An axiomatic approach to axiological uncertainty

Abstract

How ought you to evaluate your options if you’re uncertain about which axiology is true? One prominent response is Expected Moral Value Maximisation (EMVM), the view that under axiological uncertainty, an option is better than another if and only if it has the greater expected moral value across axiologies. EMVM raises two fundamental questions. First, there’s a question about what it should even mean. In particular, it presupposes that we can compare moral value across axiologies. So to even understand EMVM, we need to explain what it is for such comparisons to hold. Second, assuming that we understand it, there’s a question about whether EMVM is true. Since there are many plausible rivals, we need an argument to defend it. In this paper, I’ll introduce a representation theorem for axiological uncertainty to answer these two questions. Roughly, the theorem shows that if all our axiologies satisfy the von Neumann–Morgenstern axioms, and if the facts about which options are better than which in light of your uncertainty also satisfy these axioms as well as a Pareto condition, then these facts have a relevantly unique expected utility representation. If I’m right, this theorem at once affords us a compelling way to understand EMVM—and specifically intertheoretic comparisons—and a systematic argument for its truth.

Appendix

Let me state and prove the precise theorem. The overall argument ultimately depends not on the mathematical result in terms of abstract functions, but on the philosophical theorem in which these functions are given an extra-mathematical significance. So I shall also give a more precise formulation of the decision-theoretic explication, and then state and prove the theorem in its interpreted form. As stated above, I assume that we have a nonempty finite set of outcomes ${X}=\{x, y, \ldots z\}$, a nonempty finite set of axiologies ${T}=\{T_1, T_2, \ldots T_n\}$ with the index-set $I=\{1, 2, \ldots n\}$, and three sets of prospects, ${\mathscr {O}}=\{a: {X} \rightarrow {\mathbb {R}}_+ |\sum _{x \in {X}}a(x)=1\}$, ${\mathscr {K}}=\{{\varvec{a}}: I \times {X} \rightarrow {\mathbb {R}}_+\,|\,\sum _{x\in {X}}{\varvec{a}}(i,x)=1 \; \; \forall i \in I\}$, and ${\mathscr {Q}}=\{{\fancyscript{a}}: I \times {X} \rightarrow {\mathbb {R}}_+\,|\,\sum _{i \in I, x\in {X}}{\fancyscript{a}}(i,x)=1 \}$. For every i in I, $\succeq _i$ is a reflexive binary relation on ${\mathscr {O}}$ (representing $T_i$). $\succeq _m$ is a reflexive binary relation on ${\mathscr {K}}$ and $\widetilde{\succeq }_m$ a reflexive binary relation on ${\mathscr {Q}}$. In each case, two further relations are implied as usual: $a \succ _i b$ if $a \succeq _i b$ but not $b\succeq _i a$, and $a \sim _i b$ if $a \succeq _i b$ and $b\succeq _i a$; similarly for the relations on ${\mathscr {K}}$ and ${\mathscr {Q}}$. For any a and b in ${\mathscr {O}}$, and any $p\in [0,1]$, let $pa+(1-p)b$ in ${\mathscr {O}}$ be such that $(pa+(1-p)b)(x)=pa(x)+(1-p)b(x)$ for all x in X; similarly for the respective elements in ${\mathscr {K}}$ and ${\mathscr {Q}}$. Now for a reflexive binary relation $\succeq$ on ${\mathscr {O}}$, define the following conditions:

Transitivity$_{{\mathscr {O}}}$: for all a, b and c in ${\mathscr {O}}$, if $a\succeq b$ and $b \succeq c$, then $a \succeq c$;

Completeness$_{{\mathscr {O}}}$: for all a and b in ${\mathscr {O}}$, $a \succeq b$ or $b \succeq a$;

Independence$_{\mathscr {O}}$: for all a, b and c in ${\mathscr {O}}$ and $p \in ]0,1[$, if $a \succ b$, then $pa+(1-p)c\succ pb+(1-p)c$;

Continuity$_{\mathscr {O}}$: for all a, b and c in ${\mathscr {O}}$, if $a \succ b$ and $b \succ c$, then there exist $p, q \in ]0,1[$, such that $pa+(1-p)c \succ b$ and $b \succ qa+(1-q)c$.

