Note: This is a draft; so far, about the first half is complete. I'm posting it to Discussion for now; when it's finished, I'll move it to Main. In the mean time, I'd appreciate comments, including suggestions on style and/or format. In particular, if you think I should(n't) try to post this as a sequence of separate sections, let me know.

Summary: You want to find the truth? You want to win? You're gonna have to learn the right way to vote. Plurality voting sucks; better voting systems are built from the blocks of approval, medians (Bucklin cutoffs), delegation, and pairwise opposition. I'm working to promote these systems and I want your help.

Contents: 1. Overblown¹ rhetorical setup ... 2. Condorcet's ideals and Arrow's problem ... 3. Further issues for politics ... 4. Rating versus ranking; a solution? ... 5. Delegation and SODA ... 6. Criteria and pathologies ... 7. Representation, Proportional representation, and Sortition ... 8. What I'm doing about it and what you can ... 9. Conclusions and future directions ... 10. Appendix: voting systems table ... 11. Footnotes

1.

This is a website focused on becoming more rational. But that can't just mean getting a black belt in individual epistemic rationality. In a situation where you're not the one making the decision, that black belt is just a recipe for frustration.

Of course, there's also plenty of content here about how to interact rationally; how to argue for truth, including both hacking yourself to give in when you're wrong and hacking others to give in when they are. You can learn plenty here about Aumann's Agreement Theorem on how two rational Bayesians should never knowingly disagree.

But "two rational Bayesians" isn't a whole lot better as a model for society than "one rational Bayesian". Aspiring to be rational is well and good, but the Socratic ideal of a world tied together by two-person dialogue alone is as unrealistic as the sociopath's ideal of a world where their own voice rules alone. Society needs structures for more than two people to interact. And just as we need techniques for checking irrationality in one- and two-person contexts, we need them, perhaps all the more, in multi-person contexts.

Most of the basic individual and dialogical rationality techniques carry over. Things like noticing when you are confused, or making your opponent's arguments into a steel man, are still perfectly applicable. But there's also a new set of issues when n>2: the issues of democracy and voting. For a group of aspiring rationalists to come to a working consensus, of course they need to begin by evaluating and discussing the evidence, but eventually it will be time to cut off the discussion and just vote. When they do so, they should understand the strengths and pitfalls of voting in general and of their chosen voting method in particular.

And voting's not just useful for an aspiring rationalist community. As it happens, it's an important part of how governments are run. Discussing politics may be a mind-killer in many contexts, but there are an awful lot of domains where politics is a part of the road to winning.² Understanding voting processes a little bit can help you navigate that road; understanding them deeply opens the possibility of improving that road and thus winning more often.

2. Collective rationality: Condorcet's ideals and Arrow's problem

Imagine it's 1785, and you're a member of the French Academy of Sciences. You're rubbing elbows with most of the giants of science and mathematics of your day: Coulomb, Fourier, Lalande, Lagrange, Laplace, Lavoisier, Monge; even the odd foreign notable like Franklin with his ideas to unify electrostatics and electric flow.

One day, they'll put your names in front of lots of cameras (even though that foreign yokel Franklin will be in more pictures)

And this academy, with many of the smartest people in the world, has votes on stuff. Who will be our next president; who should edit and schedule our publications; etc. You're sure that if you all could just find the right way to do the voting, you'd get the right answer. In fact, you can easily prove that, or something like it: if a group is deciding between one right and one wrong option, and each member is independently more than 50% likely to get it right, then as the group size grows the chance of a majority vote choosing the right option goes to 1.

But somehow, there's still annoying politics getting in the way. Some people seem to win the elections simply because everyone expects them to win. So last year, the academy decided on a new election system to use, proposed by your rival, Charles de Borda, in which candidates get different points for being a voters first, second, or third choice, and the one with the most points wins. But you're convinced that this new system will lead to the opposite problem: people who win the election precisely because nobody expected them to win, by getting the points that voters strategically don't want to give to a strong rival. But when people point that possibility out to Borda, he only huffs that "my system is meant for honest men!"

So with your proof of the above intuitive, useful result about two-way elections, you try to figure out how to reduce an n-way election to the two-candidate case. Clearly, you can show that Borda's system will frequently give the wrong results from that perspective. But frustratingly, you find that there could sometimes be no right answer; that there will be no candidate who would beat all the others in one-on-one races. A crack has opened up; could it be that the collective decisions of intelligent individual rational agents could be irrational?

