Thoughts on Voting Methods

I’ve been nerd-sniped by voting theory recently. This post is a fairly disorganized set of thoughts.

Condorcet Isn’t Utilitarian

The condorcet criterion doesn’t make very much sense to me. My impression is that a good chuck of hard-core theorists think of this as one of the most important criteria for a voting method to satisfy. (I’m not really sure if that’s true.)

What the condorcet criterion says is: if a candidate would win pairwise elections against each other candidate, they should win the whole election.

Here’s my counterexample.

Consider an election with four candidates, and three major parties. The three major parties are at each other’s throats. If one of them wins, they will enact laws which plunder the living daylights out of the losing parties, transferring wealth to their supporters.

The fourth candidate will plunder everyone and keep all the wealth. However, the fourth candidate is slightly worse at plundering than the other three.

We can model this scenario with just three voters for simplicity. Here are the voter utilities for the different candidates:

Voters ABCD

D would beat everyone in a head-to-head election. But D is the worst option from a utilitarian standpoint!! Furthermore, I think I endorse the utilitarian judgement here. This is an election with only terrible options, but out of those terrible options, D is the worst.

VSE Isn’t Everything

VSE is a way of basically calculating a utilitarian score for an election method, based on simulating a large number of elections. This is great! I think we should basically look at VSE first, as a way of evaluating proposed systems, and secondarily evaluate formal properties (such as the condorcet criterion, or preferably, others that make more sense) as a way of determining how robust the system is to crazy scenarios.

But I’m also somewhat dissatisfied with VSE; I think there might be better ways of calculating statistical scores for voting methods.

Candidate Options Matter

As we saw in the example for Condorcet, an election can’t give very good results if all the candidates are awful, no matter how good the voting method.

Voting Methods Influence Candidate Selection

Some voting methods, specifically plurality (aka first-past-the-post) and instant runoff voting, are known to create incentive dynamics which encourage two-party systems to eventually emerge.

In order to model this, we would need to simulate many rounds of elections, with candidates (/​political parties) responding to the incentives placed upon them for re-election. VSE instead simulates many independent elections, with randomly selected candidates.

Candidate Selection Systems Should Be Part of the Question

Furthermore, even if we ignore the previous point and restrict our attention to single elections, it seems really important to model the selection of candidates. Randomly selected candidates will be much different from those selected by the republican and democratic parties. These democratically selected candidates will probably be much better, in fact—both parties know that they have to select a candidate who has broad appeal.

Furthermore, this would allow us to try and design better candidate selection methods.

I admit that this would be a distraction if the goal is just to score voting methods in the abstract. But if the goal is to actually implement better systems, then modeling candidate selection seems pretty important.

Utilitarianism Isn’t Friendly

Suppose I modify the example from the beginning, to make the fourth candidate significantly worse at plundering the electorate:

Voters ABCD

Candidate D is still the utilitarian-worst candidate, by 1 utilon. But now (at least for me), the condorcet-winner idea starts to have some appeal: D is a good compromise candidate.

We don’t just want a voting method to optimize total utility. We also want it to discourage unfair outcomes in some sense. I can think of two different ways to formalize this:

  • Discourage wealth transfers. This is the more libertarian/​conservative way of thinking about it. Candidates A, B, and C are bad because they take wealth from one person and give it to another person. This encourages rent-seeking behavior through regulatory capture.

  • Encouraging equitable outcomes. A different way of thinking of it is that candidates A, B, and C are terrible because they create a large amount of inequality. This could be formalized by maximizing the product of utilities in the population rather than the sum, in keeping with Nash bargaining theory. Or, more extreme, we could maximize the minimum (in keeping with Rawls).

These two perspectives are ultimately incompatible, but the point is, VSE doesn’t capture either of them. It doing so, it allows some very nasty dynamics to be counted as high VSE.

Obviously, the Condorcet criterion does capture this—but, like maximizing the minimum voter’s utility, I would say it strays too far from utilitarianism.

Selectorate Theory

This subsection and those that follow are based on reading The Dictator’s Handbook by Bruce Bueno de Mesquita. (You might also want to check out The Logic of Political Survival, which I believe is a more formal version of selectorate theory.) For a short summary, see The Rules for Rulers video by CPG Grey.

The basic idea is that rulers do whatever it takes to stay in power. This means satisfying a number of key supporters, while maintaining personal control of the resources needed to maintain that satisfaction. If the number of supporters a ruler needs to satisfy is smaller, the government is more autocratic; if it is larger, the government is more democratic. This is a spectrum, with the life of the average citizen getting worse as we slide down the scale from democracy to autocracy.

Bruce Bueno de Mesquita claims that the size of the selectorate is the most important variable for governance. I claim that VSE does little to capture this variable.

Cutting the Pie

Imagine the classic pie-cutting problem: there are N people and 1 pie to share between them. Players must decide on a pie-cutting strategy by plurality vote.

There is one “fair” solution, namely to cut a 1/​N piece for each player. But the point of this game is that there are many other equilibria, and none of them are stable under collusion.

If the vote would otherwise go to the fair solution, then half-plus-one of the people could get together and say “Let’s all vote to split the pie just between us!”.

But if that happened, then slightly more than half of that group could conspire together to split the pie just between them. And so on.

This is the pull toward autocracy: coalitions can increase their per-member rewards by reducing the number of coalition members.

Note that VSE is unable to see a problem here, because of its utilitarian foundation. By definition, a pie-cutting problem results in the same total utility no matter what (and, the same average utility) -- even if the winner wins on a tiny coalition.

VSE’s failure to capture this also goes back to its failure to capture the problem of poor options on ballots. If the fair pie-cut was always on the ballot, then a coalition of less than 50% should never be able to win. (This is of course not a guarantee with plurality, but we know plurality is bad.)

Growing the Pie

Of course, the size of the pie is not really fixed. A government can enact good policies to grow the size of the pie, which means more for everyone, or at least more for those in power.

Bruce Bueno de Mesquita points out that the same public goods which grow the economy make revolution easier. Growing the pie is not worth the risk for autocracies. The more autocratic a government, the less such resources it will provide. The more democratic it is, the more it will provide. Growing the pie is the only way a 100% democracy can provide wealth to its constituents, and is still quite appealing to even moderately democratic governments. (He even cites research suggesting that between states within the early USA, significant economic differences can be largely explained by differences in the state governments. The effective amount of support needed to win in state elections in the early USA differed greatly. These differences explain the later economic success of the northern states better than several other hypotheses. See Chapter 10 of The Dictator’s Handbook.)

Bruce Bueno de Mesquita argues that this is the reason that domocracy and autocracy are each more or less stable. A large coalition has a tendency to promote further democratization, as growing the coalition has a tendency to grow the pie further. A small coalition has no such incentive, and instead has a tendency to contract further.

VSE can, of course, capture the idea that growing the pie is good. But I worry that by failing to capture winning coalition size, it fails to encourage this in the long term.

How can we define the size of the winning coalition for election methods in general, and define modifications of VSE which take selectorate theory into account?