Say that a binary relation $\succeq$ on ${\mathscr {O}}$ is vNM-conformable if it satisfies these four conditions. The same conditions can be formulated for a binary relation $\succeq$ on ${\mathscr {K}}$, and for a binary relation $\widetilde{\succeq }$ on ${\mathscr {Q}}$. Say that such relations are vNM-conformable if they satisfy the corresponding four conditions. For some binary relations ${\succeq _i}$ on ${\mathscr {O}}$ and ${\succeq }$ on ${\mathscr {K}}$ say that $\succeq _i$ is non-uniform if there are a and b in ${\mathscr {O}}$ such that $a \succ _i b$, and that ${\succeq }$ is non-uniform if there are ${\varvec{a}}$ and ${\varvec{b}}$ in ${\mathscr {K}}$ such that ${\varvec{a}} \,\,{\succ }\, {\varvec{b}}$.

We need to assure that $\succeq _m$ and $\widetilde{\succeq }_m$ are value-consistent—in the sense that the differences between them are fully explained by the underlying probabilities in ${\mathscr {Q}}$ and not by different values concerning prospects in ${\mathscr {Q}}$ and ${\mathscr {K}}$. To that end, let ${\mathscr {Q}}^+ \subset {\mathscr {Q}}$ be the set of prospects in ${\mathscr {Q}}$ on which all axiologies have a positive probability, i.e. ${\mathscr {Q}}^+=\{{\fancyscript{a}} \in {\mathscr {Q}}\;|\, \sum _{x\in X}{\fancyscript{a}}(i,x)>0 \;\; \forall i\in I\}$. Define a function $L:{\mathscr {Q}}^+\rightarrow {\mathscr {K}}$; ${\fancyscript{a}}\mapsto L({\fancyscript{a}})$, such that for all i in I and x in X,

$$\begin{aligned} L({\fancyscript{a}})(i,x)={\fancyscript{a}}(i,x)/\sum _{y\in X}{\fancyscript{a}}(i,y). \end{aligned}$$

(5)

Intuitively, L turns a prospect in ${\mathscr {Q}}^+$ into the corresponding prospect in ${\mathscr {K}}$ in which the underlying probabilities have been scraped out. For some i in I and ${\varvec{a}}$ and ${\varvec{b}}$ in ${\mathscr {K}}$, say that ${\varvec{a}}$ and ${\varvec{b}}$agree outsidei if for all j in $I, j \ne i$, and all x in X, ${\varvec{a}}(j,x)={\varvec{b}}(j,x)$; and similarly for some ${\fancyscript{a}}$ and ${\fancyscript{b}}$ in ${\mathscr {Q}}$. For some i in I and binary relations ${\succeq }$ on ${\mathscr {K}}$ and $\widetilde{\succeq }$ on ${\mathscr {Q}}$, say that i is null if (i) for all ${\varvec{a}}$ and ${\varvec{b}}$ in ${\mathscr {K}}$ that agree outside i, ${\varvec{a}} \,\,{\sim }\, {\varvec{b}}$, and (ii) there exist ${\fancyscript{a}}$ and ${\fancyscript{b}}$ in ${\mathscr {Q}}$ that agree outside i such that ${\fancyscript{a}} \, \widetilde{\succ } \, {\fancyscript{b}}$. Now for some binary relations ${\succeq }$ on ${\mathscr {K}}$ and $\widetilde{\succeq }$ on $\mathscr {Q}$, we can define the

Consistency Axiom: For all i in I and all ${\fancyscript{a}}$ and ${\fancyscript{b}}$ in ${\mathscr {Q}}^+$, if ${\fancyscript{a}}$ and ${\fancyscript{b}}$ agree outside i and $L({\fancyscript{a}})\,\,{\succ }\, L({\fancyscript{b}})$, then ${\fancyscript{a}}\,\, \widetilde{\succ } \, {\fancyscript{b}}$. Moreover, if i is not null, then for all ${\fancyscript{a}}$ and ${\fancyscript{b}}$ in ${\mathscr {Q}}^+$ that agree outside i, if ${\fancyscript{a}}\, \widetilde{\succ }\, {\fancyscript{b}}$, then $L({\fancyscript{a}})\,{\succ }\, L({\fancyscript{b}})$.

