Earlier, I wrote a primer on voting theory. Among the things I discussed were 5 types of pathologies suffered by different single-winner voting methods. I presented these as 5 sequential hurdles for voting method design. That is, since they are in what I view as decreasing importance and increasing difficulty, you should check your voting method against each hurdle in order, and stop as soon as it fails to pass.

Then I read Eliezer's book on Inadequate Equilibria, and Scott's "Meditations on Moloch". They argue that the point of civilization is to provide mechanisms to get out of pernicious equilibria, and the kakistotropic tendencies of civilization they characterize as "Moloch" are basically cases where pernicious incentives reinforce each other. I realized that the simple two-player games such as Prisoners' Dilemma that serve as intuition pumps for game theory lack some of the characteristics of my 5 voting pathologies. So I want to go back and explain those pathologies more carefully, to help build up intuition about how multi-player, single-outcome games differ from two-player ones.

A key point here is that I'm talking about single-winner voting methods; that is, "games" where the number of possible outcomes is far less than the number of players. In this case, it's not a matter of seeking an individual advantage for yourself; the only way for you to win is for your entire faction to win equally. This means that I will not be talking about the oldest and deepest name of Moloch, which is Malthus. All the Molochs in this essay can and should be killed or (mostly) tamed.

Also note that this essay is not the one I'd write if I were only trying to recruit the rationalist community to become electoral reform activists. As an activist, I think that the most important and short-term-viable electoral reforms are in the multi-winner space: solving the problem of coordinating public goods not directly through mechanism design, but indirectly through a combination of mechanism design and representation. Some of my reasons for thinking that are contingent and have no place here. The one that's not: I think that the problem of "ain't nobody got time for all that politics" is worse than the principal-agent problem of a well-designed representative mechanism. Regardless, I think that this community would rather hear first about these names for Moloch.

In order, my pathologies — hurdles for multi-agent shared-outcome mechanism design — are:

Dark Horse

Let's say that you have a 3-candidate election using the Borda count, and your electorate has the following true utilities:

49: A9.0 B1.0 D0.0

48: A1.0 B9.0 D0.0

3: A1.0 B0.0 D9.0

Under the Borda count, each voter must give the three candidates 2, 1, and 0 points in some order. If the B voters strategize, the election might look like:

49: A2 B1 D0

48: A0 B2 D1

3: A1 B0 D2

B wins with a total of 145. The A voters might try to retaliate with a similar strategy:

49: A2 B0 D1

48: A0 B2 D1

3: A1 B0 D2

But now D wins with a total of 103, even though D was honest last preference for 97% of voters.

This "Dark Horse 2" example becomes even harder to resolve if you make it "Dark Horse 3":

34: A9.0 B2.0 C1.0 D0.0

33: A2.0 B9.0 C1.0 D0.0

33: A2.0 B1.0 C9.0 D0.0

I'll let you work it out for yourself, but the upshot is that each group has an incentive to give D the second-most points; that if one or two groups are strategic, they can profit; but if all three are strategic, all of them lose. D can win in this situation with literally zero honest support — an epically pathological result.

What does it feel like in this situation:

To win honestly? "All is right with the world."

To weakly-lose when everyone's honest? "I am slightly tempted to strategize."

To weakly-lose when the opponents are strategic? "I need to stop being a sucker, and counter-strategize."

To win strategically? "I feel a little bit guilty, but at least I won."

To strongly-lose strategically? "WTF? This system sucks. If possible, I should change it. If not, maybe I should learn my lesson and not strategize. But regardless, those other evil sneaky strategizers against me MUST learn theirs."

This is the closest to a standard prisoners dilemma of all of the voting pathologies. As with the standard prisoners dilemma, "social glue" (that is, heuristics developed through successful cooperation in iterated scenarios) can generally avoid breakdown. But it's also the easiest to avoid using mechanism design: just don't use the Borda count (or any other strictly-ranked point-based method). That is to say, don't force people to dishonestly support D in merely in order to oppose some other candidate.

