Rethinking the Decisive Voter

    This week, I’d like to dedicate a few paragraphs to rethinking the idea of the single decisive voter, and the implications of there being many decisive voters on the cost-benefit analysis of voting. In public choice, we understand that, in an election that is not decided by a tie-breaker, the decisive voter is the winning party’s (n+1) voter to the losing party’s collection of n voters. The median voter theorem teaches us that this (n+1)th voter is the median voter, the central position that politicians rush towards in order to collect the majority of voters before election day. Now, consider a scenario in which candidate L has secured the vote of all voters to the left of M (the median voter), and candidate R has secured the vote of all voters to the right of M. In this tie-breaker voting scenario, if R secures the vote of M, he wins. What happens if one voter (call him a) who had previously decided to vote for R, decides to abstain from the election? The position of M shifts left by a hair, candidate L secures the vote of M, and L wins the elections.

    So it seems that voter a held a decisive power in the election after all – and, as a matter of fact, so did every other one of R’s voters, (n + M). Ultimately, n of R’s voters were crucial for negating the preferential power of L’s n voters and shifting M to the middle. I believe this broadening of the decisive voter applies even in non tie-breaking situations: in an election where L has n voters and R has n + 1 + ε voters (ε being the number of voters away from a tie-breaking election there were), then R’s n+1 voters were all decisive, and only L’s n voters and R’s ε voters were not decisive. Should one of R’s n +1 voters abstain, one of their voters would shift from the ε category to the n +1 category. All n +1 R voters were crucial for shifting the preferences of the electorate far enough away from L that R could retain the vote of M. In this scenario, the chances of your vote being decisive increase enormously: in an election with 100 voters where R wins with 60% of the vote, the probability of your vote being decisive goes from 0.01 to 0.41, and their marginal benefit of voting is 41 times higher.

    One final note on decisive voting: in a presidential election, which is played out on the stage of the electoral college rather than the broader electorate, many more voters have a chance of being decisive. In this scenario (n + 1) electors of the winning party are decisive electors: meaning that (n+1) of the winning party’s voters in each of those decisive states can be considered decisive. The chance of being decisive in a presidential election with the electoral college now hinges on you living in a decisive state, but the chances of being decisive in those states is now significantly increased.