Comment on “Voting, Fairness, and Political Representation”

The French enlightenment philosopher Condorcet saw in the laws of chance the laws of democracy. Writing ofjuries, he explained in Essai surlaProbabilitie that counting votes aggregates the knowledge and judgment of each individual, and, because of the aggregation of individual's judgment, the collective more surely reaches the right decision than an individual. Social scientists, philosophers, mathematicians and statisticians have since found in probability powerful analytical tools for understanding democracy both its power and its perversities. Andrew Gelman cleverly highlights many of the key ideas of probability and statistics used in the study of representation. How do we define and measure democratic representation? What does fair representation look like, and how well do our elections approximate that ideal? The normal distribution holds center stage in probability models of representation. Normality is the reference distribution, the ideal against we compare reality. Tacitly, that is the comparison drawn ineyeing figure 1in "Voting, Fairness, and Political Representation." and it is the conceptual model used in Figure 2 to introduce about the median voter theorem, though normality, even unimodality, is not required for the theoretical result. Again,the idea has a verylonglineage. In 1896, E Y. Edgeworth first noted that the normal distribution should approximate the aggregate distribution of the vote. Votes are binary: yes/no, Democrat/Republican, Conservative/Labour. Each person's vote for a party can be modeled as a Bernoulli trialand the number of votes for a party will, therefore, follow the Binomial. In 1950, Kendall and Stuart expanded this theoretical result into a fuller statistical model of elections and voting that is still widely used today. When we look at a distribution of votes we are naturally drawn to the normal as the reference distribution. Imagine that there is a national division of the public between two choices, say the parties or the presidential candidates. The administrative units in which votes are counted, say, precincts or counties or districts, amount to arbitrary divisions (or samples) of the public. As a result we expect a normal distribution. Deviations signal something wrong. Figure I in Professor Gelman's essay is quite famous in political science. The picture derives from an article by David Mayhew published in 1971; it spawned an enormous literature on the "incumbency advantage." The idea is that the bi-modality evident in the picture stems from the fact that incumbents receive about a 10 percentage point higher vote margin compared to politicians of the same party running in similar districts for open seats (where no incumbent is running). Essentially, the figure is a mixture of three normal distributions. Those seats where there isa Republican incumbent; those seats where there is a Democratic incumbent; and those seats where no incumbent is running. There is also a cautionary tale in Figure I. FigureA is the graph of the Democratic shares of the vote in all statewide elections from 1978 to 2000. I've dropped the uncontested races. This figure looks much more normal; there is certainly no bimodality.Statistical analysis of the data reveals that the incumbency advantage in statewide elections is just as large as, and for some offices even larger than, US House elections. Histograms such as Figure I and Figure A are not enough. They do not display the key components of the data. Rather one needs a good model. Why Figures 1 and A differ is puzzling. The components of the variance reveal something of an answer and a problem for further contemplation. The variation explained by incumbency is the same in the two figures. The differences lie in the wider dispersion of party divisions across congressional districts than across states, and, importantly,the higher amount of idiosyncratic variation in state elections than in US House elections. My consideration of figures 1 and A suggests one avenue for inquiry if you like, a boundary that needs to be pushed outward. That avenue is the meaning of the variance in theories of representation. Many of the concepts considered by Professor Gelman concern the mean of the distribution forexample, the bias and the incumbency advantage. The year-to-yearvariation, the volatilityof the vote, is just as important; in understanding elections and representation. What normative meaning do we give to the variabilityin the vote? Does more volatility from election to election make for better or fairer representation? Such questions are largely unexamined. The binary nature of the models presents a second boundary for research on