The symmetry and complexity of elections

On personal and professional levels, the Marquis de Condorcet did like his fellow Academy member J.C. Borda. It is not clear why; maybe it was the regrettable tension that sometimes exists between abstract and applied mathematicians, or maybe Condorcet harbored jealousy over Borda beating him by over a decade in recognizing the complex and interesting mathematical issues that arise in the social sciences. Whatever the reason, rather than serving as an obscure historical footnote about the French Academy of Sciences during this prerevolutionary period of the 1780’s, their conflict still characterizes basic divisions in the voting theory field that they started. Their concern, which is as critical today as two centuries ago, was whether we may inadvertently choose badly in our elections. Only recently have we become to understand the mathematical complexity of the problem along with what are some of the answers. The story starts in 1770 when Borda worried whether Academy’s decisions reflected who they truly wanted. His concern was not whether the voters were informed or voted, but rather about how they tallied the ballots. Through a cleverly constructed example, Borda demonstrated that the Academy’s procedure was so bad that they could elect someone who they actually viewed as inferior while their election’s bottom-ranked candidate is who they find to be superior! Clearly, such a misguided procedure should have been tossed into the trash heap of history. It was not; instead we still use it to select members of the Senate, Congress, City Councils, Mayors, Assemblies, and, indirectly, the President of the USA. This highly flawed approach is the standard plurality vote where we vote for one candidate and the winner is the candidate with the most votes. Borda’s example has twelve voters trying to rank the three candidates Alice, Becky, and Candy. Suppose the preferences of five voters are Alice first, Candy second, and Becky last – denoted by Alice Candy Becky. The next four voters have the preferences Becky Candy Alice, and the last three have Candy Becky Alice. Trivially, the plurality election outcome is Alice Becky Candy with a 5:4:3 tally. To analyze this outcome, let’s play some thought games. For instance, had Becky dropped out of contention just before balloting, who should win the comparison between the topand bottom-place plurality candidates? If our elections mean what we think they do, the immediate answer is Alice. But instead, a quick count shows that Candy would beat Alice with a 7:5 vote! Compounding the contradiction is that in the other two-person races Candy also beats Becky by 8:4 and Becky defeats Alice by 7:5. As Borda pointed out, these head-to-head comparisons strongly suggest that these voters view the plurality bottom-ranked Candy as the superior choice – they choose her over anyone else in pairwise votes – while they view the plurality winner Alice – who always loses when compared with others – as the inferior option! This suggests that we could suffer elections