Bosmans and Ooghe (2013, p. 155) incorrectly state that the axioms in their theorem are independent. In fact, anonymity is redundant. The theorem may therefore be strengthened as follows.

FormalPara Theorem

A quasi-ordering $$R$$ on $${\mathbb {R}}^{n}$$ satisfies continuity, weak Pareto and weak Hammond equity if and only if it is maximin.

FormalPara Proof

Let $$R$$ be a quasi-ordering on $${\mathbb {R}}^{n}$$ that satisfies continuity, weak Pareto and weak Hammond equity. Consider two utility vectors $$u,v\in {\mathbb {R}}^{n}$$.

First, we have to show that $$u_{(1)}>v_{(1)}$$ implies $$uPv$$. Let $$i\in N$$ be such that $$v_{i} = v_{(1)}$$. Construct vectors $$w,z\in {\mathbb {R}}^{n}$$ such that

\begin{aligned} v_{i}<w_{i}<z_{i}<&z_{(2)}= z_{(3)} = \cdots =z_{(n)} \\&<u_{(1)}\le \max (u_{(n)},v_{(n)}) < w_{(2)} = w_{(3)} = \cdots = w_{(n)}. \end{aligned}

By weak Pareto, we have $$uPz$$ and $$wPv$$. By weak Hammond equity, we have $$zRw$$. Hence, we obtain $$uPv$$ using transitivity.

Second, we have to show that $$u_{(1)}=v_{(1)}$$ implies $$uIv$$. This follows from part (ii) of the proof in Bosmans and Ooghe (2013, p. 155). $$\square$$

The standard Hammond equity axiom is stronger than weak Hammond equity (given transitivity). Hence, contrary to what Bosmans and Ooghe (2013, footnote 9) state, Miyagishima’s (2010) claim that maximin is the only quasi-ordering satisfying continuity, weak Pareto and Hammond equity is correct.