Higher Order Condorcet Cycles
William S. Zwicker*
February 18, 2013
Bailey Hall 201
Refreshments will be served in Bailey Hall 204 at 4:15
Seven students form a club, and are deciding on which day of the week to hold regular meetings. Tuesday is proposed, but a majority of the seven express a strict preference for Wednesday over Tuesday. When Wednesday is suggested, a majority say they like Thursday better. When Thursday is suggested . . . and this keeps up until they are back at Tuesday. Can that happen? Yes – the pattern, called a "Condorcet cycle," has been known for over 200 years, and is linked to the general problem of designing fair rules for choosing democratically from among more than two options.
Alice, trying to make peace, suggests that meetings alternate between Tuesdays one week and Fridays the next. Hmmm . . . what if a majority prefer Wednesday to Tuesday and to Friday both? Worse yet – suppose whichever two days Alice suggests, a majority is able to name a single day they prefer to both of them. Can that happen? Yes, it can! But such an order 2 Condorcet cycle is more complex than an ordinary cycle; for a seven day week one needs at least 21 students voting (at least, we think so).
We'll discuss some of what we do know, and what we do not yet understand, about this new variant of an old idea.
* This talk is based on joint research with Davide Cervone and Chris Hardin.
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