Re: [EM] Generalizing manipulability
--- On Fri, 23/1/09, Kristofer Munsterhjelm km-el...@broadpark.no wrote: Juho Laatu wrote: I try to summarize my comments in the form of some rough definitions. A simple method requires 1) a 'simple' method to convert honest preferences into optimal votes A zero-info method requires 2) this method may not use info about other voters, but still be able to convert honest preferences into optimal votes A non-manipulable method requires 3) it is in everyone's interests to use the default method to convert honest preferences into optimal votes (I didn't cover the if everyone else uses this method case.) These definitions allow also e.g. Approval to be categorized as (close to) simple, not zero-info and non-manipulable. One more definition to point out one weakness of Approval. A decidable method requires 1) a method to convert honest preferences into an unambiguous optimal vote The point is that the there should be no lotteries that may lead also to unoptimal votes but the best vote should be found in a deterministic way. Approval fails this criterion since picking the correct number of approved candidates is sometimes tricky (when there are more than two strong candidates). Since all ranked methods are vulnerable to strategy, what constitutes an optimal vote depends on the votes of everybody else. Thus no such method can be either of the above I refer to our discussion on the possibility to meet some criteria partially. I think we too often use black and white criteria (or use the criteria in a bw way). I'd use all four criteria that I listed also as partially met criteria. One can thus define an ideal and then check how close each method gets. , and any simple method (by the definition) must also be non-manipulable, since to discover the optimal vote otherwise, you'd have to know the votes of potentially everybody else. The definitions that I gave are not necessarily good/optimal/useful. One could e.g. remove word optimal from the definition of the non-manipulable definition. One should maybe have a separate term for optimal vote at the time of voting and optimal vote at the time of counting the votes. One should also have separate terms for a method with a default vote creation method defined and for one without. These correspond to election method as part of the society (with default rules of behaviour) and vote tabulation method (that doesn't take position on how and where the ballots came from). You are welcome to propose better definitions. I don't have a perfect set available right now. The definitions you gave could be used for zero info strategy. For instance: Simple zero-info: The optimal zero-information strategy is simple to determine. Dominant zero-info: If everybody uses zero info strategy, and the method doesn't output a tie, no single voter could gain by changing his vote to something else. And there's also the usual zero-info strategy criterion: No zero-info strategy: The optimal zero information strategy is a sincere vote. No zero-info strategy implies simple zero-info. Dominant zero-info is vaguely similar to SDSC, though the latter deals with counterstrategies. Dominant zero-info may also be too strong: consider a situation where the voters produce a tie minus one vote (where a certain ballot can produce a tie); then, if the final voter prefers a candidate that would be ranked lower to one that would be ranked higher, he can construct a vote that leads to the two being tied. This last note sounds quite a lot like one-man-one-vote (=one voter can not change the end result much). I need to think more what kind of useful definition sets we might have, and which ones could be used as an (few ideal targets based) coordinate system to describe and classify the methods. That would (in theory) mean few core criteria and their refinements and estimated levels of compliance (instead of just a large set of black and white criteria). I'm not sure if it is possible to achieve anything useful though but one can always try. Juho Election-Methods mailing list - see http://electorama.com/em for list info
Re: [EM] Generalizing manipulability
Juho Laatu wrote: I try to summarize my comments in the form of some rough definitions. A simple method requires 1) a 'simple' method to convert honest preferences into optimal votes A zero-info method requires 2) this method may not use info about other voters, but still be able to convert honest preferences into optimal votes A non-manipulable method requires 3) it is in everyone's interests to use the default method to convert honest preferences into optimal votes (I didn't cover the if everyone else uses this method case.) These definitions allow also e.g. Approval to be categorized as (close to) simple, not zero-info and non-manipulable. One more definition to point out one weakness of Approval. A decidable method requires 1) a method to convert honest preferences into an unambiguous optimal vote The point is that the there should be no lotteries that may lead also to unoptimal votes but the best vote should be found in a deterministic way. Approval fails this criterion since picking the correct number of approved candidates is sometimes tricky (when there are more than two strong candidates). Since all ranked methods are vulnerable to strategy, what constitutes an optimal vote depends on the votes of everybody else. Thus no such