Condorcet Winners and Anscombes Paradox Under Weighted Binary Voting

TL;DR

The paper studies multi-issue voting with issue weights, focusing on the tension between achieving majorities on individual issues (IWM) and guaranteeing a Condorcet winner under pairwise comparisons. It establishes that determining the existence of a Condorcet winner is $co$-$NP$-hard even without weights, motivates the search for tractable domains via the single-switch condition under external weights, and provides linear-time recognition plus forbidden-substructure certificates. It then analyzes Anscombe's paradox under both external and internal weights, deriving upper and lower bounds that quantify how close majority-supported proposals can be to the IWM, and extends Wagner's Rule of Three-Fourths to weighted settings to preclude paradox in broad regimes. The results yield efficient algorithms, topological insights (Möbius-strip structure), and practical guidance on when issue-wise majorities align with robust proposal outcomes, with distinct implications for external vs. internal weighting schemes. These contributions advance understanding of robust aggregation in separable multi-issue domains and inform design of voting rules under weighted importance of topics.

Abstract

We consider voting on multiple independent binary issues. In addition, a weighting vector for each voter defines how important they consider each issue. The most natural way to aggregate the votes into a single unified proposal is issue-wise majority (IWM): taking a majority opinion for each issue. However, in a scenario known as Ostrogorski's Paradox, an IWM proposal may not be a Condorcet winner, or it may even fail to garner majority support in a special case known as Anscombe's Paradox. We show that it is co-NP-hard to determine whether there exists a Condorcet-winning proposal even without weights. In contrast, we prove that the single-switch condition provides an Ostrogorski-free voting domain under identical weighting vectors. We show that verifying the condition can be achieved in linear time and no-instances admit short, efficiently computable proofs in the form of forbidden substructures. On the way, we give the simplest linear-time test for the voter/candidate-extremal-interval condition in approval voting and the simplest and most efficient algorithm for recognizing single-crossing preferences in ordinal voting. We then tackle Anscombe's Paradox. Under identical weight vectors, we can guarantee a majority-supported proposal agreeing with IWM on strictly more than half of the overall weight, while with two distinct weight vectors, such proposals can get arbitrarily far from IWM. The severity of such examples is controlled by the maximum average topic weight $\tilde{w}_{max}$: a simple bound derived from a partition-based approach is tight on a large portion of the range $\tilde{w}_{max} \in (0,1)$. Finally, we extend Wagner's rule to the weighted setting: an average majority across topics of at least $\frac{3}{4}$'s precludes Anscombe's paradox from occurring.

Figures (3)

• Figure 1: The profile $\mathcal{P}$ in \ref{['fig:ss_a']} is single-switch because its columns can be permuted and flipped as in \ref{['fig:ss_b']} to ensure that ones on each row form a prefix or a suffix.
• Figure 2: Möbius strip of orbit $O_{\mathcal{P}'}$ for $\mathcal{P}' = (c_1, \ldots, c_{10})$. We start with a rectangular piece of paper of length 10 and write $(c_1, \ldots, c_{10})$ on the (green) front side and $(\overline{c_1}, \ldots, \overline{c_{10}})$ on the (red) backside. We then give the paper a length-wise half-turn and glue the endpoints (bold strip). This gives raise to a surface with a single continuous side.
• Figure 3: A summary of our bounds on $g_{\tilde{w}_{max}}$.

Theorems & Definitions (37)

• theorem 0
• lemma 1
• lemma 2
• theorem 2
• lemma 2
• corollary 3
• theorem 3
• theorem 4
• theorem 4
• theorem 5