Of course, the "you" in this story is the Marquis de Condorcet, and the year 1785 is when he published his Essai sur l’application de l’analyse à la probabilité des décisions rendues à la pluralité des voix, a work devoted to the question of how to acheive collective rationality. The theorem referenced above is Condorcet's Jury Theorem, which seems to offer hope that democracy can point the way from individually-imperfect rationality towards an ever-more-perfect collective rationality. Just as Aumann's Agreement Theorem shows that two rational agents should always move towards consensus, the Condorcet Jury Theorem apparently shows that if you have enough rational agents, the resulting consensus will be correct.

But as I said, Condorcet also opened a crack in that hope: the possibility that collective preferences will be cyclical. If the assumptions of the jury theorem don't hold — if each voter doesn't have a >50% chance of being right on a randomly-selected question, OR if the correctness of two randomly-selected voters is not independent and uncorrelated — then individually-sensible choices can lead to collectively-ridiculous ones. 

What do I mean by "collectively-ridiculous"? Let's imagine that the Rationalist Marching Band is choosing the colors for their summer, winter, and spring uniforms, and that they all agree that the only goal is to have as much as possible of the best possible colors. The summer-style uniforms come in red or blue, and they vote and pick blue; the winter-style ones come in blue or green, and they pick green; and the spring ones come in green or red, and they pick red.

Obviously, this makes us doubt their collective rationality. If, as they all agree they should, they had a consistent favorite color, they should have chosen that color both times that it was available, rather than choosing three different colors in the three cases. Theoretically, the salesperson could use such a fact to pump money out of them; for instance, offering to let them "trade up" their spring uniform from red to blue, then to green, then back to red, charging them a small fee each time; if they voted consistently as above, they would agree to each trade (though of course in reality human voters would probably catch on to the trick pretty soon, so the abstract ideal of an unending circular money pump wouldn't work).

This is the kind of irrationality that Condorcet showed was possible in collective decisionmaking. He also realized that there was a related issue with logical inconsistencies. If you were take a vote on 3 logically related propositions — say, "Should we have a Minister of Silly Walks, to be appointed by the Chancellor of the Excalibur", "Should we have a Minister of Silly Walks, but not appointed by the Chancellor of the Excalibur", and "Should we in fact have a Minister of Silly Walks at all", where the third cannot be true unless one of the first is — then you could easily get majority votes for inconsistent results — in this case, no, no, and yes, respectively. Obviously, there are many ways to fix the problem in this simple case — probably many less-wrong'ers would suggest some Bayesian tricks related to logical networks and treating votes as evidence⁸ — but it's a tough problem in general even today, especially when the logical relationships can be complex, and Condorcet was quite right to be worried about its implications for collective rationality.³

And that's not the only tough problem he correctly foresaw. Nearly 200 years later and an ocean away, in the 1960s, Kenneth Arrow showed that it was impossible for a preferential voting system to avoid the problem of a "Condorcet cycle" of preferences. Arrows theorem shows that any voting system which can consistently give the same winner (or, in ties, winners) for the same voter preferences; which does not make one voter the effective dictator; which is sure to elect a candidate if all voters prefer them; and which will switch the results for two candidates if you switch their names on all the votes... must exhibit, in at least some situation, the pathology that befell the Rationalist Marching Band above, or in other words, must fail "independence of irrelevant alternatives".

Arrow's theorem is far from obvious a priori, but proof is not hard to understand intuitively using Condorcet's insight. Say that there are three candidates, X, Y, and Z, with roughly equal bases of support; and that they form a Condorcet cycle, because in two-way races, X would beat Y with help from Z supporters, Y would beat Z with help from X supporters, and Z would beat X with help from Y supporters. So whoever wins in the three-way race — say, X — just remove the one who would have lost to them — Y in this case — and that "irrelevant" change will change the winner to be the third — Z in this case.

Summary of above: Collective rationality is harder than individual or two-way rationality. Condorcet saw the problem and tried to solve it, but Arrow saw that Condorcet had been doomed to fail.

3. Further issues for politics

So Condorcet's ideals of better rationality through voting appear to be in ruins. But at least we can hope that voting is a good way to do politics, right?

Not so fast. Arrow's theorem quickly led to further disturbing results. Alan Gibbard (and also Mark Satterthwaite) extended that there is no voting system which doesn't encourage voting strategy. That is, if you view an voting system as a class of games where the finite players and finite available strategies are fixed, no player is effectively a dictator, and the only thing that varies are the payoffs for each player from each outcome, there is no voting system where you can derive your best strategic vote purely by looking "honestly" at your own preferences; there is always the possibility of situations where you have to second-guess what others will do.

