A

What happens if we enlarge the class of alternatives to include a wider range of probability functions, including some with a different domain? This would strengthen the principle linking epistemic utility and rationality: it would no longer suffice, for a credence function to be deemed epistemically rational, that it expects itself to be doing better, epistemically, than credence functions with the same domain. And this stronger principle would arguably give us a more plausible theory of epistemic rationality, at least on some ways of widening the range of alternatives. Suppose an agent with a credence function defined over a collection of propositions takes herself to be doing better, epistemically, than she would be by having another credence function defined over the same collection of propositions. But suppose she thinks she would be doing better, epistemically, having a credence function defined over a smaller collection of propositions—perhaps she thinks she would be doing better, epistemically, not having certain defective concepts and thus that she would be doing better, epistemically, simply not having propositions with those concepts as constituents in the domain of her credence function. Such an agent would seem to be irrational in much the same way as an agent who thinks she would be doing better, epistemically, by assigning different credences to the propositions she assigns credence to.

Now, my interest here is not with the question what is the right principle linking epistemic utility and rationality. Rather, I am interested in understanding how strong a principle we can consistently endorse: I am interested in the kinds of constraints on epistemic utility functions that come from different views on how epistemic utility and epistemic rationality are related to one another. So I start by considering the strongest version of a principle linking epistemic utility and rationality, one that says that an epistemically rational credence function should take itself to be doing better than any other credence function, regardless of its domain. As we will see, the resulting immodesty constraint is far too strong, in that, perhaps surprisingly, it cannot be satisfied by any reasonable epistemic utility function—that this is so is a consequence of the main results in this paper (Subsection 3.1–Subsection 3.2).

I then consider different possible ways of weakening this principle, study the resulting constraints on epistemic utility functions and their relationship to one another, and establish a few characterization results for the class of epistemic utility functions satisfying these constraints (Subsection 3.3). Before concluding, I discuss (Section 4) how my results relate to recent work on the question whether epistemic utility theory is incompatible with imprecise, or ‘mushy’, credences.

Fix a collection

I will say that a real-valued function

Let

Throughout, I assume that epistemic utility functions are

By definition, for fixed

where

I will say that an epistemic utility function

I will say that

Strictly proper epistemic utility functions have been the subject of considerable interest. In discussions of how to reward a forecaster’s predictions, strictly proper functions are of interest because they reward honesty—someone whose forecasts will be rewarded using a strictly proper epistemic utility function cannot expect to do better than by reporting her true credences (

And in discussions of justifications of Probabilism—the requirement on degrees of belief functions that they satisfy the axioms of the probability calculus—strictly proper utility functions have played a starring role in a range of dominance results to the effect that probabilistic credences strictly dominate non-probabilistic credences and are never dominated by any other credence function (

One natural question to ask is how to generalize the framework of epistemic utility theory to allow for comparisons of probability functions defined over distinct algebras of propositions. And given such a generalization, an equally natural question is how to generalize the notion of (strict) propriety. Let me take each of these questions in turn.

Let

Define now

where

A

It is straightforward to define generalized epistemic utility functions that are partition-wise proper. For example, take the generalized version of the

where

If we are working with a fixed partition and only considering probability functions defined over that partition, a strictly proper epistemic utility function for that partition ensures the kind of immodesty that is allegedly a feature of epistemic rationality (

Once we relax the assumption that we are working with a fixed partition, however, partition-wise strict propriety does not suffice to ensure immodesty, nor to encourage honest reporting. To see why, first note that for any

We can now see, using the Brier score as our epistemic utility function, that any probability function that is not

Now let

and thus

An interesting question, then, is whether there are epistemic utility functions that capture the relevant kind of immodesty once we consider probability functions defined over any partition. In other words, the question is whether there are epistemic utility functions such that, for any probability function

Before turning to this question, let me introduce a few more pieces of terminology. Fix

It will be convenient to also have at our disposal three different quantities which (albeit imperfectly) summarize some of the information about how

Similarly, define the

Finally, for

Intuitively, the lower expectation of

Clearly,

with equality if

Also note that for any

so that for any

Given all of these resources, we have two ways of formulating a generalized immodesty principle.

and

and

Before turning to this question, I want to spend some time explaining why these two principles stand out among other plausible generalizations as worthy of our attention. (Those who find

One way to think about immodesty is as the claim that epistemic utility functions should make all coherent credence functions immodest in the following sense: an agent with that credence function will think her own credence function is

The literature on decision-making with imprecise probabilities contains a number of options we can make use of: rules for deciding between options whose outcomes depend on the state of the world when we do not have well-defined credences for each of the relevant states of the world.

