STAR and CB are sometimes less honest than SV

[Tom]

I’ve been working on a further generalization of generalized cumulative voting and in the process confirmed that rational STAR voting and rational CB voting are sometimes less honest than rational SV voting (i.e. min-max). The following is an example:

Utilities: A[11] B[1] C[0] D[0]
Range: [0,5]
Rational zero-information SV vote: A[5] B[0] C[0] D[0]
Rational zero-information STAR or CB vote: A[5] B[1] C[0] D[0]

The SV vote is identical to the honest vote (in which B’s score is rounded to 0) and closer than STAR/CB to the utilities fitted to the range. If we increase A’s utility, the rational STAR and CB votes will eventually become honest (i.e. min B), but not until it reaches 14 (in the case of CB) or 17 (in the case of STAR). The cause of this dishonesty is of course that, whereas in SV the question is, is increasing my influence by 1 point in the event of a BC tie (i.e. whether B or C survives being within my control) or BD tie worth decreasing my influence by 1 point in the event of a BA tie? Of course not. But in STAR (and to a lesser extent CB), that tie may occur in, for example, the runoff, where there’s no difference between 5 and 1 and 5 and 0 and all the difference in the world between 1 and 0 and 0 and 0.

Combine that with the fact that, even when STAR and CB are more honest than SV, they’re not much more honest (in fact, rational voting in both STAR and CB approaches min-max as MAX approaches infinity, the only difference being that CB’s approach is slower). For example, the STAR ballot A[5] B[3] C[2] D[0] is necessarily irrational, regardless of the voter’s preferences.

So what remains of the argument for STAR/CB once the resistance-to-strategy myth is dispelled? Resistance to irrelevant alternatives, I suppose, only irrelevance is a self-fulfilling prophecy in these systems. The final two are often part of a top cycle, in which it’s not clear that the rightful winner is even part of the cycle, much less one of those two. The Condorcet winner should obviously always win when there is one, but I’ve seen no evidence that any candidates other than Condorcet losers and candidates outside the Mutual Majority set can be safely eliminated. That’s the virtue of Tom’s method. It has all the benefits of normalization without the risks attendant to automatic elimination.

[cfrank]

Well, I think with enough thought you will realize that there is no such thing as an “honest” vote, and hence no such thing as a “more” honest voting system (i.e. one that encourages voters to be “more honest”). What you mean to say, I believe, is that these voting systems are vulnerable to different manipulation strategies, but that is obviously true to begin with.

More to the point, I think you will come to the realization that utility is not a very useful concept in this context. Scores do not represent utilities—the ballots are tools constructed with an intentional purpose. Interpersonal utility is totally meaningless, since utility is just an auxiliary construct used to predict the behavior of an individual agent. Utility is defined in terms of individual behavior, not the other way around. Once you start adding utilities from a model of one person to the utilities of the model of another person, you get objects that need to be examined very closely for any scientific functionality in terms of producing superior outcomes (something else that needs to be defined or agreed upon).

I also don’t necessarily believe it’s important to worry too much about manipulation as long as it doesn’t frequently lead to majoritarianism or otherwise bad outcomes. In my personal opinion we should all try to shift towards pragmatism regarding these systems rather than focusing on or elevating one or several abstract principles over others. Ideally our advocacy or principles should be results-driven, right?

Just my thoughts.

[Tom]

While I thank you for your faith that I will one day realize what you realize, you’re a couple decades late. I’ve been critical of both the concept of strong honesty (by which min-maxing is dishonest merely for failing to indicate all preferences) and utilitarianism in this very forum. But the fact remains that the myth that STAR and CB incentivize strong honesty and are therefore more utilitarian than SV/Approval factors heavily into arguments for them; moreover, the goal of incentivizing strong honesty is the explicit cause of STAR’s existence. Would I like these systems’ advocates to come around to our point of view? Of course. But in the meantime, I think it’s worthwhile to point out their failure to meet their own advocates’ standards.

As for your opposition to abstract principles in general, I find it over-empirical, not pragmatic. Any good statistician knows that the absence of good data (and, make no mistake, confounded “natural experiments” and, more commonly, simulations using unrealistic voter behavior, do not produce good data) puts all the more pressure on theory. Scorning theory, confusing it with idealism, being or pretending to be purely data-driven, only ensures that one’s predictions will be less reliable than if one had given theory its proper place. I welcome good data showing that elimination (which rescaling is, of course, separable from) yields better outcomes, but its theoretical vacuity prevents me from holding my breath.

