Normative uncertainty meets voting theory

Abstract

Recent attempts to provide a viable account of decision making under normative uncertainty through the use of voting procedures seem promising, but incomparable options pose a challenge. This paper focuses on the so-called Borda approach to decision making under normative uncertainty (MacAskill, Mind 125(500):967–1004, 2016; MacAskill et al., Moral uncertainty, Oxford University Press, 2020) and illustrates how an extended version of it aimed at accommodating incomplete preference orderings associated with normative theories fails. I propose a different account for proponents fond of the voter-theoretic approach that is grounded by approval voting, which sufficiently addresses the problem of option incomparability and is pragmatically compelling due to its simpler structure.

1 Introduction

Some decision problems concern uncertain empirical states of the world, like deciding whether to carry an umbrella in case of rain. Others are a bit more complicated, resting on uncertain normative contents, like deciding whether you should skip the beef sandwich for lunch today in case a moral theory that recommends against eating meat turns out to be correct.¹ In the latter instance, you might have considered a variety of normative theories, at least some of which yield opposing verdicts on what is morally permissible. As the decision maker, you find yourself uncertain which among the normative theories is correct. Faced with normative uncertainty, how should you choose?

It has come to the attention of many that classical decision theory cannot deliver on what a decision maker should do in the face of normative uncertainty. But in recent decades, there have been major efforts to fill the gap.² Accommodating normative uncertainty in decision theory has been met by some challenges, though. If the goal is to develop a quantitative expectationalist account, any effort is headed toward failure if the considered normative theories only evaluate options ordinally, i.e., options are merely regarded as ‘better than’ or ‘equal to’ other options. This challenge is sometimes called the problem of merely ordinal theories. Supposing, however, that cardinal representations of the ordinal evaluations are obtainable, it does not follow that inter-theoretic comparison is possible. This challenge is sometimes called the problem of inter-theoretic incomparability (see MacAskill 2016; Tarsney 2019).

Among those set on resolving these challenges, and providing a feasible decision model for normative uncertainty in general, some have reimagined the philosophical problem as a voting problem for the theories considered.³ The thought is that normative theories are like voters with preference orderings over the set of options available to the decision maker. MacAskill (2016) and MacAskill et al. (2020), who adopt the voter view, cleverly impose the Borda count on the ordinal rankings associated with normative theories. Their strategy provides a cardinal representation of value, thus alleviating the problem of merely ordinal theories. In addition, the Borda count fixes a common range, thus alleviating the problem of inter-theoretic incomparability. Although the solution is not perfect, it puts the decision maker in a position to evaluate the relative goodness of options through weighted votes on the options, where the weight given to each theory t is the subjective probability or credence the decision maker assigns to t being the correct theory, thereby encoding their normative uncertainty.

Despite the Borda account appearing to offer much promise in addressing the problem of normative uncertainty, this paper focuses on a limitation of the evaluative criteria, namely, option incomparability. That is, for options a and b among a set of options A, neither a is judged to be better than or equal to b, nor is b judged to be better than or equal to a according to some normative theory t considered by the decision maker. Aware of the limitation, MacAskill (2016) proposed a fix, where the decision maker considers in their evaluation a set of completions for every normative theory t. The set of completions for each normative theory t is the set of all permutations of all options preserving the order of comparable options in t’s true preference ordering. This extension makes an option’s appropriateness depend on weighted votes for all completions and normative theories t, resulting in a robust version of the original Borda account.

In this paper, I argue against the completions approach for handling option incomparability, as it yields misleading preference orderings on behalf of theories with incomplete preferences in the decision maker’s evaluation. Furthermore, extending the Borda account by completions makes the evaluative rule epistemically misleading by holding the decision maker’s credence fixed in weighting the Borda scores derived from completions. The extended Borda account also suffers a practical blow by passing the problem of option incomparability from theories to the decision maker, leaving the decision maker further unsure about what they should do. After illustrating the defects of the extended Borda account by completions, I propose a different account for proponents fond of the voter-theoretic approach that is grounded by approval voting. The subsequent Approval Rule sufficiently addresses the problem of option incomparability and is further appealing from a pragmatic perspective given its simpler structure.

