Generalized Immodesty Principles in Epistemic Utility Theory

1. Introduction

Fix a collection $W$ of possible worlds and a finite partition $\pi$ of $W$ —a collection of pairwise disjoint, jointly exhaustive subsets of $W$, which we call cells.

I will say that a real-valued function $P$ defined over $\pi$ is coherent iff for each $s\in \pi$, $P(s)\in [0,1]$, and ${\sum }_{s\in \pi }P(s)=1$. A coherent function over $\pi$ uniquely determines a probability function over the Boolean closure of $\pi$. Accordingly, and slightly abusing notation, I will refer to coherent functions over $\pi$ as probability functions over $\pi$.⁵

Let ${\mathcal{P}}_{\pi }$ denote the collection of probability functions over $\pi$. An epistemic utility function (for $\pi$) is a function $\mathfrak{u}:{\mathcal{P}}_{\pi }\times W\to \overline{ℝ}:=ℝ\cup \{-\infty ,\infty \}$ such that for each $P\in {\mathcal{P}}_{\pi },\mathfrak{u}(P,·):W\to \overline{ℝ}$ is $\pi$-measurable—where $f:W\to \overline{ℝ}$ is $\pi$-measurable iff for each $r\in \overline{ℝ},\{w:f(w)=r\}$ is in the Boolean closure of $\pi$.⁶

Throughout, I assume that epistemic utility functions are bounded above (for each $\mathfrak{u}$ there exists a finite $M$ such that ) and truth-directed in the following sense: for all $w\in W$, if (i) for any proposition $s$ in $\pi$, $P(s)$ is at least as close as $P^{\prime }(s)$ is to the truth-value of $s$ in $w$,⁷ and (ii) for some proposition $s$ in $\pi$, $P(s)$ is strictly closer than $P^{\prime }(s)$ is to $s$’s truth-value in $w$, then $\mathfrak{u}(P,w)>\mathfrak{u}(P^{\prime },w)$.⁸

By definition, for fixed $\mathfrak{u}$ and $Q\in {\mathcal{P}}_{\pi },\mathfrak{u}(Q,\cdot )$ is a discrete random variable. Accordingly, for $P\in {\mathcal{P}}_{\pi }$ I will let ${𝔼}_P[\mathfrak{u}\text{(}Q)]\text{ :=\hspace{0.17em}}{𝔼}_X{}_{~P}[\mathfrak{u}\text{(}Q,X)]$ denote the expectation of $\mathfrak{u}(Q)$ relative to $P$, so that

where $s\mapsto w_s$ is a choice function, in that for each $w$ and $s$, $w_s\in s$. The $\pi$-measurability of $\mathfrak{u}(Q,·)$ ensures that our definition does not depend on our choice function. Indeed, I will simply write $\mathfrak{u}(Q,s)$ to denote $\mathfrak{u}(Q,w_s)$, so that ${𝔼}_P[\mathfrak{u}\text{(}Q)]={\sum }_{s\in \pi }P(s)\mathfrak{u}(Q,s)$.

I will say that an epistemic utility function $\mathfrak{u}$ for $\pi$ is proper iff for each $P,Q\in {\mathcal{P}}_{\pi }$,

I will say that $\mathfrak{u}$ is strictly proper iff for each $P\ne Q\in {\mathcal{P}}_{\pi }$, the above inequality is always strict. (When $\mathfrak{u}$ is proper but not strictly proper I will sometimes say that $\mathfrak{u}$ is weakly proper.) A variety of characterization results can be found in the literature—see especially Gneiting and Raftery (2007).

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 (Brier 1950; Savage 1971). In general discussions of epistemic value, strictly proper functions are of interest because they incorporate a certain kind of immodesty—if your epistemic values are represented by a strictly proper epistemic utility function and you are rational, you will never expect any other credence function to be doing better, epistemically, than your own (Joyce 2009; Gibbard 2008; Horowitz 2014; Greaves & Wallace 2006; inter alia).⁹

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 (Joyce 1998; 2009; Leitgeb & Pettigrew 2010; Pettigrew 2016; Predd, Seiringer, Lieb, Osherson, Poor, & Kulkarni 2009).¹⁰

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.

