On the informational basis of social choice with the evaluation of opportunity sets

Abstract

This paper examines the informational basis of social choice in a broader conceptual framework. Formal welfarism is a social evaluation in which any information other than the well-being of individuals is excluded, where the notion of individual well-being can be conceived in various ways. We propose a notion of individual well-being defined over pairs of outcomes and opportunity sets from which they are chosen. The concept of consequentialism and non-consequentialism is naturally introduced by restricting individual evaluation functions over the pairs of outcomes and opportunity sets. The two formal welfarism theorems provide axiomatic characterizations of formal welfarism in the extended framework. We show that the presence of a consequentialist or a non-consequentialist affects the validity of the two formal welfarism theorems in the extended framework.

Appendix

1.1 Proofs

This part of the appendix provides the proofs of Theorems 6 and 7. We make use of the proof strategy to show the welfarism theorems on economics domains, which is proposed by Bordes et al. (2005) and Weymark (1998) and is well-known as the local approach.

We first provide a few definitions. Let $\Gamma$ be a subset of $\Omega$. Two pairs of alternatives $\Pi$ and $\Pi '$ contained in $\Gamma$ are strongly connected with respect to $\mathcal D _{i}$ (resp. $\mathcal D$) if $\Pi \cup \Pi '$ is a free triple with respect to $\mathcal D _{i}$ (resp. $\mathcal D$). Two pairs of alternatives $\Pi$ and $\Pi '$ contained in $\Gamma$ are connected with respect to $\mathcal D _{i}$ (resp. $\mathcal D$) if there exists a finite sequence of pairs contained in $\Gamma$, $\Pi ^{1},\ldots ,\Pi ^{t}$ with $\Pi ^{1}=\Pi$ and $\Pi ^{t}=\Pi '$ such that $\Pi ^{j}$ and $\Pi ^{j+1}$ are strongly connected with respect to $\mathcal D _{i}$ (resp. $\mathcal D$) for all $j\in \{1,\ldots ,t-1\}$. A Cartesian domain $\mathcal D$ is saturating for $\Gamma$ if (i) there exist at least two non-trivial pairs contained in $\Gamma$ with respect to $\mathcal D$ and (ii) any two non-trivial pairs contained in $\Gamma$ with respect to $\mathcal D$ are connected with respect to $\mathcal D$. When $\Gamma =\Omega$, we say that a domain $\mathcal D$ is saturating without referring to $\Omega$ explicitly. Note that an unrestricted domain is an example of a saturating domain.

The next result is useful to prove our results, which corresponds to a result in Weymark (1998). While his result is stated for the case $\Gamma =\Omega$, ours are applied for any subset $\Gamma$ of $\Omega$. However, his proof is still valid for any subset $\Gamma$ of $\Omega$. Therefore, we leave the readers to prove it.

Proposition 1

Given any subset $\Gamma$ of $\Omega$, suppose that a domain is common and saturating for $\Gamma$. Then, it satisfies PI on $\Gamma$, WP on $\Gamma$, and BI on $\Gamma$ if and only if it is defined in terms of a restricted RBSF for $\Gamma$, where some real-valued function $W_{|\Gamma }$ is weakly monotonic.

On the other hand, the following result is a counterpart of results in Bordes et al. (2005). The proof is similar to theirs and is omitted.

Proposition 2

Given any subset $\Gamma$ of $\Omega$, suppose that a domain is common and saturating for $\Gamma$. If an ESOFL satisfies WP on $\Gamma$, BI on $\Gamma$, and PC on $\Gamma$, then it is defined in terms of a restricted weak RBSF for $\Gamma$.

The following result shows the property of the domain $\mathcal D _{N^{*}}$ and is used to prove Theorems 6 and 7.

Lemma 1

The domain $\mathcal D _{N^{*}}$ is common and saturating for $\Gamma _{-X}$.

Proof

It is clear that the domain $\mathcal D _{N^{*}}$ is common. We now show that it is saturating for $\Gamma _{-X}$.

Condition (i) By $\# X\ge 3$, $\mathcal D _{N^{*}}$ clearly satisfies condition (i) of a saturating domain for $\Gamma _{-X}$.

Condition (ii) For any pair $\{(a,S),(b,T)\}\subseteq \Gamma _{-X}$, the pair is non-trivial with respect to $\mathcal D _{N^{*}}$ if and only if $S$ is not strictly included in $T$ and vice versa. Consider any two non-trivial pairs $\{(x,A),(y,B)\}\subseteq \Gamma _{-X}$ and $\{(z,C),(w,D)\}\subseteq \Gamma _{-X}$ with respect to $\mathcal D _{N^{*}}$. Let $a\notin A$, $b\notin B$, $c\notin C$, and $d\notin D$. We assume that $a\ne c$ and $b\ne d$. For any other cases, it is possible to show that $\{(x,A),(y,B)\}$ and $\{(z,C),(w,D)\}$ are connected with respect to $\mathcal D _{N^{*}}$ by a similar argument. Consider the following sequence of pairs: $\{(x,A),(y,B)\}$, $\{(y,B),(a,\{a,b\})\}$, $\{(a,\{a,b\}),(b,\{b,c\})\}$, $\{(b,\{b,c\}),(c,\{c,d\})\}$, $\{(c,\{c,d\}),(z,C)\}$, and $\{(z,C),$ $(w,D)\}$. Note that for any triple $\{(a,S),(b,T),(c,U)\}\subseteq \Gamma _{-X}$, it is free with respect to $\mathcal D _{N^{*}}$ if and only if each of $S$, $T$ and $U$ does not strictly include the other two opportunity sets and is not strictly included in them. Therefore, each union of adjoining pairs in the sequence is a free triple with respect to $\mathcal D _{N^{*}}$. Hence, we prove that $\mathcal D _{N^{*}}$ satisfies condition (ii) of a saturating domain for $\Gamma _{-X}$.$\square$

We are now ready to prove Theorems 6 and 7.

