Thoughts on Voting Methods

by abramdemski4 min read17th Nov 202025 comments

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Voting TheoryRationalityWorld Optimization
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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:

 Candidates
Voters  ABCD
1100001
2010001
3001001

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:

 Candidates
Voters  ABCD
11000033
20100033
30010033

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?

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Curious what you think of Consistent Probabilistic Social Choice.

My summary:

There is a unique consistent voting system in cases where the system may return a stochastic distribution of candidates!

(where consistent means: grouping together populations that agree doesn't change the result, and neither does duplicating candidates)

What is the rule? Take a symmetric zero-sum game where each player picks a candidate, and someone wins if their candidate is preferred by the majority to the other, winning more points if they are preferred by a larger majority. This game's Nash equilibrium is the distribution.

OK, I basically don't like the voting system.

Scott pointed out to me that the condorcet criterion makes more sense if we include stochastic outcomes. In the cases where the Condorcet winner is the utilitarian-worst candidate, a mixture of other candidates will win over the Condorcet winner. (So that candidate won't really be the Condorcet winner, if we include stochastic outcomes as "candidates".)

But that's not what's going on here, because this technique always selects a Condorcet winner, if there is one.

So (apparently) it's not including stochastic outcomes in the right way.

We can do better by modifying the game:

We specify a symmetric two-player zero-sum game where each player selects a distribution over candidates. You score points based on how many more votes your proposed distribution would get against the other player's. The game's Nash equilibrium (a distribution over distribution over candidates) is the output distribution.

However, I'm a bit suspicious of this, since I didn't especially like the basic proposal and this is the same thing one level up.

Since this is the unique voting system under some consistency conditions, I must not agree with some of the consistency conditions, although I'm not sure which ones I disagree with.

I agree that moving to distributions and scalar utility is a good way of avoiding Pareto suboptimal outcomes.

[+][comment deleted]8d 2

Sounds awesome!! I'll need to evaluate it to get a better idea of what's going on. 

I'm not necessarily expecting it to be utilitarian-good or good in a bargaining sense, but still, it sounds really interesting. 

That's already the case though since a random voter is always more likely to be on the winning side.

I don't know what you are saying is the case.

I believe Adam Smith is saying that your description of "the rule" sounds like something which is true of any voting method. I don't agree, but I must admit that I myself don't understand what you're  describing in that paragraph. How is the game supposed to work? Is there a more direct explanation of the distribution, rather than just a characterization as a Nash equilibria?

FPTP would be if there weren't more points awarded for winning by more votes.

Here is an example of an election.

3 prefer A > B > C

4 prefer B > C > A

5 prefer C > A > B

(note, this is a Condorcet cycle)

Now we construct the following payoff matrix for a zero sum game, where the number given is for the utility of the row player:

\ A B C

A 0 4 -6

B -4 0 2

C 6 -2 0

This is basically rock paper scissors, except that the A strategy wins twice as much when it wins as the C strategy does, and the B strategy wins 3 times as much as the C strategy does.

This game's unique Nash equilibrium picks A 1/6 of the time, B 1/2 of the time, C 1/3 of the time. So this is the probability of the candidates being elected.

Ah, I see. What I was missing from this description:

What is the rule? Take a symmetric zero-sum game where each player picks a candidate, and someone wins if their candidate is preferred by the majority to the other, winning more points if they are preferred by a larger majority. This game's Nash equilibrium is the distribution.

was understanding that we construct a two player game, not a game where the players are the voters (or even a game where the candidates are the players).

One major problem is that most voting carries only ordinal data (preference ranking), not cardinal (magnitude of preference).  And the ones that allow aggregation or magnitude are very susceptible to strategy.  This makes utilitarian optimization among uncooperative members very difficult.

STAR voting largely solves this problem, by taking the runoff idea from ordinal voting and applying it to cardinal voting.

The idea is this: your ballot is just like cardinal voting. You find the top two candidates by total score. Then you choose the candidate who would win in a pairwise election between the two, under the assumption that you'd vote for whoever you gave a higher score, and abstain otherwise.

This incentivises voters to score different candidates differently, unless they really don't care between two, because otherwise you could be throwing away your vote in the final stage.

As a result, STAR does better at collecting the utilitarian preferences, but at the cost of sometimes choosing a candidate other than the utilitarian max. (But arguably, choosing someone who would win in a pairwise election against the utilitarian max isn't too bad -- to win against the utilitarian max in a pairwise election, you have to be more of a compromise candidate, meaning STAR favors more equitable outcomes.)

I think that the intuition that D is a good compromise candidate (in the second example) is wrong, since each of the voters would prefer a random candidate out of {A,B,C} to D. In other words, a uniform lottery over {A,B,C} is a Pareto improvement on D.

Do you disagree with the wealth-transfer point? IE, it seems to me like a voting system should actively disincentivize wealth transfers to discourage the pathologicall pie-cutting disequilibria, which could destroy value in the long term via repeated tug of war over resources. 

