Wolfgang Schwarz

Decision problems without equilibria

In my recent posts on decision theory, I've assumed that friends of CDT should accept a "ratifiability" condition according to which an act is rationally permissible only if it maximises expected utility conditional on being chosen.

Sometimes no act meets this condition. In that case, I've assumed that one should be undecided. More specifically, I've assumed that one should be in an "stable" state of indecision in which no (pure) option is preferable to one's present state of indecision. Unfortunately, there are decision problems in which no act is ratifiable and no state of indecision is stable. I'm not sure what to say about such cases. And I wonder if whatever we should say about them also motivates relaxing the ratifiability condition for certain cases in which there are ratifiable options.

I have briefly motivated the ratifiability condition in this earlier post, where I complained that Ahmed (2014) presents it as an alternative to orthodox CDT. Others do this as well. Apparently "orthodox CDT" says that one should choose an act that maximises expected utility relative to one's pre-deliberation credences. "Orthodox CDT" therefore says that it's perfectly fine for you to choose an act even if, at the time of choice, you know that some other act is guaranteed to be better. I don't know anyone who has ever defended this form of CDT. It should be called "unreflective CDT" or "idiotic CDT" or "strawman CDT", not "orthodox CDT".

Anyway. Let me explain how there can be decision problems without equilibria.

On some ways of modelling states of indecision – notably, the models developed in Skyrms (1990) – one can prove that every decision problem has an equilibrium solution in the form of either a ratifiable option or a stable state of indecision. This result depends on a certain way of assessing the quality of an indecision state, so that we can determine whether it is at least as good as any of the (pure) options. Skyrms assumes that the value of an indecision state is a weighted average of the expected utility of the options between which the agent is undecided. But this isn't always correct.

The problem is especially clear if we assume, as I like to do, that indecision is a genuine type of intentional state, on a par with decisions. Like decisions, indecision states cause actions. But the connection between an indecision state and the resulting action is not deterministic. From the agent's perspective, at least, it is a matter of chance which act will come about if they are in a given state of indecision, with the chances matching the agent's inclination towards the relevant act.

The Skyrmsian model now becomes implausible in cases where someone can predict the outcome of this chancy (or quasi-chancy) process. For a simple example, consider the Death in Damascus case from Gibbard and Harper (1978).

Death in Damascus. You have a choice between going to Aleppo and staying in Damascus. Death has predicted where you will go and is awaiting you there.

This is an "unstable" decision problem because neither act is ratifiable. It seems that you should be perfectly undecided between the two options. If death has a utility of -10 and surviving 0 then Skyrms's model gives the state of indecision a score of 5. (Hence neither of the pure options look better if you're in that state: their expected utility is also 5.)

But what if Death can foresee how states of indecision get resolved – which acts they eventually cause? Then the state of indecision is arguably no better than, say, a straight decision to go to Aleppo. Either way you can be sure to die. The state should get a score of -10, not -5.

This matters for whether the state is an equilibrium. If Death can foresee how states of indecision get resolved, and you are perfectly undecided between Aleppo and Damascus, then you can be certain you will die. You don't know where Death is waiting, because you don't know which act you'll end up taking. For all you know, Death is equally likely to wait in Aleppo and in Damascus. This means that if (counterfactually) you were to decide to go to Aleppo, you would have a 50% chance of survival. That's better than guaranteed death. The indecision is unstable. Both pure options are better.

All the decision problems discussed in Ahmed (2014) are tame insofar as they all have an equilibrium, even if the relevant predictor (that figures in many of them) can foresee resolutions of indecision. But this isn't always the case. In Death in Damascus, it is not – assuming Death can foresee where you will go, no matter how the act is caused. Here no pure act is ratifiable, and no state of indecision is stable.

We can make the problem more vivid if we assume that Death is much better at foreseeing resolutions of indecision (say, 100% success rate) than at foreseeing pure choices (60% success rate). In that case, you surely don't want to remain in a state of indecision. You'd rather take a pure option with a 40% chance of survival than go into a state of indecision with a 0% chance of survival. But neither of the pure options is ratifiable. If you decide to go to Aleppo, it would be better to stay in Damascus. If you decide to stay in Damascus, it would be better to go to Damascus. What should you do?

I'm not sure.

Perhaps it would help to be less serious about "states of indecision". Skyrms sometimes talks as if being in (what I call) a state of indecision is simply a matter of having certain credences about what you will do. His model of deliberation then says that at the endpoint of deliberation you should give equal credence to both options in Death in Damascus. It doesn't say that you should be in a special intentional state that gets resolved by some chancy or quasi-chancy mechanism. Arntzenius (2008) endorses this approach. Ahmed (2014) calls it "deliberational decision theory".

If there's no process of resolving states of indecision, one might hope that we don't need to worry about cases in which someone can predict the outcome of that process.

But then I worry about something else. I want decision theory to be a highly idealised model connecting an agent's beliefs and desires with their behaviour, not just with their beliefs about what they will do. To be sure, one could add to the "deliberational" theory that an agent will initiate an act whenever they are certain (at the end of deliberation) that they will perform it. This gives us a connection to behaviour in every situation in which there is a ratifiable option. But I'd like to have more. I'd like decision theory to provide an instruction for building an ideal agent. The instruction should not fall silent whenever there is no ratifiable option. It should cover as many cases as possible.

