## Decision-making under determinism

Suppose you have a choice between two options, say, raising your arm and lowering your arm. To evaluate these options, we should compare their outcomes: what would happen if you raise your arm, what if you don't? But we don't want to be misled by merely evidential correlations. Your raising your arm might be evidence that your twin raised their arm in a similar decision problem yesterday, but since you have no causal control over other people's past actions, we should hold them fixed when evaluating your options. Similarly, your choice might be evidentially relevant to hypotheses about the laws of nature, but you have no causal control over the laws, so we should hold them fixed. But now we have a problem. The class of facts outside your causal control is not closed under logical consequence. On the contrary, if the laws are deterministic then facts about the distant past together with the laws logically settle what you will do. We can't hold fixed both the past and the laws and vary your choice.

This is the problem underlying Arif Ahmed's arguments against "Causal Decision Theory" in Ahmed 2013 and Ahmed 2014.

Arif assumes that Causal Decision Theory advises us to hold fixed the past but not the laws. It is easy to see that this leads to trouble.

Imagine you have strong evidence that some proposition L expresses
the fundamental laws of nature, where L is deterministic. As part of a
group of scientists, you are asked about your opinion on L. By raising your arm, you can signal that L is false, by lowering
that it is true. You want to give a true signal. Intuitively, you
should lower your arm. But if L is indeed true and you lower your
arm, then any possible situation in which you raise your arm, and
in which the distant past is just as it actually is, must be a
situation in which L is false. And in any such situation you signal
a *truth* if you raise your arm.

The problem can brought into sharper focus by assuming a combination of Gibbard & Harper's (1979) formulation of decision theory and Lewis's (1979, 1981) account of counterfactuals. Let P be the hypothesis that the physical state of the early universe together with L entails that you'll lower your arm. According to Lewis, if L is true and in fact you're going to lower your arm, then what would have been the case if you had raised your arm is that P would still have been true but L false, and so you would have signalled something true. Indeed, according to Lewis, in every possible situation in which you would give a true signal by raising your arm, you would also give a true signal by lowering it. (If the closest Raise worlds to w are L worlds, then w is an L worlds, and then the closest Lower worlds are not-L worlds.) So raising your arm would never yield a better outcome than lowering it. On the other hand, since you are not absolutely certain that L is true, you give non-zero credence to not-L situations in which lowering your arm would signal something true and raising something false. (Assuming that in these not-L situations, L would still be false if you were to raise your arm, which should be true at least for almost all of them.) Thus if you evaluate your options by considering what would be the case if you were to choose the options (a la Gibbard and Harper 1978), it looks like you should raise your arm.

(This is the "counterexample to Causal Decision Theory" discussed in Ahmed 2013.)

Note that if you go ahead and raise your arm, then you're signalling something of which you're confident that it is false. Imagine your colleague who hasn't raised their arm wonders, "don't you accept L, given all our evidence?" -- Your response would have to be something like this: "I do believe in L. But since I'm raising my hand, L would have been false if I hadn't raised it. So I had no choice but to signal something false. (Incidentally, you should be grateful, for if I hadn't raised my hand, your signal would have been false.)" Your colleague will hardly be satisfied with this bizarre defense.

The problem is that our Gibbard-Harper-Lewis theory wrongly assumes that the laws of nature are under our control. How can we fix this? The most obvious alternative is to change the standards for the relevant counterfactuals in such a way that the laws are privileged over the distant past. In this case, if L is true and you lower your arm, then what would have been the case if you had raised your arm is that L would still have been true but P false.

But this also looks problematic. P is a proposition about the
intrinsic physical state of the world in the distant past, and
intuitively this is not affected by your present choice. Worse, it
looks like we can recover our problem by considering a situation in
which you have strong independent evidence for P (perhaps God told
you) and in which raising your arm would signal that P is false. Here
presumably you shouldn't raise your arm. But on the revised
Gibbard-Harper-Lewis account, if you *had* raised your arm, it is
likely that P would have been false, so you would still have signalled
something true. More generally, there is no world with deterministic
laws in which lowering your arm is better than raising it: if the
closest Lower worlds are P worlds then the closest Raise worlds are ~P
worlds. On the other hand, in some deterministic worlds P is false no
matter what you do, and then you're better off raising your arm. So if
your credence is concentrated on deterministic worlds, it looks like
you should raise your hand and signal that P is false, despite your
strong evidence in favour of P.

