I wrote this last Autumn as a private “blog post” shared only with a few colleagues. I’m posting it publicly now (after mild editing) because I have some vague idea that it can be good to make things like this public.
In this post I’m going to discuss two papers regarding imprecise probability that I read this week for a Decision Theory seminar. The first paper seeks to show that imprecise probabilities don’t adequately constrain the actions of a rational agent. The second paper seeks to refute that claim.
Just a note on how seriously to take what I’ve written here: I think I’ve got the gist of what’s in these papers, but I feel I could spend a lot more time making sure I’ve understood them and thinking about which arguments I find persuasive. It’s very possible I’ve misunderstood or misrepresented the points the papers were trying to make, and I can easily see myself changing my mind about things if I thought and read more.
Also, a note on terminology: it seems like “sharpness/unsharpness” and “precision/imprecision” are used interchangeably in these papers, as are “probability” and “credence”. There might well be subtle distinctions that I’m missing, but I’ll try to consistently use “imprecise probabilities” here.
Imprecise probabilities
I imagine there are (at least) several different ways of formulating imprecise probabilities. One way is the following: your belief state is represented by a set of probability functions, and your degree of belief in a particular proposition is represented by the set of values assigned to it by the set of probability functions. You also then have an imprecise expectation: each of your probability functions has an associated expected utility. Sometimes, all of your probability functions will agree on the action that has the highest expected value. In that case, you are rationally required to take that action. But if there’s no clear winner, that means there’s more than one permissible action you could take.
Subjective Probabilities should be Sharp
The first paper, Subjective Probabilities should be Sharp, was written in 2010 by Elga. The central claim is that there’s no plausible account of how imprecise probabilities constrain which choices are reasonable for a perfectly rational agent.
The argument centers around a particular betting scenario: someone tells you “I’m going to offer you bet A and then bet B, regarding a hypothesis H”:
Bet A: win $15 if H, else lose $10
Bet B: lose $10 if H, else win $15
You’re free to choose whether to take bet B independently of whether you choose bet A.
Depending on what you believe about H, it could well be that you prefer just one of the bets to both bets. But it seems like you really shouldn’t reject both bets. Taking both bets guarantees you’ll win exactly $5, which is strictly better than the $0 you’ll win if you reject both bets.
But under imprecise probabilities, it’s rationally permissible to have some range of probabilities for H, which implies that it’s permissible to reject both bet A and bet B. So imprecise probabilities permit something which seems like it ought to be impermissible.
Elga considers various rules that might be added to the initial imprecise probabilities-based decision theory, and argues that none of them are very appealing. I guess this isn’t as good as proving that there are no good rules or other modifications, but I found it fairly compelling on the face of it.
The rules that seemed most likely to work to me were Plan and Sequence. Both rules more or less entail that you should accept bet B if you already accepted bet A, in which case rejecting both bets is impermissible and it looks like the theory is saved.
Elga tries to show that these don’t work by inviting us to imagine the case where a particular agent called Sally faces the decision problem. Sally has imprecise probabilities, maximises expected utility and has a utility function that is linear in dollars.
Elga argues that in this scenario it just doesn’t make sense for Sally to accept bet B only if she already accepted bet A—the decision to accept bet B shouldn’t depend on anything that came before. It might do if Sally had some risk averse decision theory, or had a utility function that was concave in dollars—but by assumption, she doesn’t. So Plan and Sequence, which had seemed like the best candidates for rescuing imprecise probabilities, aren’t plausible rules for a rational agent like Sally.
Should Subjective Probabilities be Sharp?
The 2014 paper by Bradley and Steele, Should Subjective Probabilities be Sharp? is, as the name suggests, a response to Elga’s paper. The core of their argument is that the assumptions for rationality implied by Elga’s argument are too strong and that it’s perfectly possible to have rational choice with imprecise probabilities provided that you don’t make these too-strong assumptions.
I’ll highlight two objections and give my view.
Objection 1:
Bradley and Steele give the label Retrospective Rationality to the idea that an agent’s sequence of decisions should not be dominated by another sequence the agent could have made. They seem to reject Retrospective Rationality as a constraint on rational decision making because “[it] is useless to an agent who is wondering what to do… [the agent] should be concerned to make the best decision possible at [the time of the decision]”.
My view: I don’t find this a very compelling argument, at least in the current context—it seems to me that the agent should avoid foreseeably violating Retrospective Rationality, and in Elga’s betting scenario the irrationality of the “reject both bets” sequence of decisions seems perfectly foreseeable.
Objection 2:
Their second objection is that Elga is wrong to think that your current decision about whether to accept bet B should be unaffected by whether you previously accepted or rejected bet A (they make a similar point regarding the decision to take bet A with vs without the knowledge that you’re about to be offered bet B).