Proof of Theorem 6

Consider the ESOFL constructed in the statement of Theorem 6. It is easy to show that $R_{V}$ is an ordering for all $V\in \mathcal D$. We now show that the ESOFL satisfies PI, WP, and BI. For all $(x,A),(y,B)\in \Omega$, it is only possible to have $V(x,A)=V(y,B)$ if $A$ does not strictly include $B$ and vice versa. But if $V(x,A)=V(y,B)$, we obtain $W_{|\Gamma _{X}}^{1}(V(x,A))=W_{|\Gamma _{X}}^{1}(V(y,B))$ and $W_{|\Gamma _{-X}}^{2}(V(x,A))=W_{|\Gamma _{-X}}^{2}(V(y,B))$. By construction, we have $(x,A)I_{V}(y,B)$, and so the ESOFL satisfies PI. Suppose $V(x,A)\gg V(y,B)$. Note that $V(x,A)\gg V(y,B)$ implies that B does not strictly include $A$ because every individual is a non-consequentialist. By construction, we have $(x,A)P_{V}(y,B)$, which implies that the ESOFL satisfies WP. Suppose $V(x,A)=V'(x,A)$ and $V(y,B)=V'(y,B)$. By construction, we have $(x,A)R_{V}(y,B)\Leftrightarrow (x,A)R_{V'}(y,B)$. Thus, the ESOFL satisfies BI.

We now show the converse. Suppose that an ESOFL satisfies PI, WP, and BI. First, consider $\Gamma _{X}$. Note that the ESOFL satisfies PI on $\Gamma _{X}$, WP on $\Gamma _{X}$, and BI on $\Gamma _{X}$. Since $\Gamma _{X}$ is free with respect to $\mathcal D _{N^{*}}$, Proposition 1 can be applied to $\Gamma _{X}$, and the ESOFL is defined in terms of a restricted RBSF for $\Gamma _{X}$, where $W_{|\Gamma _{X}}^{1}:\mathbb R ^{n}\rightarrow \mathbb R$ is weakly monotonic. Next, consider $\Gamma _{-X}$. Note that the ESOFL satisfies PI on $\Gamma _{-X}$, WP on $\Gamma _{-X}$, and BI on $\Gamma _{-X}$. By Lemma 1, $\mathcal D _{N^{*}}$ is common and saturating for $\Gamma _{-X}$. By Proposition 1, the ESOFL is defined in terms of a restricted RBSF for $\Gamma _{-X}$, where $W_{|\Gamma _{-X}}^{2}:\mathbb R ^{n}\rightarrow \mathbb R$ is weakly monotonic.

Finally, consider any pair $\{(x,A),(y,B)\}$ with $(x,A)\in \Gamma _{X}$ and $(y,B)\in \Gamma _{-X}$. Since every individual is a non-consequentialist, we have $V(x,A)\gg V(y,B)$ for all $V\in \mathcal D _{N^{*}}$. Since the two $W_{\Gamma _{X}}^{1}$ and $W_{\Gamma _{-X}}^{2}$ are weakly monotonic, we have $W_{\Gamma _{X}}^{1}(V(x,A))>W_{\Gamma _{X}}^{1}(V(y,B))$ and $W_{\Gamma _{-X}}^{2}(V(x,A))>W_{\Gamma _{-X}}^{2}(V(y,B))$. We also have $(x,A)P_{V}(y,B)$ by WP, which completes the proof.$\square$

Proof of Theorem 7

Suppose that an ESOFL satisfies WP, BI, and PC. First, consider $\Gamma _{X}$. Note that the ESOFL satisfies WP on $\Gamma _{X}$, BI on $\Gamma _{X}$, and PC on $\Gamma _{X}$. Since $\Gamma _{X}$ is free with respect to $\mathcal D _{N^{*}}$, Proposition 2 can be applied to $\Gamma _{X}$, and the ESOFL is defined in terms of a restricted weak RBSF for $\Gamma _{X}$. Next, by Lemma 1, $\mathcal D _{N^{*}}$ is common and saturating for $\Gamma _{-X}$. By Proposition 2, the ESOFL is defined in terms of a restricted weak RBSF for $\Gamma _{-X}$.

Consider any pair $\{(x,A),(y,B)\}$ with $(x,A)\in \Gamma _{X}$ and $(y,B)\in \Gamma _{-X}$. Since every individual is a non-consequentialist, we have $V(x,A)\gg V(y,B)$ for all $V\in \mathcal D _{N^{*}}$. Since the two $W_{\Gamma _{X}}^{1}$ and $W_{\Gamma _{-X}}^{2}$ are weakly monotonic, we have $W_{\Gamma _{X}}^{1}(V(x,A))>W_{\Gamma _{X}}^{1}(V(y,B))$ and $W_{\Gamma _{-X}}^{2}(V(x,A))>W_{\Gamma _{-X}}^{2}(V(y,B))$. By WP, we also have $(x,A)P_{V}(y,B)$, which completes the proof.$\square$