So "Dark Horse" is a name for a Moloch that's outstandingly evil but not particularly powerful.

Lesser evil

If you live in the US, UK, Canada, or India — or any other country that uses First Past the Post voting — you already know this Moloch well. In a system where you can only vote for one, you'd better not "waste your vote" on the option you most truly support; you must instead support the lesser evil, the least-bad of the viable options. The logical end-point is a world with only two options, each of which has far stronger incentives to make the other side look bad than to actually pursue the common good. If you're lucky, one or both of those two options will pursue the common good for the fun of it; if you're unlucky, they'll each be as corrupt as they can get away with without losing support to the other side; but either way, there's relatively little you can do about it.

Of course, I should point out that this game theory doesn't always play out exactly in real life. The US has only 2 parties that matter, but most other FPTP countries have a bit more than that, even if the top two matter more than they should. So if you want to continue to spar with the teeth of this Moloch instead of just cutting off its head, OK, you're not doomed to lose every time. Just most of the time.

In terms of election scenarios, this looks something like the following. Utilities are:

15: A9.0, B8.0, C0.0

36: A8.0, B9.0, C0.0

24: A0.0, B9.0, C8.0

25: A0.0, B1.0, C9.0

Votes are:

15+36=51: A

24+25=49: C

This is an equilibrium because, in most games where there are far more players than outcomes, almost everything is an equilibrium; no one voter could get a better outcome by changing their vote, even though the society as a whole would be far happier if they could elect B. Any A voter who moved to B would be helping C win; any C voter who moved to B would be making it easier for A to win, even if next election honest C>A voters are a majority.

I probably don't have to tell you what this one feels like, but here goes anyway:

On top of the winning coalition (15 A voters): "All is right with the world."

On the bottom of the winning coalition (36 B>A>C voters): Conflicted. On the one hand, "the lesser evil is still evil". On the other hand, "a vote for B is a vote for C". Both are true; this dilemma is inescapable without changing the voting method. Short-term incentives favor continuing to vote for A, and in fact actively suppressing discussion of A's flaws and B's ideas; but human nature favors getting mad at A and exaggerating their flaws. Either way, mind-killing is likely.

On the bottom of the losing coalition (24 B>C>A voters): Enraged. Ripe for a demagogue.

On the top of the losing coalition (25 C voters): Must... try... harder. Next time, we'll win!

This is a lesser Moloch, in that we could easily kill it by changing the voting method. Note that proportional representation can (if it's done well) be just as good at killing this Moloch as the single-winner methods discussed below! But it's still strong enough to rule over most of you who are reading these words.

Center Squeeze

OK, you say; if the Lesser Evil is enabled by the existence of wasted votes, let's fix that by moving all the votes until they're not wasted. You've just invented Instant Runoff Voting (IRV). Each voter ranks the candidates; votes are piled up by which candidate they rank first; and then, iteratively, the smallest pile is eliminated and those votes are moved to whichever remaining pile they rank highest (if any). You can stop as soon as one pile has a majority of remaining votes, because that pile is guaranteed to win.

This would solve the spoiler problem of the 2000 Florida presidential election. Here's a simplified version of utilities in that scenario (B/G/N stand of course for Bush/Gore/Nader):

490: B9.0 G1.0 N0.0 (Bush>Gore)

100: B1.0 G9.0 N0.0 (Gore>Bush)

389: B0.0 G9.0 N1.0 (Gore>Nader)

10: B0.0 G1.0 N9.0 (Nader>Gore)

6: B0.0 G0.0 N9.0 (Nader>nobody)

5: B1.0 G0.0 N9.0 (Nader>Bush)

Under FPTP, honest voting would "spoil" the election and let Bush win. But under IRV, the Nader supporters can vote honestly; when Nader is eliminated, those votes will transfer, so Gore will beat Bush 499 to 495.