[cfrank]

That’s fair. I am not pretending to be data driven, what I mean to point to is the vacuum of empirical evidence that you also address, which one could argue deserves to be supplemented with theory, but I would personally rather have the vacuum filled with the missing empirical evidence if possible. I think that simulations if done correctly can give us a lot of good evidence to work with.

Also I didn’t mean to come across as condescending in case I did, I think we seem to agree for the most part and I wouldn’t want to condescend in any case. Although, I do not find it obvious that the Condorcet winner should always win when it exists. I’ve been wrestling with that concept and my skepticism of it hasn’t given way in either direction so far. It may very well be true, but I definitely do not agree that it’s obvious. Maybe you can convince me!

There are aspects of Condorcet methods that I appreciate significantly, and other aspects that I dislike. That’s true of many other methods as well, so for me it’s a matter of balancing pros against cons, the solution of which isn’t clear to me personally. So what may be obvious to some I think is more a matter of values and ethics than a matter of logic. If my set of values is outweighed by an overwhelming margin then that’s OK with me, but I expect I’m not so radical.

[Tom]

I suppose I have a moderate view on the importance of pairwise counts. I accept the Condorcet criterion, the Condorcet loser criterion and the mutual majority criterion but reject stronger criteria like the Smith criterion (my relationship with Smith is similar to yours with Condorcet; I hope someone convinces me of its validity–it would greatly simplify things–but at present I’m unconvinced).

It isn’t that I think the best candidate is, for example, always the Condorcet winner when there is one. Rather, informed voters would never eliminate him; he can only lose when voters err in their prediction of the other votes. So a system that fails Condorcet is not bad because it sometimes eliminates Condorcet winners; it’s bad because of what that implies: it fails to adapt votes to each other. This maladaptation would be great if votes were honest, but the opposite is true; the less the voting system simulates strategy, the more strategic the voter must be.

My acceptance of the Condorcet criterion is not to be mistaken for support for pure Condorcet methods. I view their pairwise counts as simulations of strategic single-choice voting, whereas we should be simulating strategic approval voting. My acceptance of the Condorcet criterion is based on the observation that the Condorcet winner, uniquely, wins a strategic approval election he is the expected winner of.

[cfrank]

I would argue that the state of manipulation of the system does not matter if the results are consistently good. I think your explanation of why you accept the Condorcet criterion exactly illustrates that point: I think the reasoning focuses too much on the means of the election and not enough on the ends.

I understand that the Condorcet criterion tamps down on manipulation, but I just can’t help but feel like it loses too much in the process. It’s almost a cynical argument—i.e., “Why not cut out the middle man and give the voters a worse outcome to begin with? They’ll muck it up anyway.” I just don’t think that actually solves the problem. Maybe the problem can’t be solved but I don’t know about that either. I think it’s possible to get the incentives right so that people are encouraged to compromise towards a good outcome.

At the same time, it is still vulnerable to some serious pathologies like “Dark Horse Plus Three Rivals,” more generally burying. But the main indicator for me that the Condorcet criterion is less than desireable is that it implies the majority criterion. I personally do not want a majoritarian system, because I believe that over time it will encourage the formation of a small number of large political factions, which almost surely destroys the possibility of a good outcome.

[Tom]

I would argue that the state of manipulation of the system does not matter if the results are consistently good.

That’s a big if. Systems with greater voter manipulation have worse outcomes because voter manipulation is based on fallible predictions and can thus backfire, which randomizes the outcome rather than improving it. It has nothing to do with voters being stupid; it has everything to do with their not being privy, at the time of voting, to the information the system is privy to (the other votes, namely) when it, for instance, wins the Condorcet winner.

I think it’s possible to get the incentives right so that people are encouraged to compromise towards a good outcome.

The relevant question is of course whether it’s possible to incentivize an outcome better than the Condorcet winner, and I just don’t see how any deterministic voting system could do that. If there is a Condorcet winner who is not the winner, there are obviously voters who have an incentive to change their votes.

At the same time, it is still vulnerable to some serious pathologies like “Dark Horse Plus Three Rivals,” more generally burying.