First, we could say that

Alternatively, we could say that

We could instead say that

Or that

We could also say that

Finally, we could say that

For each of these ways of understanding what it is for

Fortunately, these generalizations are not logically independent of one another. To see that, start by fixing

Similarly, it follows from these observations that (6) is the strongest generalization of immodesty from among those we have considered. In short, the most we can hope for when formulating a generalized immodesty principle is essentially the requirement that all epistemic utility functions satisfy (6)—that is, universal

I begin by asking whether there are any universally

Say that an epistemic utility function

and

and

Using these definitions we can make a few simple observations. First, and most clearly, (strict) downwards propriety and (strict) upwards

For the right-to-left direction of (ii), simply note that for

which ensures

Say that an extension

and let

A consequence of the last two results is that for determining whether

But by assumption,

(resp. the above inequality is strict when

We can thus conclude that

the definition of upper expectation entails that

Now, a natural question to ask is whether there are epistemic utility functions that are both (strictly) upwards

And since

We thus have that for any

And as announced above, there just are no strictly universally

which shows that

Finally, we can strengthen Theorem 3.6 if we restrict ourselves to the class of partition-wise strictly proper epistemic utility functions.

Since by construction

Putting all of this together and using the definition of upper-expectation, we have that there is an extension

which shows that

The next question to ask is whether there are any universally

Much like in the previous section, I will define an analogue of upwards

Before asking whether there are strictly universally

So we can conclude that if

Suppose now

And since

which means

Before concluding this subsection, let me note two consequences of Lemma 3.10, which serve as counterparts to Corollary 3.4 and Fact 3.5.

The left-to-right direction now follows immediately (simply let

where the last equality follows from Lemma 3.10. We can thus conclude that

as desired. If

We have seen that there are no strictly universally

Say that an epistemic utility function

Say that a function

and say that

If

For a given local accuracy measure

The linearity of expectation ensures that if

From this we can easily derive the following characterization result.

and

Since

For the converse, fix

and

Since

Parallel reasoning shows that, for proper

As we saw in Example 1, the (generalized) Brier score is not downwards proper, but the (generalized version of the) well-known

and let

Clearly, the restriction of

To see why, note that for any

where

Since the Euclidean norm is a norm, it satisfies the triangle inequality, and thus for any

which means

We also need not look far to find an example of an upwards proper additive accuracy measure.

where

Fix now

And of course,

From Corollary 3.3 we conclude that

Note that the log score is also an additive accuracy measure with local accuracy measure l, where

Note too that

To establish the other direction, fix

Note now that

And since clearly

u(0,0) ≤ 0 (resp. u(0,0) < 0) entails that

Now, it is well-known

(resp. the above inequality is always strict). And it is well-known (see, e.g.,

(resp. the above inequality is always strict). Since by definition of

To conclude this section, let me state one final characterization result, this time for the class of upwards

and that

For the right-to-left direction, start by fixing

and

whence

Surprisingly, it follows from this that for additive accuracy measures, upwards

Using Theorem 3.14, we conclude that

According to the standard, Bayesian picture we have been taking for granted, an agent’s epistemic state can be adequately represented with a single probability function. But many think this is a mistake: on their view, an agent’s epistemic state is best represented not with a single probability function but with a

Grant that proponents of this dissenting view are right—grant, in other words, that one can be in the kind of epistemic state that is better modeled with a set of probability functions than with a single probability function.

There has been much debate around this question and it is not my purpose here to take a stance either way.

In the literature on epistemic utility theory, it is by and large taken for granted that something like the following principle captures an important relationship between epistemic utility and epistemic rationality:

D

So, much attention has been paid to the question what kinds of reasonable epistemic utility functions can be defined that allow us to compare the epistemic utility of a ‘precise’ credence function at a world with that of an ‘imprecise’ one—here we think of sets of probability functions as ‘imprecise’ or ‘indeterminate’ credence functions since for many propositions they do not determine a unique degree of credence.