1.2 Proof of Theorem 5

This part of the appendix provides the proof of Theorem 5. We first introduce a few definitions. The first notion is a strong strict preference (SSP), proposed by Roberts (1980a). Suppose that $(x,A)$ is socially ranked higher than $(y,B)$. If individual evaluations of $(x,A)$ are not as high and individual evaluations of $(y,B)$ are higher, while the same strict social ordering holds, then we say that the social ordering exhibits an SSP. Formally, given $V\in \mathcal D$, $(x,A)$ is strongly strictly preferred to $(y,B)$ in terms of $R_{V}$ for all $(x,A),(y,B)\in \Omega$, if there exists $V'\in \mathcal D$ such that for all $(z,C)\in \Omega \backslash \{(x,A),(y,B)\}$ $V'(z,C)=V(z,C)$, $V'(y,B)\gg V(y,B)$ and $V'(x,A)\ll V(x,A)$ and $(x,A)P_{V'}(y,B)$. Let $PP_{V}$ be an SSP in terms of $R_{V}$. The notion of an SSP is used to define Weak Neutrality (WN). WN requires that the social ordering using $PP_{V}$ depends only on individual evalations of the two alternatives. Formally, for all $(x,A),(y,B),(z,C),(w,D)\in \Omega$ and all $V,V'\in \mathcal D$, if $V(x,A)=V'(z,C)$ and $V(y,B)=V'(w,D)$, then $(x,A)PP_{V}(y,B)\Leftrightarrow (z,C)PP_{V'}(w,D)$ and $(y,B)PP_{V}(x,A)\Leftrightarrow (w,D)PP_{V'}(z,C)$. Given a subset $\Gamma$ of $\Omega$, we say that the ESOFL satisfies WN on $\Gamma$ if the same property holds for all $(x,A),(y,B),(z,C),(w,D)\in \Gamma$.

The following proposition is useful to prove Theorem 5.

Proposition 3

Given any $\mathcal{D }\in \fancyscript{D}_{E\cup S}\backslash \{\mathcal{D }_{E}\}$ and a subset $\Gamma$ of $\Omega$ with $\# \Gamma =4$ such that all outcomes are distinct and $\Gamma$ contains a free triple $\Delta$ with respect to $\mathcal D$, if an ESOFL defined on $\mathcal D$ satisfies WP on $\Gamma$, BI on $\Gamma$, and PC on $\Gamma$, then it is defined in terms of a restricted weak RBSF for $\Gamma$.

The way to prove Proposition 3 is to check that each step of the proof of Theorem 1 in Roberts (1980a), which is equivalent to Theorem 2 in this paper, remains valid even in the relevant environments. See also Hammond’s (1999) correction. In what follows, we consider any $\mathcal{D }\in \fancyscript{D}_{E\cup S}\backslash \{\mathcal{D }_{E}\}$ and assume that a subset $\Gamma$ satisfies the assumptions in Proposition 3.

Lemma 2

If an ESOFL defined on $\mathcal D$ satisfies WP on $\Gamma$ and BI on $\Gamma$, then for all $V\in \mathcal D$ and all $(x,A),(y,B)\in \Gamma$, $(x,A)PP_{V}(y,B)$ implies $(x,A)P_{V}(y,B)$.

Proof

Suppose that $(x,A)PP_{V}(y,B)$ for all $V\in \mathcal D$ and all $(x,A),(y,B)\in \Gamma$. By the definition of SSP, there exists $V^{1}\in \mathcal D$ such that $V^{1}(x,A)=V(x,A)-\eta ^{1}$, $V^{1}(y,B)=V(y,B)+\eta ^{2}$, and $V^{1}(a,S)=V(a,S)$ for all $(a,S)\in \Gamma \backslash \{(x,A),(y,B)\}$ where $\eta ^{1},\eta ^{2}\gg 0$, and $(x,A)P_{V^{1}}(y,B)$. Consider the profiles $V^{2}$, $V^{3}$, $V^{4}$, $V^{5}\in \mathcal D$ in Table 1, where $\eta ^{1}\gg \gamma ^{1}\gg \epsilon ^{1}\gg 0$ and $\eta ^{2}\gg \gamma ^{2}\gg 0$, and for all $i\in N$, $V_{i}^{2}(z,C)\ne V_{i}^{2}(a,S)$ for all $(a,S)\in \Gamma \backslash \{(z,C)\}$, $V_{i}^{3}(w,D)\ne V_{i}^{3}(a,S)$ and $V_{i}^{5}(w,D)\ne V_{i}^{5}(a,S)$ for all $(a,S)\in \Gamma \backslash \{(w,D)\}$, and $V_{i}^{4}(z,C)\ne V_{i}^{4}(a,S)$ and $V_{i}^{5}(z,C)\ne V_{i}^{5}(a,S)$ for all $(a,S)\in \Gamma \backslash \{(z,C)\}$. Note that it is possible to choose $\gamma ^{1}$, $\gamma ^{2}$, and $\epsilon ^{1}$ since both extreme and strong consequentialists do not impose any restriction on the strict part in terms of their evaluation functions on $\Gamma$.

Table 1 Profiles in the proof of Lemma 2

Let $\mapsto$ mean $``$implies by BI on $\Gamma$” and let $\rightrightarrows$ mean $``$implies by WP on $\Gamma$ and the transitivity of $R_{V}$.” Then we have $(x,A)P_{V^{1}}(y,B)\mapsto (x,A)P_{V^{2}}(y,B)\rightrightarrows (z,C)P_{V^{2}}(y,B)\mapsto (z,C)P_{V^{3}}(y,B)\rightrightarrows (w,D)P_{V^{3}}(y,B)\mapsto (w,D)P_{V^{4}}(y,B)\rightrightarrows (w,D)P_{V^{4}}(z,C)\mapsto (w,D)P_{V^{5}}(z,C)\rightrightarrows (x,A)P_{V^{5}}(y,B)\mapsto (x,A)P_{V}(y,B)$.$\square$

Lemma 3

If an ESOFL defined on $\mathcal D$ satisfies WP on $\Gamma$ and BI on $\Gamma$, then for all $V\in \mathcal D$, $PP_{V}$ is both transitive and asymmetric on $\Gamma$.

