Jordan Ellenberg, Princeton mathematician and longtime commenter over at Matthew Yglesias' website, has a pretty good column up over on Slate about the best campaigning strategies for both Presidential candidates to adopt in winning over the swing states crucial to an electoral college victory.

To simplify our problem, let's suppose it's the weekend before Election Day and each candidate can only schedule one more visit. We'll concede Pennsylvania to Kerry; then for Bush to win the election, he must win both Florida and Ohio. Let's say that Bush has a 30 percent chance of winning Ohio and a 70 percent chance at Florida. Furthermore, we'll assume that Bush can increase his chances by 10 percent in either state by making a last-minute visit there, and that Kerry can do the same.I have to hand it to Ellenberg: it just doesn't get much clearer than that. There's a lot more good stuff in the article than this excerpt, and I highly suggest reading it in full - his explanation of mixed strategies is also rather enlightening.If Bush and Kerry both visit the same state, then Bush's chances remain 30 percent in Ohio and 70 percent in Florida, and his chance of winning the election is 0.3 x 0.7, or 21 percent. If Bush visits Ohio and Kerry goes to Florida, Bush has a 40 percent chance in Ohio and a 60 percent chance in Florida, giving him a 0.4 x 0.6, or 24 percent chance of an overall win. Finally, if Bush visits Florida and Kerry visits Ohio, Bush's chances are 20 percent and 80 percent, and his chance of winning drops to 16 percent.

What Bush's advisers ought to notice here is that, whatever Kerry does, Bush is better off if he visits Ohio! Visiting Ohio is what game theorists call a

dominant strategy, and it makes game theory pretty easy: Bush should go to Ohio and ignore Kerry. If you run the numbers, you'll find that going to Ohio is a dominant strategy for Kerry, too, which means that if both campaigns act rationally they'll converge somewhere near Dayton and cancel each other out.The combination of the Bush and Kerry strategies is an example of a

Nash equilibrium. In general, we say that a game between two players B and K is in Nash equilibrium under the following condition:B and K would each be satisfied with their current strategy, even if they knew in advance what their opponent's strategy would be.

## Comments