Amartya Sen piled on with another depressing extension of Arrows' logic. He showed that there is no possible way of aggregating individual choices into collective choice that satisfies two simple criteria. First, it shouldn't choose pareto-dominated outcomes; if everyone prefers situation XYZ to ABC, that they don't do XYZ. Second, it is "minimally liberal"; that is, there are at least two people who each get to freely make their own decision on at least one specific issue each, no matter what, so for instance I always get to decide between X and A (in Gibbard's⁴ example, colors for my house), and you always get to decide between Y and B (colors for your own house). The problem is that if you nosily care more about my house's color, the decision that should have been mine, and I nosily care about yours, more than we each care about our own, then the pareto-dominant situation is the one where we don't decide our own houses; and that nosiness could, in theory, be the case for any specific choice that, a priori, someone might have labelled as our Inalienable Right. It's not such a surprising result when you think about it that way, but it does clearly show that unswerving ideals of Democracy and Liberty will never truly be compatible.

Meanwhile, "public choice" theorists⁵ like Duncan Black, James Buchanan, etc. were busy undermining the idea of democratic government from another direction: the motivations of the politicians and bureaucrats who are supposed to keep it running. They showed that various incentives, including the strange voting scenarios explored by Condorcet and Arrow, would tend open a gap between the motives of the people and those of the government, and that strategic voting and agenda-setting within a legislature would tend to extend the impact of that gap. Where Gibbard and Sen had proved general results, these theorists worked from specific examples. And in one aspect, at least, their analysis is devastatingly unanswerable: the near-ubiquitous "democratic" system of plurality voting, also known as first-past-the-post or vote-for-one or biggest-minority-wins, is terrible in both theory and practice.

So, by the 1980s, things looked pretty depressing for the theory of democracy. Politics, the theory went, was doomed forever to be a worse than sausage factory; disgusting on the inside and distasteful even from outside.

Should an ethical rationalist just give up on politics, then? Of course not. As long as the results it produces are important, it's worth trying to optimize. And as soon as you take the engineer's attitude of optimizing, instead of dogmatically searching for perfection or uselessly whining about the problems, the results above don't seem nearly as bad.

From this engineer's perspective, public choice theory serves as an unsurprising warning that tradeoffs are necessary, but more usefully, as a map of where those tradeoffs can go particularly wrong. In particular, its clearest lesson, in all-caps bold with a blink tag, that PLURALITY IS BAD, can be seen as a hopeful suggestion that other voting systems may be better. Meanwhile, the logic of both Sen's and Gibbard's theorems are built on Arrow's earlier result. So if we could find a way around Arrow, it might help resolve the whole issue.

Summary of above: Democracy is the worst political system... (...except for all the others?) But perhaps it doesn't have to be quite so bad as it is today.

4. Rating versus ranking

So finding a way around Arrow's theorem could be key to this whole matter. As a mathematical theorem, of course, the logic is bulletproof. But it does make one crucial assumption: that the only inputs to a voting system are rankings, that is, voters' ordinal preference orders for the candidates. No distinctions can be made using ratings or grades; that is, as long as you prefer X to Y to Z, the strength of those preferences can't matter. Whether you put Y almost up near X or way down next to Z, the result must be the same.

Relax that assumption, and it's easy to create a voting system which meets Arrow's criteria. It's called Score voting⁶, and it just means rating each candidate with a number from some fixed interval (abstractly speaking, a real number; but in practice, usually an integer); the scores are added up and the highest total or average wins. (Unless there are missing values, of course, total or average amount to the same thing.) You've probably used it yourself on Yelp, IMDB, or similar sites. And it clearly passes all of Arrow's criteria. Non-dictatorship? Check. Unanimity? Check. Symmetry over switching candidate names? Check. Independence of irrelevant alternatives? In the mathematical sense — that is, as long as the scores for other candidates are unchanged — check.

So score voting is an ideal system? Well, it's certainly a far sight better than plurality. But let's check it against Sen and against Gibbard.

Sen's theorem was based on a logic similar to Arrow. However, while Arrow's theorem deals with broad outcomes like which candidate wins, Sen's deals with finely-grained outcomes like (in the example we discussed) how each separate house should be painted. Extending the cardinal numerical logic of score voting to such finely-grained outcomes, we find we've simply reinvented markets. While markets can be great things and often work well in practice, Sen's result still holds in this case; if everything is on the market, then there is no decision which is always yours to make. But since, in practice, as long as you aren't destitute, you tend to be able to make the decisions you care the most about, Sen's theorem seems to have lost its bite in this context.

