# Noble Abstention

High voter turnout need not be good – not only is voting costly, but ignorant voters can make better candidates less likely to win.  Many have noted this last possibility, but I couldn’t find any formal models (i.e., of abstention by varying-info-quality voters) – so I made my own.  My results depend on how unequally distributed is voter info; highly unequal info means only a handful should vote, while relatively equal info means most everyone should vote (ignoring voting costs):

Let N voters simultaneously choose to abstain or vote for one of two apriori-equal candidates, based only on which action is more likely to elect the "best" candidate.  Each voter gets an independent private binary signal with a chance (1+q)/2 of accurately marking the best candidate.  Assume that if we rank voters by signal quality q (so the best voter has rank = 1, the next has rank = 2, etc.), we’ll find quality and rank are related by a power law, q = q1*rank^-power.  (Voter signal qualities q>0 are common knowledge.)

The following table shows how the number who do not abstain varies with info-quality power, for N = 10,000 voters, q1=0.1, and for two important cases.  In case 1, everyone can choose freely to vote or abstain.  In case 2, each voter chooses if to abstain, assuming no other voter abstains.

 Power 2 1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 Case 1 1 1 1 1 1 1 1 1 2 2 2 3 3 4 5 6 8 13 24 78 10000 10000 Case 2 109 135 171 221 290 389 532 739 1037 1460 2042 2404 2820 3297 3842 4472 5205 6073 7124 8437 10000 10000

The table shows that for powers above one less than 20% should vote, even if everyone else votes.  And if everyone does what they should, for powers above 1/2, almost no one votes, while for powers below 1/2 everyone votes.  (This last result holds for any N and q1.)

Similar results probably hold for correlated and non-binary signals.  I’m not sure how we could measure voter info inequality, but I’d give at least two to one odds the effective power is above 1/2, meaning few should vote.  Of course people vote not just to elect the best candidates overall, but to elect folks good for them personally, even when that hurts others.  But this isn’t something we should celebrate or encourage, and most voters see themselves as instead voting to improve society overall.

Large voter turnouts seem to me better understood as overconfidence leading to disagreement – we each think we just know better than others what is good for society.  But too few of us can be right on this for most of us to be (epistemically) rational here.  So I celebrate the noble abstainers, those willing to admit by staying home that their vote would probably just make things worse – we could use a lot more such folks.

Added 16Sep: Any model with independent signals must either have large electorates get very certain to make the right choice, or must have voter signals get very weak with high rank.  A correlated signals model would be more realistic here.

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