Proof

Since $P_{V}$ is asymmetric, it follows from Lemma 2 that $PP_{V}$ is asymmetric on $\Gamma$. We now show that $PP_{V}$ is transitive on $\Gamma$. Suppose that $(x,A)PP_{V}(y,B)$ and $(y,B)PP_{V}(z,C)$ for all $(x,A),(y,B),(z,C)\in \Gamma$. By Lemma 2, $(x,A)$, $(y,B)$, and $(z,C)$ are all distinct. By the definition of SSP, there exists $V^{1}\in \mathcal D$ such that $V^{1}(x,A)=V(x,A)-\eta ^{1}$, $V^{1}(y,B)=V(y,B)+\eta ^{2}$, and $V^{1}(a,S)=V(a,S)$ for all $(a,S)\in \Gamma \backslash \{(x,A),(y,B)\}$, where $\eta ^{1},\eta ^{2}\gg 0$ and $(x,A)P_{V^{1}}(y,B)$. Consider the profiles $V^{2}$, $V^{3}$, and $V^{4}$ in Table 2, where $\eta ^{1}\gg \gamma ^{1}\gg \epsilon ^{1}\gg 0$ and $\eta ^{2}\gg \gamma ^{2}\gg 0$, and for all $i\in N$, $V_{i}^{2}(z,C)\ne V_{i}^{2}(a,S)$ for all $(a,S)\in \Gamma \backslash \{(z,C)\}$, $V_{i}^{2}(w,D)\ne V_{i}^{2}(b,T)$ for all $(b,T)\in \Gamma \backslash \{(w,D)\}$, and $V_{i}^{3}(x,A)\ne V_{i}^{3}(a,S)$ and $V_{i}^{4}(x,A)\ne V_{i}^{4}(a,S)$ for all $(a,S)\in \Gamma \backslash \{(x,A)\}$. Note that it is possible to choose $\gamma ^{1}$, $\gamma ^{2}$, and $\epsilon ^{1}$ since both extreme and strong consequentialists do not impose any restriction on the strict part in terms of their evaluation functions on $\Gamma$.

Table 2 Profiles in the proof of Lemma 3

As in Lemma 2, let $\mapsto$ mean $``$implies by BI on $\Gamma$ and let $\rightrightarrows$ mean $``$implies by WP on $\Gamma$ and the transitivity of $R_{V}$.” Then we get $(x,A)P_{V^{1}}(y,B)\mapsto (x,A)P_{V^{2}}(y,B)\rightrightarrows (w,D)P_{V^{2}}(z,C)\mapsto (w,D)P_{V^{3}}(z,C)\rightrightarrows (x,A)P_{V^{3}}(y,B)\mapsto (x,A)P_{V^{4}}(y,B)$.

By a similar reasoning above, there exists $V^{5}\in \mathcal D$ such that $(y,B)P_{V^{5}}(z,C)$, and $V^{5}(z,C)=V(z,C)+\epsilon ^{2}$ and $V^{5}(a,S)=V(a,S)$ for all $(a,S)\in \Gamma \backslash \{(z,C)\}$, where $\epsilon ^{2}\gg 0$, and for all $i\in N$, $V_{i}^{5}(z,C)\ne V_{i}^{5}(a,S)$ for all $(a,S)\in \Gamma \backslash \{(z,C)\}$.

Consider $V^{6}$ such that $V^{6}(x,A)=V(x,A)-\epsilon ^{1}$, $V^{6}(z,C)=V(z,C)+\epsilon ^{2}$, and $V^{6}(a,S)=V(a,S)$ for all $(a,S)\in \Gamma \backslash \{(x,A),(z,C)\}$. If necessary, retake $\epsilon ^{2}$, and it follows from BI on $\Gamma$ that $(x,A)P_{V^{6}}(y,B)$ and $(y,B)P_{V^{6}}(z,C)$. We have $(x,A)P_{V^{6}}(z,C)$ by the transitivity of $R_{V^{6}}$. By the definition of SSP, we obtain $(x,A)PP_{V}(z,C)$.$\square$

Lemma 4

If an ESOFL defined on $\mathcal D$ satisfies WP on $\Gamma$ and BI on $\Gamma$, then it satisfies WN on $\Gamma$.

Proof

Consider $V$ and $V'$ such that $V(x,A)=V'(z,C)$ and $V(y,B)=V'(w,D)$. Suppose that $(x,A)PP_{V}(y,B)$. We show that $(z,C)PP_{V'}(w,D)$. For any other cases, an argument below will be used to complete the proof of Lemma 4. By the definition of SSP, there exists $V^{1}\in \mathcal D$ such that $V^{1}(x,A)=V(x,A)-\eta ^{1}$, $V^{1}(y,B)=V(y,B)+\eta ^{2}$, $V^{1}(a,S)=V(a,S)$ for all $(a,S)\in \Gamma \backslash \{(x,A),(y,B)\}$, where $\eta ^{1}$, $\eta ^{2}\gg 0$, and $(x,A)P_{V^{1}}(y,B)$. Suppose that $\{(x,A),(y,B)\}$ and $\{(z,C),(w,D)\}$ are disjoint. Since the proof is similar if this is not the case, we prove only the case.

