### One man, n votes

Buoyed by a surge of student queries of the form, “what applications does this have in real life?” I was inspired to dust off an old copy - the library’s - of John Allen Paulos’ *A Mathematician Reads the Newspaper*. By the time I finished rereading the brief aside on measuring shareholder and voter power, I had abandoned my original goal of making precalculus relevant to my pupils, and was wondering if there were any analyses online about the amounts of power held by the various states in the electoral college that went beyond the standard “wooo, gotta worry about Florida”-type punditry. Naturally, there were, and since this is about the only aspect of the US election that I can think about without wanting to claw my eyes out, I thought I’d post some of them.

To the best of my knowledge, the Banzhaf index is the standard means of measuring power of groups in block voting systems, such as the electoral college, in which each state’s vote is weighted. The Banzhaf power index for Florida, for example, is computed by considering all the state-by-state possible outcomes in the election - one outcome being the possibility that California’s 54 electoral votes go to Kerry, New York’s 33 go to Kerry, Texas’ 32 go to Bush… - and then counting the numbers of those outcomes that are swung by Florida. Here is a state-by-state list of the Banzhaf power indices for the 50 [thanks, Chris] states and DC. (The power indices in the other columns are also defined.) Florida, the largest swing state, has a power index of 0.193864, meaning that in 19.4% of possible outcomes, neither Bush nor Kerry will have enough electoral votes to win the presidency before Florida is counted. Compare this figure to the relatively small ~4.6% of total electoral votes allocated to Florida. (California, meanwhile, is critical in nearly half of all possible outcomes.) The runtime of the programs doing these computations is already pretty high (O(2^n )), but I wonder if there are any probabilistic variations on this index as applied to the electoral college. In the standard computation, for instance, an outcome that gives California’s 54 votes to Bush and Texas’ 32 to Kerry is weighted the same as the far more likely alternative. A friend of mine from Mathcamp has written some Maple routines evaluating different power indices; someone who keeps up with US politics better than I do could probably make the modification pretty easily.

Going a bit further: I haven’t yet read this detailed article about the Banzhaf power index, but it also contains an analysis of how much power each individual voter has - taking into consideration the population of the states as well as their voice in the electoral college. Despite California’s large population, its voters have the most say - each is 3.34 times as powerful as a single Montana voter. (This, I presume, makes certain assumptions - for instance, that the percentage of registered voters who actually show up is constant from state to state.)

Paulos gives a simple and dramatic example of the relative usefulness of the Banzhaf index versus more standard measurements: consider a company with three shareholders, who respectively own 49%, 35%, and 16% of the company. Although the first’s share is more than triple the third’s, all have equal voting power: in a yes/no vote, whichever side attracts at least two of the voters, carries. Consequently, all shareholders have the same Banzhaf power index - in this case, 1/2. On the other hand, if they held 51%, 35%, and 18% respectively, the first shareholder’s vote is clearly the only one that matters. His power index is 1, and the others’ are each 0.