2. Generalizing the Framework

Let $\Pi$ denote the collection of finite partitions of $W$. For $\pi ,{\pi }^{\prime }\in \Pi$, say that $\pi$ is a refinement of ${\pi }^{\prime }$ iff for each $s\in \pi$ there is $s^{\prime }\in {\pi }^{\prime }$ such that $s\subseteq s^{\prime }$. If $\pi$ is a refinement of ${\pi }^{\prime }$, I will say that ${\pi }^{\prime }$ is a coarsening of $\pi$. Of course, the refinement relation induces a partial ordering over $\Pi$, which I will denote by $⊑$, where ${\pi }^{\prime }⊑\pi$ iff ${\pi }^{\prime }$ is a refinement of $\pi$. In fact, the resulting partially ordered set constitutes a lattice, in that any subset $\mathcal{S}$ of $\Pi$ admits of an infimum (a coarsest partition that is a refinement of all elements of $\mathcal{S}$) and a supremum (a coarsening of each element of $\mathcal{S}$ that refines any other partition that coarsens each element of $\mathcal{S}$).

Define now $\mathcal{P}:={\cup }_{\Pi }{\mathcal{P}}_{\pi }$, and, for a given $P\in \mathcal{P}$, let ${\pi }_P$ denote the domain of $P$. If ${\pi }^{\prime }⊑\pi ,P\in {\mathcal{P}}_{\pi }$, and $Q\in {\mathcal{P}}_{{\pi }^{\prime }}$, I will say that $Q$ is an extension of $P$ to ${\pi }^{\prime }$ (and $P$ is a restriction of $Q$ to $\pi$) iff for each $s\in \pi$,

where $s^{\prime }$ ranges over elements of ${\pi }^{\prime }$. I will say that $P$ is a restriction of $Q$ (and $Q$ an extension of $P$) iff ${\pi }_P⊒{\pi }_Q$ and $P$ is a restriction of $Q$ to ${\pi }_P$.

A generalized epistemic utility function is a real-valued function $\mathfrak{u}$ defined over $\mathcal{P}\times W$ such that for each $\pi \in \Pi$, the restriction¹¹ of $\mathfrak{u}$ to ${\mathcal{P}}_{\pi }\times W$ is a truth-directed, epistemic utility function for $\pi$. I will say that a generalized epistemic utility function $\mathfrak{u}$ is partition-wise proper iff for each $\pi \in \Pi$, the restriction ${\mathfrak{u}}_{\pi }$ of $\mathfrak{u}$ to ${\mathcal{P}}_{\pi }\times W$ is proper. I will say that $\mathfrak{u}$ is (partition-wise) strictly proper iff ${\mathfrak{u}}_{\pi }$ is strictly proper for all $\pi \in \Pi$.

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

where $𝟙\{w\in s\}$ equals $1$ if $w\in s$ and $0$ otherwise. It is easy to check that $\mathfrak{b}$ is a generalized epistemic utility function that is partition-wise strictly proper. Indeed, for any family $\{{\mathfrak{u}}_{\pi }:\pi \in \Pi \}$ of functions such that ${\mathfrak{u}}_{\pi }$ is a partition-wise strictly proper utility function for each $\pi$, the function $\mathfrak{u}(P,w)={\mathfrak{u}}_{{\pi }_P}(P,w)$ is a generalized epistemic utility function that is partition-wise strictly proper.