The best Condorcet systems are famously strategy-resistant in general and have insignificant vulnerability to burying. For example, in an ISDA system, it’s impossible to bury successfully without putting the burying candidate in the Smith set, which gives him a shot at winning. If you’re cool with that, your preference for the buried candidate is evidently marginal, so burying is not very problematic when it occurs. More generally burying tends to improve the outcome by trigging a utilitarian tie-breaker. For example, it’s impossible to bury successfully in Condorcet/Approval without winning the approval winner, which is unaffected by burying.

As for DH3, I’m not sure it’s really appreciated how unlikely it is to occur in a Condorcet system. First, each of the 3 factions would have to consider the difference between its first and second favorite greater than the difference between its second and third favorite. Second, voters would have to (falsely) believe the DH can’t win but still (falsely) believe burying might work, despite the incompatibility of those assumptions in a good Condorcet system.

But the main indicator for me that the Condorcet criterion is less than desireable is that it implies the majority criterion. I personally do not want a majoritarian system, because I believe that over time it will encourage the formation of a small number of large political factions, which almost surely destroys the possibility of a good outcome.

Why do you believe that? A majority can force its will in any single-winner system. It sometimes fails in non-majoritarian systems, due to ignorance or foolishness, but wouldn’t that, if anything, just encourage the formation of organized factions all the more? To prevent such failures, via coordination?

[cfrank]

It is a big if, but it’s the only if I’m personally concerned with. I think you have overgeneralized some: in my opinion, systems like STAR which are not Condorcet compliant can often yield superior outcomes to Condorcet methods. I definitely understand that these economical issues do not come about due to voters who are stupid and never meant to suggest as much. At the same time, I think ignorance—i.e. a lack of information—can often be a useful tool rather than merely a force that derails the quality of a system. For example, consider the concept of the Veil of Ignorance. Risk and ignorance can be very useful for tamping out strategy, for example if manipulation strategies are likely to backfire and voters are widely aware of that fact.

About determinism, that also I personally can do without. If a nondeterministic system produces superior outcomes to a deterministic one, I don’t know why we would tie our hands behind our backs. I understand some reasoning regarding blame and voters rejecting the legitimacy of a nondeterministic system when it fails to satisfy their individual demands, but that is an issue regardless of whatever voting system is in place. Disgruntled masses will make up their own excuses anyway: “it was rigged,” “the system isn’t fair,” etc.

About majoritarianism, I think that the cost of forming large coalitions needs to outweigh its benefits. If the marginal gain of forming a large coalition is small compared with the costs, then large coalitions will not form. It needs to be inconvenient for large coalitions to form, or at least voters should be given incentives not to comply with large coalitions when it means compromising their personal interests. Global group incentives should not override local individual or small communal incentives. That may already be the case in Condorcet methods, but I think it may also be true in other methods that give superior outcomes.

I think Condorcet methods would be a huge improvement to what we have in place now, but I don’t know if it would solve some of the deeper sociological problems underlying the failure of our current system, and that it may just stir up the solution for a while until the mud re-settles. Obviously there is more to that than just majoritarianism—mainly corruption—but I think that’s a different bridge to cross.

[Tom]

At the same time, I think ignorance—i.e. a lack of information—can often be a useful tool rather than merely a force that derails the quality of a system. For example, consider the concept of the Veil of Ignorance.

The Veil of Ignorance is a not a tool, but an attempt to give utilitarianism an egoistic foundation. It’s also irrelevant, as its ignorance is ignorance of who one is, not ignorance of others’ actions.

Risk and ignorance can be very useful for tamping out strategy, for example if manipulation strategies are likely to backfire and voters are widely aware of that fact.

The STAR strategy I described in the OP is explicitly a zero-information strategy. How is it, or, for that matter, the more general rule “never separate two ordinally adjacent candidates with same sign prospective rating by more than one point” more risky than honesty?

About determinism, that also I personally can do without. If a nondeterministic system produces superior outcomes to a deterministic one, I don’t know why we would tie our hands behind our backs.

I view non-determinism as a kind of proportional representation (PR), specifically a kind of deep PR, if you will (in the sense that PR, in the usual sense, is sometimes only proportional at the electoral level, with a stable majority coalition dominating the legislature); the actual result can be viewed as a probabilistic division of the office among multiple winners. But there are alternative kinds of deep PR that are deterministic. For example, the office could be divided temporally, departmentally, budgetarily or even geographically. The advantage of a deterministic division is that it is more stable; you’re not expecting the losers of the draw to be consoled by the fact that they had a chance.