For example, generalizing some results in Schoenfield (

To see why, note that for a fixed

Now, we can first observe that in a sense my results are more general, in that they do not make any substantive assumptions about epistemic utility functions—at most, we assume that epistemic utility functions are continuous and truth-directed.

But there is a more significant difference between my results and those from the literature on imprecise probability functions. The question at the center of impossibility results for imprecise probability functions takes as given a fixed partition and asks whether there are reasonable ways of measuring accuracy or epistemic utility

P

To get a handle on what Perfection says, it helps to focus on a simple case with a two-cell partition ^{*}, than

Now, in this paper I have not trafficked in anything quite like the notion of epistemic utility relative to a partition. So it is not completely straightforward to translate Perfection into a constraint on the kind of epistemic utility functions we have been interested in. But there is a somewhat natural way to recast Perfection into a constraint on generalized epistemic utility functions in my sense. And once we see what that constraint amounts to, we will see both that it is not quite so plausible (as a constraint on generalized epistemic utility functions) and that my results do not depend on it.

Recall that in comparing the discussion of imprecise probability functions over a partition with my discussion of credence functions whose domain does not include elements of that partition, I identified a (precise) credence function defined over a coarsening

R

Now, it should be clear that my results do not depend on anything like Refinement. After all, Refinement rules out as admissible any upwards

The constraint imposed by Refinement is incompatible with thinking of some refinements as an unalloyed epistemic bad: if epistemic utility satisfies Refinement, there can be no proposition such that that you are epistemically worse off no matter what when you come to form an opinion on that proposition. Whether it be a proposition about phlogiston, or about miasma, Refinement entails that it is always in principle possible to do better, epistemically, by forming a view on that proposition.

Of course, it may be that this is the right way to think about epistemic utility, but it is certainly not

At any rate, it is not my goal here to suggest that the right way to think about epistemic utility is incompatible with Refinement. But I do want to point out that it is yet another substantive assumption about epistemic utility that is required for the impossibility results mentioned above to go through. In contrast, my results make no substantive assumptions about epistemic utility. Rather, they establish that no matter

In contexts where probability functions are stipulated to all be defined over a fixed domain, strictly proper epistemic utility functions arguably capture a certain kind of immodesty. Once we move on to contexts where probability functions are allowed to be defined over different domains, strictly proper epistemic utility functions do not capture the relevant sense of immodesty. My question was whether there was a way of characterizing immodesty in this general setting. I considered a variety of strong, generalized immodesty principles and showed that, under minimal assumptions, no epistemic utility function satisfies any of these stronger immodesty principles.

I also considered some very weak generalizations of strict propriety and showed that some of the familiar epistemic utility functions satisfy one or another of these weak immodesty principles. One interesting question left outstanding is how strong an immodesty principle can be imposed without ruling out every reasonable epistemic utility function. In particular, one interesting question is whether there are immodesty principles that distinguish among partitions—say, immodesty principles that say that for any partition of a certain kind, all credence functions defined over that partition take themselves to be doing better, in terms of epistemic utility, than any of their restrictions without thereby taking themselves to be worse than any of their extensions.

I have not, of course, argued that epistemic utility functions ought to satisfy any of these stronger immodesty principles. But it is at the very least not obvious that strict partition-wise propriety suffices to capture the sense in which epistemic rationality is said to be immodest. What else, if anything, suffices to capture that kind of immodesty is a question for some other time.

Cf. Joyce (

Cf. the principle ‘Immodest Dominance’ in Pettigrew (

Cf. Pérez Carballo (

Previous work on related issues include Carr (

Since I will be taking Probabilism for granted, we can work with these simplified definitions without loss of generality.

The standard definition of

I identify the truth-value of

Arguably, epistemic utility functions would need to satisfy additional constraints to count as genuinely epistemic ways of comparing probability functions relative to a given state of the world. For a sense of the wide range of possible constraints, see Joyce (

Strictly speaking, immodesty alone is not enough to motivate something as strong as Strict Propriety. The assumption that epistemic rationality is immodest ensures at best that at any one time, an agent’s epistemic values

See, however, Campbell-Moore and Levinstein (

I’m using ‘restriction’ here in the standard way, where the restriction of a function

Again, see Gneiting and Raftery (

This is because our assumptions ensure that

Cf. Carr (

Note that the example below suffices to show that the normalized version of the Brier score, defined by

is also not downwards proper.

