Strategy-Free Elections, And The True Measure of a Voting System

If you get involved in the debate on voting system reform, it's never long before someone brings up the concept of "strategic voting." The idea is simple enough: we all want to be as honest as possible on our ballots, but because the voting system is imperfect, we vote otherwise. This is easy to see in plurality; lots of people who claim to really want a third-party candidate end up voting for one of the two major-party candidates on election day. But it happens, to a lesser or greater extent, under all voting methods.

Actually, that's a cleverly-constructed lie of omission.

It turns out that there are a number of voting systems which are 100% strategy-free, so your honest vote will always also be your best vote. But you're not going to like them. Here's an example: voting is performed like plurality, i.e., each voter picks a single candidate. The winner is whichever candidate is named on a ballot chosen at random. Clearly, you should always vote the one candidate you think is best for the job, because there's no reason to fear that you're "throwing your vote away" or making it easier for a candidate you dislike to win. The clever part in the constructing of the lie was omitting the word deterministic, which means "no random components". There are no strategy-free deterministic voting systems.

Constructing other (and better) strategy-free methods is easy enough. So if strategy is so vitally important that it invariably comes up in every voting-system discussion, why don't we use one of them? The answer is: average performance. Random ballot voting, as your intuition probably tells you, is an absolutely terrible system. But intuition is sometimes wrong, so it's important that we can back it up with data by running computer simulations to calculate Bayesian regret. And the data shows that, based on the number of candidates competing, random ballot is two- to four-times worse than plurality voting; which we all know from experience to be a pretty bad system.

Strategy, and a voting system's susceptibility to it, are an important thing to be aware of. But immunity to strategy, even though it sounds like a great thing to strive for, isn't the goal of a voting system; if it were, we'd have an easy answer to the problem, in the form of non-deterministic voting methods. And there are a host of other reasonable-sounding things for a voting system to accomplish, many of which have been codified as voting system criterion. But, besides many of them being mutually-exclusive (i.e., you can't meet them all), using any of them as a litmus test obscures the true objective, in the same way that focusing exclusively on being strategy-free obscures the true objective. The only true measure of a voting system is it's expected performance: how well it delivers a desirable candidate to the electorate. Average performance, as measured by Bayesian regret, smooths over all the coarse edges of criterion, implicitly assessing all of them for frequency as well as impact.

Why should it matter that, for instance, approval voting fails the majority criteria, if the failure rate is vanishingly-infrequent and has minimal impact? When it performs significantly better than a host of other systems that do meet this criteria, but fail some other, equally-reasonable criteria? It shouldn't. Holding the percentage of strategic voters constant, approval voting has significantly better performance than just about any other voting method. Range voting (AKA score voting) can be even better. Which criteria are passed are secondary to that fact.

How To Decide: Where to Go For Dinner

Let me propose a situation to you: you, and eight friends of yours, all live on the same street in some city or large town. As a crazy random happenstance, you are all precisely equally distant from your nearest neighbors: two blocks away. The nine of you decide that you're going to go out for a big dinner somewhere along your street, and there are two more-or-less identically good options: you could go to ApricotBug's, over on the west end of town, or you could go to Beryl Thursday's, closer to the east end.

How do you decide? Well, let's suppose you decide to put it to a vote, since that seems most democratic. Let's further suppose that, since the restaurants are more-or-less identical, everyone simply votes for the one closest to their house. Three of you live west of ApricotBug's, four of you live east of Beryl Thursday's, and two of you live in between. Let's vote!

You can easily see that everyone west of ApricotBug's votes for it, everyone east of Beryl Thursday's votes for it, and the two in between split their votes one and one. Beryl Thursday's wins, 5 to 4.

Notice that, even though each of you used a selfish algorithm (or to put a positive spin on it, an honest, true-to-yourself algorithm) to decide how to vote (the option that minimizes your own travel time), that as a group you also voted for the globally-minimum amount of travel; it would take a total of 45 blocks traveled for everyone to get to ApricotBug's, but only 41 blocks for everyone to get to Beryl Thursday's.