2 The Borda rule

Let a normative decision situation involve a (finite) set of mutually incompatible normative theories, $T = \{1,\ldots ,n\}$, and a (finite) set of options, $A = \{a_1,\ldots ,a_m\}$. I follow MacAskill (2016) and treat the options in A as mutually incompatible and jointly exhaustive propositions, taken as sets of centered possible worlds a decision maker can make true at a particular time. A centered possible world “is a triple of a world, an agent in that world, and a time in the history of that world (2016, pp. 969–970).”⁴

On a voter-theoretic approach to accommodating normative uncertainty in decision making, we first associate each theory $i \in T$ with a (weak) preference ordering, $\succeq _i$, defined as a reflexive and transitive relation over the decision maker’s set of options, A.⁵ Each theory’s preference ordering over A indicates the comparative value of options, where ‘$a \succeq _i b$’ means that a is at least as valuable as b and that a is (weakly) preferred to b by theory $i \in T$. I write ‘$a \succ _i b$’ if $(a \succeq _i b)$ and $\lnot (b \succeq _i a)$, and ‘$a \sim _i b$’ if $(a \succeq _i b)$ and $(b \succeq _i a)$ as usual to denote the strict and indifference preference relations.

At this point, the problem of merely ordinal theories mentioned in the introduction surfaces. MacAskill (2016) and MacAskill et al. (2020) circumvent the problem by employing the Borda count, proposed in the 18th century by and named after Jean-Charles de Borda.⁶ Despite the positional scoring rule having various interpretations, the most common version takes an ordering and assigns a score to each option equal to the number of options ranked below it. That is, the most preferred option gets a score of $m - 1$ and so forth, while the least preferred gets 0, for all preference orderings.⁷ Borda voting proceeds by summing the scores for each option awarded by all voters. The Borda winner is the option with the highest total Borda score.

To see how the Borda count might bypass the problem of merely ordinal theories, consider the following decision problem.

Trolley Problem Sophie is watching as a train which is out of control hurtles toward five people working on the train track. If she flicks a switch, she will redirect the train, killing one person working on a different track. Alternatively, she could push a man onto the track, killing him but stopping the train. Or she could do nothing. She has three options available to her.

a: do nothing.

b: flick the switch.

c: push the man.⁸

Suppose that Sophie considers moral theory i, which orders the options as follows: ($b \succ _{i} a$), ($b \succ _{i} c$), and ($c \succ _{i} a$). The options, as listed above, receive scores 0, 2, and 1 under the Borda count, which preserve i’s preference ordering. Notice that the Borda count dissolves the worry of merely ordinal theories by assigning cardinal values to options in accordance with the preference orderings of normative theories. Observe further that since the range of values is dependent on |A|, and the preference orderings, for all theories $i \in T$, are defined over A, the value range is fixed for all normative theories, thereby avoiding the problem of inter-theoretic incomparability that has likewise been a serious challenge in addressing the problem of normative uncertainty.⁹

The Borda count is the first key component. The second is a representation of normative uncertainty with respect to the normative theories in T. This bit is the least novel, as many assume that the decision maker’s uncertainty is encoded by the subjective probabilities or credences they assign to theories.¹⁰ For the purposes of this paper, I assume that credence is distributed over T. Credence, in turn, is representable by a function $C : T \rightarrow [0, 1]$ satisfying: $C(i) \ge 0$ for all $i \in T$ and $\sum _{i \in T} C(i) = 1$. The credal representation of the decision maker’s belief state conveys a degree of uncertainty when the decision maker has credence less than 1 in at least one theory, implying that they are uncertain which theory is correct. Considering the trolley problem from above, for instance, it is conceivable that Sophie is less than certain that i is correct, which (strictly) prefers b to c to a, and similarly, less than certain that an alternative moral theory j is correct, which (strictly) prefers a to b to c, e.g., $C(i) = \nicefrac {2}{5}$ and $C(j) = \nicefrac {3}{5}$.

Now that we have a mechanism for cardinalizing value and a representation of the decision maker’s normative uncertainty, we can put the pieces together. The expected Borda score (or weighted vote) for each option $a \in A$ given the preference orderings of all theories $i \in T$ is defined as the sum of credence-weighted Borda scores. That is, ${{\mathbb {E}}}{{\mathbb {B}}}(a) = \sum ^n_{i = 1} C(i) Bs_i(a)$ for all $a \in A$, where $Bs_i$ is a (classical) Borda scoring function associated with theory i. Let the expected Borda scores determine the comparative appropriateness of options under the following evaluative rule.

