Fixed-point solutions to the regress problem in normative uncertainty

2.1 Choiceworthiness

A finite set \(A = \{a_{1}, \ldots , a_{|A|}\}\) of “feasible acts” presents itself. There is a finite set of “possible states” \(S = \{s_{1}, \ldots , s_{|S|}\}\) to which I assign positive probability.² I assign utilities to performing each act in each state, as represented by the utility function u(A, s), where the value assigned to each act is not necessarily independent of the alternatives in A.³ I also find myself in an overall finite epistemic position e, specifying the probabilities I assign to all relevant claims.⁴ Let us call \(\pi = \langle A, u, e \rangle \) my “choice problem”.

We will say that my utility function specifies the “objective choiceworthiness” (or simply “choiceworthiness”) of each act, conditional on each state. That is, given s, the choiceworthiness of \(a_{i}\) is \(u(A, s)_{i}\)—the ith element of the |A|-vector u(A, s). From the probabilities I assign to the states in S, therefore, I also assign probabilities to potential values of the objective choiceworthiness of each act.

Let \({\mathbb {D}}^{n}\) denote the set of all finite probability distributions in \({\mathbb {R}}^{n}\), and let some \(d(\pi ) \in {\mathbb {D}}^{|A|}\) represent the choiceworthiness distribution entailed by \(\pi \).

Many other finite probability distributions over \({\mathbb {R}}^{|A|}\) might do just as well as the chosen d at representing my finite choiceworthiness distribution. Exactly which others depends on how much structure is contained in our understanding of “utility”. If we understand utility to be a merely “ordinal” quantity, for instance, then any transformation of d that is monotonic in choiceworthiness (and constant in probability) represents the same choiceworthiness distribution. We are here assuming nothing about utility except that it at least partially orders act-state pairs from a given {feasible set \(\times \) possible set}, and that \({\mathbb {R}}\) is “rich enough” to capture any potential difference between the choiceworthiness values of particular act-state pairs—that choiceworthiness cannot be lexicographic, for instance. As discussed at the end of Sect. 2.5, these assumptions about choiceworthiness, here made implicitly, will follow from similar assumptions made explicitly about subjective choiceworthiness.

2.2 Subjective choiceworthiness

I am uncertain about acts’ choiceworthinesses. Even so, I may know that one act is the most appropriate for me to choose, given my epistemic position. As I write this, for instance, I assign high probability to the event that, if I go to the doctor, I will swiftly be cured of my back injury (an outcome I would prefer immensely to the status quo), and low probability to the roughly complementary event that, if I go to the doctor, I will waste some time and remain injured (an outcome to which I would slightly prefer the status quo). Despite this uncertainty, and all my other uncertainty, I am in fact certain that going to the doctor is the “better choice” for me right now (by far!). There is thus some scale on which the act of going to the doctor scores higher for me than the act of not going—and would score higher for anyone with the same utility function, in the same overall epistemic position, facing the same set of feasible acts. Let us call this scale “subjective chiceworthiness”.

It is my intuition that subjective choiceworthiness c, when well-defined, is fundamentally a cardinal scale. That is, I would maintain that a representation of acts’ subjective choiceworthinesses (for an agent in a given situation) in \({\mathbb {R}}^{|A|}\) would be unique at least up to affine transformation. If my feasible act set \(A = \{a_{1}, a_{2}, a_{3}\}\) consists of going to the doctor (\(a_{1}\)), going to a very slightly less competent doctor (\(a_{2}\)), or not going at all (\(a_{3}\)), then there is some important and foundational sense in which, given my epistemic position and my preferences, the distance between \(c(\pi )_{1}\) and \(c(\pi )_{2}\) is less than the distance between \(c(\pi )_{2}\) and \(c(\pi )_{3}\). It might be objected that I will always do whatever winds up being most subjectively choiceworthy; that therefore, in the absence of a specified theory of decision-making under uncertainty, no information is conveyed by postulated differences between the acts not chosen; and that c is therefore better understood as merely an ordering, or perhaps even as a choice relation. To this it might be replied that, under certain circumstances, differences in subjective choiceworthiness could bear some relationship to the subjective probability with which a subjectively sub-optimal act would become optimal upon further reflection. Or that cardinal subjective choiceworthiness takes on a clearer meaning in other situations of normative uncertainty (i.e. one might not choose the most subjectively morally choiceworthy act, and might in some sense be more blameworthy the less subjectively morally choiceworthy one’s act was)—and that it would be strange for subjective choiceworthiness to be fundamentally cardinal in one of these situations but not the other. Or that our models of the world are generally simpler when we extend our intuitions regarding quantities’ cardinality beyond the domains in which they happen to be testable—such as our intuition that temperature is generally cardinal, even on some cold, distant star that we will only discover if its temperature rises above some threshold.

