When would an agent do something different as a result of believing the many worlds theory?

As any collapse (if it does happen) occurs so 'late' that current experiments are unable to differentiate between many worlds and collapse -- it seems quite possible that both theories will continue to give identical predictions for all realisable situations, with the only difference being 'one branch becomes realised' and 'all branches become realised'.

General:

• Assuming this practical indistinguishability between the theories, I think that any utility function based on one of the theories can be directly translated into the other theory by just reinterpreting the theory-inherent probabilities. This assumes that all branches in the many worlds reasoning are weighted with their 'probability' (e.g. the Quantum Russian Roulette thought experiment hinges on counting 'I survive'-branches differently [LW · GW]¹)

More Human related:

• One relevant aspect is how natural utility maximisation feels using one of the two theories as world model. Thinking in many worlds terms makes expected utility maximisation a lot more vivid compared to the different future outcomes being 'mere probabilities' -- on the other hand, this vividness makes rationalisation of pre-existing intuitions easier.
• Another point is that most people strongly value existence/non-existence additionally to the quality and 'probability' of existence (e.g. people might play Quantum Russian Roulette but not normal Russian Roulette as many worlds makes sure that they will survive [in some branches]). This makes many worlds feel more comforting when facing high probabilities of grim futures.
• A third aspect is the consequences for the concept of identity. Adopting many worlds as world model also means that naive models of self and identity are up for a major revision. As argued above, valuing all future branch selves equally (=weighted by the 'probabilities') should make many worlds and collapse equivalent (up to the 'certain survival [in some branches]' aspect). A different choice in accounting for many worlds might not be translatable into the collapse world model.

Disclaimer:

I am still very much confused by decision theories that involve coordination without a causal link between agents such as Multiverse-wide Cooperation. For such theories, other considerations might also be important.

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¹: To be more exact, I would argue that the case for Quantum Russian Roulette becomes identical to the case for normal Russian Roulette if many world branches are weighted with their 'probabilities' and also takes into account the 'certain survival [in some branches]' bonus that many worlds gives.

I think it's more natural to ask "how might an agent behave differently as a result of believing an objective collapse theory?" One answer that comes to mind is that they will be less likely to invest in quantum computers, which will need to rely on entanglement between a large number of quantum systems that under objective collapse theories might not be maintained (depending on the exact collapse theory). Similarly, other different physical theories of quantum mechanics will result in different predictions about what will happen in various somewhat-arcane situations.

More flippantly, an agent might answer the question 'What do you think the right theory of quantum mechanics is?' differently.

[Edited to put the serious answer where people will see it in the preview]

Things like determinism and many worlds may not affect fine grained decision making, but they can profoundly impact what decision making, choice volition, agency and moral responsibility are. It is widely accepted that determinism affects freedom of choice, excluding some notions of free will. It is less often noticed that many worlds affects moral responsibility, because it removes refraining: if there is the slightest possibility that you would kill someone, then there is a world where you killed someone. You can't refrain from doing anything that is possible for you to do

There is the Quantum Russian roulette thought experiment. It was posted in LessWrong.

Many of the other comments deal with thought experiments rather than looking at the reality of how "many worlds" is USED. From my point of view as a non-physicist it seems to primarily be used as psuedo-science "woo" - a revival of mystery and awe under the cloak of scientific authority. A kind of paradoxical mysticism for non-religious people, or fans of "science-ism".

An agent might act differently from MISUNDERSTANDING many worlds theory. Or by paying more attention to it. Psychological "priming" is real ansd powerful.

The answer by TAG below is case in point. For someone committed to a belief in determinism or fatalism, having a manyworlds theory in mind may buttress that belief.

If they are put into an interferometer, someone who thinks the wavefunction has collapsed would think, while in the middle, that they have a 50/50 chance of coming out each arm, while an everettian will make choices as if they might deterministically come out of one arm (depending on the construction of the interferometer).

The difficulty of putting humans into interferometers us more or less why this doesn't matter much. Though of course "pragmatism" shouldn't stop us from applying occam's razor.

Assume you put enormous weight on avoiding being tortured and you recognize that signing up for cryonics results in some (very tiny) chance that you will be revived in an evil world that will torture you and this, absent many worlds, causes you to not sign up for cryonics. There is an argument that in many worlds there will be versions of you that are going to be tortured so your goal should be to reduce the percentage of these versions that get tortured. Signing up for cryonics in this world means you are vastly more likely to be revived and not tortured than revived and tortured and signing up for cryonics will thus likely lower percentage of you across the multiverse who are tortured. Signing up for cryonics in this world reduces the importance of versions of you trapped in worlds where the Nazis won and are torturing you.

If you use some form of noncausal decision theory, it can make a difference.

Suppose Omega flips a quantum coin, if its tails, they ask you for £1, if its heads they give you £100 if and only if they predict that you would have given them £1 had the coin landed tails.

There are some decision algorithms that would pay the £1 if and only if they believed in quantum many worlds. A CDT agent would never pay, and a UDT agent would always pay however.

It is of course possible to construct agents that want to do X if and only if quantum many worlds is true. It is also possible t construct agents that do the same thing whether it's true or false. (Eg Alpha Go)

The answer to this question depends on which wave function collapse theory you use. There are a bunch of quantum superposition experiments where we can detect that no collapse is happening. If photons collapsed their superposition in the double slit experiment, we wouldn't get an interference pattern. Collapse theories postulate a list of circumstances that we haven't measured yet when collapse happens. If you believe that quantum collapse only happens when 10^40 kg of mass are in a single coherent superposition, this belief has almost no effect on your predictions.

If you believe that you can't get 100 atoms into superposition then you are wrong, current experiments have tested that. If you believe that collapse happens at the 1gram level. Then future experiments could test this. In short, there are collapse theories in which collapse is so rare that you will never spot it. There are theories where collapse is so common that we would have already spotted it (so we know those theories are wrong), and there are theories in between. The in between theories will make different predictions about future experiments. They will not expect large quantum computers to work.

Another difference is that current QFT doesn't contain gravity. In the search for a true theory of everything, many worlds and collapse might suggest different successors. This seems important to human understanding. It wouldn't make a difference to an agent that could consider all possible theories.