Borda Rule. An option a is more appropriate than an option b if and only if a is (strictly) greater than b in expected Borda score; a is as appropriate as b if and only if a and b are equal in expected Borda score. (see MacAskill 2016, p. 989)

Applied to the Trolley Problem, the Borda score for b is 2, 1 for c, and 0 for a with respect to the preference ordering of moral theory i, whereas the preference ordering of moral theory j yields scores 2 for a, 1 for b, and 0 for c. The expected Borda scores given the credences above are: $C(i) \cdot 2 + C(j) \cdot 1 = 1.4$ for b, $C(i) \cdot 1 + C(j) \cdot 0 = \nicefrac {2}{5}$ for c, and $C(i) \cdot 0 + C(j) \cdot 2 = 1.2$ for a. What we find from Sophie’s perspective, as the decision maker, is that ‘flick the switch’ is more appropriate than ‘do nothing’, which is more appropriate than ‘push the man’, and in this case, it is the decisive choice, as it maximizes appropriateness. We can see from this illustration that the Borda Rule offers a feasible approach to choosing when uncertain about underlying normative theories.

That said, the Borda Rule faces some difficulty when a theory fails to discern between some options in terms of the standard ‘better than’ or ‘equal to’ relations. That is, there may be a theory $i \in T$ and option pair $(a,b) \in A \times A$ for which i has no well-defined preference. Call this the problem of option incomparability.

Consider the following. Simultaneous natural disasters are about to occur in your area, a flood in the valley and an avalanche from a nearby mountain. You can either save your mother from the imminent flood or your father from the impending avalanche, but you cannot save both. You are very close to both parents, neither of which is of bad character nor in bad health. Assuming that you consider both options to be equally valuable, a small improvement to one option will break the tie. Let the small improvement be that if you save your mother, then you will receive free grief counseling for the death of your father. The small improvement to the option of saving your mother should compel you to prefer saving her over saving your father given the initial assumption that the two options without the added improvement are considered equal.¹¹

However, one might reasonably reject the conclusion of the parental death case either on the grounds that the slight improvement to the option of saving your mother is not a tie-breaker or that the premise of the two initial options being equal is false. In the latter instance, the options may be incomparable. Suppose that among some normative theories you consider in determining what to do, there is at least one theory that finds the options to be incomparable. On such an occasion, it is ambiguous how the Borda Rule can handle theories with such preference orderings, as it is presumed that preference orderings are complete—that is, $(a \succeq _i b)$ or $(b \succeq _i a)$ for all $a, b \in A$ and $i \in T$. If incomplete, the problem of inter-theoretic incomparability could resurface since the Borda rankings, for all theories $i \in T$, may not be of the same length.

Anticipating this limitation, MacAskill (2016) proposed a possible solution through an extension of the Borda Rule that takes into account completions of incomplete preference orderings. For instance, suppose that for some $a, b, c \in A$ and theory $i \in T$, a and b are comparable given i’s preference ordering, $\succeq _i$, but c is incomparable with both. A ‘completion’ of the preference ordering (hypothetically) imposes comparisons of incomparable option pairs on the ordering, thus making c comparable with both a and b for theory i. Formally, let $Comp = \{\succeq | \succeq \subseteq A \times A; (x \succeq y) \vee (y \succeq x) \; \forall x, y \in A\}$ be a completion set that is the set of all complete preference orderings on $A = \{a_1,\ldots , a_m\}$ and let ${\mathcal {R}}_i \subseteq Comp$ be the set of coherent completions, consisting of all orderings that are consistent with the actual ordering of comparable options, for all preference orderings, $\succeq _i$, and theories $i \in T$ (see MacAskill et al. (2020) who employ the ‘coherent completion’ terminology).¹² Putting coherent completions to work, MacAskill suggests the following extension I will refer to as the Robust Borda Rule.