Consider the $V^{2},V^{3}\in \mathcal D$ in Table 3, where $\eta ^{1}\gg \gamma ^{1}\gg 0$ and $\eta ^{2}\gg \gamma ^{2}\gg 0$, and for all $i\in N$, $V_{i}^{2}(x,A)\ne V_{i}^{2}(a,S)$ and $V_{i}^{3}(x,A)\ne V_{i}^{3}(a,S)$ for all $(a,S)\in \Gamma \backslash \{(x,A)\}$ and $V_{i}^{2}(y,B)\ne V_{i}^{2}(b,T)$ and $V_{i}^{3}(y,B)\ne V_{i}^{3}(b,T)$ for all $(b,T)\in \Gamma \backslash \{(y,B)\}$. Note that it is possible to choose $\eta ^{1}$, $\eta ^{2}$, $\gamma ^{1}$, and $\gamma ^{2}$ since both extreme and strong consequentialists do not impose any restriction on the strict part in terms of their evaluation functions on $\Gamma$.

Table 3 Profiles in the proof of Lemma 4

As in Lemma 2, let $\mapsto$ mean $``$implies by BI on $\Gamma ''$ and $\rightrightarrows$ mean $``$implies by WP on $\Gamma$ and the transitivity of $R_{V}$”. Then, we obtain $(x,A)P_{V^{1}}(y,B)\mapsto (x,A)P_{V^{2}}(y,B)\rightrightarrows (z,C)P_{V^{2}}(w,D)\mapsto (z,C)P_{V^{3}}(w,D)$. By the definition of SSP, we obtain $(z,C)PP_{V'}(w,D)$.$\square$

Since $\Gamma$ contains a free triple $\Delta$ with respect to $\mathcal D$, we can define $P^{*}$ on the individual evaluation space $\mathbb R ^{n}$ as follows, $\alpha P^{*}\beta$ if and only if there exist $V\in \mathcal D$ and $(x,A),(y,B)\in \Delta$ such that $V(x,A)=\alpha$, $V(y,B)=\beta$ and $(x,A)PP_{V}(y,B)$. The next result shows the properties of $P^{*}$.

Lemma 5

If an ESOFL defined on $\mathcal D$ satisfies WP on $\Gamma$ and BI on $\Gamma$, then $P^{*}$ is both transitive and asymmetric on $\Gamma$ and for all $(x,A),(y,B)\in \Gamma$ and all $V\in \mathcal D$, $V(x,A)P^{*}V(y,B)$ implies $(x,A)P_{V}(y,B)$.

Proof

It follows from Lemmas 3 and 4 that $P^{*}$ is both transitive and asymmetric on $\Gamma$. Let $V(x,A)=\alpha$ and $V(y,B)=\beta$. Suppose that $\alpha P^{*}\beta$. By the definition of $P^{*}$, there exist $V'\in \mathcal D$ and $(a,S),(b,T)\in \Delta$ such that $V'(a,S)=\alpha$, $V'(b,T)=\beta$, and $(a,S)PP_{V'}(b,T)$. From Lemma 4, we have $(x,A)PP_{V}(y,B)$. It follows from Lemma 2 that $(x,A)P_{V}(y,B)$.$\square$

Consider the following subsets of the individual evaluation space $\mathbb R ^{n}$,

$$\begin{aligned} L(\alpha ^{*})=\{\alpha |\alpha ^{*}P^{*}\alpha \}; M(\alpha ^{*})=\{\alpha |\alpha P^{*}\alpha ^{*}\}; N(\alpha ^{*})=\mathbb R ^{n}\backslash \{L(\alpha ^{*})\cup M(\alpha ^{*})\}. \end{aligned}$$

Note that $L(\cdot )$ and $M(\cdot )$ are open sets by the definition of SSP. The following three results show the properties of these subsets.

Lemma 6

If an ESOFL defined on $\mathcal D$ satisfies WP on $\Gamma$ and BI on $\Gamma$, then for all $\alpha ^{*},\eta \in \mathbb R ^{n}$ with $\eta \gg 0$, $\alpha ^{*}+\eta \in M(\alpha ^{*})$ and $\alpha ^{*}-\eta \in L(\alpha ^{*})$.

Proof

We show that $\alpha ^{*}+\eta \in M(\alpha ^{*})$. It is possible to show that $\alpha ^{*}-\eta \in L(\alpha ^{*})$ by a similar argument. Let $\Delta$ be a free triple with respect to $\mathcal D$. Choose any $(x,A),(y,B)\in \Delta$ and any $V,V'\in \mathcal D$ such that $V(a,S)=V'(a,S)$ for all $(a,S)\in \Gamma \backslash \{(x,A),(y,B)\}$, $V(x,A)=\alpha ^{*}+\eta -\epsilon ^{1}$, $V(y,B)=\alpha ^{*}+\epsilon ^{2}$, $V'(x,A)=\alpha ^{*}+\eta$, and $V'(y,B)=\alpha ^{*}$, where $\epsilon ^{1},\epsilon ^{2}\gg 0$ and $\eta -\epsilon ^{1}\gg \epsilon ^{2}$. By WP on $\Gamma$, we have $(x,A)P_{V}(y,B)$. By the definition of SSP, we can get $(x,A)PP_{V'}(y,B)$. Therefore, $\alpha ^{*}+\eta \in M(\alpha ^{*})$.$\square$

Lemma 7

If an ESOFL defined on $\mathcal D$ satisfies WP on $\Gamma$ and BI on $\Gamma$, then for all $\alpha ,\alpha ',\eta ,\eta '\in \mathbb R ^{n}$ with $\eta ,\eta '\gg 0$, $\alpha \in N(\alpha ')$ implies $\alpha +\eta \in M(\alpha '-\eta ')$.