Furthermore, cardinal subjective choiceworthiness allows for the convergence results described below, and less structured interpretations of subjective choiceworthiness would not. If we are otherwise persuaded that the regress problem must have some solution or other, it is not circular to allow this observation itself to lend credibility to the concept of cardinal subjective choiceworthiness.

In any event, for the purposes of this analysis, we will understand subjective choiceworthiness (again, when well-defined) to be cardinal. We will represent it by a “subjective choiceworthiness function” \(c(\pi )\), where c assigns a real number to the subjective choiceworthiness of each of the acts in a feasible set A, for an agent with a utility function u, in epistemic position e.

Note that by having c map into the real numbers, we are assuming that all information about differences in subjective choiceworthiness (and therefore utility) can be captured by ratios of differences in real numbers. We are here explicitly assuming for subjective choiceworthiness what we provisionally assumed above for objective choiceworthiness—that, for instance, it cannot be lexicographic. Like probability theories that let us condition on probability 0 events, utility theories that let us distinguish between acts that differ infinitesimally in choiceworthiness may also be interesting to consider in light of the regress problem. However, we will not touch them here.

2.3 Metachoiceworthiness

In general, if I am to translate a choiceworthiness distribution d into a determination of how to act, I must invoke a “decision theory”: a collection of claims concerning how to evaluate acts in light of one’s choiceworthiness distribution. For example, using this terminology, one decision theory is “Expected Choiceworthiness Theory” (EC). EC is characterized by the fact that, if I am certain that it is the correct decision theory, then each act’s subjective choiceworthiness for me is its expected choiceworthiness under d.⁵ Another decision theory would be “minimum choiceworthiness”—a theory characterized by the fact that, if I am certain that it is the correct decision theory, then each act’s subjective choiceworthiness for me is its minimum possible choiceworthiness under d.

Just as I am uncertain about the true state of the world, I may also be uncertain about the correct decision theory. To come to a determination of how to act, therefore, I may have to invoke a sort of “meta decision theory” (or, “2-metatheory”): a collection of claims concerning how to respond to one’s uncertainty over decision theories.

Note that, since this is so, the decision theories (we will awkwardly call these “1-metatheories”, for ease of indexing) cannot themselves be claims about subjective choiceworthiness. This is perhaps a surprising claim, so it bears repeating: expected utility theory (for example) is not, in this language, a theory about what subjective choiceworthiness is, or even about what it ought to be “all things considered”. It is, rather, a theory about what subjective choiceworthiness “1-ought” to be, for someone with a given objective choiceworthiness distribution over his feasible set—or, a theory about what subjective choiceworthiness is for someone with a given objective choiceworthiness distribution over his feasible set, if he knows the true 1-metatheory.

Suppose, for instance, that I am faced with three feasible acts, that I assign probability to each of two 1-metatheories, \(t_{1}\) and \(t_{2}\), and that I am certain of “2-metatheory” m. The theories are such that if I were certain of \(t_{1}\), the subjective choiceworthinesses of the acts would be ordered \(a_{1} \succ a_{2} \succ a_{3}\); if I were certain of \(t_{2}\), the subjective choiceworthinesses of the acts would be ordered \(a_{3} \succ a_{2} \succ a_{1}\); and, given the probabilities I assign to \(t_{1}\) and \(t_{2}\), but my certainty about m, the subjective choiceworthinesses of the acts are in fact ordered \(a_{2} \succ a_{3} \succ a_{1}\). Although I assign probability \(\frac{1}{2}\) to \(t_{1}\), I assign no positive probability to the event that \(a_{1}\) is more subjectively choiceworthy than \(a_{2}\) from my epistemic position. The 1-metatheories’ claims, therefore, are not claims about the acts’ subjective choiceworthinesses given my empirical uncertainty, but about how the acts score on an altogether different scale. Let us call this scale “metachoiceworthiness”, or “1-metachoiceworthiness”. Of course, metachoiceworthiness must be constructed such that, if I know that an act’s 1-metachoiceworthiness is x, then the act’s subjective choiceworthiness for me is also x. We might therefore informally think of 1-metachoiceworthiness as “whatever subjective choiceworthiness is, for someone who knows the correct 1-metatheory”. But since, again, decision theories are not actually claims about subjective choiceworthiness, let us begin by thinking about 1-metachoiceworthiness on its own terms, and only afterward consider its relationship to subjective choiceworthiness.

