“Passion Over Reason? Mixed Motives and the Optimal Size of Voting Bodies,” J. Morgan & F. Vardi (2010)

Why do people vote? Voting has, as Condorcet noted centuries ago, great information aggregation properties. The classic mystery, best described in Downs’ famous 1957 paper, is basically the following: as the size of the voting polity increases, the probability of being a pivotal voter decreases. Therefore, the balance tilts for all of us toward freeriding if voting is at all costly and your benefit from determining who wins the election is not enormous (though see fellow blogger Andrew Gelman for his rebuttal based on the social benefits of voting). One way around this is in Sandroni and Feddersen’s 2006 AER. There, voters derive utility from fulfilling a duty, and that utility is endogenous to the strategies other agents choose. That is, I get utility from voting if you vote. This is a form of Harsanyi’s rule utilitarianism. You may also know this in game theory as a form of “procedural concern” – your utility in the game depends not only on the ends, but also on the means. A number of papers have taken a similar tack to try to explain stylized facts about voting.

In a new paper, Morgan and Vardy propose using similar procedural utility to investigate whether Condorcet is still right when people get “expressive” utility. Their jumping off point is Madison’s argument in Federalist Paper #58 that “passion, not reason” in votes tends to prevail when legislatures get too large (we’ll ignore for now that Madison’s “passion” works very differently than the theory in the present paper – Madison was worried that in large legislatures you just get more dumb, easily-swayed people, not that individual worries about expressing themselves to their constituency would lead to suboptimal votes).

In the present paper, there are two states, both ex ante the true one with equal probability. Each potential voter is given a signal positively correlated with the actual state. Voters all have an ex-ante expressive bias toward one of the states, and voters are all “malleable” or “stubborn” in that with some probability they are willing to ignore their bias when they receive the informative signal. When voters are unwilling to ignore their bias, they receive a fixed (potentially arbitrarily small) utility payoff from voting their bias instead of using their signal. For example, the evidence may suggest voting for one economic policy, but the home constituency may not like that particular vote. “Stubborn” voters are those who get some small amount of extra utility from voting against the evidence in a way that makes constituents back home happy. “Malleable” voters are those who, when they see the evidence, completely ignore their original bias. Note that stubborn voters may still vote for the economic policy they’re biased against: they’re just weighing the epsilon amount of utility from expressing their bias against the (potentially large) utility they will get from seeing the “good” policy enacted, multiplied of course times the probability that their vote is the critical one.

Even a small amount of extra utility from expressiveness can lead to inefficient information aggregation, particularly when there are a large number of stubborn voters. The intuition here is fairly straightforward: the probability of being pivotal decreases in the number of voters, but the benefit of voting expressively if you are stubborn is fixed regardless of how many voters there are. When there are a relatively large number of stubborn voters, an additional voter in expectation (perhaps stubborn, perhaps not) is more likely to break a tie correctly if the true state is the state with bias than they are to break a tie incorrectly if the true state is the state without bias. Tiebreaking is the only situation that concerns voters that only care about enacting the right policy. Though with even probability of ties information aggregation still occurs, as the number of voters increases, the proportion of tied votes becomes more and more tilted toward the state without bias. The marginal voter with some probability is stubborn, and since her probability of being pivotal is tiny when the number of voters is large, she will vote expressively with high probability. Voting expressively means, in expectation, voting for the state with bias, which breaks ties incorrectly much more often than it does so correctly. Information aggregation is destroyed. Indeed, as the number of voters goes to infinity, the vote is no more likely than a coin flip to choose the “correct” policy in these cases.

There are a couple other minor results of note. First, even when the level of stubbornness is low, it is not always beneficial to increase the number of voters. That is, even though as the probability of the vote aggregating information correctly goes to 1, it does not do so monotonically. In particular, when the number of voters is small, no matter what the other parameters are, everybody simply votes according to the signal they receive about the state. When the number of voters are large, even when voting is purely expressive for stubborn voters, their expressive desires are outweighed by the information aggregating votes of the malleable voters, and the Condorcet result holds. However, there is a range of voter polity size where voters whose signal is contrary to their bias play a mixed strategy, and this particular voting behavior is bad for information aggregation; I wish I could tell you why, but unfortunately the authors give very little intuition for their proofs in the current version of the paper. In any case, the results are an interesting and novel argument for smaller voting polities when information is dispersed.

http://econ.la.psu.edu/papers/JMorgan110210.pdf (Nov 2010 Working Paper)

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