Global group incentives should not override local individual or small communal incentives. That may already be the case in Condorcet methods, but I think it may also be true in other methods that give superior outcomes.

If you mean Condorcet methods are no more vulnerable to compromising than non-majoritarian methods, I’m glad we’re on the same page. But I don’t know what superior outcomes you speak of. I’m not aware of any evidence that any system has better outcomes than a system consisting of itself and a preliminary search for a Condorcet winner. It’s implausible on its face, at least for cardinal systems; failing to elect the Condorcet winner is proof that a critical subset of votes were critically influenced by irrelevant alternatives, which tends to distort cardinal but not ordinal preferences.

[cfrank]

I want to respond to each of your points in turn just to illustrate my perspective. I tried to write something about the Veil of Ignorance here, it might make what I’m thinking more clear and maybe you will find a serious issue with it:

Veil of Ignorance: Combating Manipulation with Artificial Empathy and Risk Avoidance

Hi Voting Theory people! I want to discuss the concept of the “Veil of Ignorance,” and ways that it might be used to mitigate the manipulation of voting systems via strategic voting. Discussion on Manipulation in General Manipulation of voting systems can occur in many ways that have been discussed at length on this forum. Manipulation is often described in terms of casting a “dishonest” ballot in order to gain a strategic advantage regarding the probable outcome of an election. This is a good…

How is it, or, for that matter, the more general rule “never separate two ordinally adjacent candidates with same sign prospective rating by more than one point” more risky than honesty?

Just to be clear I don’t necessarily think that STAR exemplifies the use of risk I envision. Rather it introduces competing incentives that at least mitigate the strategy of min-maxing, or more generally of neglecting pairwise preferences, which allows the system to extract more relevant information from voters.

Also about nondeterminism, I am trying to speak very generally although that might come at the cost of some meaning. Some of my thoughts about it are also elaborated a bit in the post I linked above.

If you mean Condorcet methods are no more vulnerable to compromising than non-majoritarian methods, I’m glad we’re on the same page. But I don’t know what superior outcomes you speak of

I think the word “compromise” has been used in different ways. When I used the word in a positive light, I meant it in the context of willingness to come to an agreement with others about a decision despite competing interests. In a negative light, I meant it in the sense of “the mission is compromised” if you understand me. I know that “compromising” is also a formal voting theory term so perhaps I should enhance my vocabulary to make my meaning more clear.

I suppose I am referring to simulations of voting systems, where according to the models used, systems like STAR and 3-2-1 seem to outperform Condorcet, Score, and Approval systems. That may depend on the model, but at the very least it is evidence, albeit not of the primary sort.

Also, previously you posed the question to me, “why a voting system?” And to me the answer is exactly the alternative you explained, i.e. to avoid authoritarianism. An excuse for behavior is not always a legitimate reason. I’m sorry if I sound like an old aristocrat talking about “disgruntled masses,” but I can assure you I’m not one lol. I don’t refer to the entire or even a large portion of the electorate when I refer to these disgruntled masses, again maybe my vocabulary could be honed some. I mean the fringe groups who are by definition those who are least well-off due to the outcome of the voting system. The hope is that this group is dynamic and variable, because if it is a fixed group that is disadvantaged then problems are sure to arise. There has to be oppression somewhere, unfortunately, but better in my opinion (and I don’t think it’s controversial) to have it spread out than concentrated.

[Tom]

I suppose I am referring to simulations of voting systems, where according to the models used, systems like STAR and 3-2-1 seem to outperform Condorcet, Score, and Approval systems. That may depend on the model, but at the very least it is evidence, albeit not of the primary sort.

The simulations I’ve seen show Approval outperforms both 3-2-1 and STAR under the most pragmatic assumption of 100% strategy. Are we looking at different simulations, or do you think Approval’s inferiority under honesty outweighs its superiority under strategy (that’s not very risk-averse of you)? As a fellow pragmatist, I’m sure you’ll agree that the burden of proof is on the proposer of the more complex (not to mention costlier and less known) system, and I see neither proof nor even the slightest evidence that honesty is as likely as strategy. I’ve seen attempts at evidence, non-binding polls and such, but no real evidence. I also doubt STAR or 3-2-1 would outperform Condorcet/Score (i.e. Score after ruling out a Condorcet winner) under any rate of strategy.

