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@toby-pereira yes monotonicity fails, but I think pairwise probability margins are apparently monotonic.

I think maximal lotteries are interesting too, and I wonder if it makes sense to compensate the dissatisfied majorities with electoral credit to spend in subsequent election(s).

For example, say the maximal lottery (or maybe any other Condorcet method) elects A when there is no Condorcet winner. Let beat(X) be the set of candidates that beat X in a head-to-head majority.

Then for each B in beat(A), it seems like the majority preferring B to A should be compensated in some way. My thinking is that in aggregate, that majority should be compensated by something like the smallest extra weight they would have needed to secure B as the (Condorcet) winner against each C in beat(B). This might get complicated, and there may already be literature on this kind of thing.

@cfrank I think lottery methods in general are interesting and worth looking into. Looking at the Wikipedia page, it seems interesting that this would satisfy participation but not monotonicity, which is the opposite of most Condorcet methods. So while some might criticise it for failing monotonicity (seen as easy to get in a Condorcet method), the prize is arguably better, since elections are really about voters rather than candidates.

I just learned about this now. It’s a Smith compliant method, and is participation compliant up until necessary randomization.

https://en.wikipedia.org/wiki/Maximal_lotteries

Basically, you form the majority margin matrix Mij over candidates. Then consider the two player game where each player chooses one candidate: if player 1 chooses i, and player 2 chooses j, then player 1 gets Mij and player 2 gets -Mij.

This game has at least one mixed strategy Nash equilibrium, which is a distribution over candidates. A maximal lottery selects a candidate at random according to one such mixed strategy Nash equilibrium. In this sense, a maximal lottery is a mixed “candidate” that cannot be challenged by a majority.

Occasionally maximal lotteries are non-unique, but they often are. A trivial case is when a Condorcet winner exists, in which case the maximal lottery selects the Condorcet winner as a pure strategy. Orthogonally, if there is an odd number of voters, and if preferences are strict, the maximal lottery is always unique. Otherwise, symmetries like multiple dominance components in the Smith set can induce degeneracies.

Still, any maximal lottery satisfies probabilistic notions of participation, where participation cannot reduce the chances of a preferred outcome for a voter.

I think this is nearly the “right” notion of compliance. There is only slight room in choosing which maximal lottery to implement. For example, in case of degeneracy, one could choose a maximal lottery that optimizes expected score. Or, maybe compute Jeffrey’s prior over maximal lotteries, and then sample a maximal lottery accordingly.

What thoughts do others have?

@cfrank I revisited this concept, and I'm quite sure now that participation will not hold even with careful adjustments to the method. Several simple counterexamples were discovered involving the generation of a top Condorcet cycle. C'est la vie.

@kaptain5 said in Professional Politicians skew towards altrusism? [study to read]:

In this study an interesting finding was that on average politicians we behaving altruistically (giving more to the other party than they would receive). On average over all sampled politicians were giving 49% of the prize to political opponents and giving 57% to in-group members. Compared to most other studies of the ultimatum game this is far above most people and very far above the rational expectation. The result is also remarkably consistent between countries. When the out-group and in-group results are combined this gives the result that the politicians surveyed are slightly altruistic in the game.
This one surprised me, I expected the politicians surveyed to be much more greedy.

I haven't properly read the paper but this bit is interesting, and I've been considering whether we should be surprised or not. Politicians certainly have a reputation for being greedy and corruptible. But a lot of people who go into politics probably do so for the right reasons - that they actually want to make things better for people and remove unfairness etc.

I think there is more corruption at the top, but this isn't most politicians. Plus I also think that higher-up politicians might still have some good principles, but these conflict with other things that get in the way of that - wanting to get in with the right people, make the right deals etc. So removed from politics, they still might make fair decisions. They just happen to be morally weak in certain ways.

This looks interesting. I'm not sure I've necessarily got that much to add though but I'll see what comes out...

You're looking at utility, but this is with real people rather than a simulation, so I wonder how this will work.

