EA - Casting the Decisive Vote by Toby Ord

Link to original articleWelcome to The Nonlinear Library, where we use Text-to-Speech software to convert the best writing from the Rationalist and EA communities into audio. This is: Casting the Decisive Vote, published by Toby Ord on March 27, 2023 on The Effective Altruism Forum.The moral value of voting is a perennial topic in EA. This piece shows that in any election that isn't a forgone conclusion, the chance of your vote being decisive can't be much lower than 1 in the number of voters. So voting will be worth it around the point where the value your preferred candidate would bring to the average citizen exceeds the cost of you voting.What is the chance your vote changes the outcome of an election? We know it is low, but how low?In particular, how does it compare with an intuitive baseline of a 1 in n chance, where n is the number of voters? This baseline is an important landmark not only because it is so intuitive, but because it is roughly the threshold needed for voting to be justified in terms of the good it produces for the members of the community (since the total benefit is also going to be proportional to n).Some political scientists have tried to estimate it with simplified theoretical models involving random voting. Depending on their assumptions, this has suggested it is much higher than the baseline — roughly 1 in the square root of n (Banzhaf 1965) — or that it is extraordinarily lower — something like 1 in 10^2659 for a US presidential election (Brennan 2011).Statisticians have attempted to determine the chance of a vote being decisive for particular elections using detailed empirical modelling, with data from previous elections and contemporaneous polls. For example, Gelman et al (2010) use such a model to estimate that an average voter had a 1 in 60 million chance of changing the result of the 2008 US presidential election, which is about 3 times higher than the baseline.In contrast, I’ll give a simple method that depends on almost no assumptions or data, and provides a floor for how low this probability can be. It will calculate this using just two inputs: the number of voters, n, and the probability of the underdog winning, p_u.The method works for any two-candidate election that uses simple majority. So it wouldn’t work for the US presidential election, but would work for your chance of being decisive within your state, and could be combined with estimates that state is decisive nationally. It also applies for many minor ‘elections’ you may encounter, such as the chance of your vote being decisive on a committee.We start by considering a probability distribution over what share of the vote a candidate will get, from 0% to 100%. In theory, this distribution could have any shape, but in practice it will almost always have a single peak (which could be at one end, or somewhere in between). We will assume that the probability distribution over vote share has this shape (that it is ‘unimodal’) and this is the only substantive assumption we’ll make.We will treat this as the probability distribution of the votes a candidate gets before factoring in your own vote. If there is an even number of votes (before yours) then your vote matters only if the vote shares are tied. In that case, which way you vote decides the election. If there is an odd number of votes (before yours), it is a little more complex, but works out about the same: Before your vote, one candidate has one fewer vote. Your vote decides whether they lose or tie, so is worth half an election. But because there are two different ways the candidates could be one vote apart (candidate A has one fewer or candidate B has one fewer), you are about twice as likely to end up in this situation, so have the same expected impact. For ease of presentation I’ll assume there is an even number of voters other than you, but nothing turns on this.(In real elections, you may also have to worry about probabilistic recounts, but if you do the analysis, these don’t substantivel...