This essay is the fifth in my series on the presidential candidates (see my introductory comments about methodology, and then my essays on Clinton, Edwards, and Obama). Because it concerns John McCain - the central question that it must answer is why his candidacy has had so much trouble acquiring traction.

In recent days, many answers have been offered in response to this question. I think many of them have validity. My task here will not be to give a definitive answer that wholly explains McCain's collapse. I think that no single answer can do that. Rather, my agenda here shall be twofold. First, I shall argue why one typical answer - McCain is too moderate for the Republican Party - is based upon faulty intuitions and, while not necessarily wrong, is oversimple. Second, I shall offer an alternative explanation that, while I do not think explains McCain's decline entirely, is an intuitive and compelling account that helps illuminate a good deal.

It is part of the conventional wisdom in Washington that presidential candidates have to "run to the base" in the primary, and then "run to the middle" in the general election. This conventional wisdom is partially accurate, and it is a simplified version of what is known as the median voter theorem.

I do not wish to bore you with the details of the theorem. I will just say that the median voter theorem is a solution to what has come to be known as the problem of social choice - or how we may develop reasonable and morally satisfactory rules to aggregate the preferences of individuals in society. The median voter theorem argues that, under certain conditions, the voter whose preferences are in the ideological middle of the electorate will be the winner in a simple majority rule procedure. In the case of the Republican Party primary, the median voter is a staunch conservative - hence the intuition that Republican contenders must move to the right.

To obtain the result of the theorem, we first assume that voters have "single-peaked preferences," i.e. we assume that every voter prefers one ideological position to all other positions. We also assume that there are only two candidates in the race, that candidates are free to change their stated issue positions as they like, that voters possess perfect information about these positionings, and that voters make their choices based upon one issue "dimension" (like ideology). With these assumptions, we obtain the result of the median voter theorem: the ideological position of the voter who has half of the electorate on one side and half on the other side can beat any other position in a two-way contest. Accordingly, both candidates will adopt this issue position, and thus "run to the center" of the electorate.

Graphically, it would predict the result of a general election campaign in the following way.

The line represents the issue positions one can take - as you move from the right to the left, your issue positions become more liberal. The three letters - A, B, and C - represent the ideological positions of the voters (in this simplification, our electorate consists of only three people). D and R represent the issue positions taken by the Democratic and Republican candidates. The equilibrium position of the campaign is for both candidates to take issue positions consistent with Voter B. What do I mean by equilibrium? I mean that it is the best response of the Democrat to the Republican's decision to take the median position, and vice-versa. It is a Nash equilibrium.

The median voter theorem is quite intuitive, and most people who are not trained political scientists have some notion of it that is of utility in analyzing politics. Many pundits are relying upon their intuitions of the theorem to explain McCain's troubles. In the Republican primary election, the median voter is a conservative Republican - to whom McCain refuses to "pander." One might point to the most recent Republican debate where McCain was the only candidate on stage who expressed support for immigration reform. This is a sign, they argue, that McCain is not appealing to the median Republican, and this is why he is in so much trouble.

This, however, makes use of a notion of the median voter theorem that is naïve. The median voter's position is only an equilibrium in a two candidate race. There are many more than two candidates in the race for the Republican nomination. What happens when there are more than two voters? The ideological position of the median voter ceases to be a candidate's best response to the decision of the other candidates to adopt the median voter's position. Thus, it is no longer a Nash equilibrium.

Imagine a race with three major candidates - R, G, and M - all of whom are trying to espouse an ideological position that will win them the election. We continue to assume that they are free to espouse any position they like, that there is only one issue dimension, and that voters have single-peaked preferences and perfect information. This time, let us also imagine more than three voters (any odd-numbered will do) who are spread out evenly along the left-right continuum (between the arrows, which are retained to indicate that candidates may take any issue position they like).

The goal of each candidate is to take an issue position that wins them the election. So, let them all start around the median:

Obviously, G's position is a loser. The median is not a "best response" for G, based upon what R and M have chosen. G will increase his vote share if he does the following:

But now R is sandwiched in between G and M, and can expect to lose. So R responds:

This, in turn, forces M to change his position because the median is no longer his "best response."

And G responds:

And so on - this is a cycle that would end only because the election ends the campaign.

Notice how we are slowly moving from the median voter. The median is not an equilibrium position in a three-way race. If all three candidates are at the median, all three candidates can increase their expected vote shares by moving either to the left or to the right. We therefore cannot say that those who refuse to espouse the median position are doomed to lose the election. It is simply not true. In a three-way race, a candidate could win with an ideological position that is quite far from the median. What matters is what issue positions his opponents espouse immediately before the election, and what the exact distribution of voter preferences is.

And so, theoretically speaking, it is possible for a candidate to win the Republican nomination in a multi-candidate race by taking moderate issue positions. For instance, if multiple candidates are spread from the center to the right and voters are again evenly distributed, a Republican nominee could win the GOP nomination by taking an ideological position that is more liberal than the others. Consider:

If this was the last iteration of ideological positioning before the actual election, candidate M would win with a centrist ideology! He would take all of the voters to the left of his position, and half of the voters between G and him. This would be greater than either the voters that G would win or that R would win. And so, one need not necessarily pander to the party base (i.e. appeal to the median voter in the party) to win the nomination. In this example, candidate M wins the nomination despite being more liberal than the rest of the party.

McCain's problem, therefore, is not simply that the Republican electorate will not select a moderate Republican. Depending upon the issue positions of the other candidates and the distribution of voter preferences, there are some instances in which we should expect it to. Of course, under certain circumstances, we should also expect the party not to select a moderate. Again, it depends upon how voter preferences are distributed, and how candidates position themselves. The point is that the argument that McCain's moderation is the source of his problem is oversimple and is based upon a false understanding of the median voter theorem. Tomorrow, I will offer an alternative explanation that I think is more compelling.