To state the above-mentioned Pareto condition formally, let ${\mathscr {Q}}^i \subset {\mathscr {Q}}$ be the set of prospects in which $T_i$ has a strictly positive probability, i.e. ${\mathscr {Q}}^i=\{{\fancyscript{a}} \in {\mathscr {Q}}\;| \,\sum _{x\in X}{\fancyscript{a}}(i,x)>0\}$. Define for each i in I a function $H_i:{\mathscr {Q}}^i\rightarrow {\mathscr {O}}$; ${\fancyscript{a}}\mapsto H_i({\fancyscript{a}})$, such that for all x in X,

$$\begin{aligned} H_i({\fancyscript{a}})(x)={\fancyscript{a}}(i,x)/\sum _{y\in X}{\fancyscript{a}}(i,y). \end{aligned}$$

(6)

Intuitively, $H_i$ turns a prospect ${\fancyscript{a}}$ into the prospect that ${\fancyscript{a}}$ represents, given $T_i$ (if there is such a prospect). Say that P is a probability distribution onI if P is a function $P: I \rightarrow {\mathbb {R}}_+$ with $\sum _{i \in I} P(i)=1.$ For any probability distribution P on I, let ${\mathscr {Q}}^P \subset {\mathscr {Q}}$ be the set of prospects in which P is the underlying probability distribution over axiologies, i.e. ${\mathscr {Q}}^P=\{{\fancyscript{a}} \in {\mathscr {Q}}\;|\,\sum _{x\in X}{\fancyscript{a}}(i,x)=P(i)\;\; \forall \;i\in I\}$. Now for some binary relation $\widetilde{\succeq }$ on ${\mathscr {Q}}$ and binary relations $\succeq _i$ on ${\mathscr {O}}$, define the

Pareto Condition: For any probability distribution P on I, and for all ${\fancyscript{a}}$ and ${\fancyscript{b}}$ in ${\mathscr {Q}}^P$, if $H_i({\fancyscript{a}})\sim _i H_i({\fancyscript{b}})$ for all i in I with $P(i)>0$, then ${\fancyscript{a}}\widetilde{\sim } \, {\fancyscript{b}}$; and if $H_i({\fancyscript{a}})\succeq _i H_i({\fancyscript{b}})$ for all i in I with $P(i)>0$ and $H_j({\fancyscript{a}})\succ _j H_j({\fancyscript{b}})$ for some j in I with $P(j)>0$, then ${\fancyscript{a}}\widetilde{\succ } \, {\fancyscript{b}}$.

Let me now state the decision-theoretic explications more formally. So for binary relations ${\succeq }$ on $\mathcal{K}$ and $\widetilde{\succeq}$ on $\mathcal{Q}$, a probability distribution P on I and a function $u:I \times {X}\rightarrow {\mathbb {R}}$, say that the expectation of P and u represents${\succeq }$ and$\widetilde{\succeq}$ordinally if for all ${\varvec{a}}$ and ${\varvec{b}}$ in ${\mathcal {K}}$ and all ${\fancyscript{a}}$ and ${\fancyscript{b}}$ in ${\mathcal {Q}}$,

$$\begin{aligned} {\varvec{a}}\; \succeq_{m} \; {\varvec{b}} \,\, {\text { iff}} \sum _{i\in I, x \in {X}} {\varvec{a}}(i,x)P(i)u(i,x) \geq \sum _{i\in I, x \in {X}} {\varvec{b}}(i,x)P(i)u(i,x), {\text {and}} \end{aligned}$$

(7)

$$\begin{aligned} {\fancyscript{a}}\; \widetilde{\succeq}_m \; {\fancyscript{b}} \,\, {\text { iff}} \sum _{i\in I, x \in {X}} {\fancyscript{a}}(i,x)u(i,x) \geq \sum _{i\in I, x \in {X}} {\fancyscript{b}}(i,x)u(i,x). \end{aligned}$$

(8)

For a binary relation ${\succeq _i}$ on ${\mathscr {O}}$ and a function $u: {X}\rightarrow {\mathbb {R}}$, say that the expectation ofurepresents${\succeq }_i$ordinally if for all a and b in ${\mathscr {O}}$,

$$\begin{aligned} a \,\,{\succeq _i}\,\, {b} \,\,{\text { iff }} \sum _{x \in {X}} a(x)u(x) \ge \sum _{x \in {X}} b(x)u(x). \end{aligned}$$

(9)

Furthermore, say that a function $u: {X}\rightarrow {\mathbb {R}}$represents$\succeq _i$cardinally if for all x, y, z and t in X, statement (B) (from Sect. 1) is true if and only if $(u(x)-u(y))/(u(z)-u(t))=n$. The decision-theoretic explication of intratheoretic comparisons is that if there is a function $u: {X}\rightarrow {\mathbb {R}}$, unique up to positive affine transformation, whose expectation represents $\succeq _i$ ordinally, then u represents $\succeq _i$ cardinally. Or in other words, for such a utility function to represent $\succeq _i$ cardinally just is for it to be such that it represents $\succeq _i$ ordinally and is unique up to positive affine transformation in doing so. Similarly, for a probability distribution P on I and a function $u:I \times {X}\rightarrow {\mathbb {R}}$, say that P and urepresent intertheoretic comparisons cardinally and axiological probabilities quantitatively if for all i and j in I and all x, y, z and t in X, the respective statements (A) and (C) are true if and only if $p_i=P(i)$, $p_j=P(j)$ and $(u(i,x)-u(i,y))/(u(j,z)-u(j,t))=n$. The decision-theoretic explication of intertheoretic comparisons and axiological probabilities is that if there is such a pair of P and u, with P being unique and u unique up to positive affine transformation, whose expectation represents $\succeq _m$ and $\widetilde{\succeq}_m\,$ ordinally, and if for all i in I, $u(i,\cdot )$ represents $\succeq _i$ cardinally, then P and u represent intertheoretic comparisons cardinally and axiological probabilities quantitatively.