Robust Borda Rule. An option a is more appropriate than an option b if and only if a is at least as great as b in expected Borda score, for all completions in ${\mathcal {R}}_i$ and theories $i \in T$, and a is (strictly) greater than b on some completions; a is neither more appropriate, nor less appropriate, nor equal to b in case a is (strictly) greater than b in expected Borda score on some completions but is (strictly) less than b on others. (see MacAskill 2016, p. 992)

The extended Borda approach addresses the problem of option incomparability by making all $a \in A$ comparable in all possible ways that are consistent with the actual preference orderings, $\succeq _i$, for all theories $i \in T$. Conceptually, coherent completions might be regarded as a set of possible (but counterfactual) ways a theory’s preferences could have been had they been complete, and the evaluative criteria offering something like a supervaluationist account. That is, option a is (strongly) ‘superappropriate’ compared to option b if and only if for all completions in ${\mathcal {R}}_i$, a is at least as great as b in expected Borda score, and strictly greater on some completions, for all theories $i \in T$.¹³

The proposed fix has some appeal, but it neglects the fact that incomparable options are just that, incomparable. This could be due to value incommensurability or incomparability with respect to the bearers of value.¹⁴ The parental death case is one that fits under the latter and conforms to the ‘small-improvement argument’ (see Broome, 1997; Chang, 2002). If the conclusion of the case is indeed false, that would seem to suggest that the standard trichotomy view—that is, options are comparatively either ‘better than’, ‘worse than’, or ‘equal to’ one another—is false (Chang, 2002).¹⁵

But Chang and others claim that there are ways to make options comparable by adding additional relations. If feasible, then the Borda account might not need completions. Among the proposals, Parfit (1987) claimed that some options might be “roughly equal”, but the comparison is, as the term indicates, too rough to conclude that they are equal. Chang (2002) alternatively introduced ‘parity’. She illuminates the plausibility of parity through comparisons across different domains. For example, the famous composer Mozart and painter Michelangelo achieved artistic creativity at the highest levels and “in the same neighborhood”, but their artistic works belong to different domains. Nevertheless, their skill levels are ‘on a par’ (Chang 2016, p. 193).

While adding notions like ‘roughly equal’ or ‘parity’ to our technical vocabulary might be tempting, both suffer from being intransitive (Hsieh, 2007). In the voter-theoretic setup, imposing intransitive relations on theories with incomplete preferences violates the initial transitivity assumption.¹⁶ So, the robustified extension by coherent completions appears to be the best option for handling incomplete preference orderings thus far. However, proceeding with this measure comes with significant costs. Consider again the parental death case such that $A = \{a, b\}$, where $a =$ ‘save your mother’ and $b =$ ‘save your father’. Suppose that a and b are incomparable from the perspective of some theory $i \in T$, i.e., $\lnot (a \succeq _i b)$ and $\lnot (b \succeq _i a)$. Given such incomparability, the decision maker weighs in their evaluation ${\mathcal {R}}_i = \{(a \succ _i b), (b \succ _i a), (a \sim _i b)\}$.

But on reflection, it is difficult to see the relevance of such completions, especially from the decision maker’s perspective, considering how theory i actually views the options. Indeed, the completions are not only irrelevant to the evaluation but they are misleading. The completions are misleading precisely because they are untrue of i.¹⁷ In the given case, how does theory i actually view the options? Theory i says a and b are incomparable. Completions say otherwise and consequently misrepresent i.

At first glance, one might think that completions have little or no import in the decision maker’s evaluation of options. However, untrue preference orderings are not benign. Consider normative theories 1, 2, 3 such that theory 1’s preference ordering is $(a \bowtie _1 b)$,¹⁸ theory 2’s is $(a \succ _2 b)$, and theory 3’s is $(a \succ _3 b)$ for $A = \{a, b\}$. Suppose that the credences the decision maker assigns to the theories are $C(1) = 0.501$, $C(2) = 0.25$, and $C(3) = 0.249$. Since the preference orderings of theories 2 and 3 are complete, the (coherent) completion set for each of the theories consists of their actual or true preference ordering. The preferences of theory 1, however, are incomplete. By completing 1’s preference ordering, we have ${\mathcal {R}}_1 = \{ (a\succ _1 b), (b \succ _1 a), (a\sim _1 b)\}$.¹⁹ The expected Borda scores for the options under these assumptions are given in Table 1.

Table 1 Expected Borda scores of options a and b under all coherent completions

Notice that neither a is more appropriate than b, nor is b more appropriate than a, nor are they equal given the evaluative criteria set out by the Robust Borda Rule. The result should not come as a surprise since theory 1 is afforded the most credence by the decision maker and consequently the most voting power in the evaluation.

