“Of course, it is unlikely that one’s vote decides the outcome of the election. One’s vote has an impact on the outcome only when (1) the votes of all other voters are evenly split between the two candidates, or (2) one’s preferred candidate would lose by one vote if one did not vote.
P [the probability that your vote is decisive] has been calculated in several ways. Under one approach, each voter can be viewed as picking a ball out of a bag in which p fraction of the balls are labeled candidate 1 and (1 – p) are labeled candidate. Each voter is assumed to have a prior as to what p is. If there are N voters and N is odd, then P1 for any voter is simply the probability that exactly one half of the remaining (N – 1) voters would pick a ball labeled candidate 1 and the remaining one half would pick a ball labeled candidate 2, given this voter’s prior p. P then becomes:

P = 3e^[(-2)(N-1)(p-½)^2]/[2*(2PI(N-1))^(½)]

P declines as N increases, and as p diverts from 1/2.” […]

“Voters do not decide how to vote by picking balls out of hats. On election day, it is more reasonable to assume that all voters are committed to voting for either candidate 1 or candidate 2. Each voter has some prior, p, of the fraction of the population of potential voters who are committed to candidate 1, based perhaps on preelection polls. The rational voter knows, however, that this p is measured with error. Thus, in deciding whether to vote, a rational voter must calculate the probability that her vote will make or break a tie, given p, and the inaccuracy with which it is estimated. This probability is inversely related to (Np (1-p))^(½), the standard deviation of the estimated number of people voting for candidate 1, and thus also becomes infinitesimal as N becomes large.(3)

Several people have noted that the probability of being run over by a car going to or returning from the polls is similar to the probability of casting the decisive vote.(4) If being run over is worse than having one’s preferred candidate lose, then this potential cost of voting alone would exceed the potential gain, and no rational self-interested individual would ever vote. But millions do, and thus the paradox.

There are essentially three ways around the paradox: (1) redefine the rational voter’s calculus so that the rational action is now to vote; (2) relax the rationality assumption; (3) relax the self-interest assumption. All three routes have been pursued.” […and the rest of the chapter deals with these]

From Mueller, chapter 14: The paradox of voting. Note that if you relax the also somewhat unrealistic assumption that everybody know who they’ll vote for beforehand, voting becomes more risky and thus less attractive given risk averse voters.

There’s an election coming up and it’s likely that I’ll post a bit more on related matters in the time to come. Before people start to claim in the comments section that Danes are people who care a lot about the poor and stuff and that’s why we usually have a relatively high voter turnout from an international perspective, actually some of the numbers are telling a quite different story. Dealing with the economic aspects of voting, we’re a bunch of selfish bastards compared to other countries:

“More direct comparisons with Hudson and Jones’s test of the ethical voter hypothesis are obtained in studies of economic voting, which estimate the relative weights placed on egotropic and sociotropic variables. Egotropic variables measure voter expectations regarding the effect of the government’s policies on the voter’s own income, employment status, and so on. Sociotropic variables measure voter expectations regarding the effect of the government’s policies on the economy at large, that is, on the welfare of all citizens. By linking voters’ support for the government to their answers to these sorts of questions, researchers have been able to estimate equivalents to θ in (14.4), where θ = 1 implies full weight on sociotropic variables, and θ = 0 implies full weight on the egotropic variables. Estimates of θ falling between 0.5 and 1.0 have been made for the United States, the United Kingdom, France and Germany.(20) Only Danish voters seem to conform largely to the egotropic economic man assumption in studies by Nannestad and Paldam (1996, 1997). They estimate a θ for Denmark of about 0.15.(21)”

When it comes to voting, just like in the case of lotteries it’s easy to argue the math but hard to argue the preferences. If you derive pleasure from voting, by all means vote. At least as long as the pleasure you derive from voting is relatively unrelated to the impact your vote will have on the election outcome.

