10.6 10.5 The Chair's Paradox
452
We conclude this chapter with an aspect of manipulability that is so counterintuitive that it is referred to as the chair's paradox. To illustrate the situation, we'll consider a hypothetical college in upstate New York that is trying to choose among three academic-year calendars:
- Terms: A term system (10 weeks–10 weeks–10 weeks)
- Semesters: A semester system (14 weeks–14 weeks)
- J-Plan: A January-plan system (12 weeks–4 weeks–12 weeks)
The trustees say that the issue will be decided by majority rule with three voters:
- Someone representing the administration
- Someone representing the student body
- Someone representing the faculty
The administration, however, is given tie-breaking power. That is, if each proposed calendar gets one vote, then the one the administration voted for wins. (In this presentation of the paradox, the administration is playing the role of the chair.)
The preferences are given by the following table (and notice that these preference lists exactly mirror the ballots in Condorcet's voting paradox):
| Administration | Students | Faculty | |
|---|---|---|---|
| First choice | J-Plan | Terms | Semesters |
| Second choice | Terms | Semesters | J-Plan |
| Third choice | Semesters | J-Plan | Terms |
We assume that everyone knows everyone else's preferences and that these are real preferences, even after taking into consideration how the other constituencies feel.
The goal now is to analyze the situation and to determine how each of the three will vote if they are all rational in the sense of being willing to vote strategically (i.e., to manipulate the system) if it's in their own best interest. This is really a game-theoretic analysis, and it's useful to borrow a couple of pieces of game- theoretic terminology.
First, a choice of which calendar to vote for is called a strategy. So each of the voters has three strategies at its disposal: Vote for J-Plan, vote for Terms, and vote for Semesters. The second piece of terminology arises from the observation that if everyone is rational and acting in his or her own self-interest, no one will vote for his or her least-preferred calendar. The point is that voting for either a first or second choice weakly dominates the strategies of voting for a third choice, in the sense that the former choices always yield outcomes that are either the same as or better than the latter.
With this, we can see that the administration's strategy of voting for its first choice (the J-Plan) weakly dominates its strategy of voting for its second choice (Terms). That is, if both students and faculty vote for Semesters, the outcome is Semesters regardless of how the administration votes, but otherwise the administration does strictly better by voting for the J-Plan rather than Terms. Hence, assuming that the administration is rational, we know that it will, in fact, vote for its top choice, the J-Plan.
453
Now, given that we know what the administration will do, the claim is that the faculty's strategy of voting for Semesters weakly dominates its strategy of voting for the J-Plan. That is, if the students vote for Terms, the outcome is the J-Plan regardless of whether the faculty votes for Semesters or the J-Plan. On the other hand, if the students vote for Semesters, then the faculty can secure its best outcome Semesters by voting for Semesters. Assuming that the faculty is rational, we thus know that the faculty will, like the administration, vote for its top choice, which is Semesters.
But let's see where these decisions leave the students. They know that the administration is voting for the J-Plan and that the faculty is voting for Semesters. So if they vote for Terms, then the outcome is the J-Plan— their last choice. However, if they vote for Semesters along with the faculty, then the outcome is Semesters, their second choice. There is no way that the students can secure their top choice, Terms, as the winner. If they are rational, then they will also vote for Semesters, and thus Semesters will win the election.
So why is this paradoxical? Well, the administration clearly had the most "power," but the eventual winner of the election was its least-preferred calendar! The administration would have been better off handing over the tie-breaking power to either the faculty or the students.
The Chair's Paradox THEOREM
With three voters and three candidates, the voter with tie-breaking power can, if all three voters act rationally in their own self-interest, end up with his or her least- preferred candidate as the election winner.
The chair's paradox represents only one of manipulability's first cousins, some of which involve not only the fields of mathematics and political science but psychology as well. One of the authors relates the following from his early years:
I recall a third-grade penmanship contest in which each of us had a writing sample taped to the blackboard, and the teacher, Mrs. Levy, announced that we'd get to vote for the one we thought best, with the proviso that the voter couldn't vote for his or her own paper. She also announced that if two or more were tied, we'd have a runoff among those.
I remember being torn as to which of three particular ones to vote for, all of which I thought were very good and considerably better than the rest, including my own. When the votes were counted, these three were, in fact, tied for the win, with my writing sample alone in fourth, and only one vote out of the tie.
After announcing the results, Mrs. Levy went on to say that the runoff would involve not three of us, but four, as she had decided also to vote, and she was voting for me! I don't remember the final tally, or what Mrs. Levy then said to the class, or what my three classmates, all plenty smart enough to realize what had just happened, later said to me. But I do remember sitting back and smiling—absolutely sure of the outcome—as soon as she had announced her intention to vote for me.