But what happens if Nader appeals to more voters, and 300 of the Gore>Nader voters shift to Nader>Gore? That would mean that Nader had 321 first-choice supporters, and Gore only 189. So Gore would be eliminated first, 100 of those votes would shift to Bush, and Bush would win! In this scenario, the centrist Gore was "squeezed" on both sides and prematurely eliminated, even though he could have beaten either of the others in a 1-on-1 race.

And the result is that, just like in the real election, Nader's supporters ended up helping cause the election of Bush, the candidate most of them like the least. That spoilage doesn't happen until after Nader passes 25%, but it still happens. And this problem is real; it happened in the Burlington 2009 mayoral election (though in that case, the voters whose honesty worked against them were the Republicans).

Now, Center Squeeze is a much smaller problem than Lesser Evil. If you have a choice, you'd rather run a race with a minefield between 25% and 50% of the way, than one where the minefield stretches from the beginning up to 50%. If you're skillful, maybe you can build up enough speed in the first 25% to leap over the minefield. And parties that stay under 25% can at least get more attention than those who are stuck around 0% as in Lesser Evil.

What does this one feel like?

Win, not spoiled: "All is right with the world."

Small fringe party, vote honestly, still matter: "At least I tried."

Medium fringe party, vote honestly, spoil the election: Dilemma. Some will decide to be strategic; others will say "wasn't my fault. It was the fault of those treacherous centrists who ranked the greater evil as their second choice."

Centrist, lose due to spoilage: "Huh? What happened? We're the rightful Condorcet winners, how can we lose?"

Large fringe party, win due to spoilage on the other side: "Ha! My far-off enemies were so disgusting that some of my nearby former enemies joined my cause! I deserved that."

Large fringe party, don't win: "Hmm... how can I divide my enemies?"

This Moloch is a relatively benign one, who acts to protect incumbent winners but allows dissenting voices up to a certain point. Living under its reign (as, arguably, Australia now does) involves occasional craziness but is mostly OK. Still, it can be killed.

Chicken Dilemma

This scenario actually exists in two separate versions, depending on the voting method: slippery and non-slippery slope. Both share the same underlying voter utility scenario, with two similar candidates who must team up in order to beat a third one:

35: A9.0 B8.0 C0.0 (A>B)

25: A8.0 B9.0 C0.0 (B>A)

40: A0.0 B0.0 C9.0 (C)

For the slippery slope version, let's assume the election uses approval voting: voters can approve as many candidates as they want, and the most approvals wins. If voters approve any candidate with a utility above 5.0, the ballots will be:

35+25=60: AB

40: C

A and B end up in an exact tie for first place (as Burr and Jefferson did in 1800; thus, the chicken dilemma is sometimes called the Burr dilemma). C, the candidate whom the majority opposes, has been safely defeated; but the outcome between A and B is essentially random. Incentives are clearly high for the first two groups of voters to approve only their favorite candidate. If 1 of the A>B voters votes for only A, then A wins; but then, 2 of the B voters can get B to win by switching to only B; and next 2 more A voters defect; etc. It's a slippery slope until over 20 of each group defect, and then C wins, an outcome the majority hates.

In game theory terms, this is a "chicken" or "snowdrift" game, with 2 equilibria: either the A voters stably cooperate and the B voters stably defect, so that B wins, or vice versa. But in emotional terms, neither of these equilibria feel stable: both are arguably "unfair" cases where one group is exploiting the other's cooperation. It might be "fair" if the smaller group was reliably the one to cooperate, but that's hard to coordinate in practice in cases where the sizes are similar, both sides will probably bet that they are the larger group. So in practical terms, probably the more "stable" outcomes are "both enforce cooperation, and hope there's some odd C voters who care enough to swing the election one way or the other", or "both bicker and defect".