Given these explications, the following theorem—the main philosophical theorem of this paper in its precise technical form—holds:

Expected Moral Value Theorem: Suppose that all $\succeq _i$ are vNM-conformable and non-uniform. If ${\succeq }_m\,$ and $\widetilde{\succeq }_m$ are vNM-conformable and jointly satisfy the Consistency Axiom, if ${\succeq }_m\,$ is non-uniform and $\widetilde{\succeq }_m$ satisfies the Pareto Condition with respect to $\succeq _i$, then for all ${\varvec{a}}$ and ${\varvec{b}}$ in ${\mathscr {K}}$,

$$\begin{aligned} {\varvec{a}}\,\,{\succeq }_m\, {\varvec{b}} \, {\text { iff }}\sum _{i\in I, x \in {X}} {\varvec{a}}(i,x)p_iG_i(x) \ge \sum _{i\in I, x \in {X}}{\varvec{b}}(i,x)p_iG_i(x). \end{aligned}$$

(10)

Note that there’s thus another restriction to the scope of the theorem. It’s not a fully general theorem of axiological uncertainty. It’s only a theorem concerning uncertainty about vNM-conformable and non-uniform axiologies.

Let me now prove that this theorem is true. Karni and Schmeidler (1980; see also 2016) prove

Karni and Schmeidler’s Theorem: Let ${\succeq }$ be a reflexive binary relation on ${\mathscr {K}}$ and $\widetilde{\succeq }$ a reflexive binary relation on ${\mathscr {Q}}$. Suppose that they’re both vNM-conformable and jointly satisfy the Consistency Axiom, and that ${\succeq }$ is non-uniform. Then (i) there exists a function $u:I \times {X}\rightarrow {\mathbb {R}}$ and a probability distribution P on I such that, for all ${\varvec{a}}$ and ${\varvec{b}}$ in ${\mathscr {K}}$,

$$\begin{aligned} {\varvec{a}} \,\,{\succeq }\,\, {\varvec{b}} \,{\text { iff }} \sum _{i\in I, x \in {X}}{\varvec{a}}(i,x)P(i)u(i,x) \ge \sum _{i\in I, x \in {X}}{\varvec{b}}(i,x)P(i)u(i,x), \end{aligned}$$

(11)

and for all ${\fancyscript{a}}$ and ${\fancyscript{b}}$ in ${\mathscr {Q}}$,

$$\begin{aligned} {\fancyscript{a}}\,\, \widetilde{\succeq } \,\, {\fancyscript{b}} \,\,{\text { iff }} \sum _{i\in I, x \in {X}}{\fancyscript{a}}(i,x) u(i,x) \ge \sum _{i\in I, x \in {X}} {\fancyscript{b}}(i,x) u(i,x). \end{aligned}$$

(12)

(ii) u in (i) is unique up to positive affine transformation. (iii) If i is null, $P(i)=0$, and if i is not null and there are ${\varvec{a}}$ and ${\varvec{b}}$ in ${\mathscr {K}}$ that agree outside i such that ${\varvec{a}} \,\,{\succ }\, {\varvec{b}}$, $P(i)>0$. Moreover, if for all i in I there are ${\fancyscript{a}}$ and ${\fancyscript{b}}$ in ${\mathscr {Q}}$ that agree outside i such that ${\fancyscript{a}}\, \widetilde{\succ }\, {\fancyscript{b}}$, then P in (i) is unique.