While the system should discourage utility transfers, if you build in a bias against transfers of nominal wealth, that favors certain political platforms over others.

Yes.

I actually don't think there's a good way for a voting system to distinguish between transfers and simple inequality, so I think voting systems designed with this concern in mind will actually favor wealth (/utility) transfers that reduce inequality, rather than discouraging all transfers.

Either way, though, there's inherently some political content in there, which is unfortunate in some sense.

I think that pie-cutting is usually negative-sum, because of diminishing returns and transaction costs. So, if you could make utilitarianism into a voting system it would at least ameliorate the problem (ofc we can't easily do that because of dishonesty). However, ideally what we probably want is not utilitarianism but something like a bargaining solution. Moreover, in practice we don't know the utility functions, so we should assume some prior distribution over possible utility functions and choose the voting system that minimizes some kind of expected regret.

Right, very much agreed. The thing I was trying to convey was the difference between utilitarianism and a bargaining solution.

Structural effects are, indeed, super important. Deciding what the options are (or even just what order the options are presented in: imagine United States elections, but with the general election held first (voting only for party) and then the party with the most votes picks the officeholder afterwards - you get a very different set of incentives).

Also, knowing how others are going to vote: as detailed elsewhere, if there are two close options, the exact mechanics of the election and the knowledge available to strategic voters can allow either (or neither) of them to win, even with the same set of preferences in the population.

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.

Is that still a problem if we assume there's diminishing returns? Someone with twice the pie might not be twice as happy, so a less equal distribution would have less utility.

Sure, and I think that's a realistic assumption in most cases. But we could still come up with a scenario equivalent to pie-cutting. And the behavior of many voting systems in that scenario still seems problematic to me, and argues in favor of voting systems preferring compromise candidates to utilitarian candidates in some situations.

To be more concrete, if we assume diminishing returns, we could still get something like the pie-cutting scenario given by assuming the transfers are small relative to the total, and executed in a way that's independent of wealth of the individuals (ie only based on their political party). I would defend my assertion that VSE is failing to capture a desirable feature of a voting system in that scenario.

I really enjoyed this post, especially the implication that the Condorcet criterion only makes sense in the presence of an adequate candidate selection system. Even aside from process and bureaucracy for getting on the ballot, I wonder if using different voting systems for primary vs general elections might make a difference. Like, maybe approval voting for a party primary but ranked choice for the general?

I've been thinking more about this lately, since my state just voted against a ballot measure to switch to ranked choice voting from FPTP. Everyone I know is kinda horrified by that, but I assume it mostly reflects most people not having any idea what it's about, why they should care, or how to evaluate the arguments for and against that people published. 

I've been thinking: couldn't we "essentially" get rid of the two party system, without any voting reform, by having sub-parties be more of a thing? Like how the "tea party" emerged as a type of republican. Why isn't the green party just a type of Democrat? Don't try to run a third party candidate; instead, third parties should try to get their candidates to be the candidate of choice for one of the two major parties. The green party could have its caucuses early and send its candidates to the democratic caucuses. Stuff like that. 

If things progressed in this direction, the major parties would become coalitions of minor parties. Those coalitions could shift over time. 

I think that's essentially what we had in the mid-1900s, at least in the form of party regional differences, like the conservative Southern Democrats. I do think it worked much better, since the idea of compromise and negotiation with people you disagree with seemed much easier and less tribal then than now. The Republicans only started becoming consistently conservative after the Civil Rights Act, and more so after 1994 when they found that having a single national platform won them a lot of House and Senate seats. Message unity made it a heck of a lot easier and cheaper to reach people, especially pre-internet.

As far as shifting towards coalition parties, I think that would only work if 1) we find ways to make party affiliation less tribal in general, and 2) there is still some clear axis on which each party is unified and differentiated from the other (and presumably, those unified issues are the ones that would be pursued most strongly, and become the party's de factor agenda).

Also: isn't this basically what Bernie Sanders tried to do, an independent running as a Democratic primary candidate? And the general trend towards larger primary fields in presidential races? But of course, most candidate selection power is still out of voters' hands

Personally I'd like to see the democrats focus more on state and local offices, and the judiciary, instead to ceding so much to the republicans. That would help in promoting wider variety in the next generation of its leaders, and increase engagement without having to become more ideologically unified. And you know, actually talk about these kinds of difference in party composition style.

It's pretty cool to see a sort of snapshot of your thoughts about a specific topic, even without that much structure or organization.

About the selectorate theory, do you know why just adding a penalty term on the size of the winning coalition doesn't solve the problem? It looks too naive, but I'm curious about an actual rebuttal.

Bruce Bueno de Mesquita argues that this is the reason that domocracy and autocracy

Typo

About the selectorate theory, do you know why just adding a penalty term on the size of the winning coalition doesn't solve the problem?

How do you formally measure the size, in general? 

In plurality voting, it's just the number of voters. Simple. 

In IRV, it's what? The number of voters? But not all of those voters have the winner first place in their ranking. Many might have given them near to last place. 

In score voting,???