In my undergraduate game theory course, I have a unit on game theory applied to voting. The main topic is strategic voting — that is, voting for one’s second or third choice rather than the first, to prevent an even worse result. This term, since the Democrats are pretty boring, I asked my students to rank 14 Republican candidates. Since some might not consider voting for any Republican, and some are not citizens, I did not ask for their preferences as to who should win, but their judgment as to which is the most conservative, the second-most, and so on. I asked that at least the top five be evaluated. Twenty students responded. Most ranked the minimum five, several ranked all 14 and one left several out but made a point of ranking Chris Christie last.
So who is most conservative? Well, it depends somewhat on how you count the votes. First, plurality rule — whoever gets the most first-place rankings wins. By that count, Rand Paul wins with five votes. This is a well-known fact about plurality or first-past-the-post voting: a minority who are united on one alternative can prevail over a majority who oppose that alternative but are not united on an alternative. Thus, in the recent British election, the Conservatives won a majority in Parliament with only about 37 percent of the popular vote. Pennsylvania also has plurality rule, for the most part. In some other states, if no candidate gets a majority, there is a “run-off” election with only the top two vote-getters in this second round. Since Ted Cruz came in second with four first-place votes, the runoff would have been between Cruz and Paul. Paul would have won, 10 to six (My announcement in class was based on a hasty miscount of the Cruz vote, but I believe all my errors are corrected).
In the 1700s, the Marquis de Condorcet proposed that a candidate who could beat all other candidates in two-way races should be the winner. Not all elections have a candidate who can do this, but Nobel Laureate Eric Maskin has much more recently proposed a two-stage process that would select the “Condorcet candidate” at the first round if there is one. So, do we have a Condorcet candidate in this ranking? Yes. Jeb Bush. Why does Bush defeat Paul, Cruz and the other contenders? Because Bush is listed second or third on many rankings, and on the rankings of students who chose quite differently at the first choice. This also suggests that Bush could do well if there were strategic voting.
Another way to count second and lower preferences is the “single transferable vote.” This is a multi-stage count, and at each stage the candidate with the least votes is eliminated and those who voted for him have their vote transferred to the next lower choice on their preference rankings. This is done in some elections in Australia and several other countries, and in some California cities including San Francisco. (Thanks to a student for this information.)
There is a complication since it is a little unclear how to handle ties, and while ties are uncommon in real elections, in this ranking of 14 alternatives by 20 voters, ties for last place were common. It seems pretty clear, though, that Jeb Bush is the one candidate that would survive all eliminations, regardless how ties are handled. Thus, in this case, the single transferable vote yields the Condorcet candidate — but we know that would not always be true.
In the 1700s, the Chevalier de Borda was a rival of Condorcet, and he proposed a different way of using second and lower choices: give a certain number of points to each candidate for each first place ranking, one less point for a second-place vote, and so on down to nothing for a last-place vote (a spreadsheet helps). Applying the Borda count, we get — Jeb Bush! Mitt Romney, Paul and Cruz follow in that order, but are very tightly grouped about 20 points behind Bush. Carly Fiorina beats out Christie for last place.
What does this tell us about the coming election season? Maybe not much — my students probably are not representative of the voters in Republican primaries, since they are smarter but also younger and several are from other countries so perhaps less well informed.
Passing over that, though, to the extent that their perception of conservatism is representative, and if the Republicans choose the most “conservative” candidate, it looks good for Jeb Bush. Rand Paul and Ted Cruz may do very well while the field is large, but are likely to be eliminated as it narrows. In the single transferable vote enumeration, they remain in the field until it is down to four. Strategic voting might push this elimination faster. Romney should probably stay in the race; like Bush, he has high ranks — although few for first place. If Bush should stumble, Romney would probably move into his place. But, having been nominated before, he may not want to be Plan B. Santorum, despite having “come in second” four years ago, never breaks into the top five on any count.