However, that is where the problem lies. Given theory 1’s true preferences (or lack thereof), i.e., ($a \bowtie _1 b$), 1 essentially abstains and does not realize its voting power in the evaluation. With respect to Borda voting, 1 is a “no-show.” Provided that 1 is a no-show, then by a Borda vote among those who turn up (theories 2 and 3), a is the clear Borda winner (credal weightings included).²⁰ But that is not the outcome we arrive at under the Robust Borda Rule. Instead, the rule acts as a proxy and is able to materialize $|{\mathcal {R}}_1|$ misleading votes on 1’s behalf in response to 1 lacking a preference between a and b, thus allowing theory 1 to realize its voting power via its weight (credence).

This sort of delegation, though, demonstrates how completions of incomplete preference orderings mislead given that theory 1 truly signaled abstention all along.²¹ Furthermore, the Robust Borda Rule itself is epistemically misleading by holding the decision maker’s credences fixed for theories with incomplete preferences and weights the scores derived from untrue preference orderings by the credences in the evaluation. In the above case, what the decision maker actually assigned credence to is theory 1 with the view that $(a \bowtie _1 b)$, independent of the evaluation, not $(a \succ _1 b)$ or $(b \succ _1 a)$ or $(a \sim _1 b)$.²²

The defects of the Robust Borda Rule are certainly a cause for alarm. While proponents might have overlooked the details, they surely cannot look past the consequences that untrue preference orderings engender for the account in general. The above example also delivers a practical blow to the rule by illustrating that it can leave the decision maker confused and further unsure what they should do provided that a and b are made incomparable under it. Recall that the Borda approach is supposed to fit within the expectationlist tradition, where expected value is maximized. But in the given example, there is no option that is maximally appropriate under the Robust Borda Rule.

In an attempt to resolve the problem of option incomparability, the Robust Borda Rule appears to have side-stepped the concern by passing it from the advisers (normative theories) to the consumer (the decision maker). But that is not much of a solution when considering the main objective. The defects of the proposed extension by coherent completions illustrated pose serious problems for the Borda plan.

3 The approval rule

In this section, I propose a different account for proponents fond of the voter-theoretic approach to handling normative uncertainty that appeals to approval voting (Brams & Fishburn, 1978). Approval voting is an intuitive voting rule, where voters select a subset of the candidates they approve. The candidate receiving the most approvals from voters is the winner, or if two or more candidates tie, one can be selected by random lottery. A practical advantage of approval voting is its simplicity resulting from a dichotomous preference structure—options are either approved or not.²³

In representing the approval account of normative uncertainty, let ${\mathcal {B}}_i \subseteq A$ be an approval ballot for all $i \in T$, consisting of all the options i approves. If ${\mathcal {B}}_i = \emptyset$, then i abstains from voting.²⁴ Similarly, approving all candidates, i.e., ${\mathcal {B}}_i = A$, is tantamount to abstaining as it concerns the outcome of a vote (Brams & Fishburn, 1978, p. 835).²⁵

Next, associate with each ballot ${\mathcal {B}}_i$, for all $i \in T$, a function $f_i$ on A such that

$$\begin{aligned} f_i(a) = {\left\{ \begin{array}{ll} 1 \quad \text {if} \; \; a \in {\mathcal {B}}_i, \\ 0 \quad \text {otherwise}. \end{array}\right. } \end{aligned}$$

(1)

With the approval scores given by $f_i$ for all approval ballots ${\mathcal {B}}_i$ and theories $i \in T$, we can now define the evaluative criteria:

Approval Rule. For all options $a, b \in A$, option a is more appropriate than option b if and only if a’s expected approval score, i.e., ${{\mathbb {E}}}{{\mathbb {A}}}(a) = \sum _{i=1}^n C(i)f_i(a)$, is (strictly) greater than that of b; a is as appropriate as b if and only if a’s expected approval score is equal to that of b.

One should then maximize appropriateness, just the same as on the Borda account.

