Generalized rawlsianism

Abstract

This paper proposes and characterizes a family of social choice rules, including maximin and leximin, by considering only ordinal social choice in the sense that individual utilities are ordinally measurable and ordinally comparable. This family of rules, called generalized Rawlsianism, provides a unified approach for dealing with different informational constraints on ordinal interpersonal comparisons. Rank noncomparability, which states that individual utilities under the same social ranking should always be interpersonally noncomparable, is then proposed as a new basic invariance axiom. We show that a social welfare ordering with super domain is a generalized rank hierarchy if and only if it satisfies anonymity, nonnullity, full rank noncomparability, and the Pareto monotonicity principle; together with the Pigou–Dalton principle, generalized Rawlsianism can then be fully characterized. Our characterizations depend heavily on the result that a social welfare ordering with super domain is a generalized hierarchy if and only if it satisfies nonnullity, interpersonal noncomparability, and the Pareto monotonicity principle.

Appendix A: Proofs

1.1 Proof of Theorem 1

Necessity is obvious, and we only prove sufficiency. Suppose that $W:\mathcal {O} (X)^{N }\rightarrow \mathcal {O} (X)$ satisfies IIA, nonnullity, and the Pareto monotonicity principle. Note that the Pareto monotonicity principle implies the Pareto indifference principle, and thus by the strong neutrality theorem, strong neutrality must be satisfied. By Wilson’s independence theorem and the Pareto monotonicity principle, it must be weakly dictatorial. This dictator can be denoted by $\pi$(1).

Then consider a social welfare function $W_{1}: \mathcal {O}(X)^{N\backslash \{\pi (1)\} }\rightarrow \mathcal {O}(X)$ for the society $N \backslash \{ \pi (1)\}$ such that for all ${{\varvec{R}}} \in \mathcal {O}(X)^{N}$ and for all $x, y \in X$, if $x \sim _{R_{\pi \left( 1 \right) } } y$, then$W_{1}({{\varvec{R}}}_{N\backslash \{\pi (1)\}})(x, y)=W(R_{\pi (1)}, {{\varvec{R}}}_{N\backslash \{\pi (1)\}})(x, y)=W({{\varvec{R}}})(x, y)$. It is easy to check that this social welfare function $W_{1}$ is well defined and satisfies IIA and the Pareto monotonicity principle. If $W_{1}$ is null, then $\pi$(1) must be a strong dictator in the society N. If $W_{1}$ is nonnull, then again by Wilson’s theorem and the Pareto monotonicity principle, there must be a weak dictator in the society $N \backslash \{\pi (1)\}$, which can be denoted by $\pi$(2).

Similarly, for each $k \in$ {1, $\ldots , n$-2}, we can define a social welfare function $W_{k}: \mathcal {O}(X)^{N\backslash \{\pi (1), \ldots , \pi (k)\} }\rightarrow \mathcal {O}(X)$ for the society $N \backslash$ { $\pi$(1), $\ldots$, $\pi (k)\}$ such that for all R $\in \mathcal {O}(X)^{N}$ and for all $x, y \in X$, if $x \sim _{R_{\pi \left( 1 \right) } } y, \ldots , x \sim _{R_{\pi \left( k \right) } } y$, then $W_{k}({{\varvec{R}}}_{N\backslash \{\pi (1), \ldots , \pi (k)\}})(x, y)=W({{\varvec{R}}}_{\{\pi (1), \ldots , \pi (k)\}}$, ${{\varvec{R}}}_{N\backslash \{\pi (1), \ldots , \pi (k)\}})(x, y)=W({{\varvec{R}}})(x, y)$. It is easy to check that this social welfare function $W_{k}$ is well defined and satisfies IIA and the Pareto monotonicity principle. If $W_{k}$ is null, then $\pi (k)$ must be a strong dictator in the society $N \backslash \{ \pi (1), \ldots , \pi (k-1)\}$. If $W_{k}$ is nonnull, then again by Wilson’s theorem and the Pareto monotonicity principle, there must be a weak dictator in the society $N \backslash \{ \pi (1), \ldots , \pi (k)\}$, which can be denoted by $\pi (k+1)$.

Finally, if $k=n$- 2, then $\pi (n-1)$ is a weak dictator in the society $N \backslash \{ \pi (1), \ldots$, $\pi (n-2)\}$, and let $\pi$(n) be the single member in $N \backslash \{ \pi (1), \ldots , \pi (n-1)\}$. For all R $\in \mathcal {O}(X)^{N}$ and for all $x, y \in X$, if $x \sim _{R_{\pi \left( 1 \right) } } y, \ldots , x \sim _{R_{\pi \left( {n-1} \right) } } y$, and $x \succ _{R_{\pi \left( n \right) } } y$, then by the Pareto monotonicity principle there are only two cases: (1) $x \sim _{W(R)} y$; (2) $x \succ _{W(R)} y$. In the first case, $\pi$(n-1) must be a strong dictator in $N \backslash$ { $\pi$(1), $\ldots$, $\pi$(n-2)}; in the second case, $\pi$(n) must be a strong dictator in $N \backslash$ { $\pi$(1), $\ldots$, $\pi$($n-1$)}. $\square$

1.2 Proof of Theorem 2

We only prove sufficiency. Suppose that $\succsim _{E}$ has a super domain E and satisfies anonymity, nullity, full rank noncomparability, and the Pareto monotonicity principle. Without loss of any generality, suppose $E_{0} = \{{{\varvec{u}}} \in \mathbb {R}^{N} | u_{1} \le u_{2} \le \cdots \le u_{n}$} $\subseteq E$. Then by Corollary 1, $\succsim _{E}$ must be a generalized hierarchy in $E_{0}$, and let $\pi$: {1, 2, $\ldots , k$} $\rightarrow N$ denote the hierarchical injection. For any u, v in E, there must be some r, s in $E_{0}$ such that $\forall i \in N, u_{(i)}=r_{i}$ and $v_{(i)}=s_{i}$. By anonymity, ${{\varvec{u}}} \sim _{E} {{\varvec{r}}}$ and ${{\varvec{v}}} \sim _{E}\, {{\varvec{s}}}$, and hence ${{\varvec{u}}} \succsim _{E} {{\varvec{v}}}$ if and only if r $\succsim _{E}$ s. By the generalized hierarchy in $E_{0}, {{\varvec{r}}} \succ _{E}\; {{\varvec{s}}}$ if and only if there is some $i \in$ {1, 2, $\ldots , k$}, $r_{\pi (i)} > s_{\pi (i)}$, and for all $j < i, r_{\pi (j)}=s_{\pi (j)}$, which means that u $\succ _{E}$ v if and only if there is some $i \in \{1, 2, \ldots , k\}, u_{(\pi (i))} > v_{(\pi (i))}$, and for all $j < i, u_{(\pi (j))}=v_{(\pi (j))}$. Therefore, let $K = \{ \pi (1),\pi (2),\ldots ,\pi (k)\}$, and then $\succsim _{E}$ must be a generalized rank hierarchy. $\square$