Continuing with the story: You have so much fun, you decide to do it again next month. However, a new restaurant has opened down the street, CBIM (Can't Believe It's (Already) Monday).

How does CBIM change the voting?

If we used the well-known method of plurality voting, we can see that the vote for ApricotBug's stays at 4, but the 5 votes that originally all went to Beryl Thursday's are now split 3 for it and 2 for CGIM. Which means that, now, ApricotBug's wins.

This is no longer the globally-optimal solution, and if you think about it, it's really kind of stupid that adding more options, particularly an obviously bad choice like CBIM, would ruin the vote like this.

What to Do, What to Do...

After much existential gnashing of teeth, you decide to use "instant runoff voting" (because one of you heard about it on the internet and, at first glance, it sounds pretty good!) With IRV, you can pick additional options beyond just your first choice, and if your first choice doesn't win, your vote will be moved over to the next in line.

Everyone west of ApricotBug's and your friend who lives right next to it will vote A > B > C. Also, the two friends closest to CBIM will vote C > B > A. For the three in the middle, two vote B > A > C and one votes B > C > A. (Count the distances yourself if this isn't clear to you.) So which restaurant wins now? IRV tells us to eliminate the option with the fewest first-place votes. That's CBIM. We then move those votes to the voter's second choice, which in this case is Beryl Thursday's for both voters.

Those two votes puts Beryl Thursday's back over the top, which is good, since we know that it's the globally-optimal solution, with respect to total distance traveled. So it sure seems like IRV has improved things for us. (If you're curious, CBIM would have taken 81 blocks; almost twice as bad as ApricotBug's!)

Next month rolls by, and gosh darnit, you've all gotten really attached to this big night out, so you're going to do it again! One minor change though: CBIM had a little problem at their original location (it was too popular, so it wasn't large enough) and so the owner relocated four blocks over (a total distance of 53 blocks for everyone to reach.)

No problem; we'll let IRV handle this! It's still 4 for A > B > C, and 1 for B > C > A, but now we're up to 3 for C > B > A and down to 1 for B > A > C. That's 4 first-place votes for A, 3 for C, and two for B. Uh oh...

Now we're back to the exact same problem we had before under plurality. None of you or your friends have moved. ApricotBug's and Beryl Thursday's haven't moved. The only thing that changed is a that a third fringe option has moved to a slightly-more-acceptable position. The mere presence of CBIM spoiled the vote under plurality, and now its mere presence has spoiled the election in the exact same way under IRV.

Perhaps you're thinking to yourself that, no, some of the CBIM fans will see what's going to happen, and they'll instead vote for Beryl Thursday's first. But they could have just as easily done that under plurality voting; why would IRV be different? And even if it is, for some unexplained reason, different, why should they have to choose among the lesser of two evils, between ApricotBug's and Beryl Thursday's, when in their heart-of-hearts, they want CBIM? Wasn't IRV supposed to fix that problem?

IRV Isn't a Good Idea

If you think IRV is a good idea because it will allow third parties to win elections, you are wrong; it will be just like under plurality. If you think IRV is a good idea because it fixes the spoiler problem, you are wrong; it will be just like under plurality.

In the real-world, we don't have access to simple metrics like "fewest total blocks traveled" to guide us in choosing our political leaders; we only have everyone's biased, greedy, short-sighted, and misinformed opinions. I believe that, deep down, we all want what's best for everyone, but that, obviously, opinions differ on what that is. If IRV can't properly decide which of three kitsch restaurants to hang out at in a simple, one-dimensional scenario like this, I guarantee that it won't do any better in the real world.

Coming Soon!

Next up, I'll have a dinner-vote using score voting, and then maybe we'll try a two-dimensional example, which will show the problems with Condorcet methods.

Strategy

I got into a bit of an argument discussion in the comments of this article about instant runoff voting over at Independent Political Report. I hope I was able to provide some compelling arguments to the folks there, but I stopped myself short of getting down into the weeds about strategy in different systems, and which ones are more susceptible to strategic manipulations. But I did promise to talk about it here.