Note that

Note that we can think of

To anticipate, while I will focus on these two formulations, the reason is not that I think either one of them is the best way to generalize immodesty to allow for alternatives to a credence function with different domains. Rather, it is because these two principles stand at the extreme ends of a much larger family of plausible generalizations: one is stronger and the other is weaker than any other generalization.

See, e.g., Troffaes (

Specifically, (4), below, corresponds to the fourth preference ranking listed in §5.4.3 of Halpern (

As an anonymous referee rightly points out, all of these principles violate the arguably unobjectionable principle of

It can also be seen as a direct consequence of Fact 3.2, below.

A helpful mnemonic: for

For a given

(Simply fix an enumeration of

In fact, something slightly weaker than the full continuity assumption may be all that is really needed—see Grünwald and Dawid (

I am thus implicitly assuming that accuracy measures satisfy what Joyce (

See Pérez Carballo (

The canonical reference here is Savage (

Cf. Joyce (

See, e.g., Levi (

See, e.g., Schoenfield (

I say ‘something like’ because the principle as stated is in need of clarification and arguably subject to a number of powerful objections. For one thing, we need to clarify whether the principle holds for any admissible measure of epistemic utility, or whether it needs to be understood as quantifying over all admissible ways of measuring epistemic utility—Schoenfield (

An additional assumption, worth pointing out since it may go unnoticed, is that epistemic utility functions are real-valued. For a way of thinking about epistemic utility for imprecise probabilities that does without this assumption—a view on epistemic utility on which imprecise probabilities are only partially ranked in terms of epistemic utility relative to any world—see Seidenfeld et al. (

The results in Seidenfeld et al. (

Cf. the principle Schoenfield calls ‘Boundedness’ (

Note that this view is incompatible with Extensionality—the thesis that the epistemic utility of a credence function at a world is independent of the content of the propositions it assigns credence to. Indeed, it may be that a commitment to Extensionality all but requires a commitment to Refinement.

I should add that whereas my results rely on much weaker assumption than those from the literature on imprecise credence functions, they are not stronger than them, since the conclusions they derive from their stronger assumptions are stronger than those we derive from my weaker assumptions. For instance, as mentioned above, Berger and Das show that, given their assumptions on epistemic utility functions, for any imprecise credence function there will be precise credence function with the same domain that is as good, epistemically, as the imprecise credence function relative to any world. The analogous conclusion, in my framework, would be that for any credence function

My proof strategy follows some of the reasoning in the first five sections of Grünwald and Dawid (

See, e.g., (

See, e.g., (

For helpful conversations, comments, and advice, I am grateful to Kenny Easwaran, Richard Pettigrew, Itai Sher, and Henry Swift. Special thanks to Chris Meacham who, in addition to indulging me on many conversations about the material in this paper, went through an earlier draft of the paper with great care. Last but not least, thanks are also due to two anonymous referees for this journal for their many generous and extremely helpful comments. Much of this paper was written while I was a fellow at the Center for Advanced Study in the Behavioral Sciences at Stanford University: I’m grateful to the Center for its financial support.

My proof of Lemma 3.10 will rely on a fundamental result in game theory, which I will simply state without proof.

A

The

This is the maximum payoff that player

is the best player I can do. The

In general,

We say that

If a game has a value, we say that player I has an

Similarly, we say that player II has an optimal strategy iff there is

If the game has a value and both players have an optimal strategy, the pair of optimal strategies corresponds in an intuitive way to an

Not all games have a value. Some of the foundational results in game theory allow us to characterize classes of games that have a value. I will be relying on one such result for the proof of Lemma 3.10.

Recall that a function

If

We can apply Theorem A.1 to show that, whereas many games of interest do not contain a saddle-point, if we allow players to

For any compact

Now, fix

Since each

If

I can finally present the proof of Lemma 3.10.

Again abusing notation, I will use

Note that

Our next step is to define a particular mixed extension of

Let now

and accordingly

From Corollary A.2, we know that our game

and thus that

But since

Summing up, we have a saddle point of the form (

In other words,

and

But note that (10) entails both

by definition, and

since

Hence, from (11), (12), and (13), we have that for any

as desired.