Imperfect Voting Systems

by Scott Alexander5 min read20th Jul 201288 comments


Voting TheoryGame Theory

Stalin once (supposedly) said that “He who casts the votes determines nothing; he who counts the votes determines everything “ But he was being insufficiently cynical. He who chooses the voting system may determine just as much as the other two players.

The Art of Strategy gives some good examples of this principle: here's an adaptation of one of them. Three managers are debating whether to give a Distinguished Employee Award to a certain worker. If the worker gets the award, she must receive one of two prizes: a $50 gift certificate, or a $10,000 bonus.

One manager loves the employee and wants her to get the $10,000; if she can't get the $10,000, she should at least get a gift certificate. A second manager acknowledges her contribution but is mostly driven by cost-cutting; she'd be happiest giving her the gift certificate, but would rather refuse to recognize her entirely than lose $10,000. And the third manager dislikes her and doesn't want to recognize her at all - but she also doesn't want the company to gain a reputation for stinginess, so if she gets recognized she'd rather give her the $10,000 than be so pathetic as to give her the cheap certificate.

The managers arrange a meeting to determine the employee's fate. If the agenda tells them to vote for or against giving her an award, and then proceed to determine the prize afterwards if she wins, then things will not go well for the employee. Why not? Because the managers reason as follows: if she gets the award, Manager 1 and Manager 3 will vote for the $10,000 prize, and Manager 2 will vote for the certificate.  Therefore, voting for her to get the award is practically the same as voting for her to get the $10,000 prize. That means Manager 1, who wants her to get the prize, will vote yes on the award, but Managers 2 and 3, who both prefer no award to the $10,000, will strategically vote not to give her the award. Result: she doesn't get recognized for her distinguished service.

But suppose the employee involved happens to be the secretary arranging the meeting where the vote will take place. She makes a seemingly trivial change to the agenda: the managers will vote for what the prize should be first, and then vote on whether to give it to her.

If the managers decide the appropriate prize is $10,000, then the motion to give the award will fail for exactly the same reasons it did above. But if the managers decide the certificate is appropriate, then Manager 1 and 2, who both prefer the certificate to nothing, will vote in favor of giving the award. So the three managers, thinking strategically, realize that the decision before them, which looks like “$10 grand or certificate”, is really “No award or certificate”. Since 1 and 2 both prefer the certificate to nothing, they vote that the certificate is the appropriate prize (even though Manager 1 doesn't really believe this) and the employee ends out with the gift certificate.

But if the secretary is really smart, she may set the agenda as follows: The managers first vote whether or not to give $10,000, and if that fails, they next vote whether or not to give the certificate; if both votes fail the employee gets nothing. Here the managers realize that if the first vote (for $10,000) fails, the next vote (certificate or nothing) will pass, since two managers prefer certificate to nothing as mentioned before. So the true choice in the first vote is “$10,000 versus certificate”. Since two managers (1 and 3) prefer the $10,000 to the certificate, those two start by voting to give the full $10,000, and this is what the employee gets.

So we see that all three options are possible outcomes, and that the true power rests not in the hands of any individual manager, but in the secretary who determines how the voting takes place.

Americans have a head start in understanding the pitfalls of voting systems thanks to the so-called two party system. Every four years, they face quandaries like "If leftists like me vote for Nader instead of Gore just because we like him better, are we going to end up electing Bush because we've split the leftist vote?"

Empirically, yes. The 60,000 Florida citizens who voted Green in 2000 didn't elect Nader. However, they did make Gore lose to Bush by a mere 500 votes. The last post discussed a Vickrey auction, a style of auction in which you have have no incentive to bid anything except your true value. Wouldn't it be nice if we had an electoral system with the same property: one where you should always vote for the candidate you actually support? If such a system existed, we would have ample reason to institute it and could rest assured that no modern-day Stalin was manipulating us via the choice of voting system we used.

