Game Theoretic Understanding of Presidential Elections in the United States
BY STEVE KIM
When Mitt Romney failed to make Barack Obama a one-term president, conservative leaders blasted him for not having been conservative enough. During his campaign, Romney made deliberate efforts to move left. Did Romney fail because he was too liberal for a Republican candidate, or did Romney fail despite his attempt to appear more liberal?
Hotelling’s model of spatial competition tells us that in many markets, it is beneficial for producers to make their products as similar to those of their competitors as possible. Of course, there exists a phenomenon that states the opposite: firms should differentiate their products as much as possible to maximize profit. Neither of these two opposing narratives is invalid. Which is true depends on what market we’re describing. So the question becomes: is Hotelling’s model of spatial competition applicable to the 2012 presidential election, or presidential elections in general? The answer: yes.
Let’s line up the entire American population in one single line in order of their score on the “Conservative Index:” how conservative they are on a scale of 0 to 100, with 100 being the most conservative. And let’s assume we have two candidates, A and B, running for president. Let xA be the numerical representation of candidate A’s score on the Conservative Index line. Regardless of the value of xA, candidate B is always going to position herself closer to the “center” of the spectrum. The dominant strategy – the value of xB that earns candidate B the most votes – would be a value closer to the “center” than xA by the smallest margin. (Both candidates employing the dominant strategy would mean both candidates positioning themselves at the center, which is the Nash Equilibrium of this voting game.) A winning strategy – the value of xB that wins candidate B the election – would be any xB such that lc – xAl > lc – xBl when c = 50. The reason for this is voters will vote for whichever candidate is closer to their individual position on the spectrum. For example, if candidate A positioned himself at 20, candidate B would want to position herself at 21; every voter who scored between 0 and 20 on the Conservative Index would vote for candidate A whereas every voter who scored between 21 and 100 would vote for candidate B.
Of course, the model I’ve just provided is overly simplified and contains some flaws. I didn’t fully develop the idea of the “center” in the previous paragraph. In other words, should we use the mean or the median for this so called “center?” It turns out the median is the “center” that candidates should move towards (at least game theoretically). In our model, we didn’t need to distinguish between the mean and the median because we were implicitly assuming equal distribution. As soon as equal distribution is violated, we could have the mean at 50 but the median at 2. In that scenario, moving towards the mean of 50 and away from the median of 2 will cost you votes. Second, I have assumed that candidates can freely move up and down the Conservative Index. In reality, it couldn’t be further from the truth. Every candidate’s movement along the index is restrained by the public’s perception of the candidate and his party. Third, we are ignoring the effect the distance between the position of a candidate and the position of a voter on the index could have on the probability of the voter expressing his preference or in less technical terms, casting a vote. For example, if you scored 100 on the Conservative Index, you would be indifferent between two moderate candidates, prompting you to abstain in this two-candidate scenario. The implication is that if a considerable number of your voters are at the end of the political spectrum, moving towards the center could hurt you. This potential loss is what scares the Republican leadership out of moving closer to the center.
Despite its flaws, our model nevertheless provides us with some insight into how both the Republican Party and the Democratic Party should approach the 2016 presidential election. Only twice have the Democrats won a presidential election after its candidate won two prior elections. In other words, the odds are stacked against the Democrats. Given their recent failure, the Republicans should take a chance on moving towards the center and potentially risk losing their extreme (read: backwards) supporters. The Democrats should make sure that the Republicans can’t position themselves closer to the center than themselves, which is a strategy they’ve followed for years.