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 (Lewis 1971). And in the context of elicitation, strictly proper epistemic utility functions for a given partition can be used to devise systems of penalties and rewards that ensure the kind of honest reporting of forecasts over that partition that made epistemic utility functions, or scoring rules, play the starring role in a wide body of literature.¹²

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 $Q$, our assumptions so far allow us to define the expectation of $\mathfrak{u}(Q)$ relative to any $P$ defined over a refinement of ${\pi }_Q$,¹³ and in fact, where $P_Q$ is the restriction of $P$ to the domain of $Q$, we have:

We can now see, using the Brier score as our epistemic utility function, that any probability function that is not maximally opinionated—any probability function that assigns values other than 0 or 1 to some propositions—will assign a greater expected epistemic utility to a probability function other than itself.¹⁴ (Consequently, if we do not fix a partition but allow a forecaster to choose which partition to report her forecasts on, she will expect to do better by reporting a strict restriction of her credence function as long as her credence function is not maximally opinionated.¹⁵)

Example 1. Suppose $P$ is not maximally opinionated. Let ${\pi }^{*}$ be a coarsening of ${\pi }_P$ such that the restriction $P^{*}$ of $P$ is maximally opinionated. Note now that for $s\in {\pi }^{*}$ with $P^{*}(s)\ne 0$, $\mathfrak{b}(P^{*},s)=0$, and hence that

Now let $s_0\in {\pi }_P$ be such that $P(s_0)\in (0,1)$, and let ${\pi }_P^{-}={\pi }_P\setminus \{s_0\}$. By definition,

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 $P$, $P$ ‘takes itself’ to be doing better than any other $Q\ne P$ in terms of epistemic utility. But in order to answer this question, of course, we need to make clear what it is for some probability function to ‘take itself’ to be doing better than another in terms of epistemic utility. After all, we cannot just use the familiar notion of expectation here since, in general, for given $P,Q\in \mathcal{P}$, the expectation of $\mathfrak{u}\text{(}Q)$ relative to $P$ is not well-defined.

Before turning to this question, let me introduce a few more pieces of terminology. Fix $P$ and let $\pi$ be some partition of $W$. I will denote by ${[P]}_{\pi }$ the collection of all extensions of $P$ to the coarsest common refinement of ${\pi }_P$ and $\pi$—thus, each $P^{+}$ in ${[P]}_{\pi }$ will be an extension of $P$ whose domain refines both ${\pi }_P$ and $\pi$.¹⁶ Slightly abusing notation, for a given $Q$ and $P$, I will use ${[P]}_Q$ as shorthand for ${[P]}_{{\pi }_Q}$. (Note that if $\pi$ is a refinement of ${\pi }_P$, ${[P]}_{\pi }$ is just the set of extensions of $P$ to $\pi$, and that if $\pi$ is a coarsening of ${\pi }_P$, ${[P]}_{\pi }$ is just the singleton set of the restriction of $P$ to $\pi$.)

It will be convenient to also have at our disposal three different quantities which (albeit imperfectly) summarize some of the information about how $\mathfrak{u}(P)$ and $\mathfrak{u}\text{(}Q)$ compare relative to members of ${[P]}_Q$. First, define the lower expectation¹⁷ of $\mathfrak{u}\text{(}Q)$ relative to $P$, which I denote by ${\underset{¯}{𝔼}}_P[\mathfrak{u}\text{(}Q)]$, by

Similarly, define the upper expectation of $\mathfrak{u}\text{(}Q)$ relative to $P$, which I denote by ${\overline{𝔼}}_P[\text{u(}Q)]$, by

Finally, for $\alpha \in [0,1]$, we can define the $\alpha$-expectation of $\mathfrak{u}\text{(}Q)$ relative to $P$, which I denote by ${𝔼}_P^{\alpha }[\mathfrak{u}\text{(}Q)]$, by

Intuitively, the lower expectation of $\mathfrak{u}\text{(}Q)$ relative to $P$ can be thought of as $P$’s worst-case estimate for the value of $\mathfrak{u}\text{(}Q)$; similarly, the upper expectation of $\mathfrak{u}\text{(}Q)$ relative to $P$ can be thought of as $P$’s best-case estimate for the value of $\mathfrak{u}\text{(}Q)$. (For a given $\alpha$, the $\alpha$-expectation of $\mathfrak{u}\text{(}Q)$ relative to $P$ is a weighted average of the two estimates.)