To improve matters, we can use a non-slippery-slope voting method such as 3-2-1 voting. In this method, voters rank each candidate "good", "OK", or "bad", and the winner is decided in 3 steps. First, choose 3 semifinalists, those with the most "good" ratings; then of those, choose 2 finalists, those with the fewest "bad" ratings; then of those, the winner is the one rated higher on more ballots (the pairwise winner).

(When choosing the third semifinalist, there are two additional rules. First, to avoid a clone-candidate incentive, they must not be from the same party as both of the first two or, in a nonpartisan race, do not count their "good" ratings on the same ballots as also rated the first semifinalist "good". Second, to avoid a dark horse issue, they must have at least 1/2 as many "good" ratings as the first semifinalist. If no candidate meets these criteria, then skip step 2.)

In this method, if each voter votes honestly, then all 3 will be semifinalists (eliminating any also-rans whom we left out of the scenario for simplicity); A and B will be finalists (eliminating the majority loser C); and A will win, as the honest pairwise winner between those two.

It's still possible, in this scenario, for 21 B voters to defect, rate A as "bad", and cause B to win. But if under 20 of them do so, it doesn't change the result. Thus, there's no "slippery slope". Even though "everyone cooperates" is not a strong Nash equilibrium in strict game theory terms, it is probably strong enough to endure in practical terms.

Is it possible to make a voting method without even a non-slippery chicken dilemma? Yes, we've already seen that: IRV. But since defectors in the chicken dilemma look exactly like fringe voters in center squeeze, it's impossible to fully solve the chicken dilemma like this without creating a center squeeze problem — one I'd argue is worse, at least as compared to the non-slippery CD.

What does a non-slippery CD feel like? If both sides cooperate, I'd argue that it feels basically fair to everyone involved. If the smaller side wins through strategic defection, that feels unfair, and technically it's an equilibrium; but I'd argue that human stubbornness is enough to counter-defect as a punishment, and thus iterate back to cooperation. 9 So non-slippery CD isn't really Moloch at all. And as for slippery CD... it's mean, but capricious, and can sometimes be distracted or overcome.

Condorcet Cycles

Here's the scenario. Instead of utilities, I'll just give preferences, because there's almost no way to make this one "realistic".

34: A>B>C

33: B>C>A

33: C>A>B

This scenario is so unavoidably strategic that it's at the heart of a proof of the Gibbard-Satterthwaite theorem that no (non-dictatorial) voting method can entirely avoid strategy. If one of the three groups preemptively throws their favorite under the bus and embraces their second choice, the ballots will show at least a 66% majority for that second choice, so any democratic voting method will elect that candidate. So to all three groups, this situation will feel like a dilemma between racing to signal they'll compromise first and most convincingly, or hoping that the group before them in the cycle makes the compromise.

In practice, Condorcet cycles probably happen only 1-5% of the time. This is true in the most sophisticated voter utility models I can create (hierarchical "crosscat" Dirichlet clusters in ideology/priority space), and also in empirical evidence (where cyclical preferences seem rare but not nonexistent). So this last lesser Moloch is one which can never be defeated, but which spends most of its time in the deep woods and only occasionally rampages out, doing surprisingly little damage in the process.


I set out to write this because I thought that multiplayer game theory has some fundamental differences from single-player game theory and specifically that we need to stop leaning so hard on the prisoners' dilemma. Having written it, I realize that though I touched on these issues, I spent most of the time going over more basic points of voting methods. So I'm not sure this essay is exactly what I wanted it to be, but I think what it is can still be at least somewhat useful; I hope you feel the same way.

I guess my larger point is that evolution has actually equipped us pretty well with social strategies for dealing with PD or CD, but that by that same token we humans are particularly subject to pernicious equilibria of the "lesser evil" variety. The feeling of "we all agree these aren't the best options but looking for better ones would waste energy we need to spend fighting against the worse one" (lesser evil) seems like at least as important a paradigm of Moloch as "if I weren't evil someone else would be" (tragedy of the commons/multiplayer prisoner's dilemma/dark horse). It's important to remind ourselves that mechanism design offers a way out of lesser evil (and thus also center squeeze); not just in politics, but wherever it occurs.

