Social choice, the strong Pareto principle, and conditional decisiveness

Abstract

This paper examines social choice theory with the strong Pareto principle. The notion of conditional decisiveness is introduced to clarify the underlying power structure behind strongly Paretian aggregation rules satisfying binary independence. We discuss the various degrees of social rationality: transitivity, semi-transitivity, the interval-order property, quasi-transitivity, and acyclicity.

Appendix: Proofs

Lemma 1

Let $W$ be a QSDF satisfying strong Pareto and binary independence, and let $A \subsetneq \mathcal{N }$. If $B \subseteq \mathcal{N } \setminus A$ is $A$-conditionally decisive over some pair $(x,y)$ for $W$, then it is $A$-conditionally decisive for $W$.

Proof

Let $W$ be a QSDF satisfying strong Pareto and binary independence, and let $A \subsetneq \mathcal{N }$. Suppose that $B \subseteq \mathcal{N } \setminus A$ is $A$-conditionally decisive over $(x,y)$ for $W$. We first prove the following claim.

Claim

For all $z \in X,\,B$ is $A$-conditionally decisive over $(y,z)$ for $W$.

Let $\mathbf{R} \in \mathcal{R }^{\mathcal{N }}$ be such that

$$\begin{aligned} \{ (x,y),(y,z) \} \subseteq I(R_i) \text{ for} \text{ all} i \in A ,\\ (x,y) \in P(R_i) \text{ for} \text{ all} i \in B ,\\ (y,z) \in P(R_i) \text{ for} \text{ all} i \notin A. \end{aligned}$$

Since $B$ is $A$-conditionally decisive over $(x,y)$ for $W$, we have $(x,y) \in P(W(\mathbf{R}))$. Strong Pareto implies that $(y,z) \in P(W(\mathbf{R}))$. By quasi-transitivity, $(x,z) \in P(W(\mathbf{R}))$. The preferences of individuals outside of $B \cup A$ over $x$ and $z$ are not specified, and thus, binary independence implies that $B$ is $A$-conditionally decisive over $(x,z)$ for $W$. Hence, for all $z \in X,\,B$ is $A$-conditionally decisive over $(x,z)$ for $W$.

Let $\mathbf{R}^{\prime } \in \mathcal{R }^{\mathcal{N }}$ be such that

$$\begin{aligned} \{ (x,y),(y,z) \} \subseteq I(R^{\prime }_i) \text{ for} \text{ all} i \in A ,\\ (x,z) \in P(R^{\prime }_i) \text{ for} \text{ all} i \in B ,\\ (y,x) \in P(R^{\prime }_i) \text{ for} \text{ all} i \notin A. \end{aligned}$$

Since $B$ is $A$-conditionally decisive over $(x,z)$ for $W$, we have $(x,z) \in P(W(\mathbf{R}^{\prime }))$. Strong Pareto implies that $(y,x) \in P(W(\mathbf{R}^{\prime }))$. By quasi-transitivity, $(y,z) \in P(W(\mathbf{R}^{\prime }))$. The preferences of individuals outside of $B \cup A$ over $y$ and $z$ are not specified, and thus, binary independence implies that $B$ is $A$-conditionally decisive over $(y,z)$ for $W$. Hence, for all $z \in X \setminus \{ x \},\,B$ is $A$-conditionally decisive over $(y,z)$ for $W$.

Take $z \in X \setminus \{ x \}$. Let $\mathbf{R}^{\prime \prime } \in \mathcal{R }^{\mathcal{N }}$ be such that

$$\begin{aligned} \{ (x,y),(y,z) \} \subseteq I(R^{\prime \prime }_i) \text{ for} \text{ all} i \in A ,\\ (y,z) \in P(R^{\prime \prime }_i) \text{ for} \text{ all} i \in B ,\\ (z,x) \in P(R^{\prime \prime }_i) \text{ for} \text{ all} i \notin A. \end{aligned}$$

Since $B$ is $A$-conditionally decisive over $(y,z)$ for $W$, we have $(y,z) \in P(W(\mathbf{R}^{\prime \prime }))$. Strong Pareto implies that $(z,x) \in P(W(\mathbf{R}^{\prime \prime }))$. By quasi-transitivity, $(y,x) \in P(W(\mathbf{R}^{\prime \prime }))$. The preferences of individuals outside of $B \cup A$ over $x$ and $y$ are not specified, and thus, binary independence implies that $B$ is $A$-conditionally decisive over $(y,x)$ for $W$. Hence, for all $z \in X ,\,B$ is $A$-conditionally decisive over $(y,z)$ for $W$. We thus complete the proof of the claim.