Some countries do claim to have better systems than the simple winner-takes-all approach of the United States. My own adopted homeland of Ireland uses a system called “single transferable vote” (also called instant-runoff vote), in which voters rank the X candidates from 1 to X. If a candidate has the majority of first preference votes (or a number of first preference votes greater than the number of positions to fill divided by the number of candidates, in elections with multiple potential winners like legislative elections), then that candidate wins and any surplus votes go to their voters' next preference. If no one meets the quota, then the least popular candidate is eliminated and their second preference votes become first preferences. The system continues until all available seats are full.

For example, suppose I voted (1: Nader), (2: Gore), (3: Bush). The election officials tally all the votes and find that Gore has 49 million first preferences, Bush has 50 million, and Nader has 5 million. There's only one presidency, so a candidate would have to have a majority of votes (greater than 52 million out of 104 million) to win. Since no one meets that quota, the lowest ranked candidate gets eliminated - in this case, Nader. My vote now goes to my second preference, Gore. If 4 million Nader voters put Gore second versus 1 million who put Bush second, the tally's now at 53 million Gore, 51 million Bush. Gore has greater than 52 million and wins the election - the opposite result from if we'd elected a president the traditional way.

Another system called Condorcet voting also uses a list of all candidates ranked in order, but uses the information to run mock runoffs between each of them. So a Condorcet system would use the ballots to run a Gore/Nader match (which Gore would win), a Gore/Bush match (which Gore would win), and a Bush/Nader match (which Bush would win). Since Gore won all of his matches, he becomes President. This becomes complicated when no candidate wins all of his matches (imagine Gore beating Nader, Bush beating Gore, but Nader beating Bush in a sort of Presidential rock-paper-scissors.) Condorcet voting has various options to resolve this; some systems give victory to the candidate whose greatest loss was by the smallest margin, and others to candidates who defeated the greatest number of other candidates.

Do these systems avoid the strategic voting that plagues American elections? No. For example, both Single Transferable Vote and Condorcet voting sometimes provide incentives to rank a candidate with a greater chance of winning higher than a candidate you prefer - that is, the same "vote Gore instead of Nader" dilemma you get in traditional first-past-the-post.

There are many other electoral systems in use around the world, including several more with ranking of candidates, a few that do different sorts of runoffs, and even some that ask you to give a numerical rating to each candidate (for example “Nader 10, Gore 6, Bush -100000”). Some of them even manage to eliminate the temptation to rank a non-preferred candidate first. But these work only at the expense of incentivizing other strategic manuevers, like defining “approved candidate” differently or exaggerating the difference between two candidates.

So is there any voting system that automatically reflects the will of the populace in every way without encouraging tactical voting? No. Various proofs, including the Gibbard-Satterthwaite Theorem and the better-known Arrow Impossibility Theorem show that many of the criteria by which we would naturally judge voting systems are mutually incompatible and that all reasonable systems must contain at least some small element of tactics (one example of an unreasonable system that eliminates tactical voting is picking one ballot at random and determining the results based solely on its preferences; the precise text of the theorem rules out “nondeterministic or dictatorial” methods).

This means that each voting system has its own benefits and drawbacks, and that which one people use is largely a matter of preference. Some of these preferences reflect genuine concern about the differences between voting systems: for example, is it better to make sure your system always elects the Condorcet winner, even if that means the system penalizes candidates who are too similar to other candidates? Is it better to have a system where you can guarantee that participating in the election always makes your candidate more likely to win, or one where you can be sure that everyone voting exactly the opposite will never elect the same candidate?

But in practice, these preferences tend to be political and self-interested. This was recently apparent in Britain, which voted last year on a referendum to change the voting system. The Liberal Democrats, who were perpetually stuck in the same third-place situation as Nader in the States, supported a change to a form of instant runoff voting which would have made voting Lib Dem a much more palatable option; the two major parties opposed it probably for exactly that reason.

Although no single voting system is mathematically perfect, several do seem to do better on the criteria that real people care about; look over Wikipedia's section on the strengths and weaknesses of different voting systems to see which one looks best.