Escaping Arrow's Theorem: The Advantage-Standard Model

TL;DR

The paper revisits Arrow’s Impossibility Theorem and proposes the Advantage-Standard (AS) model as a tempered weakening of Independence of Irrelevant Alternatives (IIA). By representing a collective choice rule (CCR) via a pairwise intrinsic advantage and a context-dependent standard, AS representability sits strictly between weak IIA and IIA, and is compatible with non-dictatorial CCRs. The authors prove that AS representability is equivalent to the conjunction of weak IIA and orderability, and they demonstrate this with three substantive AS representable CCRs: Gillies Covering, Ranked Pairs, and Split Cycle. These results show that Arrow-like impossibilities need not force dictatorship and that practically relevant voting rules can be reconciled with Arrowian intuition under the AS framework, potentially broadening the design space for robust social choice mechanisms.

Abstract

There is an extensive literature in social choice theory studying the consequences of weakening the assumptions of Arrow's Impossibility Theorem. Much of this literature suggests that there is no escape from Arrow-style impossibility theorems, while remaining in an ordinal preference setting, unless one drastically violates the Independence of Irrelevant Alternatives (IIA). In this paper, we present a more positive outlook. We propose a model of comparing candidates in elections, which we call the Advantage-Standard (AS) model. The requirement that a collective choice rule (CCR) be representable by the AS model captures a key insight of IIA but is weaker than IIA; yet it is stronger than what is known in the literature as weak IIA (two profiles alike on $x,y$ cannot have opposite strict social preferences on $x$ and $y$). In addition to motivating violations of IIA, the AS model makes intelligible violations of another Arrovian assumption: the negative transitivity of the strict social preference relation $P$. While previous literature shows that only weakening IIA to weak IIA or only weakening negative transitivity of $P$ to acyclicity still leads to impossibility theorems, we show that jointly weakening IIA to AS representability and weakening negative transitivity of $P$ leads to no such impossibility theorems. Indeed, we show that several appealing CCRs are AS representable, including even transitive CCRs.

Figures (7)

• Figure 1: two profiles $\mathbf{R}$ and $\mathbf{R'}$ such that $\mathbf{R}|_{\{a,b\}}=\mathbf{R}'|_{\{a,b\}}$. The first column indicates that voter $i$ has $aP(\mathbf{R}_i)b$, $bP(\mathbf{R}_i)c$, etc.
• Figure 2: logical relations between axioms
• Figure 3: a majority graph (left) with cycles $a\to b\to c\to d\to a$ and $a\to c\to d\to a$, together with its associated covering relation (right): $a$ covers $b$ because not only is $a$ majority preferred to $b$, but the only candidate majority preferred to $a$, namely $d$, is also majority preferred to $b$.
• Figure 4: an example in which $P(f_{cov}(\mathbf{R}))$ is not negatively transitive. The first graph is the majority graph for the profile $\mathbf{R}$ on the left, showing that $a\succ_\mathbf{R}b$ and $b\succ_\mathbf{R}c$, but neither $a\succ_\mathbf{R} c$ nor $c\succ_\mathbf{R} a$. The second graph shows the induced covering relation: since $a\succ_\mathbf{R}b$ but $a\not\succ_\mathbf{R}c$, it is not the case that $bP(f_{cov}(\mathbf{R}))c$. Finally, since not$aP(f_{cov}(\mathbf{R}))c$, not$cP(f_{cov}(\mathbf{R}))b$, and yet $aP(f_{cov}(\mathbf{R}))b$, $P(f_{cov}(\mathbf{R}))$ is not negatively transitive.
• Figure 5: a margin graph. The arrow from $a$ to $b$ with weight 3 indicates that 3 more voters prefer $a$ to $b$ than prefer $b$ to $a$, etc.

Theorems & Definitions (66)

• Theorem 2.1: Arrow1951
• Lemma 2.2
• Proposition 2.3
• proof
• Theorem 2.4: Murakami1968, Wilson1972
• Proposition 2.5
• Definition 3.1
• Example 3.2
• Proposition 3.3
• proof