Game Theory for Swingers
What states should the candidates visit before Election Day?
That's not the only simplification we made in crushing a real-world strategic problem down to something math could handle. Let's now try to make the model more realistic by putting Pennsylvania back in play. How should Bush and Kerry arrange their visits to maximize their chances of winning two of the big three? If we assume that each state is equally likely to tip toward either candidate, the question is simply: How should Bush allocate his travel time so that, in two out of three states, he's made more visits than Kerry? This is what game theorists call a Colonel Blotto game, and, once again, only mixed strategies can be Nash equilibria.
On the other hand, if the states have different profiles—say, Bush's chances of winning Florida, Ohio, and Pennsylvania are 80 percent, 60 percent, and 20 percent, respectively—then there is a dominant strategy. In this case, it's "spend your money in Ohio"—it turns out that it's a better idea to swing the state in the middle than to try to pick off Pennsylvania or shore up Florida. In fact, the "spend the money in the middle state" strategy is dominant whenever Bush's probabilities of victory in the three states are widely separated. (Math fans can check out my calculations for the three-state scenario here.)
Then again, Bush doesn't know the probability he'll win in Florida; all he can do is estimate this number by Bayesian inference, as I discussed two weeks ago. We also haven't taken into account Florida's 27 electoral votes, which make it a bigger prize than Pennsylvania or Ohio. Even if we did that, 47 states would still be absent from our analysis. So, don't rush to judge the candidates' real-world strategies against the math we did here; the problems they face are too hard to be hashed out in a few lines of algebra.
Jordan Ellenberg is a professor of mathematics at the University of Wisconsin. His book How Not To Be Wrong is forthcoming. He blogs at Quomodocumque.
Illustration by Robert Neubecker. Photograph of George Bush on the Slate home page by Kevin Lamarque/Reuters.