Are they voting for more abstract things in which case will you ask the participants for their honest utility rating of the options in addition to their actual votes? Or are they voting for options that directly benefit them in a clearer way - e.g. option 1 means voters A, B and C get this amount of money/chocolate etc.

It will be interesting to see equilibria emerging from repeated elections, and which methods are more stable in that respect. There is obviously the question of how relevant this would be in the real world. National elections generally take place several years apart and a lot can change in between, and so voters wouldn't be able to apply game theory in the same way as they would with multiple elections close together under the same conditions. And also I imagine people in the study are more likely to be "clued up" than the average member of the public.

So that raises a question - with multiple elections, will it just be the same conditions each time just to see how the methods behave in these ideal conditions, or will certain variables change to make it more "realistic", or possibly you'd model both? Both would be interesting in their own ways.

@kaptain5 said in I'm designing an experiment on voting systems, what would you like to see?:

My method will have human participants make the decisions for the voting blocks instead of attempting to simulate what a selfish imperfect-utility-maximizing person might do. Participants will be screened for competitive people trying to win.

Interesting for sure. It makes sense you would not want to share the full methodology if this is original research. I do have questions, I'm curious primarily about the scope of the simulation, because my observation is that often the routes to exploitation of systems are discovered by operating outside of or on the boundaries of their intended framing. For example, some questions would be, how are participants encouraged to be competitive? Will they engage in (quasi) long-run participation so that they can learn from the strategies of others? Which voting systems are you considering to test, and how large scale would this study be in terms of the number of participants?

I personally think key systems to include would be vote-for-one (obviously), approval, score, STAR, IRV, and a Condorcet method (I would suggest bottom-2-runoff). I would say it's important to keep in mind limitations of these, especially in terms of independence of clones.

I am not wholly familiar with the findings in this field, but I think it would be very helpful to see how coalitions emerge in different voting settings.
This is probably well outside the scope of what you want to investigate, but I personally would be happy to see this kind of system simulated:
https://www.votingtheory.org/forum/topic/299/pr-with-ambassador-quotas-and-cake-cutting-incentives/6

So IMO some basic questions would be about the emergence of coalitions, investigating the stability of multi-party coalitions versus duopolistic structures. Resistance to outsized influence of radical minority factions would also be good to test if possible.

In this realm, a question myself and @SaraWolk found interesting is whether "over time the two opposing factions [naturally] become more and more polarized." (see https://www.votingtheory.org/forum/topic/586/is-duopoly-more-resistant-to-fascism)

@cfrank If others have examples, simulations, or citations for my claim that "over time the two opposing factions become more and more polarized” I'd love to have those on hand too. I'm not sure I'm referencing any one thing I've read or heard in particular, but more putting multiple things together to get the big picture.

On the math alone I think you're right that the center is an important block for the two parties to court as well, but in practice I think that that incentive is outweighed by the other perverse social incentives to demonize the other side, to punish "traitors" considering switching, and so on.

Cancelling people who question the party line costs center voters, but it also discourages others from following. I think this was the subliminal Democratic Party tactic over the last decade that paid dividends for a while but then ultimately lost them the "big-tent" advantage and the presidency. I'm not saying it was an intentional strategy. There are big cultural forces at play here. That's obviously my own personal opinion.

Back on topic: As the narrative gets dominated by two polarized factions and the middle is silenced, the real middle (the center of public opinion) almost ceases to be a part of the political spectrum because it's doesn't actually map on to the left, right, and swing voter boxes.

Identifying and presenting consensus win-win policy and then getting it passed is the goal. We need to incentivise and empower that one way or another.

@SaraWolk I agree with you. When I say majoritarian, I only mean in the sense of Lijphart’s ontology, because you’re right that ultimately it isn’t even actually majoritarian but largely an illusion of it.

I’m curious about examples of this: “the center-squeeze effect ensures that over time the two opposing factions become more and more polarized.” I feel you know more than I do in this area, but it seems possible to me that there could be a stable polarization that doesn’t necessarily explode. In principle, as long as there is a large enough population of centrists, if one party leans too far in one direction, naively I would imagine the other could gain more power by appealing to those centrists than by appealing to the fringes.