In any event, the elusiveness of subjective choiceworthiness is not restricted to “order 1”. Just as I may be uncertain as to the correct 1-metatheory, I may be uncertain as to the correct 2-metatheory; I may therefore have to appeal to a “3-metatheory”; and the 2-metatheories are therefore making claims not about acts’ subjective choiceworthiness given beliefs about their 1-metachoiceworthiness, but about acts’, say, “2-metachoiceworthiness” given beliefs about their 1-metachoiceworthiness. So our regress begins.

2.4 k-metachoiceworthiness

Let us call choiceworthiness “0-metachoiceworthiness”, choiceworthiness distributions “0-metachoiceworthiness distributions”, and decision theories “1-metatheories”. The concepts of k-choiceworthiness, k-metachoiceworthiness distributions, and k-metatheories can then together be defined recursively.

Let us denote acts’ relative k-metachoiceworthiness by the two-place relation \(\succ _{\pi ,k}\).

Note that, strictly speaking, if we want our k-metatheories to make k-metachoiceworthiness claims over finite act-sets of arbitrary size, we would have to say that a k-metatheory is a family of functions \(\{t_{k}^{n}\}\) from \({\mathbb {D}}^{n}\) to \({\mathbb {R}}^{n}\), with one for each \(n \in {\mathbb {N}}\). For simplicity, however, we will take \(n = |A|\) as given and interpret our project only as an attempt to find criteria under which the subjective choiceworthinesses of any n acts will be well-defined—with the understanding that identical reasoning would apply to any other n.

We can now define a few aditional terms.

Let \(|d_{t_k}|\) and \(|d_T|\) denote the number of k-metatheories and hierarchies, respectively, to which I assign positive probability.

Let \(\vec {c_k} \in {\mathbb {R}}^{|d_{t_k}||A|}\) represent the claims made by my \(|d_{t_{k}}|\)k-metatheories about the k-metachoiceworthinesses of the |A| acts in A. Let \(\vec {p_{k}} \in \Delta ^{|d_{t_{k}}|-1}\) represent the probabilities I assign to these k-metatheories. We can now represent my k-metachoiceworthiness distribution by \(d_{k} = \langle \vec {c_{k}},\vec {p_{k}}\rangle \).⁶

2.5 The relationship of k-metachoiceworthiness to subjective choiceworthiness

Upon introducing the cardinal subjective choiceworthiness function \(c(\pi )\) above, we placed no restrictions on what it could be. Now that we have documented the emergence of an elaborate web of concepts concerning \(\pi \), however, we can consider how it relates to c.

Recall that k-metachoiceworthiness claims are defined so that, if I know that an act’s k-metachoiceworthiness for me is x, the act’s subjective choiceworthiness for me is x. Let us now introduce a compatible, minimally restrictive principle with which one’s subjective choiceworthiness function might comply in the face of uncertainty about an act’s k-metachoiceworthiness.

Note that if I accept the Dominance Principle, it follows immediately that my subjective choiceworthiness for an act \(a_i\) is well-defined whenever \(|\cap _{k \in {\mathbb {N}}}[\min ([\vec {c_k}]_i), \max ([\vec {c_k}]_i)]|=1\). That is, whenever exactly one number lies in the ranges of “admissible” (not dominated) k-metachoiceworthiness values, across all k, for an act, that number must be the act’s subjective choiceworthiness.

Note also that any claim about subjective choiceworthiness itself, such as the Dominance Principle, in some sense takes on both a positive and a normative interpretation. One could interpret the Principle normatively as asserting that one’s subjective choiceworthiness always ought to obey the above pattern. In this case, if one accepts the Principle, one’s subjective choiceworthiness also does obey it, since to hold that an act should be ranked highly for someone in your epistemic position is simply another way to say that it is highly subjectively choiceworthy. Alternatively, one could interpret the Principle positively as asserting that, as a matter of fact, subjective choiceworthiness always obeys the above pattern. If one accepts this claim (and that “ought implies can”), one must also accept that subjective choiceworthiness always ought to obey the above pattern. Either way, if one accepts the Principle, one cannot assign positive probability to k-metatheories that claim that the k-metachoiceworthiness of an act lies outside the admissible range imposed by one’s \(k^{\prime }\)-metachoiceworthiness distribution for the act for lower orders \(k^{\prime } < k\).