method can be either of the above, and any simple method (by the definition) must also be non-manipulable, since to discover the optimal vote otherwise, you'd have to know the votes of potentially everybody else. The definitions you gave could be used for zero info strategy. For instance: Simple zero-info: The optimal zero-information strategy is simple to determine. Dominant zero-info: If everybody uses zero info strategy, and the method doesn't output a tie, no single voter could gain by changing his vote to something else. And there's also the usual zero-info strategy criterion: No zero-info strategy: The optimal zero information strategy is a sincere vote. No zero-info strategy implies simple zero-info. Dominant zero-info is vaguely similar to SDSC, though the latter deals with counterstrategies. Dominant zero-info may also be too strong: consider a situation where the voters produce a tie minus one vote (where a certain ballot can produce a tie); then, if the final voter prefers a candidate that would be ranked lower to one that would be ranked higher, he can construct a vote that leads to the two being tied. Election-Methods mailing list - see http://electorama.com/em for list info
Re: [EM] Generalizing manipulability
On Tue, Jan 20, 2009 at 10:57 AM, Kristofer Munsterhjelm km-el...@broadpark.no wrote: Perhaps. My point is not this. I explicitly said that I didn't know the zero info strategy (not sure). But also note that what I'm talking about is /zero info strategy/, i.e. how you'd vote if you were stuck on Mars with the candidates (who had broadcast systems with which to run their campaigns), and then you all traveled back to Earth just before the vote. The zero-info strategy may be something else than mean cutoff (again, *I don't know!*), but it may also just be lousy because the method has a bad zero-info strategy and voters have to know how others are likely to vote. I wonder would zero info allow some knowlegde of the electorate. (I guess not :) ). Not knowing anything about poll results, I think most voters could split a set of candidates into no hopers/crazies and possible winners. This would be based purely on the type of candidates who were competitors in previous elections. You could then use the mean strategy to determine the threshold, but only include possible winners. Anyway maybe a non-manipulable method requires 1) a simple method to convert honest preferences into valid votes 2) this method may not use info about other voters 3) If everyone else uses this method, then it is in your interests to also use this method One possible subjective aspect would be what simple means. Methods that require some knowledge of polls to work would fail this definition. However, most people have little problems with plurality and use the standard strategy quite effectively. I think the concept of requiring a zero-info strategy to be optimal is a clean way of saying that voters who have access to more information should not have an advantage (be able to manipulate). Perhaps also, the zero info strategy should be reasonably easy to understand. The criterion could perhaps be relaxed a little by allowing publicly available information to be used rather than it being purely zero-info (and that the method is somewhat resistant to inaccuracies in that info). Also, perhaps if the partial info strategy was only 'slightly' less effective than the optimal strategy under perfect info, then that would be OK too. Election-Methods mailing list - see http://electorama.com/em for list info
Re: [EM] Generalizing manipulability
--- On Thu, 22/1/09, Raph Frank raph...@gmail.com wrote: Anyway maybe a non-manipulable method requires 1) a simple method to convert honest preferences into valid votes 2) this method may not use info about other voters 3) If everyone else uses this method, then it is in your interests to also use this method One possible subjective aspect would be what simple means. One could also drop the requirement of simplicity (since non-manipulability doesn't necessarily require that) and keep it as a separate requirement. Methods that require some knowledge of polls to work would fail this definition. Non-manipulability could also allow use of this knowledge. However, most people have little problems with plurality and use the standard strategy quite effectively. I think the concept of requiring a zero-info strategy to be optimal is a clean way of saying that voters who have access to more information should not have an advantage (be able to manipulate). If there is a simple non zero-info strategy that all can easily use (as in Plurality) that could still be classified as non-manipulable (if otherwise ok). Perhaps also, the zero info strategy should be reasonably easy to understand. I already noted that simplicity could be a separate requirement / criterion. The criterion could perhaps be relaxed a little by allowing publicly available information to be used rather than it being purely zero-info Difficult to define what the public info is. But it would be good to have criteria that can be met more or less fully. (and that the method is somewhat resistant to inaccuracies in that info). Also level of tolerance against inaccuracy, risk of backfiring of the strategy, required level and difficulty of coordination of the strategy, frequency of the vulnerability etc. would be good parameters. Also, perhaps if the