"Strategic reallocation of political effort" and the additional factor of "strategic reallocation of voting that takes into account other people's strategic reallocation of voting effort" seems both very complicated to calculate and likely to actually matter to what happens in real elections. I would expect quibbles with your conclusions in this area.
You have one sentence that handles the issue, but I'm not entirely sure how you handled it because your sentence involves two pathenthicals, two double negatives, and ambiguity inducing self reference to "this fact". Here is the sentence:
When we take into differences between election cycles, usually another 1-2 orders of magnitude, the value of voting in a "safe" jurisdiction in an election which is not close winds up negligible (if your reaction to this fact is not independent of others').
Here is an attempted rewrite that I think restates the same thing with less ambiguity:
From one election cycle to another, fluctuating global factors account for 1-2 orders of magnitude difference in first order tie estimates. In these situations the value of voting in a "safe" jurisdiction is negligible unless many other people in your "safe" jurisdiction reason identically so that the safe status functions as a secondary global factor that causes the probability of a tie to increase in districts where the "naive" probability of a tie is low.
Assuming this re-writing captures the same basic idea, I think the issue of self-awareness induced ties can be analyzed in terms of the number of people who think of voting as "siding with a winning or losing side" versus "a costly duty to act in a publicly beneficial way". Voters who think of voting as a costly duty seem potentially subject to self-awareness induced ties. Voters who side with predicted winners seem likely to push dynamics away from these sorts of ties.
This suggests small scale experiments and real world polling where voters are measured to see whether they vote according to one, both, or neither of these dynamics and the numbers who do so are used to refine election predictions.
The historical data already take into account the rough current distribution of such voters, and the efforts of national political organizations that try to put money into competitive races. If arguments like mine become more widespread in the future, they will change matters.
This post explicitly limits itself to causal decision theory to help avoid these issues, but I'll discuss them in a future post on decision theory complications. The second parenthetical was an acknowledgment that there is more to say on it.
Experiments and studies like the ones you suggest do seem like they would be helpful in navigating those complications.
Follow-up to: Politics as Charity
Can we think well about courses of action with low probabilities of high payoffs?
Giving What We Can (GWWC), whose members pledge to donate a portion of their income to most efficiently help the global poor, says that evaluating spending on political advocacy is very hard:
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.
Disclaimer
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 [2] 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 [3].
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.
[1] At least for two-boxers. More on one-boxing decision theorists at a later date.
[2] 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.
[3] Although for very low values, the possibility that our models are fundamentally mistaken looms progressively larger. See Ord et al.
[4] Including other relevant sorts of competitiveness, e.g. California is typically a safe state in Presidential elections, but there are usually competitive ballot initiatives.