Personally I favor the no-collapse interpretation of quantum physics. It does away with the quantum measurement problem (thus resolving the apparent inconsistency within quantum mechanics), removes mysterious nonlocality, and makes quantum physics compatible with general relativity, at least in principle. Thus no-collapse is an attractive way to understand quantum mechanics, at least from the perspective of its implications for physics. However, no-collapse interpretations seem to have disturbing philosophical implications. Because quantum states don’t collapse every possible outcome of events in some sense happens; it only seems to us that one particular outcome occurs because each outcome has its own version of our consciousness observing that outcome, it isn’t the case that our consciousness is somehow outside of all this looking in. But if every possible outcome does happen this may seem to take away the significance of choices, because no matter what I choose to do everything that can happen will happen. What then is the point of trying to make one event occur instead of another?
To unravel this problem we first have to understand what choices are under determinism in general; some of the confusion here arises from the fact that we seem forced into one outcome or another, and this in part stems from the deterministic nature of physics. Even in classical determinism choices may seem ineffective, since there is no possibility for events to deviate from the course they have been set in since the beginning of time. This apparent contradiction between meaningful choices and determinism is resolved by realizing that our choices are not something that float outside the physical world and try to prod events in one direction or another. Rather making a choice is simply being in a particular internal state. Obviously that internal state is purely physical, and so it will have a real effect on the way events transpire. Of course that internal state wasn’t arbitrary, but it was determined mostly by past internal states, past choices. So we are perfectly free to choose whatever we want, and our choices do have a real effect on events, all we have to give up is the idea that who we are is in some way random. The choices I will make in the future are fully determined by who I am now and the experiences I will have between now and then, but I don’t think that is a problematic proposition.
Now this brings us somewhat closer to tackling the quantum case, but before we address it directly we must clarify a bit what we desire from our choices. Usually we simply say that we desire a particular result to occur. That is of course true, but I think there is a bit of missing detail. Specifically I would say that what we really desire is for a particular result to occur and for that result to have an effect on / be experienced by ourselves and/or other people. Of course normally there is no need to express this bit of detail, since the result of most of our choices will at least be experienced by us. But it will be important in dealing with the quantum case.
Finally I must elaborate a bit on how no-collapse interpretations account for apparent probabilities. Let us suppose that some particle is in a superposition where standard quantum mechanics says that upon observing it there is a 2/3 probability that it will collapse into being determinately in state A and 1/3 probability that it will collapse into being determinately in state B. Now under the no-collapse interpretation this superposition will not collapse when observed, but rather the observer will be put into a superposition of observing it to be in state A and observing it to be in state B. But how can we account for the fact that after measuring many such superpositions we will think that it has a 2/3rds probability of being in state A (this is in fact the experimental outcome)? Obviously when we are put into a superposition there are two distinct consciousnesses that can be legitimately said to be our future selves, one which observes A and one which observes B. And obviously which future consciousness we find ourselves to have will be arbitrary (obviously this is from our current perspective, and not the person we will be). But since it is arbitrary shouldn’t we expect the probability of finding ourselves in either one to be 1/2, and if not, why not? Of course clearly something is wrong here, since we know empirically the expected probability is not 1/2. One answer is to discard the idea that before being put into a super-position we have a single consciousness. Instead we can think of the pre-superposition state as being a superposition of infinitely many identical worlds. Thus there are infinitely many consciousnesses in the pre-superposition, but since they are all the same they are effectively one consciousness. But when put into a superposition a fraction of those worlds is “assigned” to each part of the superposition. In our case 2/3 would become the part of the superposition where we observe A and 1/3 where we observe B. Of course pre-superposition we can’t know which our consciousness is destined to become “assigned” to, and so there will be an apparent probability of 2/3 of observing A, and 1/3 of observing B.
With these ideas it is now relatively easy to make sense of choices under the no-collapse interpretation. To make a choice is to be in a particular internal state, which has a deterministic effect on the progression of events. Obviously in the quantum case this effect can’t be to bring about one outcome instead of the other. However it can make one outcome seem more likely, by increasing the perceived probability of ending up in part of the superposition that reflects that outcome occurring. And since we desire for that result to be experienced by us and by other people it makes sense to make such choices, since they will increase the chance that other people and us will find ourselves experiencing that outcome. (Or, in other words, more of the consciousnesses that are currently ours and others will end up experiencing that result.) Thus choices are still meaningful, and still important, even in a quantum world where everything happens.