Your vote makes a difference only when it is pivotal. Now don’t worry, I am not bringing this up to sort through the tired old arguments about whether you should go to the polls today. You should! That settled, let’s talk about what it implies for how you should vote once you get there.
Because if your vote only makes a difference when it breaks a tie (or makes a tie), then when it comes time to decide how to vote, you might as well assume your vote will be pivotal. And ask yourself how would you vote if your vote was going to make or break a tie.
Be careful. This is not the same as the question “How would you vote if you were the dictator?” Indeed quite often your vote should not be the vote you would cast if yours was the only vote. That’s because when your vote is pivotal you learn something that a dictator doesn’t. You learn that all of the other voters were (almost) perfectly split and and that implies something very specific about the other voters and what they must know about the candidates (or propositions) on the ballot.
Quite often that information is crucial for determining how you want to vote. Let me give you a simple example. Judges are almost always re-elected. Pretty much the only time a judge is voted off the bench is if that judge is completely incompetent. Now you haven’t bothered to read anything about the judges on your ballot. You know nothing about them individually but you know that most judges are doing just fine and should be re-elected.
If you were the dictator (an uninformed dictator!) you would vote yes for every judge. But things turn completely upside-down in an election when you factor in the information you learn from your vote being pivotal. Since all competent judges are easily re-elected, the only way it could have happened that all the other voters are split is that this judge is not competent! Knowing that, and knowing that your vote will decide whether an incompetent judge is re-elected, you should vote no. Against every judge.
Now, the smart readers of this blog have already thought one step ahead and noticed that this logic is self-defeating. Because if everyone figured this out, then everyone is voting against every judge and then every judge is voted down, not just the incompetent ones. Here’s where the theory takes one of two paths, use your judgement.
First you might not believe that the electorate in general is as sophisticated as you are. The vast majority of voters don’t understand the logic of pivotalness and they are naively voting the way they would if they were dictators. In that case, the argument I have laid out works as written and you should vote against every judge.
On the other hand you might believe that a signifcant fraction of voters do understand the strategic subtleties of voting. Then we have an equilibrium to find. For starters we take as given that the judge himself and all of his friends will vote for him. So he has a head start. Now there’s a small group of do-gooders who have read up on this judge and know whether he is competent. They vote as if they are dictators, with good reason now because they are informed. They vote yes if he is competent and no if he is not.
The rest of us know nothing. Until, that is, we take into account what we can infer from being pivotal. And if it were just the informed and the judge’s friends who were voting then what we can infer is that enough of the informed are voting no to counteract the judge’s head start. That is, the judge is incompetent.
In equilibrium none of us uninformed voters vote yes. Because if any of us are voting yes, then effectively the judge has an even bigger head start and that makes it even worse news that the no votes caught up with the head start. But not all of us vote no. Some of us do, but most of us abstain. Enough of us that it remains a valid inference that a pivotal vote means that enough of the informed voted no to make it optimal for us to vote no.
This is the logic of The Swing Voter’s Curse.
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November 1, 2010 at 11:14 pm
kittyreporter
Everyone’s vote counts. If someone thinks their vote won’t matter, they are wrong. Everyone’s vote adds up. That’s why it is crucial for people to vote. Some elections are won by a mere 27 votes — that’s 27 more people that decided to vote that day. Every vote is pivotal. Every vote matters.
November 2, 2010 at 7:59 am
dog reporter
Cats never tell the whole story. This is another example of kitty reporting. You’re vote doesn’t matter, don’t listen to the cat party!
November 2, 2010 at 9:04 am
twicker
Methinks this is the key (and the point at which most economic analyses of voting — at least, the ones I’ve seen — break down):
” if your vote only makes a difference when it breaks a tie (or makes a tie), then when it comes time to decide how to vote, you might as well assume your vote will be pivotal. And ask yourself how would you vote if your vote was going to make or break a tie.”
So: we start with the assumption that only tie-breaking votes make a difference. I would argue that this is a false assumption.
If my vote breaks a tie, then, yes, it absolutely makes a difference. Further, if my vote adds to a margin of victory (for my person), or reduces the margin of victory (of the competition), it *also* makes a difference — because I vote to express my opinion. The stronger the expression of opinion (the stronger the vote for my candidate/position and against my opposition), the better off I am.
If I am anti-incumbent, then I definitely prefer that the incumbent get walloped with a 60%-40% defeat instead of a mere 50%+1/50%-1 near-tie. If I am in *favor* of the incumbent, then I definitely prefer a 60%-40% victory (to quiet the opposition) instead of a mere 50%+1/50%-1 squeaker (which gives more voice to my opposition).