What about Gibbard's theorem on strategy? Here, things are not so easy. Yes, Gibbard, like Sen, parallels Arrow. But while Arrow deals with what's written on the ballot, Gibbard deals with what's in the voters head. In particular, if a voter prefers X to Y by even the tiniest margin, Gibbard assumes (not unreasonably) that they may be willing to vote however they need to, if by doing so they can ensure X wins instead of Y. Thus, the internal preferences Gibbard treats are, effectively, just ordinal rankings; and the cardinal trick by which score voting avoided Arrovian problems no longer works.

How does score voting deal with strategic issues in practice? The answer to that has two sides. On the one hand, score never requires voters to be actually dishonest. Unlike the situation in a ranked system such as plurality, where we all know that the strategic vote may be to dishonestly ignore your true favorite and vote for a "lesser evil" among the two frontrunners, in score voting you never need to vote a less-preferred option above a more-preferred option. At worst, all you have to do is exaggerate some distinctions and minimize others, so that you might end giving equal votes to less- and more-preferred options.

Did I say "at worst"? I meant, "almost always". Voting strategy only matters to the result when, aside from your vote, two or more candidates are within one vote of being tied for first. Except in unrealistic, perfectly-balanced conditions, as the number of voters rises, the probability that anyone but the two a priori frontrunner candidates is in on this tie falls to zero.⁷ Thus, in score voting, the optimal strategy is nearly always to vote your preferred frontrunner and all candidate above at the maximum, and your less-preferred frontrunner and all candidates below at the minimum. In other words, strategic score voting is basically equivalent to approval voting, where you give each candidate a 1 or 0 and the highest total wins.

In one sense, score voting reducing to approval OK. Approval voting is not a bad system at all. For instance, if there's a known majority Condorcet winner — a candidate who could beat any other by a majority in a one-on-one race — and voters are strategic — they anticipate the unique strong Nash equilibrium, the situation where no group of voters could improve the outcome for all its members by changing their votes, whenever such a unique equilibrium exists — then the Condorcet winner will win under approval. That's a lot of words to say that approval will get the "democratic" results you'd expect in most cases.

But in another sense, it's a problem. If one side of an issue is more inclined to be strategic than the other side, the more-strategic faction could win even if it's a minority. That clashes with many people's ideals of democracy; and worse, it encourages mind-killing political attitudes, where arguments are used as soldiers rather than as ways to seek the truth.

But score and approval voting are not the only systems which escape Arrow's theorem through the trapdoor of ratings. If score voting, using the average of voter ratings, too-strongly encourages voters to strategically seek extreme ratings, then why not use the median rating instead? We know that medians are less sensitive to outliers than averages. And indeed, median-based systems are more resistant to one-sided strategy than average-based ones, giving better hope for reasonable discussion to prosper. That is to say, in a simple model, a minority would need twice as much strategic coordination under median as under average, in order to overcome a majority; and there's good reason to believe that, because of natural factional separation, reality is even more favorable to median systems than that model.

There are several different median systems available. In the US during the 1910-1925 Progressive Era, early versions collectively called "Bucklin voting" were used briefly in over a dozen cities. These reforms, based on counting all top preferences, then adding lower preferences one level at a time until some candidate(s) reach a majority, were all rolled back soon after, principally by party machines upset at upstart challenges or victories. The possibility of multiple, simultaneous majorities is a principal reason for the variety of Bucklin/Median systems. Modern proposals of median systems include Majority Approval Voting, Majority Judgment, and Graduated Majority Judgment, which would probably give the same winners almost all of the time. An important detail is that most median system ballots use verbal or letter grades rather than numeric scores. This is justifiable because the median is preserved under any monotonic transformation, and studies suggest that it would help discourage strategic voting.

Serious attention to rated systems like approval, score, and median systems barely began in the 1980s, and didn't really pick up until 2000. Meanwhile, the increased amateur interest in voting systems in this period — perhaps partially attributable to the anomalous 2000 US presidential election, or to more-recent anomalies in the UK, Canada, and Australia — has led to new discoveries in ranked systems as well. Though such systems are still clearly subject to Arrow's theorem, new "improved Condorcet" methods which use certain tricks to count a voter's equal preferences between to candidates on either side of the ledger depending on the strategic needs, seem to offer promise that Arrovian pathologies can be kept to a minimum.

