October 26, 2005

This week, the Department of Mathematics and Statistics welcomed Andrew Gelman of Columbia University as this year’s Arnold Dresden Lecture Series speaker. The Lecture Series invites its participants to give two lectures over the course of two days, so that one is accessible to a general audience and the other can be more specialized for primarily math majors. Before a full house on Monday and Tuesday, Gelman gave an engaging, occasionally sarcastic, and often amusing talk on the application of mathematical and statistical models in the social sciences.

A graduate of MIT and later Harvard, Gelman was by name a mathematician (i.e., such was his major, along with physics). However, as he said in an interview, he “wasn’t so interested in pure math, and looked for ways to do math within the social sciences.” Now at Columbia, he teaches statistics as a member of both the statistics and political science departments.

The general-audience talk on Monday was entitled “Mathematical versus Statistical Models in Social Science” and aptly consisted of many examples of such, and reasons as to why they may not always accurately predict real-life situations. Gelman first offered the concept of political representation, and questioned what it meant to be “represented.” He gave possible criteria of representation, namely, having one’s vote matter, whether or not one’s political views were represented, and whether or not one’s representatives resembled one’s own demographics, as well as ways to measure or model them.

In light of this last point, Gelman had a story: “Someone once said to me, ‘If only we had more economists in Congress, they wouldn’t be so stupid.’ Well, I looked it up and there are two economists in Congress but there is a much smaller proportion of the population that consists of economists–it’s something like one-tenth of one percent–so, there are too many economists already.”

Gelman went on to present a research problem, that of unequal representation. As it stands now, small states are over-represented and get too large a share of federal funding. California, on the other hand, should get forty-two senators.

Another prominent example of the use of models in social sciences was trench warfare and the prisoner’s dilemma. Essentially, what it boils down to is that if two opponents are in a stand-off, the worst scenario for them is if both of them shoot, so they should not shoot. However, the stable situation is indeed if they both shoot, so they will. Gelman refuted this interpretation of the problem by pointing out that there is no real payoff to shoot because the action poses a risk regardless of what the opponent does. Therefore not shooting is the real stable situation, which commanders manipulate in order to force soldiers into combat.

Gelman also examined voting, a subject that carried over into his Tuesday lecture, when he spoke on “Coalitions, Voting Power, and Political Instability.” Coalitions basically are formed in order to increase one’s voting power, that is, how decisive one’s vote is in an election. The kicker is that if it occurs to everyone to join a coalition, then everyone is worse off than before in terms of voting power.

Some examples of coalition sizes that optimize voting power that Gelman gave included one of four in an electorate of ten, and one of fourteen in an electorate of one-hundred. However, the conjecture is that coalition-forming becomes unstable for coalitions larger than three members.

As interesting as voting power models may be, fitting them into real-life data remains an open problem. Such models predict sometimes quite erroneous situations. One in particular claims that larger states are endowed with higher voting power, a statement with which Gelman has grievances.

“I don’t agree…this actually made me hate mathematical models in social sciences,” remarked Gelman.

A more realistic model, said Gelman, would show that voting power is higher in states where the election was close, not where the population is great.

The bottom line? “Models give insights–but they must be looped back to real data…Models are filling a gap where people may not have thought before, but some people take them too seriously.”