So: all votes count — because I want the collective expression of my opinion to be as strong as possible, NOT just barely above 50% (much as I would prefer to have $50,000 with which to buy a $15k car, instead of just $15,001).
Surplus matters.
Same thing for the judicial election: if I am in support of the system as it currently stands, then, *unless I have information that the judge is incompetent,* I want to express that support — and vote in favor of the judge. I particularly want to do this if I suspect that people opposed to the current system will be coming out strongly. Unless I possess information in opposition to this specific judge, I should vote to favor the system I favor (i.e., in favor of the judge) — both to ensure the opposition does not win and to send a social signal to the opposition (and to my fellow system-supporters) that the system has strong, significant support.
In all cases, I win by expressing my opinion in concert with others. Whether or not I am the tie-breaking vote is nearly irrelevant.
November 2, 2010 at 9:10 am
Right Wing-nut
So, we should vote 100% of the time based on an assumption that we KNOW to be false 99.99% of the time?
You assumptions regarding how a judge might be close to 50% is also deeply flawed. Perhaps there is a political wave this year, and you oppose the wave. (Pick 2008 or 2010) Perhaps the opponent is highly recognizable, comes in with wads of cache, has a particularly catchy campaign, or has the support of a political heavyweight for the wrong reasons. Yes, I have examples in mind.
Your reasoning only works if the informed voters break heavily against the incumbent. But there are many other ways to split the vote.
November 2, 2010 at 9:42 am
Right Wing-nut
My voting strategy does vary for judges, just not like you say:
RLGNWD. (Travis county–it’s a whole ‘nuther Texas) Whereas I will skip over an unopposed D in other races, I specifically DO NOT want a judge elected with 60% of the vote of the other races. I cannot be held responsible for the failure of my party to field a candidate.
November 2, 2010 at 11:56 am
Noto
I forwarded this to my wife, who was convinced by your logic and decided to vote “No” on all the judges on the ballot. Annoyingly, the time it took her to vote “No” on what must have been at least 30 judges made us miss out on our favorite breakfast sandwiches (which was going to be our reward to ourselves for being good citizens).
November 2, 2010 at 4:21 pm
Joe
In the part of the scenario where you describe that the judge and all his friends will vote for him, and consider that as part of a head start in determining whether or not the judge is competent, you have forgotten that the other candidates and their friends will also cast votes in their own favor. As such, neither candidate will have a head start and the dictatorial votes of the savvy but uninformed voters will not, as you suggest, be negated.
Remember, you are not voting ‘yes’ or ‘no’ on a particular candidate. You are distributing ‘yes’ votes to the candidates you prefer.
November 2, 2010 at 4:29 pm
Joe
Well, I take some of that back. A Californian friend of mine just informed me that they do, in fact, vote ‘Yes’ or ‘No’ on judges. My apologies for being a bit too local-centric to myself.
November 2, 2010 at 9:51 pm
Right Wing-nut
They’re called “retention elections”. I find them odious.
November 3, 2010 at 9:19 am
k
I don’t understand how you learn your vote is pivotal.
If I were the judge I would go out and try to convince people their vote is pivotal and represent the situation as if they should vote yes. But some people are not going to listen…you may argue….then the incentive for me to do this is all the greater.
November 3, 2010 at 10:22 pm
Scott
You have to assume that your vote is pivotal, you don’t know it. And as Right Wing-nut said, we KNOW that this assumption will be wrong almost every single time.
November 3, 2010 at 11:35 pm
jeff
scott, since your vote only makes a difference when it is pivotal, there is no need to worry about the cases when it is not. that’s what i mean when i say “you might as well assume” that it is pivotal. It is optimal to vote as if your vote was always pivotal.
Here is an analogy. Suppose that we are going to pick one person at random. First, everybody is asked to guess who that person will be. Then the person picked will win a prize but only if he guessed correctly who was picked. How would you guess? It is a one-in-a-million chance that you will be picked BUT nevertheless you should always guess that it is you that will be picked because only when you are picked does your guess matter.
November 4, 2010 at 7:21 am
twicker
@jeff,
I’m still not sure I accept the first assumption: that my vote *only* makes a difference when it is pivotal.
For example: if you and 499 other people volunteer for a non-profit (say, American Red Cross), you probably see all of you as making a difference. Now, your participation may not be what would be considered “pivotal” — if only the other 499 people showed up, they would still get an awful lot done. However, I suspect that your perception is that your marginal contribution still makes a difference.