Besides, it's not even clear that the "deliberational" approach really avoids the problem. Suppose you're in the Death in Damascus scenario where Death only has a 60% success rate at predicting your acts, and you know that Death's sister is watching you and will strike you down immediately if you end up unsure about what you will do. Shouldn't you plunge for one of the pure acts rather than deliberate yourself into uncertainty? And yet none of these acts is ratifiable.

I think we need to say something about equilibrium-free decision problems. And I'm not sure what we should say. (I briefly touched on this here.)

One option is to stick with the requirements of stability and ratifiability and infer that in such a case, everything is rationally forbidden. You're in a rational dilemma.

Another option is to restrict the requirements to problems with equilibria. If there's no equilibrium, we might say that decision theory falls silent. Nothing is rationally forbidden, nothing is rationally permitted.

Alternatively, one might say that the norms of practical rationality become indeterminate in such a case. It is not determinately permitted to go to Aleppo nor is it determinately forbidden.

Alternatively, one might try to find some backup rule that tells us which of the non-equilibrium solutions are permissible and which aren't.

Some of these options might suggest that we could also weaken the equilibrium requirement for certain decision problems in which there are ratifiable options or stable states of indecision.

For example, it is tempting to say that in the version of Death in Damascus where Death is much better at predicting acts that result from indecision than at predicting straight decisions, you should make a straight choice. Doesn't matter which, but you should choose an unratifiable act. That gives you a 40% chance of survival, whereas indecision would mean certain death. True, whatever option you choose, you'll think that the other option would have given you a 60% chance of survival. But you also know that the same consideration would detract you from the other option if you were to choose it.

Now consider another variation on Death in Damascus. This time, there is a ratifiable act, but it looks terrible.

Dionysos in Damascus. You have a choice between going to Aleppo, staying in Damascus, and going to Homs. Dionysos is likely to have predicted what you will do, even if it results from a state of indecision. If he predicted that you'd go to Aleppo he will meet you there and engage you in a somewhat unpleasant orgy. If he predicted that you'd stay in Damascus he will meet you in Damascus and engage you in the same kind of orgy. If he predicted that you'd go to Homs, he will soon release a toxic gas into the air of Aleppo and Damascus that will kill all their inhabitants. In any case, Alastor (a different God, and for reasons we don't need to get into) is waiting for you in Homs and will torture you savagely if you show up there.

Your decision matrix might look like this, where 'P-Aleppo' means that Dionysos has predicted that you'll go to Aleppo.

(I assume that you prefer the torture in Homs over dying from toxic gas. I also assume – although this is inessential – that you care somewhat about the lives of the other people in Aleppo and Damascus.)

Without the third column and the third row, this is just the original Death in Damascus case, except with milder stakes. That problem has no equilibrium. This remains true in the extended problem: going to Aleppo and staying in Damascus are not ratifiable, and no state of indecision between these two options is stable.

The third option, however, is ratifiable. If you decide to go to Homs, you can be confident that Dionysos has predicted your escape from the orgy and that he is about to release the toxic gas in his fury, killing everyone in Aleppo and Damascus. And then you are really better off going to Homs.

I think this is the only equilibrium, but I'm not entirely sure. At any rate, there is no stable state of indecision in which you are less than 60% inclined towards going to Homs. That's bad enough.

Intuitively, you'd be crazy to go to Homs. Much better to endure the orgy in Aleppo or Damascus. Or so one is tempted to think.

In fact, I'm not sure what to make of this case. You could defend a decision for Homs much like we two-boxers defend our decision to two-box. "Yes", you might say, "choosing the other options would be better news, but my aim is not to bring about good news. Like any rational person, I will go to Homs. Dionysos has predicted that I'm rational, and so he has prepared the gas in Aleppo and Damascus. Nothing I can do about that. It's already all set up. The choice Dionysos has given me is between being tortured in Homs and dying in one of the other places. I wish Dionysos had thought that I'm irrational. Then he would have given me better options. Alas I'm rational, and he knows that I am."

So maybe we should bite the bullet and say that you should go to Homs.

But maybe we should instead allow you to choose an unratifiable act. Above I suggested that if Death is especially good at predicting resolutions of indecision, then you should either decide to go to Aleppo or to stay in Damascus, because the alternative – remaining undecided – is much worse and because you can recognise that the reasons against going to whichever city you choose would equally apply if you were to choose the other city. One might argue that this is true for Dionysos in Damascus as well. You should either decide to go to Aleppo or to stay in Damascus (or remain undecided between these options), because the alternative – going to Homs (or being inclined towards Homs) – is much worse and because you can recognise that the reasons against going to whichever of Aleppo or Damascus you choose would equally apply if you were to choose the other city.

Or maybe we should say that we should evaluate not only our actual, fine-grained options, but also more coarse-grained possibilities, represented by disjunctions of options. The expected utility of a disjunction is simply the average of the expected utility of the disjuncts. In Dionysos in Damascus, Aleppo ∨ Damascus has much greater expected utility than the alternative, conditional on being "chosen" (i.e., conditional on Aleppo ∨ Damascus). In that sense, the disjunction is ratifiable. Now one might suggest that we should first identify all ratifiable disjunctions (including disjunctions with one element, and including the disjunction between all options), then select the best of them, and then restrict our attention to the decision problem in which these are the only options. (Any other options are forbidden.) If a state of indecision between the options in the remaining problem is stable then this is the unique answer to the original problem. If not, any of the pure options is permitted.

I vaguely remember Sobel having once proposed something like this. Anything else I should read on these issues?