Proof

Let $\Delta =\{(x,A),(y,B),(z,C)\}$ be a free triple with respect to $\mathcal D$. Then there exists $V\in \mathcal D$ with $\alpha +\eta \gg V(x,A)\gg V(y,B)\gg \alpha$ and $\alpha '\gg V(z,C)\gg \alpha '-\eta '$. Suppose $(z,C)P_{V}(y,B)$. Choose any $V'\in \mathcal D$ such that $V'(a,S)=V(a,S)$ for all $(a,S)\in \Gamma \backslash \{(y,B),(z,C)\}$, $V'(y,B)=V(y,B)-\eta ^{1}\gg \alpha$, and $\alpha '\gg V'(z,C)=V(z,C)+\eta ^{2}$, where $\eta ^{1},\eta ^{2}\gg 0$. By the definition of SSP, $(z,C)PP_{V'}(y,B)$. By the definition of $P^{*}$, $V(z,C)+\eta ^{2}P^{*}V(y,B)-\eta ^{1}$. By Lemma 6, $\alpha 'P^{*}V(z,C)+\eta ^{2}$ and $V(y,B)-\eta ^{1}P^{*}\alpha$. By the transitivity of $P^{*}$, $\alpha 'P^{*}\alpha$, which implies that $\alpha '\in M(\alpha )$. If $\alpha '\in M(\alpha )$, then $\alpha ' \notin N(\alpha )$. By the definition of $N(\cdot )$, we have $\alpha \notin N(\alpha ')$. Hence, if $\alpha \in N(\alpha ')$, then $\alpha '\notin M(\alpha )$, which implies $(y,B)R_{V}(z,C)$. Then WP on $\Gamma$ implies $(x,A)P_{V}(y,B)$ and so if $\alpha \in N(\alpha ')$, then $(x,A)P_{V}(z,C)$ by the transitivity of $R_{V}$. We can show that $(x,A)P_{V}(z,C)$ implies $\alpha +\eta P^{*}\alpha '-\eta '$ by a similar reasoning above and so if $\alpha \in N(\alpha ')$, then $\alpha +\eta P^{*}\alpha '-\eta '$, which implies $\alpha +\eta \in M(\alpha '-\eta ')$.

$\square$

Lemma 8

If an ESOFL defined on $\mathcal D$ satisfies WP on $\Gamma$, BI on $\Gamma$, and PC on $\Gamma$, then

• (a) For all $\alpha ,\alpha ',\epsilon \in \mathbb R ^{n}$ with $\epsilon \gg 0$, if $\alpha +\eta \in M(\alpha '+\epsilon )$ for all $\eta \gg 0$, then $\alpha \in M(\alpha ')$;

• (b) There exist no $\alpha ,\alpha ^{*},\gamma \in \mathbb R ^{n}$ with $\gamma \gg 0$ such that $\alpha ,\alpha +\gamma \in N(\alpha ^{*})$.

Proof

• (a) Given $\epsilon \gg 0$, let $\epsilon '\gg 0$ be specified as in the statement of PC on $\Gamma$. Choose any $\eta \gg 0$ such that $\eta \ll \epsilon '$. Since $\alpha +\eta \in M(\alpha '+\epsilon )$ implies $\alpha +\eta P^{*}\alpha '+\epsilon$, there exist $\bar{V}\in \mathcal D$ and $(x,A),(y,B)\in \Delta$ such that $\bar{V}(x,A)=\alpha +\eta$, $\bar{V}(y,B)=\alpha '+\epsilon$ and $(x,A)PP_{\bar{V}}(y,B)$, where $\Delta$ is a free triple with respect to $\mathcal D$. By the definition of SSP, there exists $V\in \mathcal D$ such that $\alpha +\eta \gg V(x,A)$ and $V(y,B)\gg \alpha '+\epsilon$ and $V(a,S)=\bar{V}(a,S)$ for all $(a,S)\in \Gamma \backslash \{(x,A),(y,B)\}$, and $(x,A)P_{V}(y,B)$. By PC on $\Gamma$, there exists $V'\in \mathcal D$ such that $V'(x,A)\ll V(x,A)-\epsilon '$ and $V'(y,B)\gg V(y,B)-\epsilon$, and $(x,A)P_{V'}(y,B)$. However, we have $V'(x,A)\ll \alpha +\eta -\epsilon '\ll \alpha$ and $V'(y,B)\gg \alpha '$. Hence we have $\alpha P^{*}\alpha '$, which implies $\alpha \in M(\alpha ')$.

• (b) Suppose that $\alpha +\gamma \in N(\alpha ^{*})$. By the definition of $N(\cdot )$, we have $\alpha ^{*}\in N(\alpha +\gamma )$. Choose any $\gamma '\gg 0$ such that $\gamma '\ll \gamma$. It follows from Lemma 7 that $\alpha ^{*}+\eta \in M(\alpha +\gamma ')$ for all $\eta \gg 0$. Part (a) of Lemma 8 implies that $\alpha ^{*}\in M(\alpha )$. Therefore, we have $\alpha \notin N(\alpha ^{*})$. $\square$

Define $I^{*}$ as $\alpha I^{*}\alpha '\Leftrightarrow \alpha \in N(\alpha ')$ for all $\alpha ,\alpha '\in \mathbb R ^{n}$. To show that $N(\alpha ^{*})$ is an equivalence set, it must be shown that $I^{*}$ is an equivalence relation.

Lemma 9

If an ESOFL defined on $\mathcal D$ satisfies WP on $\Gamma$, BI on $\Gamma$, and PC on $\Gamma$, then $I^{*}$ is an equivalence relation.

