## Betting on collapse (EDC, ch.6)

Chapter 6 of Evidence, Decision and Causality presents another alleged counterexample to CDT, involving a bet on the measurement of entangled particles.

The setup is Bohm's version of the Einstein, Podolsky, Rosen experiment, as described in Mermin (1981) (see esp. pp.407f.).

We have prepared a "source" S that, when activated, emits two entangled spin 1/2 particles, travelling towards causally isolated detectors A and B. The detectors contain Stern-Gerlach magnets whose orientation is controlled by a switch with three settings (1, 2, 3). When the switches on the two detectors are on the same setting, the magnets have the same orientation. Detector A flashes 'y' if the measured spin is along the magnetic field and 'n' otherwise. Detector B uses the opposite convention, flashing 'n' if the measured spin is along the magnetic field.

Quantum mechanics tells us that if the switches are on the same setting, then they always display the same result – 'y' or 'n', in equal proportion – because the spin state of the entangled particles will collapse in opposite ways. If the switches are on different settings, they display the same reading with probability 1/4.

Now consider the following schematic offer.

Case 1. You pay me some amount $z. We set the switches on both detectors to 3. If the detectors display the same reading on the next run, you win$1. How much would you pay for this bet?

Intuitively, you should pay up to $1. After all, you know that if the detectors are on the same setting then they will display the same reading. Arif argues that CDT might not deliver this intuitive verdict. He assumes a K-partition formulation of CDT (as in Skyrms (1984)), where the "states" are hypotheses about the chance of relevant outcomes conditional on acts. We therefore need to ask about the chance of the detectors showing the same reading conditional on you accepting the bet (for, say$z=$0.90), in a situation where both switches are set to 3. I would have thought that the chance is 1. Arif, however, points out that accepting the bet arguably doesn't make it the case that the detectors have the same reading. If the relevant conditional chances are supposed to measure degree of causal influence, then the chance of 'same reading' conditional on 'accept the bet' may be considerably less than 1. And then CDT says you should not accept the bet. I'll return to this dubious argument below. Arif doesn't put much weight on it. Let's assume, he says, that the chance of 'same reading' conditional on 'accept the bet' really is 1. Then there is a different case in which CDT gives the wrong verdict: Case 2. First, you have to choose one of three joint settings for the detectors: (i) detector A is in position 1, B is in 2 (ii) detector A is in position 1, B is in 3, (iii) detector A is in position 2, B is in 3. Next, you can bet on 'same reading' or on 'different reading'. If you bet on 'same reading' and win, you get$2. If you bet on 'different reading' and win, you get $1. Recall that if the detectors are on different settings then quantum mechanics tells us that the probability of getting the same reading is 1/4. It therefore seems clear that you should bet on 'different reading' (irrespective of what you choose in the first stage). The 'different reading' bet has expected payoff$.75, compared to $.50 for the 'same reading' bet. Arif suggests that CDT yields a different verdict. In the assumed K-partition formulation of CDT, we need to consider the chance of various combinations of readings conditional on setting the switches and accepting a bet. By assumption, the detectors are causally isolated. From this, Arif infers that "the chance of either reading on either receiver is independent of the reading on the other receiver, even given the settings on both receivers" (p.150). For example, what is the chance of getting 'y' from both A and B upon setting A to 1 and B to 2? According to Arif, that chance equals the product of the chance of getting 'y' from A and the chance of getting 'y' from B, both conditional on the relevant setting. Moreover, the reading on each detector is independent of the setting on the other. So, the chance of getting a 'y' from A upon setting A to 1 and B to 2 equals the chance of getting a 'y' from A upon setting A to 1. Finally, recall our assumption that the chance of 'same reading' upon A=1 and B=1 is 1. Since the readings are independent, it follows that the conditional chance of each joint reading on each setting is either 0 or 1. It then also follows that CDT endorses some bet on 'same reading'. See pp.151-154 for the details. I won't go through the details. The problem should be obvious to anyone familiar with the EPR paradox. Arif's assumptions about the chances are precisely the assumptions refuted by John Stewart Bell in 1964. Arif acknowledges this in footnote 3 on p.151. He points out that one could still accept the relevant assumptions if one denies that chances correspond to long-run frequencies. He also says that this is "what [chance] should represent if conditional chance matters to Causal Decision Theory". So here's the situation. We've assumed a formulation of CDT in terms of conditional chance. If we interpret the relevant notion of chance in accordance with quantum physics, we get the intuitively right result. Arif says that proponents of CDT should not interpret chance in this manner. Rather, they should invoke a different conception of chance that comes apart from both long-run frequencies and the probabilities of quantum physics. Why on earth should they do this? Because, Arif thinks, proponents of CDT should want conditional chances to measure degree of causal influence. Since there is no direct causal link between the readings of the two detectors, for example, proponents of CDT should claim that the chance of getting 'y' on one detector is independent of the chance of getting 'y' on the other. I doubt that there's any coherent concept of "chance" that measures degree of causal influence. Suppose I set both switches to 3, as in Case 1. I run the experiment but don't yet tell you the result. First I offer you a bet for$0.90 that pays \$1 if the detectors show the same reading. What is the chance of 'same reading' conditional on 'buy the bet'? Well, there's no causal influence between the bet and the outcome. Should we therefore say that the conditional chance is 1/2? It would then be 1/2 both conditional on 'buy the bet' and on 'reject the bet'. And so the unconditional chance of 'same reading' would also be 1/2. But the experiment is already over and the detectors are showing the same reading – as they must, given that the switches are in the same position. In what sense is the chance of that outcome still 1/2?