Clearly,

with equality if ${\pi }_P⊑{\pi }_Q$, in which case

Also note that for any $\alpha \in [0,1]$,

so that for any $\alpha \in [0,1]$, we have:

Given all of these resources, we have two ways of formulating a generalized immodesty principle.¹⁸ Say that an epistemic utility function $\mathfrak{u}$ is universally $u$-proper iff for each $P\ne Q$,

and strictly universally $u$-proper iff the above inequality is always strict. Say that it is universally $l$-proper iff for each $P\ne Q$,

and strictly universally $l$-proper iff the above inequality is always strict. The two generalized immodesty principles I will consider are (strict) universal $u$-propriety—the claim that all epistemic utility functions must be (strictly) universally $u$-proper—and (strict) universal $l$-propriety—the claim that all epistemic utility functions must be (strictly) universally $l$-proper. My question will be whether there are any epistemic utility functions that satisfy any of these principles.

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 $u$-propriety and $l$-propriety independently interesting are welcome to skip to the next section.)

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 choice-worthy—and perhaps uniquely so—among alternative credence functions she could have and relative to that epistemic utility function. When the alternatives all have a well-defined expectation, and on the assumption that an option is choice-worthy if it maximizes expected utility, immodesty thus understood amounts to the claim that any epistemic utility function should be proper or strictly proper. So in order to formulate generalizations of immodesty to the case where alternative credence functions lack a well-defined expectation, we need to consider alternative ways of identifying when a credence function is choice-worthy among a given set of alternatives.

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.¹⁹ Each of them can be used to formulate a way to understand what it is for a credence function to take itself to be choice-worthy when the alternatives include all credence functions regardless of their domain, and accordingly to formulate a generalized immodesty principle.²⁰

First, we could say that $P$ takes itself to be choice-worthy iff it has greater expectation relative to all members of ${[P]}_Q$:

Alternatively, we could say that $P$ takes itself to be choice-worthy iff there is no other option that gets greater expectation relative to all members of ${[P]}_Q$:

We could instead say that $P$ takes itself to be choice-worthy iff

Or that $P$ takes itself to be choice-worthy iff

We could also say that $P$ takes itself to be choice-worthy iff

Finally, we could say that $P$ takes itself to be choice-worthy iff for a given $\alpha \in (0,1)$,

For each of these ways of understanding what it is for $P$ to take itself to be choice-worthy, we could have a generalized version of weak propriety. Now, any objection to using one of the above principles—the detailed formulation of the principles to the more general decision-theoretic setting need not concern us here—can arguably be used to object to a particular way of making precise the fully general version of immodesty.²¹ But since it remains largely an open question whether any of the objections to the above principles are decisive, I want to remain neutral as to which is the best way of characterizing a fully general immodesty principle.

Fortunately, these generalizations are not logically independent of one another. To see that, start by fixing $\mathfrak{u}$ and noting that the supremum and infimum in the definitions of upper and lower expectations can be replaced with a maximum and a minimum. (This follows from the fact that $\{{𝔼}_{P^{+}}[\mathfrak{u}\text{(}Q)]:P^{+}\in {[P]}_Q\}$ is compact in $ℝ$.²²) It follows from this and the observation in (3) that (5) and (7) are equivalent to each other; that (4), (6), and (8) are equivalent to each other; and that (6) entails (7). As a result, (7) is weaker than all of (4), (5), (6), and (8). Further, since for a fixed $P$, any counterexample to (7) is itself a counterexample to (6), we have that the weakest form of immodesty we could hope for is given by (7): if $\mathfrak{u}$ does not satisfy (7) for all $P$, it cannot satisfy any of the other generalizations.

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 $u$-propriety; but at the very least, we want a generalized immodesty principle equivalent to the claim that all epistemic utility functions satisfy (7)—that is, universal $l$-propriety. The question now is whether there are epistemic utility functions satisfying either of these generalizations.

3. On Some Generalized Immodesty Principles

I begin by asking whether there are any universally $u$-proper epistemic utility functions. The answer, perhaps unsurprisingly, is no, at least if we restrict our attention to strictly partition-wise proper epistemic utility functions.