One of the commenters at IPR asked the following:

Wouldn’t a 100% strategic vote be identical for all voting systems? With range voting, you could just give someone a 10 out of 10 and everyone else a 0 out of 10 or not vote for them.

The short answer to the first question is "no", they're not all the same, and to the second, while you could vote that way (often called "bullet voting"), it's usually not going to be your best strategy. But these questions (well, one question and one statement) are blind to another strategic consideration, one that takes place outside the voting both and long before the election. We'll cover that at the end.

Strategy for Score Voters

While the strategy for different voting methods are different, they do all share some similarities. Most of them involve first figuring out who the two front-runners are, since that's the "real" fight and each voter is going to want to maximize the effect their vote will have on its outcome. You do that by reading the polls. Polling, and the feed-back effect polls have on opinion and voter turnout, is a whole other subject, but for now, let's assume that polls will nearly, but not perfectly reflect voter's opinions (after all, if they were perfect, we could just skip the vote entirely.) Additionally, there's a chance that there may not be two clear front runners; there may be a three-or-more-way tie for first, or a clear front runner and tie for second. We'll discuss that complication later too; for now, assume we can determine two clear front runners.

For score voting, once you've figured out who the front runners are, you vote like this: Give your most-preferred candidate among the two front runners the maximum score. If there are any candidates you prefer more than that candidate, give them the maximum score too. To the other front-running candidate, give the minimum score. If there are any candidates you prefer less than that candidate, give them the minimum score too. If there are any candidates between these extremes, give them a score scaled linearly between them. Most of the time this means that most voters will give most candidates one of the two extreme scores. In other words, highly-strategic score voters look a lot like voters in approval voting elections, but not exactly; we'll come back to that.

The practical upshot of this is, you can always give your honestly-most-preferred candidate the maximum possible score, while still having the greatest possible effect on the "real" fight between the two front-runners. This may not seem important right now, but keep it in mind, because we'll come back to it too, soon.

Strategy for Plurality Voters

Strategy under plurality should be familiar to everyone. Like score, you determine who the two front-runners are. You vote for whichever of the two you prefer. If there's another candidate you prefer more, then too bad; if you vote for them, you might accidentally give the election to your least-preferred candidate.

Theoretically, if the spread between the two front-runners is wide enough, you could throw away the tiny effect you could have on that contest, and toss your vote to your honest-favorite as a show of moral support. It's strategically not the best plan, but in that situation even the best plan isn't very good for you (no one you like has a chance to win!), so maybe you don't care, maybe communicating your dissatisfaction with both likely winners is more important to you; but this line of thinking feeds into the outside-the-ballot-box strategies which we'll discuss later. So keep it in mind (I know, it's a lot to keep in mind; stay with me!)

Strategy for Instant Runoff Voters

Again, determine the top-two. Among those two, rank your favorite first, and your least favorite last. Rank all the others in between those two. You will note that, in contrast to what IRV advocacy groups claim, it is not in your best interest to rank your honest-favorite first (unless they are in the top two), but rather to rank them second. This is because (again, in contrast to advocates claims) it is possible that your honest favorite could be a spoiler for your preferred top-two candidate, an issue I discussed at length in my posts on non-monotonicity (part I and part II).

Now, if the spread between the top-two candidates is large, or if support for your honest-favorite candidate is low (less than 25% if there are three candidates in the race, and progressively less and less if there are more candidates), you can safely communicate your dissatisfaction with both likely winners buy ranking your honest-favorite first. The percentage values are different, but the logic is the same as under plurality: if your honest vote doesn't really matter, you can afford to be honest. But if the "real" contest is close, your best bet is to betray your favorite, and rank them lower. In this way, highly-strategic instant-runoff voters look very similar to plurality voters.

Instant Runoff is to Plurality as Score is to Approval

Strategic IRV looks like plurality, and strategic score looks like approval; it's an interesting result, and one that is well supported by Warren Smith's Bayesian regret study (representative result graphic). And the values aren't just close, they are exactly the same; a surprising result that Smith went on to prove mathematically.