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16 comments, sorted by Click to highlight new comments since: Today at 8:42 PM
Note that proportional representation is just as good at killing this Moloch as the single-winner methods discussed below!

At least partially: my country uses the D'Hondt method, which is a proportional representation system, but the lesser evil Moloch still pops up when choosing between which parties to vote for. In particular, it makes it quite hard for new parties to establish themselves, since people will feel that they're wasting their votes if they support a small party which they favor but which doesn't get enough votes for anyone to be elected.

Good point. Edited.

Some acknowledgement of https://en.wikipedia.org/wiki/Arrow%27s_impossibility_theorem seems appropriate in this series - some of these aren't Moloch and misaligned incentives, but a fundamental impossibility in defining "correct" WRT aggregation of preferences.

Followed by an exploration of cardinal voting - if everyone just writes their utility function down, we just maximize U. Simple!

I did talk about Arrow's Theorem in my first post (again, can't link right now, but it's the "Voting Theory Primer for Rationalists").

Also, I acknowledge in the last two pathologies here that they can't all be solved. That is: if you actively punish CD defection so as to make cooperation a strong equilibrium, you allow center squeeze and so create pernicious 2-party equilibria. And no strong equilibrium can exist in case of a Condorcet cycle; that's essentially the basis of Arrow's theorem and thus also Gibbard and Satterthwaite's.

But Arrow's theorem only applies to ranked (ordinal) methods. Which is why I focus more on G-S. And you yourself acknowledge that when you mention cardinal voting.

The defects of cardinal voting are obvious: it becomes a pointless game of who can say the biggest number. If you limit the range of numbers allowed, it becomes score voting; which strategically reduces to approval voting; which I do discuss above.

Arrow’s theorem was mentioned in the previous post in the series.


I'm also a fan of 3-2-1 voting, and I think it has another strong advantage - it's the one I could most easily see explaining to my friends across the political spectrum, having them understand how it works and its potential advantages quickly, and leave thinking it might be worth a shot and maybe even discussing it with their friends. Some anecdata: I live in a state where ranked choice voting failed to pass in the recent election. A few years ago, before anyone knew ranked choice would even be on the ballot, I tried to explain how it worked, and was met with a few types of dismissals: many thought it was too complicated and couldn't follow along*, while one friend's reply was "no, in an election you should just get one vote, and that's that". I'm not sure exactly what he meant by that, or if it was even his true rejection, but it was an interesting response.

But with 3-2-1, I feel like I could explain it to the same people and many would immediately get it and have a positive impression of it and actually remember it again later. Why? Because now I can point out how the candidate on the other side who they can't stand is gonna get knocked out in round 2. Like, not even in the finals... in round 2! Because obviously way more people hate that other party than my party, and then we just have knock off some Libertarian or whoever, and we'll win every time! And even once they realize a different pool of candidates might emerge and change the dynamic, at least that terrible candidate who they're thinking about right now would never win.

Plus, the name itself is very memorable, underscores its simplicity, and is very chantable (for better or worse - I feel somewhat uncomfortable pointing this out, but it seems relevant to a discussion about political systems).

*to be clear, these are smart, reasonable people who would easily understand the concept given enough time. it felt like they were trying to play out elections in their head, realized it was taking too long to figure out during a normal conversation span (understandably), so just figured "forget it" and changed the subject.

Thank you, Jameson. Is the difference between Dark Horse and Chicken the plurality strength (or weakness) of the sincere Condorcet Loser? Does vulnerability to Dark Horse specifically require the method to be "slippery"?

I believe the answers is "yes" to both questions, but I'm not 100% on the second one.