By applying the claim again, it follows that if $B$ is $A$-conditionally decisive over $(y,z)$ for $W$, then $B$ is $A$-conditionally decisive over $(z,w)$ for $W$. Hence, we have established that $B$ is $A$-conditionally decisive over $(z,w)$ for any choice of distinct alternatives $z$ and $w$. $\square$

Proof of Theorem 1

Let $A \subsetneq \mathcal{N }$. We now prove that $\mathcal D _W(A)$ satisfies the four properties of an ultrafilter.

• (i) Let $\mathbf{R} \in \mathcal{R }^{\mathcal{N }}$ be such that

  $$\begin{aligned} \big [ (x,y) \in I(R_i) \text{ for} \text{ all} i \in A \big ] \text{ and} \big [ (x,y) \in P(R_i) \text{ for} \text{ all} i \notin A \big ]. \end{aligned}$$

  Strong Pareto implies that $(x,y) \in P(W(\mathbf{R}))$. Hence, $\mathcal{N } \setminus A \in \mathcal D _W(A)$. Similarly, strong Pareto implies that $\emptyset \notin \mathcal D _W(A)$.

• (ii) Suppose that $B \in \mathcal D _W(A)$ and $B \subseteq B^{\prime } \subseteq \mathcal{N } \setminus A$. Let $\mathbf{R} \in \mathcal{R }^{\mathcal{N }}$ be such that

  $$\begin{aligned} \big [ (x,y) \in I(R_i) \text{ for} \text{ all} i \in A \big ] \text{ and} \big [ (x,y) \in P(R_i) \text{ for} \text{ all} i \in B^{\prime } \big ]. \end{aligned}$$

  Since $B \subseteq B^{\prime }$, we have

  $$\begin{aligned} \big [ (x,y) \in I(R_i) \text{ for} \text{ all} i \in A \big ] \text{ and} \big [ (x,y) \in P(R_i) \text{ for} \text{ all} i \in B \big ]. \end{aligned}$$

  Since $B \in \mathcal D _W(A)$, it follows that $(x,y) \in P(W(\mathbf{R}))$. Thus, $B^{\prime } \in \mathcal D _W(A)$.

• (iii) Take any three distinct alternatives $x,y, z \in X$. Suppose that $B \in \mathcal D _W(A)$ and $B^{\prime } \in \mathcal D _W(A)$. Let $\mathbf{R} \in \mathcal{R }^{\mathcal{N }}$ be such that

  $$\begin{aligned} \big [ (x,y) \in I(R_i) \text{ and} (y,z) \in I(R_i) \big ]&\text{ for} \text{ all} i \in A ,\\ (x,y) \in P(R_i)&\text{ for} \text{ all} i \in B,\\ (y,z) \in P(R_i)&\text{ for} \text{ all} i \in B^{\prime }. \end{aligned}$$

  Note that $(x,z) \in P(R_i)$ for all $i \in B \cap B^{\prime }$. Since $B,B^{\prime } \in \mathcal D _W(A)$, it follows that $(x,y) \in P(W(\mathbf{R}))$ and $(y,z) \in P(W(\mathbf{R}))$. Quasi-transitivity, which is implied by transitivity, yields

  $$\begin{aligned} (x,z) \in P(W(\mathbf{R})). \end{aligned}$$

  The preferences of individuals outside of $(B \cap B^{\prime }) \cup A$ over $x$ and $z$ are not specified, and thus, binary independence implies that $B \cap B^{\prime }$ is $A$-conditionally decisive over $(x,z)$ for $W$. Hence, $B \cap B^{\prime } \in \mathcal D _W(A)$.

