How Likely Are Two Voting Rules Different?

Ziqi Yu, Lirong Xia, Qishen Han, Chengkai Zhang
Proceedings of the Forty-first Conference on Uncertainty in Artificial Intelligence, PMLR 286:4806-4825, 2025.

Abstract

We characterize the maximum likelihood that two voting rule outcomes are different and that the winner of one voting rule is the loser of another (implying that they are {\em drastically different}) on positional scoring rules, Condorcet winner/loser, Copeland, Ranked Pairs, and STV (Single Transferable Vote) under any fixed number of alternatives. The most famous problem in this scope is strong Borda’s paradox, in which the winner of the plurality rule is the Condorcet loser. Under mild assumptions, we show that the maximum likelihood that different rules are drastically different is $\Theta(1)$ except for a few special cases, demonstrating the difference between these rules. We also prove that two scoring rules with linear independent scoring vectors have different winners with probability $\Theta(1)$, no matter how similar they are. Our analysis adopts the {\em smoothed social choice framework} \cite{xia2020smoothed} and can be applied to a variety of statistical models, including the standard impartial culture (IC).

Cite this Paper

BibTeX
@InProceedings{pmlr-v286-yu25a, title = {How Likely Are Two Voting Rules Different?}, author = {Yu, Ziqi and Xia, Lirong and Han, Qishen and Zhang, Chengkai}, booktitle = {Proceedings of the Forty-first Conference on Uncertainty in Artificial Intelligence}, pages = {4806--4825}, year = {2025}, editor = {Chiappa, Silvia and Magliacane, Sara}, volume = {286}, series = {Proceedings of Machine Learning Research}, month = {21--25 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v286/main/assets/yu25a/yu25a.pdf}, url = {https://proceedings.mlr.press/v286/yu25a.html}, abstract = {We characterize the maximum likelihood that two voting rule outcomes are different and that the winner of one voting rule is the loser of another (implying that they are {\em drastically different}) on positional scoring rules, Condorcet winner/loser, Copeland, Ranked Pairs, and STV (Single Transferable Vote) under any fixed number of alternatives. The most famous problem in this scope is strong Borda’s paradox, in which the winner of the plurality rule is the Condorcet loser. Under mild assumptions, we show that the maximum likelihood that different rules are drastically different is $\Theta(1)$ except for a few special cases, demonstrating the difference between these rules. We also prove that two scoring rules with linear independent scoring vectors have different winners with probability $\Theta(1)$, no matter how similar they are. Our analysis adopts the {\em smoothed social choice framework} \cite{xia2020smoothed} and can be applied to a variety of statistical models, including the standard impartial culture (IC).} }
Endnote
%0 Conference Paper %T How Likely Are Two Voting Rules Different? %A Ziqi Yu %A Lirong Xia %A Qishen Han %A Chengkai Zhang %B Proceedings of the Forty-first Conference on Uncertainty in Artificial Intelligence %C Proceedings of Machine Learning Research %D 2025 %E Silvia Chiappa %E Sara Magliacane %F pmlr-v286-yu25a %I PMLR %P 4806--4825 %U https://proceedings.mlr.press/v286/yu25a.html %V 286 %X We characterize the maximum likelihood that two voting rule outcomes are different and that the winner of one voting rule is the loser of another (implying that they are {\em drastically different}) on positional scoring rules, Condorcet winner/loser, Copeland, Ranked Pairs, and STV (Single Transferable Vote) under any fixed number of alternatives. The most famous problem in this scope is strong Borda’s paradox, in which the winner of the plurality rule is the Condorcet loser. Under mild assumptions, we show that the maximum likelihood that different rules are drastically different is $\Theta(1)$ except for a few special cases, demonstrating the difference between these rules. We also prove that two scoring rules with linear independent scoring vectors have different winners with probability $\Theta(1)$, no matter how similar they are. Our analysis adopts the {\em smoothed social choice framework} \cite{xia2020smoothed} and can be applied to a variety of statistical models, including the standard impartial culture (IC).
APA
Yu, Z., Xia, L., Han, Q. & Zhang, C.. (2025). How Likely Are Two Voting Rules Different?. Proceedings of the Forty-first Conference on Uncertainty in Artificial Intelligence, in Proceedings of Machine Learning Research 286:4806-4825 Available from https://proceedings.mlr.press/v286/yu25a.html.

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