There has to be oppression somewhere, unfortunately, but better in my opinion (and I don’t think it’s controversial) to have it spread out than concentrated.

I definitely agree with this. I think we pay too much attention to efficiency and not enough to fairness, especially in single-winner theory, which is why I prefer geomean surplus utility as the measure of an outcome.

[AssetVotingAdvocacy]

I

reject

the Smith criterion

A reason to support Smith if you support Condorcet is that if all but one member of the Smith set drops out of the election, then the remaining member becomes a CW. So enforcing Smith and ISDA appear to take you the closest theoretically possible to IIA compliance that a majoritarian method can accomplish.

[Tom]

I don’t follow. When there’s no CW, IIA is necessarily violated, and I don’t know that it’s impossible for a system to make up for its dependence on Smith-dominated alternatives by being less dependent on other irrelevant alternatives than ISDA systems, or having other advantages.

My main reason for accepting Condorcet is that the CW is uniquely stable; he alone wins the election he’s the expected winner of. In the absence of a CW, each member of the Smith set is unstable by definition. Is the Smith set as a whole stable? That is, would repeatedly eliminating the stable winner of a vote on which subset of the remaining candidates to eliminate, until no stable winner remained, always reduce the set of candidates to a subset of the Smith set? Maybe, I’ve just never seen an attempt to prove it.

An alternative proof I’d accept would be: if voters engaged in repeated approval voting, their approval threshold being the weighted average candidate, each candidate’s weight being equal to his votes on the previous ballot to the nth power; for the largest n for which the balloting converges, is the winner necessarily in the Smith set?

[AssetVotingAdvocacy]

would repeatedly eliminating the stable winner of a vote on which subset of the remaining candidates to eliminate, until no stable winner remained, always reduce the set of candidates to a subset of the Smith set?

A proof to this effect would depend on whether the voters are guaranteed somehow to reach a point where at least one member of the Smith set remains uneliminated (or alternatively, the Copeland winner has all of his pairwise-winning rivals in the Smith set eliminated). At that point, it would hinge on whether the voters realize that the remaining major candidate in the Smith set is a CW, and from there you’d have to connect the proof to what you’ve already proven about CW’s winning.

[Tom]

A proof to this effect would depend on whether the voters are guaranteed somehow to reach a point where at least one member of the Smith set remains uneliminated (or alternatively, the Copeland winner has all of his pairwise-winning rivals in the Smith set eliminated). At that point, it would hinge on whether the voters realize that the remaining major candidate in the Smith set is a CW, and from there you’d have to connect the proof to what you’ve already proven about CW’s winning.

I’m afraid that might be over my head. In retrospect, I might ought to have put “winner” in quotes, since we don’t normally associate winning with being eliminated. Maybe clarifying that my “subset” does not imply a proper subset would have helped as well. But I’ll continue under the assumption you got my meanings.

Isn’t starting with no candidates eliminated in itself a guarantee that a point (the first one) will be reached where at least one member of the Smith set remains uneliminated?

I don’t understand the phrase “the remaining member of the Smith set”. How did we get from “at least one” to exactly one? Or are you referring to the parenthesis about the Copeland winner? I don’t understand why it would have to be a Copeland winner; would a different sole remaining member of the Smith set be impossible?

[AssetVotingAdvocacy]

I should not have used the phrase “at least”, sorry. To be very clear, I’m referring to a point where you have some uneliminated member of the Smith set who pairwise beats all other remaining candidates; this could happen when all but one of the Smith set members are eliminated, or theoretically it could happen with just one elimination i.e. the Copeland winner pairwise beats all but one of the candidates, and that one candidate (who has to be in the Smith set) gets eliminated. Or it could happen any other way, but it’s a point where you have a CW among the remaining candidates who is a member of the original Smith set.

[Tom]

OK, I think I understand the scenario now. When you say the voters are guaranteed to reach it, do you mean they necessarily will or they believe they necessarily will? And about it hinging on whether they realize the CW is a CW, do you mean they realize it before he’s a CW? Because, once he’s a CW, his victory is assured whether the voters know he’s a CW or not (I assume that’s what you mean about connecting to the proof about CWs winning, but just making sure).

Are you saying that, if you could prove the candidates would be reduced to either a subset of the Smith set or a set containing only one member of the Smith set, you would effectively be proving it would be reduced to a subset of the Smith set, because a set containing only one member of the Smith set will in turn be reduced to a subset of the Smith set (that member)?