Posts Tagged ‘Decision theory’

December 2, 2008

I have a little paper writing up the supertask puzzle I posted recently. I’ve added a second puzzle that demonstrates the same problem, but doesn’t use the axiom of choice (it’s basically just a version of Yablo’s paradox), and I’ve framed the puzzles in terms of failures of the deontic Barcan formulae.

Anyway – if anyone has any comments, I’d be very grateful to hear them!

Uncertainty in the Many Worlds Theory

September 2, 2008

I’ve been thinking a bit more about probability in the Everettian picture, and I’m tentatively beginning to settle on a view. Obviously, there is a lot literature on this topic, and, as always, I have read very little of it.

There are roughly two problems. The incoherence problem is: how do we make sense of probability at all in the Everettian picture. Firstly, there is no uncertainty because we know all there is to know about the physical state of the world – roughly, everything happens in some branch. Secondly, how can there be probability if there is no uncertainty? The second problem is the quantitative problem: how do we get the probabilities of a branch happening to accord with the Born rule. A lot of progress has been made on the second problem (e.g. Deutsch, Wallace) by factoring probabilities out of the expected utility equation from decision theory, but work is still needed to answer the incoherence problem (after all, decision theory only tells us how to act when we are uncertain about the future.)

There are several ways to respond to the incoherence problem. We can deny the connection between uncertainty and probability (Greaves), or we can try and make sense of subjective uncertainty even when we know the complete physical state of the world. Here are some options:

1. Self-locating uncertainty: even if you know everything there is to know about the physical state of the universe, you can still be uncertain about where you are located in that world (for example, in Lewis’s omniscient God’s example.) (Saunders & Wallace.)
2. Caring measure: there is no subjective uncertainty when you know the complete state of the world, however you can make sense of decision theory by interpreting probabilities as degrees to which you care about your future selves. (Greaves.)
3. Branches as possible worlds: if the Everettian treats branches as possible worlds, they’d be in a similar situation to a modal realist. You know everything there is to know about the state of the possibility space, yet you can still be uncertain which world actually obtains. On this view precisely one of the branches is actual, and our uncertainty is about which one this is. (From the comments of my last post, it seems that Alastair holds the branches as worlds view – but I don’t know what he’d make of this interpretation of probability, or the idea of a single actual branch.)
4. Uncertainty due to indeterminateness. This is the view I’m toying with at the moment. Here’s the analogy: we may know everything there is to know about the physical state of the world, including how many hairs Fred has and other relevant facts, but we may still be uncertain about whether Fred is bald. This is because it may be vague whether Fred is bald.

Self-locating uncertainty. The rough idea for option (1) is to treat people, and other objects, as linear (non-branching) four dimensional worms. The branching structure of the universe ensures that, if people can survive branching at all, they overlap each other frequently in such a way that, before a branching, there will be many colocated people, that share a common temporal part until the branching. To see how self-locating uncertainty arises out of this, consider Lefty and Righty – two colocated people who will shortly split along two different branches. Since Lefty is in an epistemically indistinguishable situation from Righty, he should be uncertain as to which person he his, even though he knows every de dicto fact about the world. These cases are familiar. Take Perry’s example of two people lost in a library. They both have a map with a cross where each person is, and they know all the physical facts about the library. But they may still be uncertain as to which cross represents them (provided they are both in rooms that are indiscernable from each other from the inside.)

This certainly provides a solution to the incoherence problem, but I can’t see how it will extend to an answer to the quantitative problem. My worry is based on a principle due to Adam Elga: you should assign equal credence to subjectively indistinguishable predicaments within the same world (see my earlier post.) The temporal slices of Lefty and Righty at a time just before the branch are identical, so they must be in the same (narrow) mental states, and thus must represent subjectively indistinguishable predicaments with in the same world. So according to Elga’s principle, Lefty must be 50% sure he’s Lefty, and 50% sure he’s Righty, and similarly for Righty. After all Lefty and Righty are receiving exactly the same evidence – even if God told Lefty he was Lefty, he should remain uncertain because he knows that Righty would have received exactly the same evidence, and it could easily have been him that is wrong. In essence, the problem with self-locating uncertainty is Lefty should not proportion his credences to the Born rule, but to the principle of indifference – for that is the principle according to which you should proportion your self-locating beliefs. (Note also that the Born rule, and the principle of indifference appear to be incompatible on first looks.)

Caring measure. On this view there is no subjective uncertainty to be made sense of. Rather than many colocated worms, you can just have one person stage with multiple temporal counterparts. However, one can still make sense of probability in terms of the degrees to which you care about each of your branches. Expected utility is really just the sum of the utilities of your branches proportioned to how much you care about each of those branches. Unfortunately this requires treating setting your caring degrees according to the Born rule as a primitive principle of rationality.

This aside, my main problem with this approach is that I can’t see how it gets us the statistical predictions that quantum mechanics makes. The relation between frequencies, chances and credences is clear to me, but I can’t see how the caring measure will explain the statistical data. (You might think there is also a problem with indifference, because you should care just as much about all your branches – I’m not so convinced by this version of the principle though (anyone seen ‘The Prestige‘?))

Branches as worlds. Although this initially looks like probability will be just like probability for the modal realist, it is not so simple. For the modal realist there is exactly one actual world, and uncertainty is just uncertainty about which world is actual, even though no-one is uncertain about what the whole possibility space looks like. However, it is not analogous – for the modal realist the actual world is specified indexically as that maximally connected region of space-time are a part of. For the Everettian, no such specification is possible since all the worlds are connected, and we overlap multiple worlds. The only way would be to take being the actual branch as metaphysically primitive – which does not sound attractive to me at all.

Uncertainty due to indeterminateness. Like the self-locating belief proposal, this proposal tries to make sense of uncertainty even when the agent has a complete physical description of the world at hand. Here’s the analogy: I may have a complete description of the world, down to the finest details of, including the number of hairs on Fred’s head, the way we use English, and still be uncertain as to whether Fred is bald, if Fred is a borderline case of being bald.

Will this analogy carry over to talk about the future in EQM? Why are sentences about the future indeterminate in branching time, rather than always true, (or always false, or whatever.) Here’s why. Our everyday time talk has a temporal logic of linear time, for example we find ‘tomorrow p and tomorrow ~p’ inconsistent, and so on. (This might be because our histories are always linear?) Thus, supposing our tense talk gets given a Kripke frame type interpretation, this frame must be a linear order. However, there are many different (maximal) linearly ordered sets of times for our temporal talk to latch on to – each branch will do (note: I’m not assuming the quantum state is fundamentally cut up at the world joints, nonetheless, these world things make better interpretations than the gruesome none-worlds.) Since there are many candidate linear orders to make our tense talk true, we can supervaluate them out, but keep our ordinary temporal logic without having to select a special branch. On this supervaluationist approach, many sentences will come out indeterminate – which allows us to assign them non-trivial probabilities, even when we know that at the metaphysical level (and in the meta-language for English) all the possibilities are actualised. But this is in just the same sense as we know that some admissible interpretations of English make Fred bald, and others don’t.

Of course, I haven’t said anything about the quantitative problem. It’s not clear that you can just lift the decision theoretic answer and combine it with any old answer to the incoherence problem. This is for the same reason the self-locating uncertainty proposal failed: there may be other principles that govern the evolution of credences in self-locating propositions/credences in indeterminate propositions/whatever your answer to the IQ is, that override the probabilities we need for quantum mechanics.