There are a whole bunch of ways in which infinities make decision theory go screwy. I’ve written about some of those ways here. This post is about a thought experiment in which infinities make ethics go screwy.

*WaitButWhy* has a great description of the thought experiment, and I recommend you check out the post. I’ll briefly describe it here anyway:

Imagine two worlds, World A and World B. Each is an infinite two-dimensional checkerboard, and on each square sits a conscious being that can either be very happy or very sad. At the birth of time, World A is entirely populated by happy beings, and World B entirely by sad beings.

From that moment forwards, World A gradually becomes infected with sadness in a growing bubble, while World B gradually becomes infected with happiness in a growing bubble. Both universes exist forever, so the bubble continues to grow forever.

The decision theory question is: if you could choose to be placed in one of these two worlds in a random square, which should you choose?

The ethical question is: which of the universes is morally preferable? Said another way: if you had to bring one of the two worlds into existence, which would you choose?

# On spatial dominance

At every moment of time, World A contains an infinity of happiness and a finite amount of sadness. On the other hand, World B always contains an infinity of sadness and a finite amount of happiness.

This suggests the answer to both the ethical question and the decision theory question: World A is better. Ethically, it seems obvious that infinite happiness minus finite sadness is *infinitely better* than infinite sadness minus finite happiness. And rationally, given that there are always infinitely more people outside the bubble than inside, at any given moment in time you can be sure that you are on the outside.

A plot of the bubble radius over time in each world would look like this:

In this image, we can see that no matter what moment of time you’re looking at, World A *dominates* World B as a choice.

# On temporal dominance

But there’s another argument.

Let’s look at a person at any given square on the checkerboard. In World A, they start out happy and stay that way for some finite amount of time. But eventually, they are caught by the expanding sadness bubble, and then stay sad forever. In World B, they start out sad for a finite amount of time, but eventually are caught by the expanding happiness bubble and are happy forever.

Plotted, this looks like:

So which do you prefer? Well, clearly it’s better to be sad for a finite amount of time and happy for an infinite amount of time than vice versa. And ethically, choosing World A amounts to dooming every individual to a lifetime of finite happiness and then infinite sadness, while World B is the reverse.

So no matter which position on the checkerboard you’re looking at, World B dominates World A as a choice!

# An impossible synthesis

Let’s summarize: if you look at the spatial distribution for any given moment of time, you see that World A is infinitely preferable to World B. And if you look at the temporal distribution for any given position in space, you find that B is infinitely preferable to A.

Interestingly, I find that the spatial argument seems more compelling when considering the ethical question, while the temporal argument seems more compelling when considering the decision theory question. But both of these arguments apply equally well to both questions. For instance, if you are wondering which world you should choose to be in, then you can think forward to any arbitrary moment of time, and consider your chances of being happy vs being sad in that moment. This will get you the conclusion that you should go with World A, as for any moment in the future, you have a 100% chance of being one of the happy people as opposed to the sad people.

I wonder if the difference is that when we are thinking about decision theory, we are imagining ourselves in the world at a fixed location with time flowing past us, and it is less intuitive to think of ourselves at a fixed time and ask where we likely are.

Regardless, what do we do in the face of these competing arguments? One reasonable thing is to try to combine the two approaches. Instead of just looking at a fixed position for all time, or a fixed time over all space, we look at *all space and all time*, summing up total happiness moments and subtracting total sadness moments.

But now we have a problem… how do we evaluate this? What we have in both worlds is essentially a +∞ and a -∞ added together, and no clear procedure for how to make sense of this addition.

In fact, it’s worse than this. By cleverly choosing a way of adding up the total amount of the quantity *happiness – sadness*, we can make the result turn out *however we want*! For instance, we can reach the conclusion that World A results in a net +33 *happiness – sadness* by first counting up 33 happy moments, and then ever afterwords switching between counting a happy moment and a sad moment. This summation will *eventually* end up counting all the happy and sad moments, and will conclude that the total is +33.

But of course, there’s nothing special about +33; we could have chosen to reach any conclusion we wanted by just changing our procedure accordingly. This is unusual. It seems that both the total expected value and moral value are undefined for this problem.

The undefinability of the total *happiness – sadness* of this universe is a special case of the general rule that you can’t subtract infinity from infinity. This seems fairly harmless… maybe it keeps us from giving a satisfactory answer to this one thought experiment, but surely nothing like this could matter to real-world ethical or decision theory dilemmas?

Wrong! If in fact we live in an infinite universe, then we are faced with *exactly* this problem. If there are an infinite number of conscious experiencers out there, some suffering and some happy, then the total quantity of *happiness – sadness* in the universe is undefined! What’s more, a moral system that says that we ought to increase the total happiness of the universe will return an error if asked to evaluate what we ought to do in an in infinite universe!

If you think that you should do your part to make the universe a happier place, then you must have some notion of a total amount of happiness that can be increased. And if the total amount of happiness is unbounded, then there is no sensible way to increase it. This seems like a serious problem for most brands of consequentialism, albeit a very unusual one.

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