We can tighten the knot. Assume you have strong independent evidence for both L and P. By raising or lowering your arm you can signal whether you accept or reject their conjunction. (We may also assume that your evidence is not misleading: L and P are in fact true.) Intuitively, you should signal acceptance by lowering your arm. But note that raising your arm is (not just counterfactually, but logically) incompatible with L and P. Thus on the supposition that you raise your arm, the conjunction of L and P must be false, no matter how exactly the supposition works -- even if it works not subjunctively but by indicative conditionalization, as "Evidenital Decision Theory" suggests. In general, on the supposition that you raise your arm it is logically guaranteed that you signal truly. Not so under the supposition that you lower your arm. So on this way of evaluating options you should raise your arm.

What shall we make of all this? Perhaps the lesson is that we shouldn't evaluate options by looking at possible situations in which you choose them. Then we don't face the choice of holding fixed either the laws or the past.

Consider
Savage's 1954
formulation of Causal Decision Theory. Here the space of possibilities
is partitioned into *states* which together with any option
determine an *outcome*. In this framework, we could take L &
P as a state, relative to which the option *Raise* leads to the
outcome *Signal falsely* (in all three problems mentioned above)
while *Lower* leads to *Signal truly*. Since most of your
credence lies on the L & P state, and you want to signal truly,
Savage's theory then says that you should choose *Lower*.

Of course, more needs to be said about what makes this the correct
representation of your decision problem. In particular, why is the
option *Raise* adequately represented by a function that maps the
L & P state (which logically entails that you signal truly!) to
the outcome *Signal falsely*?

One somewhat attractive way of rendering this more plausible is to
assume that (necessarily) the laws of nature specify (implicitly or
explicitly) the results of various possible
"interventions". Then the causal structure represented by L
& P might entail that (1) you will lower your hand, but also that
(2) if the *Lower* event were replaced by a *Raise* event,
a *Signal falsely* event would occur, where (2) is a
"interventionist counterfactual" in the style of
(say) Pearl
2000.

Obviously, the hypothetical *Signal falsely* event is not a
causal consequence of the *Raise* intervention. The
counterfactual (2) is a non-causal counterfactual. But we might still
suggest that in the L & P world, the
"variable" *Signal truly* is fixed to equal the
"variable" *Lower*, due to the robust convention
that *Lower* means to signal L and the fact that L is indeed
true. Fortunately, the interventionist counterfactuals required for
this application all have quite specific antecedents, representing an
agent's choice. So we don't need to enter the tricky issue of how to
interpret interventionist counterfactuals with unspecific
antecedents.

We might relabel and generalize the interventionist counterfactuals as special kinds of "conditional chance" statements, making contact with Skyrms's 1984 formulation of Causal Decision Theory. Here the expected utility of an option A is computed as

EU(A) = \sum_K Cr(K) \sum_C Ch_K(C / A) V(C),

where K ranges over complete chance hypotheses and C over
outcomes. Now intuitively, the laws alone don't fix the chances -- we
also need boundary conditions. L and P together certainly do fix the
chances. What do they say about the chance of *Signal falsely*
conditional on *Raise*? Presumably, *Raise* will have chance
zero, so the conditional chance is not defined by the usual ratio
formula. But conditional chances are better taken as primitive
anyway. And then, by the same sketchy reasoning as above --
that *Raise* means *Signal not-L* and that L is true -- we
might argue that *Ch_K(Signal falsely / Raise) = 1*, where K is
the chance hypothesis captured by L & P. Since most of your
credence goes to this hypothesis, Skyrms's formula says that you
should *Lower*.

(The problem of unspecific antecedents emerges here as the problem that conditional chances are only defined for rather specific conditions. Again, we can plausibly avoid this problem.)

It would be nice if we could be a little less sketchy. Skyrms in
fact offers an informative analysis of chance and conditional
chance. On this account, the chance of A at world w relative to a
given agent is the agent's prior credence Cr_0 conditional on w's cell
within a certain partition (which is determined by symmetries in
Cr_0). Accordingly, the conditional chance of B given A at w is Cr_0
of B conditional on the conjunction of w's cell and A. To
get *Ch_K(Signal falsely / Raise) = 1*, we would need to assume
that the cell of the L & P world contains *Raise* worlds that
verify L. But then those worlds won't verify P, and we'll run into
similar problems as with our revised Gibbard-Harper-Lewis account.

Another aspect of these proposals that bothers me is that they make
use of outcomes that are either incomplete or inconsistent. Return to
the interventionist counterfactual: if *Raise* then *Signal
falsely*. One wants to ask what else would would be the case
if *Raise*. Would L still be true? Would P still be true?
Would *Raise* be true? If the answer is `yes' each time, then
contradictory things would be true (for L & P
entails *Lower*). But how do we assign values to contradictory
outcomes? Or, how do we calculate the value of option A under
condition K if the counterfactual consequences are not closed under
conjunction? (A similar problem arises for Lewis because he rejects
the Limit Assumption.)