Competing on the Line

Competing on the Line^[3]

Like spam, political competition can be analyzed using the Nash equilibrium concept. Assume that two politicians compete for votes, and ideologically, the voters are equally distributed across a line from 0 to 100. Voters at 0 are far left, those at 50 are moderate, and those at 100 are far right. Assume that two politicians, labeled George and Al, stake out positions somewhere alone this line. Further assume that each voter will vote for the candidate who is ideologically closest to him. What is the Nash equilibrium?

Could Al = 0 and George = 50 be Nash? No. In this case Al would regret his choice since George would win by getting all the voters from 45 to 100. George would also regret his choice in this game because if Al = 40, George should pick 41 to maximize the number of votes he gets. Indeed, unless the candidates take near identical positions we don’t have a Nash equilibrium. If Al takes X, George would always want to be right next to him at either X+1 or X-1 to get as many voters as possible.

The only Nash equilibrium occurs where both candidates take positions almost at 50. If one candidate wants to be at 52, the other could win by choosing 51. Competing on the line forces both candidates to the extreme center.

The logic of the line is applicable to business. Imagine that there is a town that consists entirely of one long road. Two gas stations must choose where to locate. Assume that customers will always go to the closest station. The gas stations will both locate at the population center of town next to each other for this is the only Nash equilibrium. If one gas station were a little off center, its competitor would be able to take more than half of the customers by locating right next to it on the center side.

^[3]See Dixit and Nalebuff (1991), 248–250.