Likewise, when people go to a UNC-Duke basketball game, I don’t believe they are likely to see themselves a “pivotal.” But they (and everyone else there, it seems) do believe that their *marginal* contribution helps — it “makes a difference” (and they enjoy the feeling of expressing their opinion along with a bunch of other like-minded people, and *knowing* that a significant portion of these others share their opinion). And, the more people their side has (the larger their majority, or the more-vocal their minority), the more I would suggest they perceive that they make a difference — even though, the more people attend, the less the relative contribution of any one person.
It’s undeniable that, if your vote is pivotal, then your vote makes a difference. It just appears to me that, as in other cases of jointly-expressed opinion or action, expressions *beyond* the pivotal point *also* make a difference. IMHO, your non-pivotal vote makes a difference, in the sense that it increases the expression of your opinion and can serve to energize your allies and suppress your opponents, making it more likely that your opinion has more force.
November 4, 2010 at 10:23 pm
Scott
I also have to disagree with the assumption that your vote only matters when it is pivotal. I think your analogy misses the point, so let me present a counter-analogy. Everyone is asked to pick a number, either 0 or 1. We’re still going to randomly pick one person, but this time, a prize is given to everyone who chose the same number as the person we pick. There is no dominant strategy, as long as we have no information about what number other people pick. However, if there is a poll of a small sample, where most of the sample plans to choose 1 rather than 0, my best strategy would be to choose 1. If I adamantly refuse to choose 1 as a matter of personal principle, my best strategy is to not waste my time and just go home.
That’s what we often see in elections: people whose opinions match the majority of polls are excited and more likely to vote, while those who disagree are more likely to stay home and not vote at all. We get satisfaction by being on the winning side, even if we’re not the pivotal voter, but if we don’t vote, then we lose our ability to claim that we really are on the winning side.
November 4, 2010 at 10:49 pm
jeff
Scott, that’s probably a pretty good description of how other people vote (although probably not when the vote to retain judges.) But is that what you would recommend to someone? Is that a principle that you would use to decide how you should vote?
In any case, if that is really how people vote for judges then the prescription is unchanged. Almost all judges are re-elected. So when people have no specific knowledge of the judge’s competience, they jump on the bandwagon and vote to retain. You are almost certain not to be pivotal. But when you are that means that a lot of people know he is incompetent and indeed so incompetent that they were motivated to go against their bandwagon instincts and vote no.
November 4, 2010 at 10:59 pm
jeff
@twicker, i agree with you. When I vote I should weigh the huge impact i will have in the tiny probability event I am pivotal versus the tiny impact I will have in the almost certain event that my vote just adds or subtracts 1 from some very large vote margin.
Still, here are two thoughts. First, in the case of retention of judges, I don’t think anybody is taking any notice of the actual vote totals, so here I think that pivotalness really is the only thing that matters. Second, even in Presidential elections when pivotalness matters much less, there is an element of information and inference that still plays a role. Conditional on candidate A having a large margin over candidate B, what does that tell me about which candidate I want to add my vote to?
Actually there is a model like this. It’s Ronny Razin “Signalling and Election Motivations…” Here is a link: http://www.jstor.org/pss/1555491
November 3, 2010 at 9:27 pm
Scott
You’re missing a group in the second path: the others in the “significant fraction” who are following the same strategy as you. They will vote against the judge if they follow your logic. So you have the judge and his/her friends who vote for the judge in all cases, and they are balanced out by those who follow your logic who vote against the judge in all cases. The fact that I am the pivotal voter therefore tells me nothing about how the informed group has voted.
More generally, I second what twicker says.
November 4, 2010 at 11:24 am
Donald A Coffin
When I votedin Illinois, we had the annoying judicial retention votes. No opposing candidate, just “Should this judge remain in office?” My policy was always to vote no, because I do not believe that judges should be elected. My hope was that enough people would vote to oust sitting judges that Illinois would move to (fixed-term, like maybe 7-10 years) judicial appointments.
November 8, 2010 at 12:08 pm
Jonathan Weinstein
I used the pivotal-voter principle for my votes on judicial retention, but with a different conclusion. I remembered reading that there were campaigns to oust some judges based on their liberal opinions (e.g. on gay marriage.) I thought the judges most likely to be pivotal were targets of these campaigns, which I disagree with, so I voted yes for everyone, even not knowing for sure if anyone really was such a target.
Reading up the next day, it turns out the campaigns I was thinking of were only in Iowa, not Illinois as far as I can tell. Oh well, I probably did no harm!
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