of people with disease B. Thus, it was no coincidence that both ratios were 24 in our hypothetical.

Based on the above equation, we can also see that the proportional chance principle results in the same amount of funding per person for the two diseases.

S_A / S_B = N_A / N_B

S_A / N_A = S_B / N_B [The funding per person for a given disease X is the amount of spending on curing people of that disease, denoted by S_x, divided by the number of people with that disease, denoted by N_x.]

Let's see if we can address the second issue noted above. Instead of allocating the same amount of funding per person to each disease regardless of the cost of curing people for each disease, let's allocate funding such that the funding per person for a given disease is inversely proportional to the cost of curing a person of that disease (meaning that the funding per person for a given disease goes down as the cost of curing a person of that disease goes up). I'll call this the proportional funding principle.

Here are our variables (same as above):

N_A = number of people who have disease A [100,000]

N_B = number of people who have disease B [2,400,000]

C_A = cost of curing a person of disease A [$5,000]

C_B = cost of curing a person of disease B [$30,000]

S_T = total amount of money to be spent [$100,000,000]

S_A = amount of money to be spent curing people of disease A [to be determined]

S_B = amount of money to be spent curing people of disease B [to be determined]

Here are our equations:

3: S_A + S_B = S_T [This is the same as Equation 1.]

4: (S_A / N_A) / (S_B / N_B) = C_B / C_A [On the left side, we have the ratio of the funding per person for disease A to the funding per person for disease B. The funding per person for a given disease is found by dividing the amount spent curing people of that disease by the number of people with that disease. On the right side, we have the ratio of the cost of curing a person of disease B to the cost of curing a person of disease A.]

Now let's solve for S_B.

S_A + S_B = S_T [Equation 3 from above]

S_A = S_T - S_B

(S_A / N_A) / (S_B / N_B) = C_B / C_A [Equation 4 from above]

(S_A / S_B) / (N_A / N_B) = C_B / C_A

S_A / S_B = (N_A / N_B) * (C_B / C_A)

(S_T - S_B) / S_B = (N_A / N_B) * (C_B / C_A) 

S_T - S_B = S_B * (N_A / N_B) * (C_B / C_A) 

S_T = S_B + S_B * (N_A / N_B) * (C_B / C_A) 

S_T = S_B * (1 + (N_A / N_B) * (C_B / C_A))

S_B = S_T / (1 + (N_A / N_B) * (C_B / C_A))

Plugging in numbers, we get:

S_B = 100,000,000 / (1 + (100,000 / 2,400,000) * (30,000 / 5,000))

S_B = 80,000,000

S_A = 100,000,000 - 80,000,000 = 20,000,000

Thus, we have the following allocation:

We will spend $20,000,000 on helping the 100,000 people with disease A (with each cure costing $5,000).

We will spend $80,000,000 on helping the 2,400,000 people with disease B (with each cure costing $30,000).

We cure 4,000 people of disease A ($20,000,000 / $5,000), which is 1/25 of people with disease A (4,000 / 100,000).

We cure 2,667 people of disease B ($80,000,000 / $30,000), which is 1/900 of people with disease B (2,667 / 2,400,000).

This means that the chance of a person with disease A being cured is 36 times higher (1/25 / 1/900).

There are three things worth noting about this allocation.

1. The chance of a person with disease A being cured is 36 times higher (1/25 / 1/900). 36 is the square of 6, which is how many times more it costs to cure a person with disease B ($30,000 / $5,000). Is it a coincidence the chance of being cured is inversely proportional to the square of the cost of being cured? Let's check by attempting to derive the proportional funding principle from a squared version of the proportional chance principle.

(S_A / (C_A * N_A)) / (S_B / (C_B * N_B)) = (C_B / C_A)² [This is the same equation used for the proportional chance principle except that the two sides are switched and C_B / C_A is squared.]

(S_A / (C_A * N_A)) / (S_B / (C_B * N_B)) = C_B² / C_A²

C_A² * (S_A / (C_A * N_A)) = C_B² * (S_B / (C_B * N_B))

C_A * S_A / N_A = C_B * S_B / N_B

(N_A / (C_A * S_B)) * C_A * S_A / N_A = (N_A / (C_A * S_B)) * C_B * S_B / N_B

S_A / S_B = (N_A * C_B) / (N_B * C_A)

S_A / S_B = (N_A / N_B) * (C_B / C_A)

(S_A / S_B) / (N_A / N_B) = C_B / C_A

(S_A / N_A) / (S_B / N_B) = C_B / C_A

We arrive at the equation for the proportional funding principle, showing that the square of the cost of curing one person being inversely proportional to the chance of being cured was no coincidence. Under the proportional funding principle, a person with a disease that costs y times more to cure will be y² times less likely to be cured.

2. We have solved the second issue from above since the allocation now does consider the cost of curing a person of disease A and the cost of curing a person of disease B, meaning that relative cost-effectiveness is now a factor in the allocation.

3. However, we have not solved the first issue from above since the new allocation still cures significantly fewer people (6,667 under the new allocation (4,000 + 2,667) vs. 20,000 when maximizing the number of people cured).

For those of you who think that a weighted lottery system is compatible with effective altruism, are you comfortable with an allocation that cures significantly fewer people? Would you support the proportional funding principle or do you think a different principle should be used? And can you explain more about why you support a weighted lottery system?