This theorem immediately implies that if ${\succeq }_m\,$ and $\widetilde{\succeq }_m$ are vNM-conformable and jointly satisfy the Consistency Axiom and ${\succeq }_m\,$ is non-uniform, then there exists a function $u: I \times {X} \rightarrow {\mathbb {R}}$ and a probability distribution P on I such that, for all ${\varvec{a}}$ and ${\varvec{b}}$ in ${\mathscr {K}}$,

$$\begin{aligned} {\varvec{a}} \,\,{\succeq _m}\,\, {\varvec{b}}\,\, {\text { iff }} \sum _{i\in I, x \in {X}}{\varvec{a}}(i,x)P(i)u(i,x) \ge \sum _{i\in I, x \in {X}}{\varvec{b}}(i,x)P(i)u(i,x), \end{aligned}$$

(13)

and for all ${\fancyscript{a}}$ and ${\fancyscript{b}}$ in ${\mathscr {Q}}$,

$$\begin{aligned} {\fancyscript{a}}\,\, \widetilde{\succeq }_m \,\, {\fancyscript{b}}\,\, {\text { iff }} \sum _{i\in I, x \in {X}}{\fancyscript{a}}(i,x)u(i,x) \ge \sum _{i\in I, x \in {X}}{\fancyscript{b}}(i,x)u(i,x), \end{aligned}$$

(14)

with u being unique up to positive affine transformation. Clearly, this is consistent with the assumptions that all $\succeq _i$ are vNM-conformable and non-uniform and that $\widetilde{\succeq }_m$ satisfies the Pareto Condition with respect to the relations $\succeq _i$. What remains to be proved is the uniqueness of P, given these assumptions. So for some i in I, consider the set ${\mathscr {Q}}^i_{p=1} \subset {\mathscr {Q}}$ of prospects in which the probability of $T_i$ is 1, i.e. ${\mathscr {Q}}^i_{p=1}=\{{\fancyscript{a}}\in {\mathscr {Q}}\,|\,\sum _{x\in X}{\fancyscript{a}}(i,x)=1\}$. This set is obviously isomorphic to ${\mathscr {O}}$. All ${\fancyscript{a}}$ and ${\fancyscript{b}}$ in ${\mathscr {Q}}^i_{p=1}$ agree outside i. Moreover, given that all $\succeq _i$ are non-uniform, there are ${\fancyscript{a}}$ and ${\fancyscript{b}}$ in ${\mathscr {Q}}^i_{p=1}$ such that $H_i({\fancyscript{a}}) \succ _i H_i({\fancyscript{b}})$ and thus—since $\widetilde{\succeq }_m$ satisfies the Pareto Condition—${\fancyscript{a}} \widetilde{\succ } \, {\fancyscript{b}}$. So it follows from condition (iii) of Karni and Schmeidler’s Theorem that P is unique.

This means that we can apply the decision-theoretic explications. Consider first the explication of intratheoretic comparisons. Note that for all i in I, the function $u(i,\cdot )$ on X is such that for all ${\fancyscript{a}}$ and ${\fancyscript{b}}$ in ${\mathscr {Q}}^i_{p=1}$,

$$\begin{aligned} {\fancyscript{a}}\,\, \widetilde{\succeq }_m \,\, {\fancyscript{b}}\,\, {\text { iff }}\sum _{x\in X}u(i,x) {\fancyscript{a}}(i,x) \ge \sum _{x\in X}u(i,x) {\fancyscript{b}}(i,x). \end{aligned}$$

(15)

The Pareto Condition immediately implies that for all ${\fancyscript{a}}$ and ${\fancyscript{b}}$ in ${\mathscr {Q}}^i_{p=1}$,

$$\begin{aligned} {\fancyscript{a}}\,\, \widetilde{\succeq }_m \,\, {\fancyscript{b}} \,\,{\text { iff }}\,\, H_i({\fancyscript{a}})\succeq _i H_i({\fancyscript{b}}). \end{aligned}$$

(16)

And since for all x in X and all ${\fancyscript{a}}$ in ${\mathscr {Q}}^i_{p=1}$, $H_i({\fancyscript{a}})(x) = {\fancyscript{a}}(i,x)$, $u(i,\cdot )$ is a function on X whose expectation represents $\succeq _i$ ordinally in the sense of (9). Given the relevant uniqueness of u, the decision-theoretic explication of intratheoretic comparisons implies that $u(i,\cdot )$ represents $\succeq _i$ cardinally. So we can apply the explication of intertheoretic comparisons and axiological probabilities. Given (13) and (14), the expectation of P and u represents $\succeq _m$ and $\widetilde{\succeq}_m$ ordinally. And given the relevant uniqueness of P and u, our explication thus implies that P and u represent intertheoretic comparisons cardinally and axiological probabilities quantitatively. We can thus interpret P(i) in (13) as representing the evidential probability $p_i$ of the theory that’s represented by the value function $G_i=u(i,\cdot )$. The Expected Moral Value Theorem follows. $\square$