Conditions on social-preference cycles

Abstract

Since Condorcet discovered the voting paradox in the simple majority rule, many scholars have tried to investigate conditions that yield “social-preference cycles”. The paradox can be extended to two main approaches. On the one hand, Kenneth Arrow developed a general framework of social choice theory; on the other hand, direct generalizations of the paradox were offered. The motivation and surface meaning of the two approaches are different, as are the assumed background conditions. In this paper, we investigate the relationship between the two approaches by taking a close look at two works, Ferejohn and Fishburn (J Econ Theory 21:28–45, 1979) and Schwartz (J Econ Theory 137:688–695, 2007).

Appendix: Ferejohn and Fishburn’s result

In this section, we briefly review Ferejohn and Fishburn’s (1979) framework. As noted before, Ferejohn and Fishburn (1979) restrict their attention to a special class of CCRs. First, they consider a collection of CCRs that satisfy Arrow’s IIA. They term them binary decision rules (BDRs). Furthermore, they introduce the concept of constitution and necessitate a connection between a CCR and constitution.

Remember that $O :=\{(A,B):A \subseteq N, B \subseteq N, A \cap B = \emptyset \}$. Let $\hat{X}:= \{(x,y) \in X \times X:x \ne y\}$, and $2^O$ be the family of all subsets of $O$. A constitution $C$ is a mapping from $\hat{X}$ to $2^O$ that satisfies

$$\begin{aligned} \forall (x,y) \in \hat{X},\quad \forall (A,B) \in O,\quad (A,B) \in C(x,y) \Leftrightarrow (B,A) \notin C(y,x). \end{aligned}$$

Each $f$ induces a $C$ defined by

$$\begin{aligned}&\forall (x,y) \in \hat{X}, \quad (A,B) \in C_f(x,y) \Leftrightarrow \exists \pi \in {\mathcal {D}}\nonumber \\&\quad \text{ such } \text{ that } x \succ _{\pi } y \quad \text{ and } \quad \pi (x,y)=(A,B). \end{aligned}$$

(3)

Each $C$ induces an $f_C$ defined by

$$\begin{aligned} \forall \pi \in {\mathcal {D}},\quad \forall (x,y) \in \hat{X},\quad x \succ _{\pi }^C y \Leftrightarrow \pi (x,y) \in C(x,y). \end{aligned}$$

(4)

The original condition proposed by Ferejohn and Fishburn (1979) is as follows.

Condition R1:

$\forall T \ge 2$, $\forall$ distinct $a^1,\dots ,a^T \in X$, $\forall (A_1,B_1),\dots ,(A_T,B_T) \in O$, if $(A_t,B_t) \in C(a^t ,a^{t+1}), t=1,\dots ,T-1$, while $(A_t,B_t) \in C(a^T ,a^{1})$, then

$$\begin{aligned}&\text{ either } \quad A_t \not \subseteq \bigcup _{s \ne t}^{T}B_s\quad \text{ for } \text{ some } t\in \{1,\dots ,T \}, \end{aligned}$$

(5)

$$\begin{aligned}&\text{ or } \quad B_t \not \subseteq \bigcup _{s \ne t}^{T}A_s \text{ for } \text{ some } \quad t\in \{1,\dots ,T \}. \end{aligned}$$

(6)

Note that this is the condition for a constitution, not for a CCR.

Theorem 6

(Ferejohn and Fishburn 1979, Theorem 3) Suppose that $f$ and $C$ are related as in (3) and (4). Then, $f$ that satisfies IIA is acyclic if and only if R1 holds for $C$.

By restricting their attention to a class of CCRs, Ferejohn and Fishburn (1979) show an interesting relationship between acyclicity and R1. On the other hand, we and Schwartz (2007) do not impose the background restrictions—IIA, (3) or (4).