A decomposition of strategy-proofness

Abstract

Strategy-proofness has been one of the central axioms in the theory of social choice. However, strategy-proofness often leads to impossibility results. We find that strategy-proofness is decomposed into three axioms: top-restricted AM-proofness, weak monotonicity, and individual bounded response. We present possibility results by dropping individual bounded response from strategy-proofness. One of the results supports the plurality rule which is one of the most widely used rules in practice.

Appendix: An axiomatic foundation of scoring rules

For the reader who hesitates to restrict analyses to scoring rules without reasons, we offer an axiomatic foundation of our scoring rules. To use an earlier result (Young 1975), consider social choice correspondences in the variable population model. Let ${\mathbb {N}}=\{1, 2, \dots , \}$ be the set of potential agents. Let F be a typical notation for a social choice correspondence. Then, for each nonempty finite set $V\subset {\mathbb {N}}$, F maps $\varvec{R}_V = (R_i)_{i\in V}$ to a nonempty subset of X. To avoid additional notations and complicated arguments, we introduce the first four axioms less formally than before.¹⁵ Anonymity means a symmetric treatment of the agents. Neutrality means a symmetric treatment of the alternatives. Consistency means that for each disjoint $V, V'\subset {\mathbb {N}}$, if $F(\varvec{R}_V)\cap F(\varvec{R}_{V'})\ne \emptyset$, then $F(\varvec{R}_V, \varvec{R}_{V'}) = F(\varvec{R}_V)\cap F(\varvec{R}_{V'})$. Continuity means that for each disjoint $V, V'\subset {\mathbb {N}}$, if $f(\varvec{R}_{V})=\{x\}$, then for a sufficiently large integer k, $f(\underbrace{\varvec{R}_V, \varvec{R}_V, \dots , \varvec{R}_{V},}_{\text {k times}} \varvec{R}_{V'})=\{x\}$. Young (1975) characterizes “scoring rules” by these four axioms. Since “scoring rules” in Young (1975) is different from ours, we need two more axioms. Weak monotonicity means that for each $V\subset {\mathbb {N}}$ and each $\varvec{R}_{V}$, if $x\in F(\varvec{R}_{V})$ and some agent $i\in V$ raises the position of x in $R_i$, then x is still in the social outcome. Least selectivity means that for some $V\subset {\mathbb {N}}$ and some $\varvec{R}_V$, $f(\varvec{R}_{V})\ne X$. These six axioms characterize the multi-valued version of our scoring rules.

Proposition

Let F be a social choice correspondence. Then, the following statements are equivalent:

• (i) F satisfies anonymity, neutrality, consistency, continuity, weak monotonicity, and least selectivity.

• (ii) F is a scoring rule, i.e., there are $s_1, s_2, \dots , s_n$ such that $s_1\ge s_2\ge \dots \ge s_m$ and $s_1 > s_m$, and for each $V\subset {\mathbb {N}}$ and each $\varvec{R}_V$,

  $$\begin{aligned} F(\varvec{R}_V)=\left\{ x\in X \mid \sum _{i\in V} s_{\rho _{R_i}(x)} \ge \sum _{i\in V} s_{\rho _{R_i}(y)}\quad \;\text {for each } y\in X\right\} . \end{aligned}$$

  (1)

Proof

(ii) $\Rightarrow$ (i) is easy. We prove (i) $\Rightarrow$ (ii). Since F satisfies anonymity, neutrality, consistency, and continuity, by Young (1975), F is such that Eq. (1) holds. Thus, it suffices to show that $s_1\ge s_2 \ge \dots \ge s_m$ and $s_1 > s_m$. (Young (1975) does not include these conditions in the definition of scoring rules.) First, we show $s_1\ge s_2 \ge \dots \ge s_m$. Suppose $s_k < s_{k+1}$ for some k. Let $V=\{1, 2, \dots , m\}$ and $\varvec{R}_V$ be the one in Table 7.

Table 7 $\varvec{R}_V$ in the proof of Proposition

Then, $f(\varvec{R}_{V})=X$. Raise the position of $x_{k+1}$ in agent 1’s preference. Then, the score of $x_{k+1}$ decreases while the score of $x_k$ increases. Thus, $x_{k+1}$ is excluded from the social outcome by agent 1’s change of his preference. This is a contradiction to weak monotonicity. Thus, $s_1 \ge s_2 \ge \dots \ge s_m$.

Finally, we show $s_1 > s_m$. Suppose $s_1\le s_m$. $s_1 \ge s_2 \ge \dots \ge s_m$ and $s_1\le s_m$ together imply $s_1 = s_2 = \dots = s_m$. Then, for each $V\subset {\mathbb {N}}$ and each $\varvec{R}_V$, $f(\varvec{R}_{V})=X$. This is a contradiction to least selectivity. $\square$

By tie-breaking the value of F in Proposition by choosing the alternative with the least index, we have scoring rules in Theorems 3.2 and 3.3. The difference between the fixed population and the variable population models is not essential. Our results hold also in the variable population model.