• (iv) Suppose that $B \subseteq \mathcal{N } \setminus A$ and $B \notin \mathcal D _W(A)$. Since $B \notin \mathcal D _W(A)$, there exist $(x,y) \in X \times X$ and $\mathbf{R} \in \mathcal{R }^{\mathcal{N }}$ such that

  $$\begin{aligned} \big [ (x,y) \in I(R_i) \text{ for} \text{ all} i \in A \big ] \text{ and} \big [ (x,y) \in P(R_i) \text{ for} \text{ all} i \in B \big ], \end{aligned}$$

  but $(x,y) \notin P(W(\mathbf{R}))$. Completeness implies that $(y,x) \in W(\mathbf{R})$. Let $\mathbf{R}^{\prime } \in \mathcal{R }^{\mathcal{N }}$ be such that

  

  By construction, $(y,z) \in I(R^{\prime }_i) \text{ for} \text{ all} i \in A$. Binary independence implies that $(y,x) \in W(\mathbf{R}^{\prime })$. Since $\mathcal{N } \setminus A \in \mathcal D _W(A)$, we have $(x,z) \in P(W(\mathbf{R}^{\prime }))$. Transitivity implies that $(y,z) \in P(W(\mathbf{R}^{\prime }))$. Note that

  $$\begin{aligned} \big [ (y,z) \in I(R^{\prime }_i) \text{ for} \text{ all} i \in A \big ] \text{ and} \big [ (y,z) \in P(R^{\prime }_i) \text{ for} \text{ all} i \in \mathcal{N } \setminus (A \cup B) \big ]. \end{aligned}$$

  The preferences of individuals outside of $\mathcal{N } \setminus B$ over $y$ and $z$ are not specified, and thus, binary independence implies that $\mathcal{N } \setminus (B \cup A)$ is $A$-conditionally decisive over $(y,z)$ for $W$. By Lemma 1, it follows that $\mathcal{N } \setminus (B \cup A) \in \mathcal D _W(A)$. The proof is complete. $\square$

Proof of Theorem 2

In Theorem 1, we do not need rationality requirements to show properties (i) and (ii). Thus, (i) and (ii) can be proved in a similar manner. Moreover, we require only quasi-transitivity to show (iii), and thus, $\mathcal D _W(A)$ satisfies property (iii). Hence, it suffices to show (iv). Suppose that $B \subseteq \mathcal{N } \setminus A$ and $B \notin \mathcal D _W(A)$. Since $B \notin \mathcal D _W(A)$, there exist $(x,y) \in X \times X$ and $\mathbf{R} \in \mathcal{R }^{\mathcal{N }}$ such that

$$\begin{aligned} \big [ (x,y) \in I(R_i) \text{ for} \text{ all} i \in A \big ] \text{ and} \big [ (x,y) \in P(R_i) \text{ for} \text{ all} i \in B \big ], \end{aligned}$$

but $(x,y) \notin P(W(\mathbf{R}))$. Let $\mathbf{R}^{\prime } \in \mathcal{R }^{\mathcal{N }}$ be such that

Binary independence implies that $(x,y) \notin P(W(\mathbf{R}^{\prime }))$. Since $\mathcal{N } \setminus A \in \mathcal D _W(A)$, it follows that $(x,z) \in P(W(\mathbf{R}^{\prime }))$ and $(z,w) \in P(W(\mathbf{R}^{\prime }))$. Semi-transitivity implies that $(y,z) \in P(W(\mathbf{R}^{\prime }))$ or $(x,y) \in P(W(\mathbf{R}^{\prime }))$. Since $(x,y) \notin P(W(\mathbf{R}^{\prime }))$, it follows that $(y,z) \in P(W(\mathbf{R}^{\prime }))$. Binary independence and Lemma 1 together imply that $\mathcal{N } \setminus B \in \mathcal D _W(A)$. $\square$