I want to reiterate this: the correct strategy for IRV gives identical results to plurality voting when all voters follow it. So the only potential improvements we could see under IRV would come from voters choosing to vote honestly instead of strategically. At this point, IRV advocates claim that voters will be more honest; but there's no evidence to support that. Indeed, since this strategic favorite-betrayal is the leading reason for two-party dominance in a government, and all nations that have used IRV have seen their elections stay two-party dominated, we can presume that favorite-betrayal is, in real life, common. And that honesty is relatively rare.

Thinking Outside the (Voting) Box

Voter strategy is just part of the equation though; candidates and political parties have strategies too, under the umbrella of "strategic nomination". Under plurality, for instance, parties absolutely must make sure that they have only one candidate in the race: if they run more than one, the candidates will likely split the vote, and so neither will win. This is why we have party-primaries. In addition, recently parties have realized that they can support candidates who are similar to their opponents, in order to split their vote and ensure their own success (for instance, Republicans buying ads for Green party candidates to defeat their Democratic rival), a tactic so underhanded that even some politicians find it despicable!

IRV fares better, but only slightly better: if there's only one strong opposition party, then running two or more identical candidates can't cause the party any problems. However, since under our current system there is only one strong opposition party, and since all the candidates from the same party are quite similar, IRV advocates have stretched this truth to claim that, under IRV, any number of parties can run as many candidates as they want, without suffering any repercussions, and therefore third parties should help them get IRV enacted. But this claim isn't true; it only seems true given the context of our current two-party environment, and a false appeal that what's true for two must be true for three or more. The truth is, IRV will continue to keep us locked in to a two-party system; although you may be allowed to vote for your favorite among a large homogeneous selection of Democratic and Republican clones. Anything riskier, and they would find themselves in the same vote-splitting catastrophes that plurality suffers from.

There's also consideration of the "nursery effect". Political parties, as a general rule, don't spring up over-night; they grow slowly over several election cycles. Under plurality, they grow and grow until they spoil an election by splitting a vote, and then voters abandon them in droves (the Green Party received 4% as many votes in 2004 as they did in 2000, for instance). Under IRV, the pattern will be the same. Parties will grow, they will spoil elections, and they will be abandoned. The only difference is that they will grow larger before they spoil an election (at least 25% rather than Nader's paltry 2.7%.)

It's that critical point, where a party has grown large enough to be a spoiler, that we must focus on. How well an election method performs in "run of the mill" elections is completely unimportant; in such elections, any method that's ever been seriously proposed will work as well as any other. It's performance in messy, close, near-tie elections that matters; these are the times when polling will flounder, and determining the top-two contenders will be hard. And at these critical points, plurality and IRV will often punish honest voting, electing the wrong candidate because of a spoiler, and pushing voters back towards the two well-known parties.

But what about score and approval voting? Under these methods, any number of parties actually could run any number of candidates without fear; no need for primaries. There can't be any spoilers, and so nothing will force a "lesser of two evils" decision and keep us locked in to a two-party system. Instead, third parties would grow, and grow, until they... win. And in those messy, close, near-tie elections, your best strategy is... honesty.

Strategy Strategy

So which method is "most prone to strategic manipulation"? You've read the strategies, judge for yourself. But know that there's no objective data to back up claims one way or the other, and intelligent people on both sides have written similar descriptions and they all still disagree about what it means. But here's the thing: It doesn't matter. That question is a mere distraction, taking attention away from the actual important question: which method gives us the best results? If you think two-party domination is definitively bad then the answer is clear: score voting or approval voting. If you examine the only marginally objective measure of effectiveness, Bayesian regret, then the answer is clear: even if you cling to the (unsubstantiated) claim that IRV encourages more honesty than score or approval, score and approval still perform better (likely because they avoid two-party domination). This is the only way that new parties (and the fresh ideas they represent) can grow without being cut-off at a critical point; so that they can continue to grow, get elected, and put their ideas into practice. IRV can't do that. (In fact it's been proven that no rank-order method can.)

Mu

Strategy is the wrong question. It leads to pointless, subjective, and unsubstantiated arguments. Effectiveness is what matters. An end to two-party dominance is what matters. And score voting is the only logical answer.