September 3, 2011 - Posted by | Books, Economics, politics


  1. When I saw the equation P = 3e^[(-2)(N-1)(p-½)^2]/[2*(2PI(N-1))^(½)], I was instantly reminded of the lyrics of the brilliant Keynes vs Hayek rap: “That simple equation, too much aggregation… ignores human action and motivation.”

    As to the rational voter paradox: as with any complex phenomenon, you can statistically attribute the causality in just about infinite ways, but IMHO bounded rationality is a big part of the explanation. If I remember correctly, even classical economic models assume rationality only on average, in large enough groups of economic agents – the “irrationality” is assumed to cancel out. Whether it actually does, and whether an “irrational” vote still contains valuable information (just as “stupid” bids and asks nevertheless help price discovery in markets) are interesting questions.

    This leads, if you allow the digression, into a discussion of who should have the right to vote. I am out of the mainstream here (shocker!), so ignore and/or disagree at will. I am not a fan of democracy, and I certainly think universal suffrage (the one man – one vote variety) is a very bad idea, at least in the context of a government with virtually unlimited power over individuals. Some level of personal investment, “skin in the game”, should be present, although it is admittedly hard to quantify. Such difficulty, however, does not justify just waiving it – otherwise, you end up with massive redistribution of wealth by the government. In our geek terms, if you do not know what the optimal distribution of a variable is, just assuming uniform distribution is not a very good solution, and the struggling welfare states in Europe and America are good examples, with history providing plenty more (Rome, anyone?). This is probably the place to throw in a reference to the Prussian three class system. Yes, 3 is just as random as 1 for number of classes, but it’s an improvement – going to infinite (i.e. just assigning people votes based on taxes) is the logical extension. The thornier question is are taxes the only factor that votes should be allocated based on. I’d say no – but probably the most important one. I can see some proxies for g and emotional intelligence being very helpful, as well as non-monetary contributions to society – military service and scientific contributions, come to mind. The general idea is simple enough though – you get to control the output of the system in proportion to the inputs you contribute other than your pulse and opinion. [Corollary: you do not get to vote while serving time in jail (Europe, I am looking at you), but you do get to after you have paid your debt to society (USA, you’re not off the hook).]

    Comment by Plamus | September 3, 2011 | Reply

  2. “That simple equation, too much aggregation… ignores human action and motivation.” – the first part of the quote is from the first couple of pages in a chapter of 30 pages. The book has much more on which factors might explain the voting behaviour and this is far from the only chapter dealing with voting behaviour. The reason why I don’t go into more detail about those other factors is that I don’t really care much about those reasons. I really don’t see any convincing arguments for why they should ever enter my utility function once it’s established that all votes I’ll ever cast during my lifetime will with a very, very high probability be completely irrelevant to the election results. It’s the same reason why I try very hard to not let political developments bother me as much anymore more generally; it’s best not to spend time worrying about stuff you can’t change and in the political system my opinion doesn’t matter and nothing I can say or do will make any difference.

    In response to your comments about the optimal vote weighting rules, I’d have to say that I’m sure the voting systems that are in place most places are, in the eyes of perhaps even a high majority of voters, given some ex-ante agreed-upon sets of decision rules taking into account the preference orderings (of individuals asked) over expected outcomes of the voting systems considered, and given the inclusion of a significant number of alternatives in the analysis, not the optimal ones. That’s disregarding procedural factors, which people also generally have strong opinions about, and the inclusion of which would only muddle the waters even further. There’s no good reason why the extant systems should be optimal neither – I guess this is another place where the application of the lesswrong ‘functional vs. optimal’ approach to evaluating outcomes is useful. All that said, I’m also sure that nothing I’d have to say about this subject would ever be remotely relevant for which system would/will in fact prevail. So why would I spend my time thinking about that stuff? It’s not like there isn’t plenty of other stuff to think about.

    Comment by US | September 3, 2011 | Reply

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