## What are our options? (again)

In decision theory, the available options are often glossed informally as the acts the agent can perform, or the propositions she can make true. But this yields implausible results in cases where an agent has doubts about what she can do.

For example, assume Bob suspects that the button in front of him functions as a light switch, as in fact it does. Then Bob can turn on the light by pressing the button. But if he is not certain that the button is a light switch, decision theory should consider the consequences of pressing the button if it has some other function. So turning on the light by pressing the button should not count as an option.

Hedden 2012 presents a similar counterexample:

Jane is hiking along the trail when she comes to a raging creek. She is in fact able to ford the creek head ..., head upstream to look for an easier crossing, or turn back. Among these three things that she is able to do, fording the creek has highest expected utility ... But she has serious doubts about whether she can in fact ford the creek. After all, the water is up to mid-thigh and flowing fast. Ought Jane ford the creek? I suggest that the answer is 'no.'

Hedden wrongly attributes the thus refuted definition of options to Jeffrey and Lewis. Here is Lewis's definition, from Lewis 1981 (which Hedden actually quotes):

Suppose we have a partition of propositions that distinguish worlds where the agent acts differently ... Further, he can act at will so as to make any one of these propositions hold, but he cannot act at will so as to make any proposition hold that implies but is not implied by (is properly included in) a proposition in the partition. The partition gives the most detailed specifications of his present action over which he has control. Then this is the partition of the agent's alternative options.

Let's begin with the light switch case, because it is a little easier. Is turning on the light by pressing the button an option for Bob? By assumption, it is something Bob can do. That is, he can act at will so as to make hold the proposition that he turns on the light by pressing the button. But that is not enough to make the proposition an option, on Lewis's definition. There are are further conditions. In particular, there's the condition that the options form a partition and that they are maximally detailed propositions the agent can make true.

A natural way of dividing up the relevant possibilities for Bob looks as follows. ("Light Switch" means that the button in front of Bob is a functioning light switch.)

Light switchNot light switch
PressA1A2
Not pressA3A4

A1 is the set of possible worlds where Bob turns on the light by pressing the button. If that is a cell of Bob's option partition, what are the other cells? It is not the case that Bob turns on the light by pressing the button, i.e., the union of A2, A3, and A4? Arguably that's too unspecific. For Bob can make true not just A2 v A3 v A4, but also A3, i.e., the proposition that he does not press the button which functions as a light switch. Certainly any claim which A1 has to being an option is shared by A3. But if A1 and A3 are options, then there must be one or more further options covering A2 and A4, otherwise the options don't form a partition. But any proposition located entirely in A2 v A4 is a proposition Bob cannot make true at will, for he cannot make it the case that the button (which is in fact a light switch) is not a light switch. So on Lewis's account, turning on the light by pressing the button (A1) is not an option.

For parallel reasons, fording the creek does not come out as an option for Jane. There are regions in logical space where Jane's strength is sufficient to ford the creek and others where it isn't. Jane fords the creek is (more or less) the intersection of Jane tries to ford the creek and Jane is strong enough to ford the creek. If that is one of Jane's options, then her alternative options should also include the fact that she is strong enough to ford the creek. But then the options won't form a partition. There would have to be at least one further option which entails that Jane is not strong enough to ford the creek, but that's not something Jane can make true.

The real problem for Lewis's definition is not that it counts as options things that shouldn't count, but rather that it does not count as options things that should count. In many situations (including the two we've looked at), nothing at all qualifies as an option on Lewis's definition.

In abstract, the problem is that if P is any proposition the agent can make true and Q is any fact over which the agent has no control, then the agent can generally make true P & Q. The requirement that options be the most detailed propositions the agent can make true then implies that all the agent's options entail Q. So they don't form a partition.

I raised this problem a few years ago. At the time I claimed that if an agent can make P true then she can automatically make true P & Q for any fact Q that is outside the agent's control. But that could be resisted: suppose we read 'the agent can phi' as something like 'the agent would phi if she intended to phi'. Then it is conceivable that an agent can make true P but not P & Q: at the closest worlds where the agent intends to make P true she succeeds, but at the closest worlds where she intends to make P&Q true she fails, perhaps because she focuses her effort on a misguided attempt to ensure the truth of Q and neglects to do what it would take to make true P. But such cases seem unusual. At any rate, there are many relevant Q for which an ability to make true P carries over to an ability to make true P & Q. If Bob intended to press the button he would succeed; if he intended to turn on the light by pressing the button he would succeed as well.

If you look back at the diagram above, the challenge is to explain why Bob's options are { A1 v A2, A3 v A4 } rather than { A1, A2 v A3 v A4 } or { A1 v A2 v A4, A3 } or { A1 v A4, A2 v A3}. Each of these is a partition of propositions that Bob can make true. (You may be skeptical about the last case; I'll get back to it in a moment.) None of them only contains propositions that specify in maximal detail what Bob can make true.

In the old blog post, I suggested the following fix. What we need is a restriction on how detailed options can get, to rule out options like A1. Perhaps what's wrong with A1, compared to A1 v A2, is that it is in a certain sense not fully under Bob's control. Bob has control over whether he presses the button, but not over whether the button is a light switch.

Let's say that an agent has full control over a proposition A if, for any non-trivial doxastically possible proposition B that is entailed by A, the agent can make B true and she can make B false. Now we can follow Lewis and demand that an agent's options should be the partition that "gives the most detailed specifications of his present action over which he has [full] control."

This correctly rules out A1 as an option: A1 entails that the button is a light switch, which Bob can't make false.

Unfortunately, I now think my proposal is too weak: it does not seem to rule out the partition { A1 v A4, A2 v A3 }. A1 v A4 is the proposition that Bob either presses the button and the button is a light switch or he doesn't press the button and the button is not a light switch; in other words: Bob presses the button iff it is a light switch. Admittedly, this is a curious proposition, but it seems to be a proposition Bob can make true.

To flesh this out, suppose Bob will get a price if he presses any button that is a light switch and no button that is not a light switch. In the current round of the game, there is only this one button, and Bob has good but inconclusive reason to believe that the button is a light switch. So when Bob intends to press the button, he also intends to press all and only the buttons that are light switches, thereby winning the price. So he intends to make true A1 v A4, and he succeeds. What more could we demand for him to be able to make true A1 v A4?

If A1 v A4 is a proposition Bob can make true, then the condition I put forward seems to allow it as an option: as far as I can tell neither A1 v A4 nor A2 v A3 entails any non-trivial doxastically possible proposition that is outside Bob's control.