Proof of Theorem 3

It suffices to show that $\mathcal D _W(A)$ has property (iv). Suppose that $B \subseteq \mathcal{N } \setminus A$ and $B \notin \mathcal D _W(A)$. Since $B \notin \mathcal D _W(A)$, there exist $(x,y) \in X \times X$ and $\mathbf{R} \in \mathcal{R }^{\mathcal{N }}$ such that

$$\begin{aligned} \big [ (x,y) \in I(R_i) \text{ for} \text{ all} i \in A \big ] \text{ and} \big [ (x,y) \in P(R_i) \text{ for} \text{ all} i \in B \big ], \end{aligned}$$

but $(x,y) \notin P(W(\mathbf{R}))$. Let $\mathbf{R}^{\prime } \in \mathcal{R }^{\mathcal{N }}$ be such that

Binary independence implies that $(x,y) \notin P(W(\mathbf{R}^{\prime }))$. Since $\mathcal{N } \setminus A \in \mathcal D _W(A),(x,z) \in P(W(\mathbf{R}^{\prime }))$ and $(w,y) \in P(W(\mathbf{R}^{\prime }))$. The interval-order property implies that $(x,y) \in P(W(\mathbf{R}^{\prime }))$ or $(w,z) \in P(W(\mathbf{R}^{\prime }))$. Since $(x,y) \notin P(W(\mathbf{R}^{\prime }))$, it follows that $(w,z) \in P(W(\mathbf{R}^{\prime }))$. Binary independence and Lemma 1 together imply that $\mathcal{N } \setminus B \in \mathcal D _W(A)$ $\square$

Proof of Theorem 4

It is obvious that properties (i) and(ii) are satisfied. Moreover, we employ only quasi-transitivity to show (iii) in Theorem 1; thus, $\mathcal D _W(A)$ satisfies property (iii) for all QSDFs. $\square$

Proof of Theorem 5

Let $W$ be an SDF satisfying strong Pareto, and let $A \subsetneq \mathcal{N }$.

It suffices to show that $\mathcal D _W(A)$ has property (v). By way of contradiction, suppose that there exist $K \in \mathbb{N }$ and $B^0,B^1,\dots ,B^K \in \mathcal D _W(A)$ such that $\bigcap _{k \in \{0,\dots ,K\}}B^k =\emptyset$. Let $x^0, x^1,\dots ,x^K$ be distinct alternatives in $X$. Since $X$ is an infinite set, we can choose them for any finite number $K$. Let $\mathbf{R} \in \mathcal{R }^N$ be such that

$$\begin{aligned} \{(x^0,x^1),(x^1,x^2),\dots ,(x^{K-1},x^K) \} \subseteq I(R_i)&\text{ for} \text{ all} i \in A,\\ (x^K,x^{0}) \in P(R_i)&\text{ for} \text{ all} i \in B^0,\\ (x^{0},x^{1}) \in P(R_i)&\text{ for} \text{ all} i \in B^1,\\ \qquad \qquad \qquad \vdots \\ (x^{K-1},x^{K}) \in P(R_i)&\text{ for} \text{ all} i \in B^K. \end{aligned}$$

We demonstrate that there is such a preference profile $\mathbf{R}^*$. By the assumption that $\bigcap _{k \in \{0,\dots ,K\}}B^k =\emptyset$, it follows that for every individual $i \in \bigcup _{k \in \{0,\dots ,K\}}B^k$, there exists $k \in \{ 0,\dots , K \}$ such that $i \notin B^k$. Therefore, each individual’s preferences specified above contain no strict-preference cycle, and thus, there exists the preference profile $\mathbf{R}^* \in \mathcal{R }^N$. Since $B^k \in \mathcal D _W(A)$ for all $k \in \{0,\dots ,K\}$,

$$\begin{aligned} (x^{k-1},x^k) \in P(W(\mathbf{R})) \text{ for} \text{ all} k \in \{1,\dots ,K\} \text{ and} (x^K, x^0)\in P(W(\mathbf{R})). \end{aligned}$$

This contradicts acyclicity. $\square$