With this embarrassment of riches of systems to choose from, how should we evaluate which is best? Well, at least one thing is a clear consensus: plurality is a horrible system. Beyond that, things are more controversial; there are dozens of possible objective criteria one could formulate, and any system's inventor and/or supporters can usually formulate some criterion by which it shines.

Ideally, we'd like to measure the utility of each voting system in the real world. Since that's impossible — it would take not just a statistically-significant sample of large-scale real-world elections for each system, but also some way to measure the true internal utility of a result in situations where voters are inevitably strategically motivated to lie about that utility — we must do the next best thing, and measure it in a computer, with simulated voters whose utilities are assigned measurable values. Unfortunately, that requires assumptions about how those utilities are distributed, how voter turnout is decided, and how and whether voters strategize. At best, those assumptions can be varied, to see if findings are robust.

In 2000, Warren Smith performed such simulations for a number of voting systems. He found that score voting had, very robustly, one of the top expected social utilities (or, as he termed it, lowest Bayesian regret). Close on its heels were a median system and approval voting. Unfortunately, though he explored a wide parameter space in terms of voter utility models and inherent strategic inclination of the voters, his simulations did not include voters who were more inclined to be strategic when strategy was more effective. His strategic assumptions were also unfavorable to ranked systems, and slightly unrealistic in other ways. Still, though certain of his numbers must be taken with a grain of salt, some of his results were large and robust enough to be trusted. For instance, he found that plurality voting and instant runoff voting were clearly inferior to rated systems; and that approval voting, even at its worst, offered over half the benefits compared to plurality of any other system.

Summary of above: Rated systems, such as approval voting, score voting, and Majority Approval Voting, can avoid the problems of Arrow's theorem. Though they are certainly not immune to issues of strategic voting, they are a clear step up from plurality. Starting with this section, the opinions are my own; the two prior sections were based on general expert views on the topic.

5. Delegation and SODA

Rated systems are not the only way to try to beat the problems of Arrow and Gibbard (/Satterthwaite).

Summary of above:

6. Criteria and pathologies

do.

Summary of above:

7. Representation, proportionality, and sortition

do.

Summary of above:

8. What I'm doing about it and what you can

do.

Summary of above:

9. Conclusions and future directions

do.

Summary of above:

10. Appendix: voting systems table

Compliance of selected systems (table)

The following table shows which of the above criteria are met by several single-winner systems. Note: contains some errors; I'll carefully vet this when I'm finished with the writing. Still generally reliable though.

	Major­ity/ MMC	Condorcet/ Majority Condorcet	Cond. loser	Mono­tone	Consist­ency/ Particip­ation	Rever­sal sym­metry	IIA	Cloneproof	Poly­time/ Resolv­able	Summ­able	Equal rankings allowed	Later prefs allowed	Later-no-harm­/ Later-no-help		FBC:No favorite betrayal
Approval[nb 1]	Ambig­uous	No/­Strategic yes[nb 2]	No	Yes	Yes[nb 2]	Yes	Ambig­uous	Ambig.­[nb 3]	Yes	O(N)	Yes	No		[nb 4]	Yes
Borda count	No	No	Yes	Yes	Yes	Yes	No	No (teaming)	Yes	O(N)	No	Yes	No		No
Copeland	Yes	Yes	Yes	Yes	No	Yes	No (but ISDA)	No (crowding)	Yes/No	O(N2)	Yes	Yes	No		No
IRV (AV)	Yes	No	Yes	No	No	No	No	Yes	Yes	O(N!)­[nb 5]	No	Yes	Yes		No
Kemeny-Young	Yes	Yes	Yes	Yes	No	Yes	No (but ISDA)	No (teaming)	No/Yes	O(N2)­[nb 6]	Yes	Yes	No		No
Majority Judg­ment[nb 7]	Yes[nb 8]	No/­Strategic yes[nb 2]	No[nb 9]	Yes	No[nb 10]	No[nb 11]	Yes	Yes	Yes	O(N)­[nb 12]	Yes	Yes	No[nb 13]	Yes	Yes
Minimax	Yes/No	Yes[nb 14]	No	Yes	No	No	No	No (spoilers)	Yes	O(N2)	Some variants	Yes	No[nb 14]		No
Plurality	Yes/No	No	No	Yes	Yes	No	No	No (spoilers)	Yes	O(N)	No	No		[nb 4]	No
Range voting[nb 1]	No	No/­Strategic yes[nb 2]	No	Yes	Yes[nb 2]	Yes	Yes[nb 15]	Ambig.­[nb 3]	Yes	O(N)	Yes	Yes	No		Yes
Ranked pairs	Yes	Yes	Yes	Yes	No	Yes	No (but ISDA)	Yes	Yes	O(N2)	Yes	Yes	No		No
Runoff voting	Yes/No	No	Yes	No	No	No	No	No (spoilers)	Yes	O(N)­[nb 16]	No	No[nb 17]	Yes[nb 18]		No
Schulze	Yes	Yes	Yes	Yes	No	Yes	No (but ISDA)	Yes	Yes	O(N2)	Yes	Yes	No		No
SODA voting[nb 19]	Yes	Strategic yes/yes	Yes	Ambig­uous[nb 20]	Yes/Up to 4 cand. [nb 21]	Yes[nb 22]	Up to 4 candidates[nb 21]	Up to 4 cand. (then crowds) [nb 21]	Yes[nb 23]	O(N)	Yes	Limited[nb 24]		Yes	Yes
Random winner/ arbitrary winner[nb 25]	No	No	No	NA	No	Yes	Yes	NA	Yes/No	O(1)	No	No			Yes
Random ballot[nb 26]	No	No	No	Yes	Yes	Yes	Yes	Yes	Yes/No	O(N)	No	No			Yes