My view: it’s true that, because in Elga’s betting scenario the outcomes of the bets are correlated, knowing whether or not you previously accepted bet A might well change your inclination to accept bet B, e.g. because of risk aversion or a non-linear utility function. But to me it seems right that for an agent whose decision theory doesn’t include these features, it would be irrational to change their inclination to accept bet B based on what came before—and Elga was considering such an agent. So I think I side with Elga here.
Summary and some thoughts
In summary, in Subjective Probabilities should be Sharp, Elga illustrates how imprecise probabilities appear to permit a risk-neutral agent with linear utility to make irrational choices. In addition, Elga argues that there aren’t any ways to rescue things while keeping imprecise probabilities. In Should Subjective Probabilities be Sharp?, Bradley and Steele argue that Elga makes some implausibly strong assumptions about what it takes to be rational. I didn’t find these arguments very convincing, although I might well have just failed to appreciate the points they were trying to make.
I think it basically comes down to this: for an agent with decision theory features like Sally’s, i.e. no risk aversion and linear utility, the only way to avoid passing up opportunities like making a risk-free $5 by taking bet A and bet B is if you’re always willing to take one side of any particular bet. The problem with imprecise probabilities is that they permit you to refrain from taking either side, which implies that you’re permitted to decline the risk-free $5.
The fan of imprecise probabilities can wriggle out of this by saying that you should be allowed to do things like taking bet B only if you just took bet A—but I agree with Elga that this just doesn’t make sense for an agent like Sally. I think the reason this might look overly demanding on the face of it is that we’re not like Sally—we’re risk averse and have concave utility. But agents who are risk averse or have concave utility are allowed both to sometimes decline bets and to take risk-free sequences of bets, even according to Elga’s rationality requirements, so I don’t think this intuition pushes against Elga’s rationality requirements.
It feels kind of useful to have read these papers, because
I’ve been kind of aware of imprecise probabilities and had a feeling I should think about them, and this has given me a bit of a feel for what they’re about.
It makes further reading in this area easier.
It’s good to get an idea of what sort of considerations people think about when deciding whether a decision theory is a good one. Similarly to when I dug more into moral philosophy, I now have more of a feeling along the lines of “there’s a lot of room for disagreement about what makes a good decision theory”.
Relatedly, it’s good to get a bit of a feeling of “there’s nothing really revolutionary or groundbreaking here and I should to some extent feel free to do what I want”.
Two papers I read on imprecise probabilities
I wrote this last Autumn as a private “blog post” shared only with a few colleagues. I’m posting it publicly now (after mild editing) because I have some vague idea that it can be good to make things like this public.
In this post I’m going to discuss two papers regarding imprecise probability that I read this week for a Decision Theory seminar. The first paper seeks to show that imprecise probabilities don’t adequately constrain the actions of a rational agent. The second paper seeks to refute that claim.
Just a note on how seriously to take what I’ve written here: I think I’ve got the gist of what’s in these papers, but I feel I could spend a lot more time making sure I’ve understood them and thinking about which arguments I find persuasive. It’s very possible I’ve misunderstood or misrepresented the points the papers were trying to make, and I can easily see myself changing my mind about things if I thought and read more.
Also, a note on terminology: it seems like “sharpness/unsharpness” and “precision/imprecision” are used interchangeably in these papers, as are “probability” and “credence”. There might well be subtle distinctions that I’m missing, but I’ll try to consistently use “imprecise probabilities” here.
Imprecise probabilities
I imagine there are (at least) several different ways of formulating imprecise probabilities. One way is the following: your belief state is represented by a set of probability functions, and your degree of belief in a particular proposition is represented by the set of values assigned to it by the set of probability functions. You also then have an imprecise expectation: each of your probability functions has an associated expected utility. Sometimes, all of your probability functions will agree on the action that has the highest expected value. In that case, you are rationally required to take that action. But if there’s no clear winner, that means there’s more than one permissible action you could take.
Subjective Probabilities should be Sharp
The first paper, Subjective Probabilities should be Sharp, was written in 2010 by Elga. The central claim is that there’s no plausible account of how imprecise probabilities constrain which choices are reasonable for a perfectly rational agent.
The argument centers around a particular betting scenario: someone tells you “I’m going to offer you bet A and then bet B, regarding a hypothesis H”:
Bet A: win $15 if H, else lose $10
Bet B: lose $10 if H, else win $15
You’re free to choose whether to take bet B independently of whether you choose bet A.
Depending on what you believe about H, it could well be that you prefer just one of the bets to both bets. But it seems like you really shouldn’t reject both bets. Taking both bets guarantees you’ll win exactly $5, which is strictly better than the $0 you’ll win if you reject both bets.
But under imprecise probabilities, it’s rationally permissible to have some range of probabilities for H, which implies that it’s permissible to reject both bet A and bet B. So imprecise probabilities permit something which seems like it ought to be impermissible.
