Imagine that arrayed around a big circle are millions of people who are asked each day whether they intend to vote for George Bush or John Kerry. Assume that these people have an initial favorite, randomly choosing Bush or Kerry, but that they are very conformist and decide daily to consult some of their immediate neighbors. After polling the people on either side of them, they adjust their vote to conform with that of the majority of their neighbors.

How many people each voter consults varies from day to day and is determined by the fact that they are "lazy statisticians." They expand their samples of adjacent voters only as much as necessary and reduce them as much as possible, wishing always to conform with minimum exertion.

There are various ways to model this general idea, but let's assume the following specific rule (which can be made more realistic). If one day a voter, say Henry, polls the X people on either side of him, the next day he expands his sample to the X+1 people on either side of him. If the percentage favoring the two candidates in this expanded sample is different than it is when he polls only the X people on either side of him, he expands his sample still further.

On the other hand, if the percentage favoring the two candidates is the same in the expanded sample as it is when he polls only the X people on either side of him, Henry decides that he might be working too hard. In this case he reduces his sample to the X-1 people on either side of him. If the percentage favoring the candidates is the same in this smaller sample, he reduces the sample still further.

Every voter updates his or her favorite daily and interacts with other voters according to these same rules.

Epstein's model showed that the result of all this consulting is a little surprising. After several days of this sequential updating of votes, there are long arcs of solid Bush voters and long arcs of solid Kerry voters and between these there are small arcs of very mixed voters. After a short while, voters in the solid arcs need consult only their immediate neighbors to decide how to vote and almost never change their votes. Voters between the solid arcs need to consult many people on either side of them and change their vote quite frequently.

Although Epstein didn't apply his model to voting but to more automatically followed social norms, the idea of extending it to voting is seductive. People do tend to surround themselves with others of like mind and generally only those at the borders between partisans, the so-called swing voters, are open to much change. His major point, which I'm distorting a little here by casting his model into an electoral framework, is that social norms, often a result of nothing more than propinquity, make it unnecessary to think much — about what to wear, which side of the road to drive on, when to eat, etc.

To the considerable extent that voting is — at least for many — an unthinking emulation of those with whom they associate, the model helps explain the near uniformity of the political opinions of their friends. (Rush Limbaugh's depressingly telling phrase "ditto heads" applies to many on both sides of the political spectrum.)

When there's some sort of shock to the system, Epstein's model suggests something else rather interesting. If a large number of voters change their vote suddenly for some reason (say a terrorist attack or environmental catastrophe), the changed voter preferences soon settle down to a new equilibrium just as stable with solid Bush, solid Kerry, and mixed border areas, but located at different places around the circle. The model thus shows how political allegiances can sometimes change suddenly, but then settle quickly into a new and different segmentation just as rigidly adhered to as the old.

This model should be easy to simulate, since it's basically a type of cellular automata, similar to, but somewhat more complicated than, John Conway's well-known game of Life.

(In a cellular automata there is a grid of cells, and the state of each individual cell is determined by the state of the cells surrounding it and the application of certain rules. In "Life," for instance, a cell which is "on" or "alive" at Time=0 stays alive for Time=1 if it has 2 or 3 neighbors in the 8 cells surrounding it. If there are less than 2 it dies of loneliness, and if there are more than 3 it dies from overcrowding. A cell which is empty ("dead" or "off") at Time=0 will remain empty at Time=1 unless it has exactly 3 neighbors in the surrounding 8 cells. Using these very simple rules, all sorts of fascinating patterns and amazing "lifeforms" can be seen. A snug 5-cell unit can burst out into an almost infinite pattern, and "spaceships" can "fly" across the grid. I spent many hours fooling around with "Life" when I was in high school, and actually discovered one or two novelties -- such as the superstring -- in the process.

Cellular automata like "Life" may look like games, but they can be surprisingly powerful. The genius mathematician John von Neumann showed that a properly designed one can reproduce itself, or can be made to be a Universal Turing Machine, which means that a cellular automata of sufficient complexity and with the right rules can be made to model any digital computer. [Update: In fact, even a very simple one like "Life" will support a Universal Turing Machine, so great complexity is not required.])

Programming the Epstein/Paulos Theory of Voter Conformity should prove to be relatively simple -- and I rather think that it's already been done, as Paulos' descriptions are very visual, suggesting that he's seen the model run as a simulation.

Addenda: It would be interesting to see this theory combined in some way with Malcolm Gladwell's idea that there are a few different kinds of people who are instrumental in spreading ideas. In The Tipping Point he calls them "connectors" (who know lots of people), "mavens" (who are trusted and influential) and "salesmen" (who can convince other people):

Mavens are data banks. They provide the message. Connectors are social glue: they spread it. ... Salesmen [have] the skills to persuade us when we are unconvinced of what we are hearing ...

In terms of the cellular automata, certain cells could be designated as connectors, mavens and salesmen, randomly spread throughout the grid (at densities to be reasonably determined), and rules would be different for cells bordering on these special units, in a way that roughly models Gladwell's idea. Connectors, for instance, would have a larger circumference of influence than mavens, but the influence of mavens would carry more weight. The influence of salesmen would be wider than mavens but less intense, and stronger than connectors but with less scope.

With these adjustments, things might spread through the grid in a lumpier manner than described by Paulos.