Ignorance and Naivete in Large Elections

We consider a two-alternative election with voluntary participation and nearly common interests in which voters may acquire information about which alternative is best. Voters may be rational or naive in the sense of being able, or not, to update their beliefs about the state of the world conditioning on the behavior of others. We show that there is full information equivalence if all voters are rational and there is arbitrarily cheap information. Electoral participation converges to zero if and only if information is costly for all voters. Per contra, if some voters are naive, participation remains bounded way from zero, and full information equivalence requires that information is free for some voters. Increasing the number of naive voters is bad for information aggregation if naive voters are few, but may be good if there are already many.