Suppose you were forced to choose between killing someone today, and killing someone a century from now. Which would you choose?

To be clear: the two people are exactly the same - equally happy, healthy, etc. And the effects on others are the same, and there is no uncertainty involved. The only difference between the two murders is when they occur.

It seems hard to give a justification for why one is better than the other, and in the landmark Stern Review on Climate Change the eponymous Nicholas Stern said as much. In his interrogation by Parliament, he stated that to choose one over the other is to "discriminate between people by date of birth," a position that is "extremely hard to defend".

There is a lot of controversy about whether he made the right decision, mostly motivated by the fact that even slightly different decisions on how we "discount" the future can cause huge differences in how we respond to threats which will kill people in the future, like climate change.

A remarkable proof by Peter Diamond shows that, under some reasonable assumptions, we should indeed "discriminate by date of birth," and choose to kill the person a century from now.

## Diamond's Proof

The full proof (and several others) can be found in his paper The Evaluation of Infinite Utility Streams, but I'll present a simplified version here.

First, some notation. We denote welfare over time as a list, e.g. $(1,2,3)$ indicates that at time 1 all sentient persons have utility 1, and time 2 they have utility 2 and so forth. Because time is infinite, these lists are infinitely long. We denote infinite repetition with "rep", e.g. $1_{rep}$ is the list $(1,1,1,\dots)$. These lists are given variable names - I use $u,v$ for finite lists and $X,Y$ for infinite lists - and they are compared with the standard inequality symbols ($>,\geq$).

There are four assumptions:

- If $u\geq v$ then $u_{rep} \geq v_{rep}$. I.e. if some finite list of utilities $u$ is better than some other finite list $v$, then repeating $u$ for all of eternity is better than repeating $v$ for all of eternity.
- If $u\geq v$ then $(u,X)\geq (v,X)$. I.e. if $u$ is better than $v$, starting off the world with $u$ is better than starting things off with $v$, given that the rest of time is equal.
- If $X\geq Y$ then $(u,X)\geq (u,Y)$. I.e. if some infinite state of affairs $X$ is better than $Y$, starting them both off with $u$ won't change that.
- If $u$ is the same as $v$ except some people are better off (and no one is worse off), then $u > v$. (This is sometimes known as Pareto efficiency.)

**Proof:**The proof actually isn't that complicated, but it looks intimidating because the notation is probably unfamiliar.

Suppose, for the sake of contradiction, that $(1,2)_{rep}\geq (2,1)_{rep}$. By (A4), $(2,2,(1,2)_{rep}) > (2,1)_{rep}$ since $(2,2,(1,2)_{rep})$ is the same as $(1,2)_{rep}$, except with people being better off at $t = 1$. By rearrangement, $(2,2,(1,2)_{rep})=(2,(2,1)_{rep})$ and $(2,1)_{rep}=(2,(1,2)_{rep})$ so $(2,(2,1)_{rep}) > (2,(1,2)_{rep})$. But we had assumed that $(1,2)_{rep}\geq (2,1)_{rep}$ which by (A3) means that $(2,(1,2)_{rep}) \geq (2,(2,1)_{rep})$. We've reached a contradiction, meaning that $(1,2)_{rep} < (2,1)_{rep}$.

This means that $(1,2) < (2,1)$, for if the opposite were true, $(1,2)_{rep} \geq (2,1)_{rep}$ by (A1). Therefore, by (A2), $(1,2,X) < (2,1,X)$, meaning that if we could shift happiness from year two to year one, we should.

We should value the happiness of those born earlier more than those born later, and kill the person living a century from now.

## Discussion

My girlfriend pointed out to me that the reason why we interpret this theorem to mean that people born earlier matter more is because we assume time has a beginning but no end. If we assumed the opposite, then people born later would matter more.

This only holds for infinitely long lists. Eventually the universe will have too high entropy to support sentient beings, so any list of welfare over time must be finite. In a finite list, just take the sum of the utilities. (1, 2) repeated a billion times has the same utility as (2, 1) repeated a billion times.

ReplyDeleteAdditionally, one or more of your four assumptions may be unreasonable. It may not make sense to compare two sets that both have infinitely high utility.

Michael: Yep, great point! This is the most common objection to Diamond's theorem.

DeleteTo expand even further: if the universe has a point at which it stops supporting life, the list won't be finite - it will just have a finite number of non-zero elements. This suggests a generalization of your objection: if $\leq$ need be total only over *convergent* sequences (instead of all sequences), then the theorem doesn't hold (as $u_{rep}$ may not be comparable to $v_{rep}$).

It's not obvious that this is a legitimate restriction though. For example, it seems like a good "utility algorithm" would tell us that $(1,2,3,\dots)\leq(2,2,3,\dots)$, despite these being non-convergent and both of infinite utility.

Interesting. I think I would object to strict inequality in A4 for infinite lists, in which case the proof doesn't go through. This is just the standard point that 1 + infinity is not strictly bigger than infinity.

ReplyDeleteWe need good solutions to infinite ethics, and temporal discounting is one, but it's not my favorite.

Good point.

DeleteI would remark that in an infinite vector space, the standard distance metric is the supremum, in which case $||\vec{x}-\vec{x+1}||_{infty}$ does indeed equal one.

But our intuitions are poor - who knows if sup is really the best metric for infinite ethics?