The results of the Iowa Caucuses showed Donald Trump in second place, with 24% of the vote, behind Ted Cruz with 28 and just ahead of Marco Rubio with 23%. The difference in terms of numbers, is about 6,000 votes. In other words, if 3000 Iowans would switch their votes from Cruz to Trump, the outcome would have changed. Pundits, and the GOP establishment, seem to view this result as containing the seeds of destruction for the Donald. They point out that part of his campaign persona is that he's a "winner" and yet, in Iowa, he didn't win. What can game theory say about the GOP presidential race?
Coordination and Duverger's Law
Duverger was a French philosopher in the field of politics. He noted that, in winner take all elections (sometimes call first past the post), there is a strong tendency for just two candidates to receive large vote shares. From this, he concluded that such voting rules tend to produce two party systems as in the US and, at the time, the UK. In proportional representation systems, many parties get votes.
Note that the Iowa caucus is actually proportional representation, at least to an extent. Multiple candidates can collect delegates in Iowa, but there is overrepresentation of delegates among the top vote getters.
Duverger's Law, it turns out, can be understood using game theory. Here's the idea: The main reason that people vote is to help their candidate to get elected. Let's say that there are three candidates, A, B, and C. All voters have rankings over these candidates and, within these rankings, can feel different levels of passion for each. In other words, you and I might both rank the candidates A > B > C, but I feel very strongly for A whereas you are close to indifferent between A and B.
Now for whom should you vote if solely motivated by the outcome of the election? One possible answer is to vote truthfully choosing A if that is your top choice, or B, or C if those are on top. But now suppose a poll has been taken. It shows that C leads narrowly over B while A trails badly behind. Since I rank the candidates ABC this is very bad news. My least favorite candidate is ahead while my candidate trails badly.
So how should I vote? Since I only care about election outcomes, I should switch my vote from A to B. In a very real sense, A is a wasted vote for a voter who cares about outcomes. Of course, all A voters reason in a similar fashion and so A's vote share dwindles ever lower, a death spiral of switching away. Notice that there is a "snowball" nature to this logic very similar to the information cascade--once my candidate's chances grow sufficiently dim, my love for that candidate no longer influences my vote.
So from this, we can conclude that Carly and Jeb and all the others in the single digits in Iowa are effectively doomed. Votes for these candidates will be seen as purely "wasted" and so will dry up.
These votes make up about 15% of the votes in Iowa, probably similar to national rates as well. Where will they go?
Back to the Donald. From his perspective, he benefited from Iowa by strongly affirming what the polls showed--that a vote for the Donald is not a wasted vote. Thus, the missing 15% view him as plausible. But my suspicion is that they mostly will go elsewhere. The Donald is a polarizing figure, you love him or you hate him. He benefits from the passion of his supporters, they provide energy in getting themselves and others out to vote. But this same polarization makes him an unlikely second choice for voters whose first choice was someone else.