If we really want the cells in the K-partition to track facts about causal dependence, it would be better to replace the conditional chances by something like counterfactuals with chancy consequents.

But then CDT still gets both Cases right. In Case 1, the chance of getting the same reading is 1 and would be 1, whatever you decide to do. In Case 2, we need to ask what would happen if, say, you set switch A to 1, switch B to 2, and bet on 'same reading'. If you did all that, what would be the chance of getting the same reading on the two detectors? If the relevant chances are the chances of quantum physics, CDT delivers the intuitively correct result. I see no reason to think that any other notion of chance should figure in this version of CDT.

In effect, Arif assumes that proponents of CDT must assume a kind of locality that is generally thought to be incompatible with quantum mechanics. Locality is, of course, a very tempting and natural idea. It looks especially natural in Arif's description of the EPR scenario because Arif doesn't mention any of the quantum-mechanical details of the setup, nor does he explain what quantum mechanics actually says about the setup, except that it predicts such-and-such long-run frequencies.

According to collapse versions of quantum mechanics – Arif explicitly sets aside the Everett interpretation – if you set both detectors to 3 and activate the source, the system comprised of the two particles enters into a superposition state. At the point of measurement, this superposition state collapses. The collapse is indeterministic but guaranteed to result in opposite spin states, so that the two detectors will show the same reading. Even though the two detectors are causally isolated from one another, and the state of the system prior to measurement did not settle the reading on either detector, the reading at one detector is guaranteed to match the reading at the other.

What would happen if you're in Case 2 and you were to set the detectors to (say) 1 and 2? Upon measurement, the superposition state would collapse in such a way that we'd get an identical reading with chance 1/4 and a different reading with chance 3/4. These "chances" correspond to long-run frequencies. They play the role of chance in best systems accounts of chance and laws. As a consequence – see Schwarz (2014) – you should be 25% confident that we would get the same reading on the supposition that you were to set the detectors to 1 and 2. You should be 25% confident that you'd win a bet on 'same reading', and 75% confident that you'd win a bet on 'different reading'. You should therefore choose the bet on 'different reading'.

Unless I missed something, there is nothing to see here.

Mermin, N. David. 1981. “Quantum Mysteries for Anyone.” The Journal of Philosophy 78 (7): 397–408. doi.org/10.2307/2026482.
Schwarz, Wolfgang. 2014. “Proving the Principal Principle.” In Chance and Temporal Asymmetry, edited by A. Wilson, 81–99. Oxford: Oxford University Press.
Skyrms, Brian. 1984. Pragmatics and Empiricism. Yale: Yale University Press.