The Empirical Frequency of a Pivotal Vote

Empirical distributions of election margins are computing using data on 16,577 U.S. Congressional and 40,036 state legislator election returns. One of every 100,000 votes cast in U.S. elections, and one of every 15,000 votes cast in state elections, "mattered" in the sense that they were cast for a candidate that officially tied or won by one vote. Very close elections are more rare than a binomial model predicts. The evidence also suggests that recounts, and other margin-specific election procedures, are quite relevant determinants of the frequency of a pivotal vote. Although moderately close elections (winning margin of less than ten percentage points) are more common than landslides, the distribution of moderately close U.S. election margins is approximately uniform. In contrast, the distribution of state legislature election margins is clearly monotonic, with closer margins more likely, except for very close and very lopsided elections. The frequency of one vote elections is compared with total votes, and with the frequencies suggested by some theoretical models of voting. Roughly one of every 30,000 elections with 100,000 votes are decided by one vote. For elections with 5,000 or 20,000 votes, the frequencies are 1/1500 or 1/6000, respectively. This inverse relationship between election size and the frequency of one vote margins is found in two data sets over a wide range of election sizes, with the exception of the smallest U.S. elections for which the frequency increases with election size.