A spatial analogue of May’s Theorem

Abstract

In a spatial model with Euclidean preferences, we establish that the geometric median satisfies Maskin monotonicity, anonymity, and neutrality. For three agents, it is the unique such rule.

Appendix: Independence of the axioms

We now show that the axioms are independent. First, consider the dictatorship social choice rule $\varphi _d$ such that $\varphi _d(Z)=z_1$ for all $Z\in X^N$. It is clear that this rule satisfies neutrality and Maskin monotonicity but violates anonymity.

Next, consider the social choice rule $\varphi _c$ such that, for each Z in $X^N$, $\varphi _c(Z)$ is the unique solution to

$$\begin{aligned} \min _{x\in X}\sum _{i=1}^n\left( z_i-x\right) ^2. \end{aligned}$$

(4)

The solution to (4) is the mean (or centroid) of the ideal points. Clearly $\varphi _c$ satisfies anonymity. It is also easy to see that $\varphi _c$ satisfies neutrality.

However, $\varphi _c$ does not satisfy Maskin monotonicity. Consider the profile of n agents $Z=(\text {e}_1,- \text {e}_1,0,\ldots ,0)$ with $\text {e}_i$ being the ith standard basis vector in $\mathbb {R}^d$. It follows that $\varphi _c(Z)=0$. Now $Z'=(\frac{1}{2}\text {e}_1,- \text {e}_1,0,\ldots ,0)\in MT(Z,\varphi _c(Z))$ by Lemma 3.1 but $\varphi _c(Z')\ne 0=\varphi _c(Z)$.

Finally consider the social choice rule $\varphi _m$ such that $\varphi _m(Z)$ solves

$$\begin{aligned} \min _{x\in X}\sum _{i=1}^n|z_i\cdot \text {e}_j-x\cdot \text {e}_j| \end{aligned}$$

for each $j\in \{1,\ldots ,d\}$. That is, $\varphi _m$ selects the coordinate-wise median of the ideal points. It is obvious $\varphi _m$ satisfies anonymity. It is also fairly easy to see that $\varphi _m$ satisfies Maskin monotonicity. To see this, simply apply the proof technique used in Lemma 3.2 (see Brady and Chambers 2015) for each coordinate. However, it is a well known fact that the coordinate-wise median does not satisfy neutrality as it is not equivariant with isometries.