partial info strategy was only 'slightly' less effective than the optimal strategy under perfect info, then that would be OK too. Yes. I think too often we ignore the difference between failing some criterion in some rare cases and failing it regularly and in a way that allows strategic manipulation of the election. I try to summarize my comments in the form of some rough definitions. A simple method requires 1) a 'simple' method to convert honest preferences into optimal votes A zero-info method requires 2) this method may not use info about other voters, but still be able to convert honest preferences into optimal votes A non-manipulable method requires 3) it is in everyone's interests to use the default method to convert honest preferences into optimal votes (I didn't cover the if everyone else uses this method case.) These definitions allow also e.g. Approval to be categorized as (close to) simple, not zero-info and non-manipulable. One more definition to point out one weakness of Approval. A decidable method requires 1) a method to convert honest preferences into an unambiguous optimal vote The point is that the there should be no lotteries that may lead also to unoptimal votes but the best vote should be found in a deterministic way. Approval fails this criterion since picking the correct number of approved candidates is sometimes tricky (when there are more than two strong candidates). Juho Election-Methods mailing list - see http://electorama.com/em for list info
Re: [EM] Generalizing manipulability
OK, Range votes are just votes. But voters do have also opinions. They can be presented as ratings. If the voter casts a vote with the intention that it reflects her opinions as accurately as possible, then I'd call that vote sincere (and in most cases not strategic). If the voter casts a vote that is intended to optimize the result of the election from her point of view, then I'd call that vote strategic (at least if it deviates from the sincere opinion). I hope we can agree on some (whatever) common terminology that would apply to all methods. The names of the definitions are not important but their stability and usefulness is. Juho --- On Tue, 20/1/09, Abd ul-Rahman Lomax a...@lomaxdesign.com wrote: At 01:38 AM 1/18/2009, Juho Laatu wrote: I don't quite see why ranking based methods (Range, Approval) would not follow the same principles/definitions as rating based methods. The sincere message of the voter was above that she only slightly prefers B over A but the strategic vote indicated that she finds B to be maximally better than A (or that in order to make B win she better vote this way). That is an *interpretation* of a Range vote. In fact, they are just votes, and the voter casts them according to the voter's understanding of what's best. This has been part of my point: Range votes don't indicate preference strength, as such. Consider Approval, which is a Range method. If the voter votes A=BC=D, what does this tell us? We can infer some preferences from it, to be sure, and those preferences are probably accurate, because Approval never rewards a truly insincere vote. But does this vote indicate that the voter has no preference between A and B, nor between C and D? Of course not! Now, a Range vote. But the voter votes Approval style. What does this tell us about the voter preferences? *Nothing more and nothing less.* The voter chose to vote that way for what reason? We don't know!!! They are votes, not sentiments. Voters may choose to express relative preference, in Range, with some fineness of expression, but they may also choose not to make refined expressions, and all these votes are sincere, i.e., they imply no preferences that we cannot reasonably infer from them with a general understanding that the voter had no incentive to show preferences opposite to the actual. (Now, there is a kind of insincere voting that voters may engage in, but it isn't really rewarded, and voters will only do it when they expect it to be moot. And they may do this kind of insincere voting with any method whatever.) Election-Methods mailing list - see http://electorama.com/em for list info
Re: [EM] Generalizing manipulability
On Jan 18, 2009, at 5:13 PM, Juho Laatu wrote: --- On Mon, 19/1/09, Jonathan Lundell jlund...@pobox.com wrote: - Why was the first set of definitions not good enough for Approval? (I read rank as referring to the sincere personal opinions, not to the ballot.) vi ranks, and vi is by definition the ballot. That's why the second definition introduces o. OK. I should say that is the way I'd like to read it. I'd like to take another shot at that. Steve's first definition: Let X denote the set of alternatives being voted on. Let N denote the set of voters. Let V(X,N) denote the set of all possible collections of admissible votes regarding X, such that each collection contains one vote for each voter i in N. For all collections v in V(X,N) and all voters i in N, let vi denote i's vote in v. Let C denote the vote-tallying function that chooses the winner given a collection of votes. That is, for all v in V(X,N), C(v) is some alternative in X. Call C manipulable by voter strategy if there exist two collections of votes v,v' in V(X,N) and some voter i in N such that both of the following conditions hold: 1. v'j = vj for all voters j in N-i. 2. vi ranks C(v') over C(v). The idea in condition 2 is that voter i prefers the winner given the strategic vote v'i over the winner given the sincere vote vi. This definition is stronger than *requiring* that vi be any particular ordering--in particular i's sincere preferences. That's very neat. Notice also that we get away with it because the ballot in this case is expressive enough to represent i's sincere preference ranking. That's not true for an approval ballot, which is why the second definition needs to introduce a separate preference order o. Finally, the definition says nothing about how voter i might go about *finding* v'i, or even how to discover for any particular ballot profile whether v'i exists. Election-Methods mailing list - see http://electorama.com/em for list info