Naivety aside, I think you’re probably right. The problem is that even if there is a population of centrists, if the representatives aren’t held accountable to them, then the parties themselves seem to have no good reason not to polarize once they’ve duopolized the political market. I’m really just curious about what causes that—is polarization actually steadily preferred? Or is it just a matter of time and drift before one party tips over the edge? It seems like the opposite of Hotelling’s law.

I also don’t think multiparty systems or PR for instance alone would solve the problem, there are examples of both systems falling to authoritarianism, and resistance depends on many contingent factors that ultimately bring about your main point, which is preventing extremists from gaining leverage or control.

Hopefully we can get some technical voting reform and see whether things change. Approval would be great.

@cfrank I have been thinking about the same questions. I think that the safeguard to fascism is ensuring checks against polarizing factions taking control.

In FPTP two party domination, the center-squeeze effect ensures that over time the two opposing factions become more and more polarized and this gives the illusion of majoritarianism, but as we know, the electability bias from voters having to vote for the frontrunner on their side can wildly inflate the perceived popularity of those frontrunners. In practice the moderate and third party voices are silenced and we see super polarizing candidates like Donald Trump (who initially only had some 25% of the vote in the 2016 primary) winning decisive control of their party. His own party has little they can do if they don't like his leadership, and the opposing party also has no leverage whatsoever if they can't beat him head to head. This last presidential election we saw both parties put forward candidates with record low approval ratings, but nobody else had a chance of challenging them at the same time. This is textbook polarized entrenched two party domination.

Meanwhile, I'm not convinced that a multi-party system on it's own will address any of that and it could make it worse. Just as choose one ballots can create a center-squeeze in FPTP, they can do the same in PR, resulting in a donut of polarized factions represented and little to no representation for the middle. In an election where the quota to win is 10% for example, a candidate could theoretically be strongly opposed by 90% of the electorate. Meanwhile, other factions could win with their standard-bearer also preferred by 10% of the electorate, but also strongly supported by many more and only strongly opposed by a slim minority.

When some winners are hyper-polarizing and others are not, it not only allows for the rise of dangerous factions who are more likely to bring about civil war, it also creates a lopsided and unstable winner-set. That's not the idealized definition of proportionality we're aiming for even though it would technically pass PR criteria. Theoretically, we should be able to do better.

The magic of a more expressive ballot or especially a 5 star ballot is that voters can show not only who their favorites are, but how much they like and dislike candidates from other factions. In an ideal system, this data could then be used to:
a) ensure that factions who deserve a seat at the table get one, and
b) ensure that candidates or factions that are seen as dangerous and harmful by others are not platformed when better alternatives exist.

In single-winner STAR, voters who are in the minority who are not going to get their favorites elected still have a strong vote against their worst case-scenario in the runoff. This is a massive check on authoritarianism and fascism. This is amazing and we don't need to switch to PR to get this windfall.

And, in a top shelf 5 STAR-PR system, theoretically we could do the same, using scores to identify which candidates and factions meet quota rules, while using scores and runoffs or preference data to identify the most polarizing and most opposed candidates.

At best, PR systems boast that legislatures where all perspectives are represented at the table. These legislatures are more likely to put forward more broadly acceptable legislation, but at their worst, they can give extremist factions massive leverage to cause stagnation or tear the system apart from the inside. For example, a super polarizing candidate like Israel's Ben Gvir who was elected with only 3.5% of the vote has the power to make or break the majority coalition and call a new election with a vote of no confidence if Netanyahu defies him. At worst, PR systems can give small polarizing factions extremely disproportionate leverage. Some of this can only be reformed with governmental system reforms such as higher quotas, but some of it can be fixed with the voting method itself.

Again, I think with a more expressive ballot and a hybrid ordinal and cardinal (STAR) approach we can do better.