Voting by the Numbers

Yes, the presidential election season is upon us again, and that’s good news for assistant mathematics professor Karl-Dieter Crisman as he continues exploring several theories related to voting. In early January, he’ll be presenting a talk entitled, “Symmetry in Voting Theory: The Borda-Kemeny Spectrum and Beyond” in Boston at a special session of the American Mathematical Society joint meeting with the Mathematical Association of America, the largest annual mathematics meeting in the world. Below is an introduction of his talk (intended for nonspecialists):

“Why Math and Voting? One of the myriad applications of math is in aggregation of preferences. Economists, psychologists, and political scientists all need our tools to talk about this. In my city, we just elected new councilors-at-large, and of course, the primaries and caucuses for the next presidential race are almost upon us. So it’s always a hot topic for the public as well as undergrads. Plus, relatively elementary math yields big results.

Most American elections use one rule to aggregate our preferences – the plurality vote. If you get more first-place votes than anyone else, you win! But there are many other ways to aggregate voter preferences. This talk will discuss a surprising relation between two relatively prominent ones in voting theory circles.

(Arrow’s Theorem) There is no voting rule that satisfies all typical ideas of ‘fairness.’ However, much of the (non-mathematical) debate in voting theory revolves around whether certain axioms of fairness are really red herrings. So my talk focuses on two methods. In both, each voter lists all n candidates in her preferred order: 1.) The Borda Count: Points are assigned based on each ranking – say 0 for last, 1 for second-last, up to n − 1 for first. The candidate with the most total points wins. 2.) The Kemeny Rule: For each possible (full ranking) outcome, check for how many pairwise votes it disagrees with each voter’s ranking, and add up these disagreements. The (highest ranked candidate in the) ranking with the fewest points wins.”

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