Follow-up to: Politics as Charity
Can we think well about courses of action with low probabilities of high payoffs?
Such changes could have enormous effects, but the cost-effectiveness of supporting them is very difficult to quantify as one needs to determine both the value of the effects and the degree to which your donation increases the probability of the change occurring. Each of these is very difficult to estimate and since the first is potentially very large and the second very small , it is very challenging to work out which scale will dominate.
This sequence attempts to actually work out a first approximation of an answer to this question, piece by piece. Last time, I discussed the evidence, especially from randomized experiments, that money spent on campaigning can elicit marginal votes quite cheaply. Today, I'll present the state-of-the-art in estimating the chance that those votes will directly swing an election outcome.
Politics is a mind-killer: tribal feelings readily degrade the analytical skill and impartiality of otherwise very sophisticated thinkers, and so discussion of politics (even in a descriptive empirical way, or in meta-level fashion) signals an increased probability of poor analysis. I am not a political partisan and am raising the subject primarily for its illustrative value in thinking about small probabilities of large payoffs.
Two routes from vote to policy: electing and affecting
In thinking about the effects of an additional vote on policy, we can distinguish between two ways to affect public policy: electing politicians disposed to implement certain policies, or affecting  the policies of existing and future officeholders who base their decisions on electoral statistics (including that marginal vote and its effects). Models of the probability of a marginal vote swaying an election are most obviously relevant to the electing approach, but the affecting route will also depend on such models, as they are used by politicians.
The surprising virtues of naive Fermi calculation
One objection comes from modeling each vote as a flip of a biased coin. If the coin is exactly fair, then the chance of a tie goes with 1/(sqrt(n)). But if the coin is even slightly removed from exact fairness, then the chance of a tie rapidly falls to neglible levels. This was actually one of the first models in the literature, and recapitulated by LessWrongers in comments last time.
However, if we instead think of the bias of the coin itself as sampled from a uniform distribution, then we get the same result as Schwitzgebel. In the electoral context, we can think of the coin's bias as reflecting factors with correlated effects on many voters, e.g. the state of the economy, with good economic results favoring incumbents and their parties.
Fermi, meet data
How well does this hold up against empirical data? In two papers from 1998 and 2009, Andrew Gelman and coauthors attempt to estimate the probability a voter going into past U.S. Presidential elections should have assigned to casting a decisive vote. They use standard models that take inputs like party self-identification, economic growth, and incumbent approval ratings to predict electoral outcomes. These models have proven quite reliable in predicting candidate vote share and no more accurate methods are known. So we can take their output as a first approximation of the individual voter's rational estimates .
... the 1952-1988 elections. For six of the elections, the probability is fairly independent of state size (slightly higher for the smallest states) and is near 1 in 10 million. For the other three elections (1964, 1972, and 1984, corresponding to the landslide victories of Johnson, Nixon, and Reagan [incumbents with favorable economic conditions]), the probability is much smaller, on the order of 1 in hundreds of millions for all of the states.
probabilities a week before the 2008 presidential election, using state-by-state election forecasts based on the latest polls. The states where a single vote was most likely to matter are New Mexico, Virginia, New Hampshire, and Colorado, where your vote had an approximate 1 in 10 million chance of determining the national election outcome. On average, a[n actual] voter in America had a 1 in 60 million chance of being decisive in the presidential election.
It is possible to make sensible estimates of the probability of at least some events that have never happened before, like tied presidential elections, and use them in attempting efficient philanthropy.
 At least for two-boxers. More on one-boxing decision theorists at a later date.
 There are a number of arguments that voters' role in affecting policies is more important, e.g. in this Less Wrong post by Eliezer. More on this later.
 Although for very low values, the possibility that our models are fundamentally mistaken looms progressively larger. See Ord et al.
 Including other relevant sorts of competitiveness, e.g. California is typically a safe state in Presidential elections, but there are usually competitive ballot initiatives.