Finally, note that the framework outlined here differs from other approaches to subjective choiceworthiness in the following respect. Some other approaches [e.g. that of MacAskill (2016a)] begin with the normative theories in all their diversity; work through problems of intertheoretic comparability; and then try to define subjective choiceworthiness with no more structure than necessary. On some accounts, this minimal structure allows only for a binary classification of acts into the “permissible” and the “impermissible” [as recommended, for instance, by Barry and Tomlin (2016)]. The above approach, by contrast, begins by assuming that subjective choiceworthiness is a single cardinal scale, and it characterizes k-metachoiceworthiness claims, and the k-metatheories that make them, in terms of the subjective choiceworthinesses that they would induce if they were known. This approach has the cost of assuming cardinal subjective choiceworthiness, but it has the benefit of immediately giving all my k-metachoiceworthiness claims both unit and level comparability, without requiring any further assumptions.

Thus, from a cardinal definition of subjective choiceworthiness, we also get a cardinal definition of utility, without having to assume it explicitly. By similar reasoning, we also get cardinal definitions of k-metachoiceworthiness for all k. Note that we are not taking the Von Neumann–Morgenstern approach of defining my utility function so that it represents the choices I would make if I were maximizing expected utility; indeed, our project is to explore how far I can stray from certainty about expected utility theory while still knowing how I subjectively ought to act.

3.1 Completeness

A “partial k-metatheory” would be one that makes claims about the k-metachoiceworthinesses of some acts under some (\(k-1\))-metachoiceworthiness distributions, but not of all acts under all (\(k-1\))-metachoiceworthiness distributions. A partial decision theory of “strict dominance”, for instance, claims that \(a_{i} \succ _{\pi ,1} a_{j} \iff u(A, s)_{i} > u(A, s)_{j}\)\(\forall s \in S\), and makes no other claims at all. That is, it claims that an act \(a_{i}\) is more 1-metachoiceworthy than an act \(a_{j}\) if and only if \(a_{i}\) is more objectively choiceworthy than \(a_{j}\) in all the states to which I assign positive probability, and it is silent about acts’ relative 1-metachoiceworthinesses in all other cases.

Conversely,

One condition for the results below is that I assign positive probability only to complete decision theories. Believing that the true decision theory is complete is, I think, reasonably well motivated by the sense that, just as I know acts’ 0-metachoiceworthiness (i.e. objective choiceworthiness) if I know the true state, I should be able to know acts’ 1-metachoiceworthiness if I know the true decision theory (and so on up the hierarchy). In any event, we will remove partial decision theories from consideration so as to separate the regress problem from the problems of incomparability that can plague normative uncertainty in their own right.

Note that the framework laid out in Sect. 2 does not allow us to assign positive probability to the “nihilistic decision theory” (one that makes no claims about acts’ 1-metachoiceworthinesses under any choiceworthiness distribution). Since a decision theory is a collection of claims determining what acts’ subjective choiceworthinesses would be for me if I knew how to respond to my empirical uncertainty, and since my subjective choiceworthiness is already defined in the degenerate case of empirical certainty, all my decision theories at least claim that an act’s subjective choiceworthiness is its objective choiceworthiness, when my objective choiceworthiness distribution is degenerate.

3.2 Continuity

We will say that

To be called continuous, \(t_{k}\) need not respond continuously to the probability one assigns to some (\(k-1\))-metachoiceworthiness claim. A more formal definition of continutity, as we are using the term here, is given in the “Appendix”.

A second condition, necessary for only the first of the results below, is that I assign positive probability only to continuous decision theories.

3.3 The Analog Principle

MacAskill (2014) argues that, when we are facing both empirical and normative uncertainty over a set of acts, there is a sense in which we should treat our empirical and normative uncertainty “analogously”. If I am uncertain which act is objectively best, it may seem unlikely that the appropriate response to my uncertainty would depend on the reason (i.e., empirical or normative) for my uncertainty—especially upon considering that I might have uncertainty about how to behave without even knowing the reason for my uncertainty.