Notice the similarity to the Borda Rule. We can make the connection more precise as follows. Setting aside abstention, approval voting yields two indifference classes: ${\mathcal {B}}_i$ and $(A\backslash {\mathcal {B}}_i)$ for all $i \in T$ (see Maniquet & Mongin, 2015). The two indifference classes induce a complete ordering $\succeq _i$ on ${\mathcal {P}}_i = {\mathcal {B}}_i \cup (A\backslash {\mathcal {B}}_i)$ for all ${\mathcal {B}}_i$ and $i \in T$ such that

• $a \succ _i b$   $\Leftrightarrow$   $a \in {\mathcal {B}}_i$ and $b \in (A\backslash {\mathcal {B}}_i)$,

• $a \sim _i b$   $\Leftrightarrow$   $a, b \in {\mathcal {B}}_i$ or $a, b \in (A \backslash {\mathcal {B}}_i)$.

On the Borda account, the scores given to $m \ge 2$ options are restricted to the set, ${\mathfrak {B}} = \{0,\ldots ,m - 1\}$, and the expected Borda score defined as ${{\mathbb {E}}}{{\mathbb {B}}}(a) = \sum ^n_{i=1} C(i)Bs_i(a)$ for all $a \in A$ and theories $i \in T$, where $Bs_i : A \rightarrow {\mathfrak {B}}$. The approval approach, on the other hand, scores the $m \ge 2$ options in A via the approval scoring function whose range, $\{0, 1\}$, is a subset of ${\mathfrak {B}}$, for all approval ballots ${\mathcal {B}}_i$, complete orderings $(\succeq _i,{\mathcal {P}}_i)$, and theories $i \in T$. Under these assumptions, the Approval Rule is a special case of the Borda Rule. Put succinctly, $(\succeq _i, {\mathcal {P}}_i)$ is a complete ordering for all ballots ${\mathcal {B}}_i$ and $i \in T$; scores are restricted to a subset of the Borda scoring range and in line with their ordinal ranks for all $(\succeq _i, {\mathcal {P}}_i)$, ${\mathcal {B}}_i$, and $i \in T$; the approval scores of options, for all $(\succeq _i, {\mathcal {P}}_i)$, ${\mathcal {B}}_i$, and $i \in T$, are weighted by the decision maker’s credence in i; the appropriateness of options is determined by the sum of credence-weighted approval scores.²⁶

An advantage of the Approval Rule, however, is that it is able to sufficiently overcome the problem of option incomparability. Consider again the parental death case from earlier. Suppose that a and b are incomparable according to some theory $i \in T$. While i lacks a preference between the options in a pairwise comparison, it could be that the options individually meet the standard of approval by the lights of i. Thus, ${\mathcal {B}}_i = \{a, b\}$ and $(A \backslash {\mathcal {B}}_i) = \{ \}$. If, however, neither option is approved, then both belong to $(A \backslash {\mathcal {B}}_i)$, where neither a nor b is considered acceptable or i suspends judgment on the matter and abstains. So, we have the opposite of the first case. As it should be obvious, that is not a problem for the rule. And if a meets the standard of approval but b does not (or vice versa), an approval-based preference is induced, where a is strictly preferred to b provided that $a \in {\mathcal {B}}_i$ and $b \in (A \backslash {\mathcal {B}}_i)$, implying that $f(a) > f(b)$. Option incomparability, therefore, does not pose a challenge for the Approval Rule. This is due to approval preferences being based on a threshold of acceptance (Pacuit, 2019), not pairwise comparisons, where each option is categorically judged on its own merit by each theory. In addition, the rule does not mislead since it only takes actual approval preferences as inputs and does not conjure untrue variations.

Although the approval account gets around the problem of option incomparability, an important question remains: what is the standard of approval for all normative theories considered? A natural answer is permissibility. That is, an option is approved by a theory i iff the option is permissible by the lights of i. Consequently, all options that are judged by i as impermissible are not approved. Fixing the standard as permissibility also yields a practical benefit: the decision maker is only required to identify the option(s) each theory considers (im)permissible, nothing more. This latter point brings to light a concern that tends to be overlooked in the literature, namely, the decision maker might not only be uncertain of which normative theory is correct, but they might also be uncertain of how each theory orders the available options. The approval standard of permissibility eliminates the need for the decision maker to identify the correct and complete preference orderings in their evaluation (if such orderings exist).

