## Overlapping acts

I'm currently teaching a course on decision theory. Today we discussed
chapter 2 of Jim Joyce's *Foundations of Causal Decision Theory*,
which is excellent. But there's one part I don't really get.

Joyce mentions that Savage identifies acts with functions from states to outcomes, and that Jeffrey once suggested representing such functions as conjunctions of material conditionals: for example, if an act maps S1 to O1 and S2 to O2, the corresponding proposition would be (S1 → O1) & (S2 → O2). According to Joyce, this conception of acts "cannot be correct" (p.62). That's the part I don't really get.

Joyce points out that if S1 actually obtains, then all it takes to
make true (S1 → O1) & (S2 → O2) is to make true O1. So
if I can make true (S1 → O1) & (S2 → O2) then I can also
make true (S1 → O1) & (S2 → O3) for *any* O3, even
if I would be completely unable to secure O3 under state S2.

But why is that a problem? I agree that we should not count all propositions of the form (S1 → O1) & (S2 → O3) as available acts -- otherwise we'll wrongly conclude that I ought to choose one of those "acts" where O3 is especially valuable. But Jeffrey's proposal only entails that we should allow for such acts if we count as an act any proposition the agent can make true or false at will. This is something Jeffrey actually does say on occasion, although I think never as his considered view. Anyway it's clearly untenable, and the present example illustrates why. For if S1 obtains and the agent has control over the truth-value of S1 & O1, then she also has control over (S1 → O1) & (S2 → O3), for arbitrary O3. Yet we don't want to count all such propositions as available acts. The lesson is that not anything the agent can make true or false at will is an available act. The lesson is not that acts can't be represented as conjunctions of material state-outcome conditionals.

Joyce says that on Jeffrey's interpretation, "there is no distinction to be made" between (S1 → O1) & (S2 → O2) and (S1 → O1) & (S2 → O3), provided S1 actually obtains (p.63). But why not? Even if S1 obtains, these are different propositions.

Still, I can see two potential problems with Jeffrey's proposal, neither of which is mentioned by Joyce. They arise in cases where (S1 → O1) & (S2 → O2) and (S1 → O1) & (S2 → O3) are actually both available.

The first problem is that if S1 obtains and the agent ensures that O2 comes about, it looks like she has simultaneously performed both acts: she has made both propositions true. But it may well be that only one of them maximizes expected utility. So we'd have to say that it is OK to choose an act that doesn't maximize expected utility as long as one chooses an act that maximizes expected utility, which sounds strange. Worse, it now becomes harder to work backwards from an agent's choices to her beliefs and desires. If (S1 → O1) & (S2 → O2) has greater expected utility than (S1 → O1) & (S2 → O3), then the agent's preference relation should rank the former higher than the second; but if the agent's choice simultaneously makes both propositions true, her choice doesn't reveal her preference.

That looks like a minor problem. The other problem is a little more serious but arguably only arises in conditional formulations of decision theories like Jeffrey's (and Joyce's) where we consider the probability of outcomes conditional on acts. Suppose an agent faces the following kind of problem, in Savage form.

S1 | S2 | |
---|---|---|

A1 | O1 (.1) | O2 (.4) |

A2 | O1 (.4) | O3 (.1) |

The numbers in the cells represent the agent's unconditional probability for each outcome. So conditional on act A1, outcome O2 is a lot more likely than outcome O1. But if we identify A1 with (S1 → O1) & (S2 → O2), then the bottom left cell is part of A1 and so the conditional probability of O2 given A1 comes out as 4/9. That's clearly the wrong result.

So in the end I agree that it's probably not a good idea to identify acts with conjunctions of material state-outcome conditionals, especially in conditional decision theories. But I suspect I've missed some more reasons.