In the context of the regress problem, one might likewise argue that we should treat our empirical and k-metatheoretic uncertainty analogously. More formally:

One might wonder if it matters whether my beliefs about the k-metatheories are correlated across different orders \(k^{\prime }\) (as of course they are—very strongly!—if I accept the Analog Principle), or whether they are correlated my beliefs about the state of the world. In fact, it does not. A k-metatheory is simply a function of my (k-1)-metachoiceworthiness distribution; a k-metatheory’s output therefore does not depend on the probability that it is the true k-metatheory, nor on its probability conditional on some state or \(k^{\prime }\)-metatheory.

A final condition, necessary only for the first of the results below, is that I accept the Analog Principle.

4.1 Intuition

In the context of the framework above, the commonness of well-defined subjective choiceworthiness is not surprising. If I assign positive probability to a finite number of theories, and they disagree about how subjectively choiceworthy some act should be for me, there will be a minimum and a maximum to that range of values. In the face of that uncertainty, my subjective choiceworthiness should lie somewhere in the interior of the range. Where, exactly? I will assign positive probability to different answers as to where it should lie, producing a smaller range. And so on. Given a few other assumptions (either the continuity of my theories and my acceptance of the Analog Principle, or the possibility of transfinite hierarchies), this process will not “get stuck” by shrinking the range of potential subjective choiceworthiness values for each act merely from a larger range to a smaller range. Instead, the process is guaranteed ultimately to shrink said range to a single point.

In other words, nothing very counterintuitive falls out of the mathematical setup of the problem. The point of this exercise is simply to formally illustrate a coherent framework whereby our intuitions about normative uncertainty—including about the infinite regress that it threatens—can be reconciled with the understanding that, at the end of the day, we make norm-guided decisions.

With that said, the convergence results can be stated as follows.

4.2 Natural hierarchies

The proof can be found in the “Appendix”.

This example does not imply the potentially concerning conclusion that one might not be able to infer acts’ relative subjective choiceworthinesses, from one’s choiceworthiness distribution and one’s hierarchy distribution, in finite time. Indeed, since the subjective choiceworthiness of each act is well-defined under the above conditions, and is a logical consequence of one’s finite choice problem, we know from the completeness of first-order logic that the value of \(c(\pi )_{i}\) can be determined for all i by some finite proof—such as the one given above.

The example does, however, demonstrate that there may be facts about subjective choiceworthiness that are not discovered by the “iterate a few times and hope my uncertainty is more or less resolved” algorithm. In particular, therefore, I believe it demonstrates that fixed-point solutions to the regress problem need not take the form either of convergence after finite k or of monotonic decreases in the set of maximally k-choiceworthy acts—as hoped for in Sepielli (2014), and as claimed by Tarsney (2017, p. 239).

4.3 Transfinite hierarchies

Suppose that the k-metatheories to which I assign positive probability are all complete and compatible with the Dominance Principle, but that they are not continuous, or that I reject the Analog Principle. It is then possible that my subjective choiceworthiness for some act \(a_{i}\) does not converge, even after infinite steps. That is, though \(\lim _{k \rightarrow \infty } \min ([\vec {c_{k}}]_{i})\) and \(\lim _{k \rightarrow \infty } \max ([\vec {c_{k}}]_{i})\) do both exist (by the fact that the respective sequences in k are monotonic and bounded), \(\lim _{k \rightarrow \infty } \min ([\vec {c_{k}}]_{i}) < \lim _{k \rightarrow \infty } \max ([\vec {c_{k}}]_{i})\). In such a situation, it seems, someone could still claim that there is a “right way” for me to act. Someone could claim that my subjective choiceworthiness for \(a_{i}\) should be the average of \(\lim _{k \rightarrow \infty } \min ([\vec {c_{k}}]_{i})\) and \(\lim _{k \rightarrow \infty } \max ([\vec {c_{k}}]_{i})\), for example.

If I assign positive probability to competing theories of how to act in situations like the above, I must appeal to a theory about how to act in the face of this uncertainty. So our regress extends beyond the natural numbers, into the transfinite ordinals.

