Abstract

In 1785, Condorcet established a frustrating property of elections and majority rule: it is possible that, no matter which candidate you pick as the winner, a majority of voters will prefer someone else. You might have the brilliant idea of picking a small set of winners instead of just one, but how do you avoid the nightmare scenario where a majority of the voters prefer some other candidate over all the ones you picked? How many candidates suffice to appease a majority of the voters?  In this talk, I will answer this question. Along the way, we will roll some dice—both because the analysis involves randomness and because of a connection to the curious phenomenon of intransitive dice, that has delighted recreational and professional mathematicians alike, ever since Martin Gardener popularized it in1970.