The first link in this post should go ^ here to your voting theory primer. Instead, for me, it links here:


Thanks, fixed. I also added a paragraph at the end.


A bit off topic but curious about your reaction. In the above it's clear that the assumption is that everyone does in fact vote. Does anything change if that assumption is relaxed?

If so, what if the non-votes are counted as "against" votes and the threshold for winning is set to a true majority of all voters not merely that that do actually vote?

Your criticism of IRV is on point. A revolutionary might be happy to see the center squeezed so some kind of radical or reactionary ekes out a win, just to shake things up somehow, but I would not be.

Approval voting's main real advantage is in implementation: like FPTP, it is a one-pass system requiring only a running tally of votes, one data point per candidate. But despite the rumors based on inapt metrics, it is highly tactical. It is "inherently tactical"-- even an honest voter must make a tactical choice in a three candidate race. You approve the Good Guy, but do you also approve of the OK Guy, just to minimize the remote chance the Burn Everything Guy wins?

3-2-1 was news to me, and it looks pretty good. I would not hesitate to follow the consensus on this if the alternative were bikeshedding that meant FPTP wins from inertia. (The solution to that, of course, is for a committee to use a fancy voting scheme to vote on which fancy voting scheme to endorse, but only after the group has voted on which fancy voting scheme to use for that vote as well, and so on.) 3-2-1 is also inherently tactical, but only when there are four or more candidates, and I suppose there usually won't be four or more contenders with a chance worth worrying about.

I remain a fan of Condorcet with IRV as fallback, for reasons we don't even seem to disagree about, just a matter of degrees of preference I suppose. It isn't inherently tactical. It's also hard for me to imagine a realistic large-scale scenario under uncertainty where even a dishonest voter could expect to gain by being tricky. Your criticism that this system is less simple seems mild. Implementation-wise, offhand the system seems about as practical as 3-2-1 or any other ranked choice system, because once you step beyond a one-pass system to something you'd rather use a computer to manage, all of these systems are practical to the computer.

Condorcet is good. The one fundamental sense in which 3-2-1 is better is a better resistance to dark horse pathology, especially in the context of combined delegated and tactical voting. In Condorcet, in a highly-polarized situation, somebody 90% of everybody's never heard of might be the Condorcet winner because each side rates them above the other. In 3-2-1, that person never makes it to the top 3.

This is not a strong argument, but it's the one I have.

As regards IRV, it's definitely worse than either.

Links added.

In your explanation of the Chicken Dilemma, you say that "'everyone cooperates' is not a strong Nash equilibrium in strict game theory terms" (or something like that, I apologize if I phrased it differently), and I disagree with that assertion. In games of Stag Hunt, everyone cooperates is the Nash equilibrium. And it is ultimately the ideal state of a social system, but it tends not to be stable in real societies.

Using the concept of Public Goods Games best illustrates how that works: In any given collaboration, actors can find themselves with the opportunity to engage in selfish or prosocial behavior. Healthy mature adults in cooperative groups gravitate towards cooperation as long as their cooperation is mirrored by their peers. The problem is that there are two equilbria in public goods games that makes them inherently unstable: The most efficient outcomes are both all-defect and all-cooperate.

Further, it only takes a small amount of cheaters to destroy cooperation. Mechanism design has to include some form of punishment to keep systems stable. Lack of punishment in our existing slate of voting systems is the key problem. Given enough time and enough incentives to cheat, systems ultimately destabilize without the presence of negative sanctions, and they do so predictably, as in the case of Duverger's Law regarding FPTP.

It might be helpful to keep that in mind when further exploring voting systems to understand how they work in the real world over the long term.

Stag hunt has two equilibria and only the good one is strong. Prisoner's dilemma has only 1 bad equilibrium. But here we're talking asymmetrical Snowdrift/Chicken, where both the bad and good equilibria are strong, but, if there's uncertainty about which is which, the best outcome is non-equilibrium mutual cooperation.