The definitions given in Sects. 2 and 3 can generally be reinterpreted so that k (or, \(\kappa \)) is any ordinal, not just any natural number. Two, however, will require slight tweaks:

Note that this definition of a higher-order metatheory is strictly more general than the original, even with respect to finite k, because it allows k-metatheories to be functions of one’s beliefs not only about (\(k-1\))-metachoiceworthiness but about \(k^{\prime }\)-metachoiceworthiness at all lower orders \(k^{\prime } < k\). The following result thus holds, as Theorem 1 does not, regardless of whether we want to allow for this possibility.

We can now state the following:

The proof can be found in the “Appendix”.

4.4 The infectiousness of stubbornness

Let us call a \(\kappa \)-metatheory “compromising” if it is compatible with the (strong) Dominance Principle, and “stubborn” if it is not. Expected Choiceworthiness Theory and risk-weighted variants of it are examples of “compromising” theories. Minimax Theory, according to which an act’s metachoiceworthiness is its minimum possible choiceworthiness, is an example of a “stubborn” theory. However, it is compatible with the Weak Dominance Principle.

Both the theorems above demonstrate that, when all the decision theories (or \(\kappa \)-metatheories) to which I assign positive probability are compromising (along with some other conditions), the range of potential subjective choiceworthiness values for each act shrinks to point. In both cases, this is demonstrated roughly by the fact that, when the range of potential subjective choiceworthiness values for some act is a non-degenerate interval at some order k (or \(\kappa \)), the application of even higher-order metatheories shrinks this interval by increasing its minimum.

One might notice that, strictly speaking, neither of the proofs requires that all my decision theories (or \(\kappa \)-metatheories) be compromising. Suppose, for example, that I reject the Dominance Principle, but accept the Weak Dominance Principle. Suppose further that just one decision theory (or one\(\kappa \)-metatheory for each \(\kappa \)) to which I assign positive probability is “stubborn”—or, that all the stubborn theories to which I assign positive probability are one-sidedly pessimistic (like Minimax) or optimistic (like Maximax) at each order. Then our proofs can go through with only slight modifications. We just need to shrink our interval exclusively from the top or from the bottom, at a given order, to avoid asking concessions of our stubborn theories.

The plausibility of stubborn theories, however, poses two challenges for this project in general.

First, stubborn theories are “infectious”: they can determine our behavior regardless how little positive credence we give them. Suppose I am deciding between two acts, \(a_{1}\) and \(a_{2}\). I assign probability 0.99 to a state in which \(a_{1}\) has objective choiceworthiness 1, and probability 0.01 to a state in which \(a_{1}\) has objective choiceworthiness 0. Act \(a_{2}\) has objective choiceworthiness 0.0001 in both states. Furthermore, I assign probability 0.99 to Expected Choiceworthiness, and probability 0.01 to Minimax, at every order of the hierarchy. It would be deeply counterintuitive to conclude that, from such a position, \(a_{2}\) is more subjectively choiceworthy than \(a_{1}\). It would be perhaps even more counterintuitive to conclude that the acts’ subjective choiceworthinesses were equal, if \(a_{2}\)’s objective choiceworthiness were known to be 0. But of course we do reach both conclusions, as repeated applications of my distribution over decision theories bring \(a_{1}\)’s higher-order metachoiceworthiness arbitrarily close to 0—according even to the sequence of “Expected Choiceworthiness”-analogous theories.

Second, stubborn theories can clash with each other. If we are going to give some weight to Maximin, at every order, it seems only fair to give some weight to Maximax at every order as well. But if we do, then each act’s range of potential subjective choiceworthiness values never shrinks at all; \(\min ([\vec {c_{\kappa }}]_{i}) = \min ([\vec {c_{0}}]_{i})\), and \(\max ([\vec {c_{\kappa }}]_{i}) = \max ([\vec {c_{0}}]_{i})\), for all \(\kappa \).

