A quest for fundamental theorems of social choice

Abstract

We explore the frontier between possibility and impossibility results by analyzing different combinations of “pro-socialness” and “consistency” conditions. This exercise delivers stronger versions of four classical impossibility theorems, and offers a thorough understanding of connections among them. We also characterize social choice functions that are “independent of irrelevant alternatives,” which makes evident that the fundamental difficulty of social choice lies in “pairwise” consistency requirements. We also introduce a concise pedagogical approach to classical impossibility theorems.

Appendix: An alternative proof of Arrow’s theorem and the Gibbard–Satterthwaite theorem

While some of the proofs below are well-known, we include them in order to demonstrate an alternative approach of proving classical impossibility theorems, which starts from the Hansson–Denicolò theorem. By contrast, Yu (2013) avoids the extra step of establishing the Muller–Satterthwaite theorem, and derives them from Theorem 4, which reveals the fundamental difficulty to be pairwise consistency requirements. But the definition of (CM) involves one more attributive than (IIA) does. As a result, Yu (2013) demands a higher level of mathematical maturity. When we compare these two approaches for teaching purposes, the tradeoff between succinctness and ease thus depends on the audience.

First, Arrow’s impossibility theorem (stated in Sect. 6) is a straightforward consequence of the Hansson–Denicolò theorem as shown by Denicolò (1993).

Proof of Arrow’s theorem

A SCF $F^R$ is derived from $R$ in that for any $\vec {\succ }$, $a_iR(\vec {\succ })a_j$ for every $j\ne i$ is necessary and sufficient for $F^R(\vec {\succ })=a_i$, i.e., it selects the alternative ranked highest by $R$. $F^R$ obviously satisfies (WP) and (IIA), so the Hansson-Denicolò theorem says that $F^R$ presents a social choice dictator $n$. She is a social preference dictator too. If $a_i\succ _n a_j$, individual $n$ rules $F^R_{ij}(\vec {\succ })=a_i$, so $a_iR(\vec {\succ })a_j$ by (SP-AIIA). $\square$

Based on the Hansson–Denicolò theorem, we prove the Muller–Satterthwaite theorem (stated in Sect. 5) directly, which is a popular pathway to the theorem of Gibbard (1973) and Satterthwaite (1975).¹⁹ Denicolò (1993) does not offer this development.

Proof of Muller–Satterthwaite:

To show (IIA), let $F$ satisfy (M) and $\vec {\succ }$ and $\vec {\succ }'$ agree on $\{a_i, a_j\}$. Suppose $F(\vec {\succ })=a_i$ and $F(\vec {\succ }')=a_j$. By (M), $F_{ij}(\vec {\succ })$ should be both $a_i$ and $a_j$, a contradiction.

Suppose that (WP) is violated, i.e., for some profile $\vec {\succ }$, $F(\vec {\succ })=a_j$, where $a_i$ dominating $a_j$. By (M), $F_{ij}(\vec {\succ })=a_j$. By (O), there exists $\vec {\succ }'$ such that $F(\vec {\succ })=a_i$. By (M), $F_{ij}(\vec {\succ })=a_i$, a contradiction. $\square$

When we replace the preferences of individual $n$ in $\vec {\succ }$ with $\succ '_n\in \mathcal {P}$, denote the new profile $(\succ '_n,\vec {\succ }_{-n})$. Strategy-proofness ensures truth-telling by always switching to a worse alternative for an individual if only her misreport alters the outcome.

Definition 25

(SP). A SCF $F$ is strategy-proof if $F(\succ '_n,\vec {\succ }_{-n})\ne F(\vec {\succ })$ implies $F(\vec {\succ })\succ _n F(\succ '_n,\vec {\succ }_{-n})$ for every $n$, every $\vec {\succ }$, and every $\succ '_n$.

Theorem 11

(Gibbard–Satterthwaite). If $F$ satisfies (O) and (SP), then it is dictatorial.

Proof

It is a consequence of Theorem 5 and the sufficiency part of Lemma 10. $\square$

Muller and Satterthwaite (1977) demonstrate the equivalence of (SP) and (M). Our proof of the claim that (M) implies (SP) is shorter than the original version.

Lemma 10

$F$ satisfies (M) if and only if it satisfies (SP).²⁰

Proof

Sufficiency: Let $F(\vec {\succ })=a_i$ and $\vec {\succ }'$ maintain $a_i$’s positions in $\vec {\succ }$. Define $\vec {\succ }''\equiv (\succ '_1,\vec {\succ }_{-1})$, then $\vec {\succ }=(\succ _1,\vec {\succ }''_{-1})$. Suppose $a_j=F(\vec {\succ }'')\ne a_i$. (SP) demands $a_i\succ _1 a_j$ and $a_j\succ '_1 a_i$, so $\vec {\succ }'$ lowers $a_i$’s positions relative to $a_j$ for individual $1$, contradicting the assumption. So $F(\vec {\succ }'')=a_i$. Following the same logic, we can update $\vec {\succ }$ to $\vec {\succ }'$ one by one without altering the choice.

Necessity: Suppose $a_i=F(\succ '_n,\vec {\succ }_{-n})\succ _nF(\vec {\succ })=a_j$. Consider $\succ ''_n$ with $a_i\succ ''_na_j\succ ''_na_k$ for every $k\notin \{i,j\}$. We have $(\succ ''_n,\vec {\succ }_{-n})$ maintains $a_i$’s position in $(\succ '_n,\vec {\succ }_{-n})$ and $a_j$’s position in $\vec {\succ }$, but $F(\succ ''_n,\vec {\succ }_{-n})$ cannot be both $a_i$ and $a_j$. $\square$