Elga considers various rules that might be added to the initial imprecise probabilities-based decision theory, and argues that none of them are very appealing. I guess this isn’t as good as proving that there are no good rules or other modifications, but I found it fairly compelling on the face of it.
The rules that seemed most likely to work to me were Plan and Sequence. Both rules more or less entail that you should accept bet B if you already accepted bet A, in which case rejecting both bets is impermissible and it looks like the theory is saved.
Elga tries to show that these don’t work by inviting us to imagine the case where a particular agent called Sally faces the decision problem. Sally has imprecise probabilities, maximises expected utility and has a utility function that is linear in dollars.
Elga argues that in this scenario it just doesn’t make sense for Sally to accept bet B only if she already accepted bet A—the decision to accept bet B shouldn’t depend on anything that came before. It might do if Sally had some risk averse decision theory, or had a utility function that was concave in dollars—but by assumption, she doesn’t. So Plan and Sequence, which had seemed like the best candidates for rescuing imprecise probabilities, aren’t plausible rules for a rational agent like Sally.
Should Subjective Probabilities be Sharp?
The 2014 paper by Bradley and Steele, Should Subjective Probabilities be Sharp? is, as the name suggests, a response to Elga’s paper. The core of their argument is that the assumptions for rationality implied by Elga’s argument are too strong and that it’s perfectly possible to have rational choice with imprecise probabilities provided that you don’t make these too-strong assumptions.
I’ll highlight two objections and give my view.
Objection 1:
Bradley and Steele give the label Retrospective Rationality to the idea that an agent’s sequence of decisions should not be dominated by another sequence the agent could have made. They seem to reject Retrospective Rationality as a constraint on rational decision making because “[it] is useless to an agent who is wondering what to do… [the agent] should be concerned to make the best decision possible at [the time of the decision]”.
My view: I don’t find this a very compelling argument, at least in the current context—it seems to me that the agent should avoid foreseeably violating Retrospective Rationality, and in Elga’s betting scenario the irrationality of the “reject both bets” sequence of decisions seems perfectly foreseeable.
Objection 2:
Their second objection is that Elga is wrong to think that your current decision about whether to accept bet B should be unaffected by whether you previously accepted or rejected bet A (they make a similar point regarding the decision to take bet A with vs without the knowledge that you’re about to be offered bet B).
My view: it’s true that, because in Elga’s betting scenario the outcomes of the bets are correlated, knowing whether or not you previously accepted bet A might well change your inclination to accept bet B, e.g. because of risk aversion or a non-linear utility function. But to me it seems right that for an agent whose decision theory doesn’t include these features, it would be irrational to change their inclination to accept bet B based on what came before—and Elga was considering such an agent. So I think I side with Elga here.
Summary and some thoughts
In summary, in Subjective Probabilities should be Sharp, Elga illustrates how imprecise probabilities appear to permit a risk-neutral agent with linear utility to make irrational choices. In addition, Elga argues that there aren’t any ways to rescue things while keeping imprecise probabilities. In Should Subjective Probabilities be Sharp?, Bradley and Steele argue that Elga makes some implausibly strong assumptions about what it takes to be rational. I didn’t find these arguments very convincing, although I might well have just failed to appreciate the points they were trying to make.
I think it basically comes down to this: for an agent with decision theory features like Sally’s, i.e. no risk aversion and linear utility, the only way to avoid passing up opportunities like making a risk-free $5 by taking bet A and bet B is if you’re always willing to take one side of any particular bet. The problem with imprecise probabilities is that they permit you to refrain from taking either side, which implies that you’re permitted to decline the risk-free $5.
The fan of imprecise probabilities can wriggle out of this by saying that you should be allowed to do things like taking bet B only if you just took bet A—but I agree with Elga that this just doesn’t make sense for an agent like Sally. I think the reason this might look overly demanding on the face of it is that we’re not like Sally—we’re risk averse and have concave utility. But agents who are risk averse or have concave utility are allowed both to sometimes decline bets and to take risk-free sequences of bets, even according to Elga’s rationality requirements, so I don’t think this intuition pushes against Elga’s rationality requirements.
It feels kind of useful to have read these papers, because
I’ve been kind of aware of imprecise probabilities and had a feeling I should think about them, and this has given me a bit of a feel for what they’re about.
It makes further reading in this area easier.
It’s good to get an idea of what sort of considerations people think about when deciding whether a decision theory is a good one. Similarly to when I dug more into moral philosophy, I now have more of a feeling along the lines of “there’s a lot of room for disagreement about what makes a good decision theory”.
Relatedly, it’s good to get a bit of a feeling of “there’s nothing really revolutionary or groundbreaking here and I should to some extent feel free to do what I want”.