It does not feel as though I can fully rule out stubborn theories. Thus, despite all our progress, I am still left with the original motivating question: how do I so regularly wind up with well-defined subjective choiceworthiness? One encouraging thought is the observation that, though stubbornness is infectious in one sense, there is a sense in which compromise is infectious as well. For example, suppose I assign positive probability to stubborn \(\kappa \)-metatheories (or even, only to stubborn \(\kappa \)-metatheories) at almost all \(\kappa \), but assign positive probability only to compromising theories at a relatively sparse class of orders—at the limit ordinals, perhaps. Then, even though there is a sense in which I believe in stubborn theories “almost everywhere” up the hierarchy, the scattered all-compromising orders will still force my subjective choiceworthiness range for each act down to a point. (A minimally modified version of the proof of Theorem 2 presented in the “Appendix” will hold so long as our class \(\Gamma \) of “all-compromising” orders is such that, for every set M of ordinal numbers, there is an ordinal number \(\gamma \in \Gamma : \gamma > \mu \,\forall \,\mu \in M\).) Similar reasoning applies to the case of merely natural hierarchies. In short, my hierarchy distribution can handle a lot of stubbornness; as long as an all-compromising order comes along every now and then to shrink my subjective choiceworthiness range for each act, there are reasonable conditions under which subjective choiceworthiness will generally be well-defined.

4.5 Rescaling

MacAskill (2014) offers the following example of undefined subjective choiceworthiness.

Suppose an agent faces a choice problem:

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She must choose among acts \(A = \{a_1, a_2\}\). She assigns probability \(\frac{18}{23}\) to a state \(s_1\) in which \(a_1\) has objective choiceworthiness 0 and \(a_2\) has objective choiceworthiness 1, and probability \(\frac{5}{23}\) to a state \(s_2\) in which \(c_0(a_1) = 4\) and \(c_0(a_2) = 0\).

In evaluating the acts at order 1, she assigns probability \(\frac{18}{23}\) to Expected Choiceworthiness Theory (\(t_1\)), and probability \(\frac{5}{23}\) to “Square Root Theory” (\(t_2\)), according to which an act’s 1-metachoiceworthiness is the expectation of the square root of the difference between its objective choiceworthiness and the objective choiceworthiness of the least objectively choiceworthy act in A. Thus

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On MacAskill’s reading of the problem, “transformations of each individual choice-worthiness function by an absolute value are permissible, and transformations of all choice-worthiness functions by a multiplying factor are permissible” (222). Thus he produces

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As we can see, after rescaling, the 1-metachoiceworthiness distribution of \(a_1\) is precisely what the 0-metachoiceworthiness distribution of \(a_2\) had been, and the 1-metachoiceworthiness distribution of \(a_2\) is precisely what the 0-metachoiceworthiness distribution of \(a_1\) had been. Therefore, if we obey the Analog Principle—that is, if our distribution over k-metatheories is the same at every order—and if we rescale after every step, our k-metachoiceworthiness distribution for the acts will flip forever between that of “Order 0” and that of “Order 1, rescaled”, without converging.

To my mind, however, k-metachoiceworthiness claims are intuitively characterized by the property that, if one believes them, they define one’s subjective choiceworthiness. If an agent faces empirical uncertainty over two acts’ objective choiceworthiness as represented above, we want to say that, in the event that she learns the truth of \(s_2\), act \(a_1\)’s subjective choiceworthiness for her is 4. In precisely the same language, I think, we want to say that in the event that she learns the truth of Expected Choiceworthiness Theory (but does not learn the true state), \(a_1\)’s subjective choiceworthiness for her is \(\frac{20}{23}\)—and likewise all up the hierarchy. To keep these claims “in line”, the framework of Sect. 2 permits real-valued representations of the subjective choiceworthiness values and k-metachoiceworthiness distributions associated with a given choice problem to be rescaled only in conjunction, not independently.

Furthermore, if this is the right way to think about k-metachoiceworthiness, then Square Root Theory (SR) is, as stated, incoherent. SR does not specify which real-valued representations of objective choiceworthiness to use as inputs, so its claims should be independent to rescaling. But they are not. Using our agent’s 0-metachoiceworthiness distribution as represented above, SR claims that the 1-metachoiceworthiness of \(a_1\) is \(\frac{20}{23}\), and EC claims that the 1-metachoiceworthiness of \(a_1\) is lower (just \(\frac{10}{23}\)). But if we had represented her 0-metachoiceworthiness distribution differently,

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we would conclude that EC claims that the 1-metachoiceworthiness of A is \(\frac{5}{23}\) for her, and that SR claims the same.

To ensure that an act’s true k-metachoiceworthiness for an agent be independent of the scale she arbitrarily uses to represent her (\(k-1\))-metachoiceworthiness distribution, all our k-metatheories have to be “affine” (unique up to affine transformation). Though this condition closes the door to “Square Root Theory”, it permits a wide array of other risk-averse theories, including Buchak’s REU Theory and the risk-averse theory presented in Proposition 4.1.