Despite the advantages of the approval account, one might object that the dichotomous preference structure can result in a loss of information. Indeed, the preference orderings admitted by the Borda Rule might yield pairwise differences between all options, whereas the approval approach regards all (non)approved options as equal, which may not be true for all theories. For instance, suppose ${\mathcal {B}}_i =\{a, b\}$ for some $a, b \in A$ and $i \in T$. The approval account holds that i is indifferent toward a and b, but under the Borda account, i might strictly prefer a to b or vice versa. In the parental death case, for example, suppose that saving your mother and saving your father are both approved options, for all theories considered, but in a pairwise comparison, all theories slightly prefer saving your mother to saving your father. Under the Approval Rule, however, the options might be equally appropriate. You may then end up saving your father, which is considered to be worse than saving your mother by all the theories.²⁷

But if all theories in the latter instance permit saving your father, then it must be that all theories admit that saving your mother is permissible but not normatively required.²⁸ Otherwise, saving your father would not be permitted by the theories. Consider a standard consequentialist theory, for example. It would not permit saving your father if saving your mother is strictly better in expectation since it only permits options that are maximal in expected value. Consider also a naïve Kantian theory. If saving your mother is normatively required, while saving your father is not, and the two options cannot both be taken, then saving your father would not be permissible to begin with. If, on the other hand, both are permissible, neither is better than the other.

Any normative theory that permits and thus approves both options, while ranking saving your mother above saving your father, must admit then that saving your mother is permissible, though not normatively required. A difference in value from, say, the Borda perspective could be due to all the considered theories viewing the option of saving your mother as supererogatory. If not, it is difficult to see what kind of normative theory would say that given some options a and b, both options are permissible, yet a is normatively required. That would be incoherent on standard consequentialist views, the naïve Kantian view, and views admitting to the supererogatory. In fact, it appears incoherent for any feasible normative theory to say that an option is permissible, yet one is normatively required to choose an alternative.

So, while the approval structure does fail to recognize the betterness of supererogatory acts by regarding all approved options as equal, the structure has no effect on the permissibility of options by the lights of any theory. And I take it that what the decision maker primarily is after anyway is determining what they are at least permitted to do, given their uncertainty of which theory is correct and consequently, which (unique) option is normatively required (if such an option exists). If the decision maker were certain of the correct theory and the normatively required option, then it would be obvious that they should choose that option. Their uncertainty, however, stands in the way, making the problem difficult. That is the obstacle in this exercise, after all.

Related to the first objection, one might worry that something similar to the problem of merely ordinal theories might surface. An ordinal evaluation can provide the decision maker with an idea of which options are better than others, but not how much better. Since approval scoring is binary, i.e., 1s and 0s, we similarly lose the realization of differences in value magnitude between options. However, such differences might be important if an option is considered to be morally catastrophic by some theory and the decision maker aims to minimize their moral risk. For instance, suppose that i approves of eating meat, while j does not. But j not approving the option does not express how strongly j disapproves of it in the given framework. That is, zero does not represent the magnitude of badness j attributes to the option. If it is possible to represent how bad the option is to j by a real number, that number will likely be far below zero.²⁹

For a decision maker concerned with minimizing their moral risk of choosing an extremely bad option, knowing j’s real valuation of eating meat would be important. Admittedly, this is a limitation of the approval approach, but it too is a limitation of the Borda approach since the minimum value is also zero on the standard Borda count. However, both approaches fix a common value range to resolve the problem of inter-theoretic incomparability. In the case of i and j above, if represented instead by value functions, inter-theoretic comparison is probably not possible. So, we would end up back where we started. The voter-theoretic approach offers a way forward, but as I mentioned in the introduction, it is not perfect. That said, the decision maker’s risk attitude would be an interesting feature to incorporate into the study of normative uncertainty, but unfortunately, that cannot be explored here. What I might reiterate, possibly from a risk neutral perspective, is that the permissible options according to all normative theories contemplated by the decision maker matter the most. And once the credal weightings are factored in, we get some meaningful differences in value magnitude.

4 Conclusion

Although the Borda approach to decision making under normative uncertainty appears to be promising, I illustrated that option incomparability remains a problem. I brought to light some serious defects of the completions attempt to salvage the Borda plan that should convince us that completions are not the answer. I then offered a different account grounded by approval voting. The Approval Rule sufficiently overcomes the problem of option incomparability and provides a feasible voter-theoretic approach to decision making under normative uncertainty.