"Yes/No", in a column which covers two related criteria, signifies that the given system passes the first criterion and not the second one.

11. Footnotes

¹ When I call my introduction "overblown", I mean that I reserve the right to make broad generalizations there, without getting distracted by caveats. If you don't like this style, feel free to skip to section 2.

 

² Of course, the original "politics is a mind killer" sequence was perfectly clear about this: "Politics is an important domain to which we should individually apply our rationality—but it's a terrible domain in which to learn rationality, or discuss rationality, unless all the discussants are already rational." The focus here is on the first part of that quote, because I think Less Wrong as a whole has moved too far in the direction of avoiding politics as not a domain for rationalists.

 

³ Bayes developed his theorem decades before Condorcet's Essai, but Condorcet probably didn't know of it, as it wasn't popularized by Laplace until about 30 years later, after Condorcet was dead.

 

⁴ Yes, this happens to be the same Alan Gibbard from the previous paragraph.

 

⁵ Confusingly, "public choice" refers to a school of thought, while "social choice" is the name for the broader domain of study. Stop reading this footnote now if you don't want to hear mind-killing partisan identification. "Public choice" theorists are generally seen as politically conservative in the solutions they suggest. It seems to me that the broader "social choice" has avoided taking on a partisan connotation in this sense.

 

⁶ Score voting is also called "range voting" by some. It is not a particularly new idea — for instance, the "loudest cheer wins" rule of ancient Sparta, and even aspects of honeybees' process for choosing new hives, can be seen as score voting — but it was first analyzed theoretically around 2000. Approval voting, which can be seen as a form of score voting where the scores are restricted to 0 and 1, had entered theory only about two decades earlier, though it too has a history of practical use back to antiquity.

 

⁷ OK, fine, this is a simplification. As a voter, you have imperfect information about the true level of support and propensity to vote in the superpopulation of eligible voters, so in reality the chances of a decisive tie between other than your two expected frontrunners is non-zero. Still, in most cases, it's utterly negligible.

 

⁸ This article will focus more on the literature on multi-player strategic voting (competing boundedly-instrumentally-rational agents) than on multi-player Aumann (cooperating boundedly-epistemically-rational agents). If you're interested in the latter, here are some starting points: Scott Aaronson's work is, as far as I know, the state of the art on 2-player Aumann, but its framework assumes that the players have a sophisticated ability to empathize and reason about each others' internal knowledge, and the problems with this that Aaronson plausibly handwaves away in the 2-player case are probably less tractable in the multi-player one. Dalkiran et al deal with an Aumann-like problem over a social network; they find that attempts to "jump ahead" to a final consensus value instead of simply dumbly approaching it asymptotically can lead to failure to converge. And Kanoria et al have perhaps the most interesting result from the perspective of this article; they use the convergence of agents using a naive voting-based algorithm to give a nice upper bound on the difficulty of full Bayesian reasoning itself. None of these papers explicitly considers the problem of coming to consensus on more than one logically-related question at once, though Aaronson's work at least would clearly be easy to extend in that direction, and I think such extensions would be unsurprisingly Bayesian.