Many-worlds implies the future matters more

The idea is that there is a conserved amount of reality-stuff, and each time the world branches, the branches are thinner; they individually contain less reality-stuff than previous generations. You value a branch according to the state of the world in that branch, but then you give a weight to that branch in your calculations which depends on its thickness. This is analogous with expected utility calculations in which utility and probability enter as separate factors, except that "amount of reality-stuff" replaces probability.

One way this would make sense is if "amount of reality-stuff" was simply "number of copies of that world". Then you would be treating, and valuing, each world equally, but some worlds would be duplicated in excess of others, and so that type of world would count for more in your decision-making, not because you valued it more, but simply because it existed more often in the part of the multiverse downstream from the present moment.

Incidentally, this way of thinking would also apply to a Bohmian multiverse, in which there are no splitting worlds, but in which the Bohmian histories sometimes converge and sometimes diverge (that is, while always remaining separate self-contained worlds, sometimes they approach each other in configuration space, and sometimes they move apart). In this framework, the quantum probability density of a configuration would be the density with which the Bohmian histories cluster around that point in configuration space, and the wavefunction works as in the Copenhagen interpretation, it is a way for someone in an individual Bohmian world to reason under uncertainty. (Though if you chose to identify with the whole ensemble of your subjective duplicates throughout the Bohmian multiverse, you could to some extent preserve the MWI perspective whereby you are causally responsible for a whole ensemble of worlds, and not just for the one that this instance of you inhabits.)

One indication that "conservation of reality-stuff" may be the right principle, rather than "worlds increasing in number all the time", is that quantum mechanics only indicates relative frequencies of different outcomes. It can tell you that outcome A is twice as likely as outcome B, but it doesn't tell you the absolute number of A-worlds and B-worlds. If you had a world-splitting model, you would be free to invent your own extra law of nature about how the absolute number of worlds changes over time - bearing in mind that identical worlds can merge, so the number of worlds can decrease as well as increase. So long as the relative frequencies match quantum theory, it would be impossible to tell whether an equal-probability split in two directions produced 2 worlds, 2 billion worlds, or (2 times infinity) worlds; but according to your moral calculus, this unobservable absolute number of worlds would be extremely significant.

Returning to what people actually say about MWI, there is a tendency for MWI advocates not to talk of literally duplicated worlds, but to nonetheless treat the "quantum measure" of a world in exactly the same way that you would treat the number of duplicates of a world. If one branch has greater "measure" than another branch, then there's more of it, or it's more real, or some similar phrase will be employed. The fundamental reason for this is to be able to explain why observed probabilities are not uniform. If you break up the wavefunction into a set of basis functions (that are the "worlds") and you then treat the basis functions equally, that means that each possible world exists equally and should be of equal probability. But some things happen more often than others, to a degree that is described by the measure, so people are forced to say that some worlds count for more than others.

The reason that people don't normally justify this unequal treatment, as due to unequal numbers of duplicates being produced at branchings, is that the wavefunction contains no such phenomenon. You could postulate that the ontologically correct decomposition of the wavefunction into individual worlds always assigns equal amounts of measure to all the individual worlds, but this would involve postulating extra structure in the theory, which MWI advocates are loath to do.

Personally I find MWI advocates to be shockingly indifferent to the details of how worlds split. If the notion of world is to be taken seriously, it ought to be a mathematically exact notion. They could look for turning points (in the calculus sense) in the wavefunction, in order to identify objective boundaries and objective transitions from one world to two, e.g. when one local minimum splits into two, but once again, no-one ever follows that line of inquiry. In general, the many-worlds interpretation has a similar psychological function to the Copenhagen interpretation, namely, it's a fuzzy concept that sounds like it might make sense, so it allows users of quantum mechanics to get on with their lives and not worry about foundations. And so whole decades can pass without physicists being forced to confront the question of what the state of the unobserved electron is, or of exactly when it is that one world becomes two.

If nothing else, you can't control the words, just the measure. Every possible world exists, but they all have different measure.

We don't really know what the deal with Born's rule is, but I strongly suspect that whatever reason there is for it is a good reason to value worlds of higher measure. Maybe there are more worlds with higher measure, and there's just so many worlds that it just looks like one continuous wave (sort of like how a water ripple is made of discrete atoms). Maybe it's something nobody thought of yet.

I'm not confident that my memory here is correct, but what do you think they think instead?

Unless I misunderstood Eliezer's intention (the quote is Greg Egan's originally).

I believe he's saying he'll upvote any logic or evidence that demonstrates that this idea doesn't add up to normality.

Reality, yes, normality, maybe not. There is at least an interpretation of "adding up to normality" rattling around in my head that seems to have been inspired by Eliezer's post that would imply we would make the same choices in MWI as in an unbranching universe. I am not so sure about this.