Condorcet cycle elections with influential voting blocs
Gabriel Gendler
TL;DR
This work analyzes Condorcet-cycle elections with three candidates under the weakened anonymity notion of transitive anonymity (TA). Using a Kalai-style Boolean-function approach on slices and an extension to the cube, it shows that TA permits non-Borda SWFs when $|V| oty3$, but enforces Borda on electorates with sizes divisible by three, with near-Borda behavior otherwise. The paper develops a 3-tuple framework of relative SWFs, leverages intersecting families fixed by transitive groups, and employs Johnson-scheme eigenanalysis to establish tight bounds on how close TA-SWFs must be to Borda. It provides explicit constructions of strongly non-Borda TA-SWFs for $|V|=3k+1,3k+2$ and proves that these are nonetheless almost Borda, with quantitative 96% alignment in decisive cases. The results connect social choice with Fourier-analytic and spectral techniques on slices, offering a rigorous map from TA, MIIA, N, and PR to near-Borda outcomes in a range of electorate sizes.
Abstract
A Condorcet cycle election is an election (often called a Social Welfare Function, or SWF) between three candidates, where each voter ranks the three candidates according to a fixed cyclic order. Maskin showed that if such a SWF obeys the MIIA condition, and respects the complete anonymity of each voter, then it must be a Borda election, where each voter assigns two points to their preferred candidate, one to their second preference and none to their least preferred candidate. We introduce a relaxed anonymity condition called ``transitive anonymity'', whereby a group $G$ acting transitively on the set of voters $V$ maintains the outcome of the SWF. Elections across multiple constituencies of equal size are common examples of elections with transitive anonymity but without full anonymity. First, we demonstrate that under this relaxed anonymity condition, non-Borda elections do exist. On the other hand, by modifying Kalai's proof of Arrow's Impossibility Theorem, which employs methods from the analysis of Boolean functions, we show that this can only occur when the number of voters is not a multiple of three, and we demonstrate that even these non-Borda elections are very close to being Borda.