Maudlin on Wallace on the Problem of Probability

In this post I will give a quick summary of how Wallace deals with the problem of probability in the Everett interpretation followed by Maudlin’s response and finishing with some questions/discussion points.

In the Everett interpretation there is no collapse of the wavefunction during measurement, instead the state vector of the universe evolves unitarily always. Standard quantum mechanics primarily makes statistical predictions of the probabilities of various measurement outcomes following a measurement via the Born rule. The problem of probability is, hence, the problem of recovering the correct probabilistic predictions of quantum mechanics.

Wallace’s solution is that the branch weight ie the modulus-squared amplitude of the components of the universe’s state vector should be interpreted as objective probability in the Everett interpretation. He shows that, following from some rationality assumptions, a (rational) agent will use the branch weight as a subjective probability for making decisions. Then according to a doctrine of functionalism—that if some object is treated just like a particular type of thing then we should take it to actually be that type of thing—he reasons that we should interpret branch weight as objective probability as well.

Maudlin’s review starts by praising Wallace’s book’s thoroughness and consideration; goes on to complain about Wallace’s claim that the Everett interpretation is a `literal’ reading of quantum mechanics; raises objections to physical ontology being based on patterns in the wavefunction; and then moves onto what I will consider: the problem of probability.

According to Maudlin, the functionalism that Wallace employs is a double-edged sword since it also implies that if branch weight is not treated by rational agents as a subjective probability then nor should the interpreter of quantum mechanics consider it to be an objective probability. He then goes on to argue that a rational agent would not necessarily use branch weights as subjective probabilities (although I don’t believe he goes so far as to show that such an agent would not use them as such).

Maudlin takes issue with one of Wallace’s axioms: that the set of `rewards’ between which an agent may doesn’t allow for any considerations of several (classically mutually exclusive) things happening simultaneously in different branches. In particular an agent cannot consider the possibility of performing a quantum experiment in order to get two simultaneous outcomes (albeit in different branches). Maudlin suggests that if one takes the Everett interpretation seriously one should certainly consider in one’s decision-making that after a branching event several outcomes happen simultaneously. With this axiom false (according to Maudlin) the edifice upon which Wallace’s representation theorem rests falls, and Wallace cannot claim branch weights to be rational subjective probabilities nor, via functionalism, objective probabilities.

Discussion questions/points:

Wallace seems to imply (p148 first complete paragraph) that symmetry principles are more applicable to probabilities for an Everettian. Does his derivation of the Born rule in a special case in section 4.13 bear this out? It’s not obvious to me that his argument really requires Everettianism.

Maudlin’s claim that a rational agent might value two results occurring simultaneously (on different branches) over either one result occurring could be countered if Wallace could argue that such a preference would just not be rational. Consider someone who regularly paid for quantum experiments in order to get the `study both history and physics’ style outcomes that Maudlin describes. It seems to me that such a person might quickly come to decide that they are wasting their money after having conducted several such experiments and still only experienced one outcome each time. Would this behaviour be irrational or is it, as Maudlin seems to believe, an arrational and valid preference?

Maudlin draws a distinction between betting in the normal sense and betting on Everettian quantum experiments: `No wagering or betting is involved, so the thorny issue of decision-making under uncertainty never arises.’ (p801 final sentence of 2nd paragraph) This is reminiscent of the `probability requires genuine uncertainty/indeterminacy’ objection that Wallace deals with in section 4.2. Is it the same? or related? is it as easily dismissed?