Proof

It is clear that $I^{*}$ is reflexive and symmetric. Suppose that $I^{*}$ is not transitive. Then there exist $\alpha ,\alpha ',\alpha ''\in \mathbb R ^{n}$ such that $\alpha I^{*}\alpha '$, $\alpha 'I^{*}\alpha ''$ and $\alpha P^{*}\alpha ''$. Since $M$ is open, there exist $\eta ^{1},\eta ^{2}\gg 0$ such that $\alpha -\eta ^{1}P^{*}\alpha ''+\eta ^{2}$. By part (b) of Lemma 8, we have $\alpha -\eta ^{1}\in L(\alpha ')$, and therefore $\alpha 'P^{*}\alpha -\eta ^{1}$ and $\alpha '' +\eta ^{2}\in M(\alpha ')$, and therefore $\alpha ''+\eta ^{2}P^{*}\alpha '$. However, since $P^{*}$ is transitive, we have a contradiction.$\square$

We are now ready to prove Proposition 3.

Proof of Proposition 3

It follows from Lemmas 5 and 9 that $R^{*}$ is an ordering on $\mathbb R ^{n}$ by defining $\alpha R^{*}\alpha '\Leftrightarrow \alpha \notin L(\alpha ')$. Since $L$ and $M$ are open, their complements $\{\alpha |\alpha R^{*}\alpha '\}$ and $\{\alpha |\alpha 'R^{*}\alpha \}$ are closed, which implies that $R^{*}$ is continuous. By the representation theorem of Debreu (1954), there exists a continuous real-valued function $W$ with the property that $\alpha R^{*}\alpha '$ if and only if $W(\alpha )\ge W(\alpha ')$. Since $\alpha \gg \alpha '$ implies $W(\alpha )>W(\alpha ')$, $W$ is weakly monotonic. Thus, the proof of Proposition 3 is completed. $\square$

Next, we prove a similar result to Claim 1 in Bordes et al. (2005).

Lemma 10

Suppose that $W^{1}:\mathbb R ^{n}\rightarrow \mathbb R$ and $W^{2}:\mathbb R ^{n}\rightarrow \mathbb R$ are continuous and weakly monotonic real-valued functions. Unless $W^{1}=W^{2}$, there exist $\alpha =(\alpha _{1},\ldots ,\alpha _{n}) ,\beta =(\beta _{1},\ldots ,\beta _{n}) \in \mathbb R ^{n}$ with $\alpha _{i}\ne \beta _{i}$ for all $i\in \{1,\ldots ,n\}$, and $W^{1}(\alpha )>W^{1}(\beta )$ but $W^{2}(\beta )>W^{2}(\alpha )$.

Proof

Suppose that $W^{1}\ne W^{2}$. There exist $\alpha =(\alpha _{1},\ldots ,\alpha _{n}),\gamma =(\gamma _{1},\ldots ,\gamma _{n})\in \mathbb R ^{n}$ such that $W^{1}(\alpha )>W^{1}(\gamma )$ but $W^{2}(\gamma )\ge W^{2}(\alpha )$. Since $W^{2}$ is weakly monotonic, we have $W^{2}(\gamma +\epsilon )>W^{2}(\gamma )$ for all $\epsilon \gg 0$ and so $W^{2}(\gamma +\epsilon )>W^{2}(\alpha )$. But the continuity of $W^{1}$ implies that $W^{1}(\alpha )>W^{1}(\gamma +\epsilon )$ for all small enough $\epsilon \gg 0$. Therefore, we have the result for $\beta =\gamma +\epsilon$ when $\epsilon =(\epsilon _{1},\ldots ,\epsilon _{n})\gg 0$ is small enough and $\alpha _{i}\ne \beta _{i}=\gamma _{i}+\epsilon _{i}$ for all $i\in \{1,\ldots ,n\}$.$\square$

Finally, we prove Theorem 5.

Proof of Theorem 5

Consider any domain $\mathcal{D }\in \fancyscript{D}_{E\cup S}\backslash \{\mathcal{D }_{E}\}$. We assume that there exists at least one extreme consequentialist. It is possible to prove Theorem 5 by a similar procedure below even if every individual is a strong consequentialist. Suppose that an ESOFL defined on $\mathcal D$ satisfies WP, BI, and PC. We now show that the ESOFL is defined in terms of a restricted weak RBSF for every non-trivial pair. Note that for any pair $\{(a,S),(b,T)\}\subseteq \Omega$, the pair is non-trivial with respect to $\mathcal D$ if and only if $a\ne b$. Since $\# X\ge 4$, there exist at least two non-trivial pairs with respect to $\mathcal D$. Consider any two non-trivial pairs $\{(x,A),(y,B)\}$ and $\{(z,C),(w,D)\}$ with respect to $\mathcal D$. We now assume that x, y, z, and w are all distinct. The proof is similar for any other cases.

Since $\# X\!\ge \! 4$, for any non-trivial pair $\{(a,S),(b,T)\}$ with respect to $\mathcal D$, there exist at least two alternatives $\{(c^{1},U^{1}),(c^{2},U^{2})\}\subseteq \Omega$ such that the triple $\{(b,T),(c^{1},U^{1}),(c^{2},U^{2})\}$ is free with respect to $\mathcal D$. Consider the following sequence of subsets of $\Omega$, $\Gamma ^{1},\ldots ,\Gamma ^{6}$ such that $\Gamma ^{1}\!=\!\{(x,A),(y,B),(z,B^{1}),(w,B^{2})\}$, $\Gamma ^{2}\!=\!\{(z,B^{1}),(w,B^{2}),(x,B^{3})\}$, $\Gamma ^{3}\!=\!\{(w,B^{2}),(x,B^{3}),(y,B^{4}),(z,C^{1})\}$, $\Gamma ^{4}\!=\!\{(y,B^{4}),(z,C^{1}),(w,C^{2}),(x,C^{3})\}$, $\Gamma ^{5}\!=\!\{(w,C^{2}),(x,C^{3}),(y,C^{4})\}$, and $\Gamma ^{6}\!=\!\{(x,C^{3}),(y,C^{4}),(z,C),(w,D)\}$, where $\{(y,B),(z,B^{1}),(w,B^{2})\}$, $\{(z,B^{1}),(w,B^{2}),$ $(x,B^{3})\}$, $\{(w,B^{2}),(x,B^{3}),(y,B^{4})\}$, $\{(z,C^{1}),(w,C^{2}),(x,C^{3})\}$, $\{(w,C^{2}),(x,C^{3}),$ $(y,C^{4})\}$, and $\{(x,C^{3}),(y,C^{4}),(z,C)\}$ are free triples with respect to $\mathcal D$.

Then, $\Gamma ^{1}$, $\Gamma ^{3}$, $\Gamma ^{4}$, and $\Gamma ^{6}$ each contains a free triple with respect to $\mathcal D$. By Proposition 3, the ESOFL is defined in terms of a restricted weak RBSF for $\Gamma ^{i}$, $i\in \{1,3,4,6\}$. $\Gamma ^{2}$ and $\Gamma ^{5}$ are free triples with respect to $\mathcal D$. Therefore, Proposition 2 can be applied to $\Gamma ^{2}$ and $\Gamma ^{5}$, and the ESOFL is defined in terms of a restricted weak RBSF for $\Gamma ^{i}$, $i\in \{2,5\}$.

Let $W_{|\Gamma ^{1}}^{1}$ and $W_{|\Gamma ^{2}}^{2}$ be the two continuous and weakly monotonic real-valued functions applied to $\Gamma ^{1}$ and $\Gamma ^{2}$, respectively. Unless $W_{|\Gamma ^{1}}^{1}\ne W_{|\Gamma ^{2}}^{2}$, there exist $\alpha ,\beta \in \mathbb R ^{n}$ with $\alpha _{i}\ne \beta _{i}$ for all $i\in \{1,\ldots ,n\}$ such that $W_{|\Gamma ^{1}}^{1}(\alpha )>W_{|\Gamma ^{1}}^{1}(\beta )$ but $W_{|\Gamma ^{2}}^{2}(\beta )>W_{|\Gamma ^{2}}^{2}(\alpha )$ by Lemma 10. Because $\Gamma ^{1}$ and $\Gamma ^{2}$ contain a common pair $\{(z,B^{1}),(w,B^{2})\}$ and both extreme and strong consequentialists impose no restriction on the strict part in terms of their evaluation functions on $\{(z,B^{1}),(w,B^{2})\}$, there exists $V\in \mathcal D$ with $V((z,B^{1}))=\alpha$ and $V((w,B^{2}))=\beta$. By the definitions of $W_{|\Gamma ^{1}}^{1}$ and $W_{|\Gamma ^{2}}^{2}$, $W_{|\Gamma ^{1}}^{1}(\alpha )>W_{|\Gamma ^{1}}^{1}(\beta )$ implies that $(z,B^{1})P_{V}(w,B^{2})$, and $W_{|\Gamma ^{2}}^{2}(\beta )>W_{|\Gamma ^{2}}^{2}(\alpha )$ implies that $(w,B^{2})P_{V}(z,B^{1})$, which is a contradiction. Therefore, we have $W_{|\Gamma ^{1}}^{1}=W_{|\Gamma ^{2}}^{2}$. For any other $W_{|\Gamma ^{i}}^{i}$ and $W_{|\Gamma ^{i+1}}^{i+1}$, $i\in \{2,3,4,5\}$, we can show that $W_{|\Gamma ^{i}}^{i}=W_{|\Gamma ^{i+1}}^{i+1}$ by a similar procedure above. Thus, we have $W_{|\Gamma ^{i}}^{i}=W_{|\Gamma ^{j}}^{j}$ for all $i,j\in \{1,\ldots ,6\}$.

Any one of these restricted weak RBSFs for $\Gamma ^{i}$ can be employed as a common restricted weak RBSF for every non-trivial pair. Let $W$ be the continuous and weakly monotonic real-valued function. We now prove that it also applies to any trivial pair $\{(x,A),(y,B)\}$ with respect to $\mathcal D$. Note that for any pair $\{(a,S),(b,T)\}\subseteq \Omega$, the pair is trivial with respect to $\mathcal D$ if and only if $a=b$. Let $\{(x,A),(y,B)\}$ be any trivial pair with respect to $\mathcal D$. Let $V\in \mathcal D$ be any profile with $V(x,A)=\alpha$ and $V(y,B)=\beta$. Note that for all $i\in N$, if $\mathcal D _{i}=\mathcal D _{e}$, then we have $V_{i}(x,A)=V_{i}(y,B)$. Suppose $W(V(x,A))>W(V(y,B))$. Then, because $W$ is continuous, by the intermediate value theorem, there exists a $\lambda \in (0,1)$ such that $W(\alpha )>W(\gamma )>W(\beta )$, where $\gamma =(1-\lambda )\alpha +\lambda \beta$. Consider $V'\in \mathcal D$ and $(z,C)\in \Omega$ such that $\{(x,A),(z,C)\}$ and $\{(y,B),(z,C)\}$ are non-trivial pairs with respect to $\mathcal D$, and $V'(x,A)=\alpha$, $V'(y,B)=\beta$ and $V'(z,C)=\gamma$. Since the ESOFL is defined in terms of a restricted weak RBSF for every non-trivial pair, we obtain $(x,A)P_{V'}(z,C)$ and $(z,C)P_{V'}(y,B)$. By the transitivity of $R_{V'}$, we have $(x,A)P_{V'}(y,B)$. By BI, we have $(x,A)P_{V}(y,B)$.$\square$