Re: [EM] Generalizing manipulability
--- On Mon, 19/1/09, Jonathan Lundell jlund...@pobox.com wrote: On Jan 18, 2009, at 5:13 PM, Juho Laatu wrote: --- On Mon, 19/1/09, Jonathan Lundell jlund...@pobox.com wrote: - Why was the first set of definitions not good enough for Approval? (I read rank as referring to the sincere personal opinions, not to the ballot.) vi ranks, and vi is by definition the ballot. That's why the second definition introduces o. OK. I should say that is the way I'd like to read it. I'd like to take another shot at that. Steve's first definition: Let X denote the set of alternatives being voted on. Let N denote the set of voters. Let V(X,N) denote the set of all possible collections of admissible votes regarding X, such that each collection contains one vote for each voter i in N. For all collections v in V(X,N) and all voters i in N, let vi denote i's vote in v. Let C denote the vote-tallying function that chooses the winner given a collection of votes. That is, for all v in V(X,N), C(v) is some alternative in X. Call C manipulable by voter strategy if there exist two collections of votes v,v' in V(X,N) and some voter i in N such that both of the following conditions hold: 1. v'j = vj for all voters j in N-i. 2. vi ranks C(v') over C(v). The idea in condition 2 is that voter i prefers the winner given the strategic vote v'i over the winner given the sincere vote vi. This definition is stronger than *requiring* that vi be any particular ordering--in particular i's sincere preferences. That's very neat. Notice also that we get away with it because the ballot in this case is expressive enough to represent i's sincere preference ranking. That's not true for an approval ballot, which is why the second definition needs to introduce a separate preference order o. Finally, the definition says nothing about how voter i might go about *finding* v'i, or even how to discover for any particular ballot profile whether v'i exists. Yes, this is neat in the sense that there is no need to explain what the sincere opinion of the voter is and how the strategic vote will be found. A definition that would cover also Approval and other methods with simple ballots at one go would be nice too. Although it is sometimes difficult to say what a sincere vote in Approval is (could be e.g. to mark all candidates that one approves) I think it is quite natural to assume that each voter has some preferences (order), and that strategies mean deviation from simply voting as one feels and not considering the technical details of the method, the impact of how others are expected to vote and how one could get better results out (by e.g. voting or nominating candidates in some particular way). Juho Election-Methods mailing list - see http://electorama.com/em for list info
Re: [EM] Generalizing manipulability
At 01:38 AM 1/18/2009, Juho Laatu wrote: I don't quite see why ranking based methods (Range, Approval) would not follow the same principles/definitions as rating based methods. The sincere message of the voter was above that she only slightly prefers B over A but the strategic vote indicated that she finds B to be maximally better than A (or that in order to make B win she better vote this way). That is an *interpretation* of a Range vote. In fact, they are just votes, and the voter casts them according to the voter's understanding of what's best. This has been part of my point: Range votes don't indicate preference strength, as such. Consider Approval, which is a Range method. If the voter votes A=BC=D, what does this tell us? We can infer some preferences from it, to be sure, and those preferences are probably accurate, because Approval never rewards a truly insincere vote. But does this vote indicate that the voter has no preference between A and B, nor between C and D? Of course not! Now, a Range vote. But the voter votes Approval style. What does this tell us about the voter preferences? *Nothing more and nothing less.* The voter chose to vote that way for what reason? We don't know!!! They are votes, not sentiments. Voters may choose to express relative preference, in Range, with some fineness of expression, but they may also choose not to make refined expressions, and all these votes are sincere, i.e., they imply no preferences that we cannot reasonably infer from them with a general understanding that the voter had no incentive to show preferences opposite to the actual. (Now, there is a kind of insincere voting that voters may engage in, but it isn't really rewarded, and voters will only do it when they expect it to be moot. And they may do this kind of insincere voting with any method whatever.) Election-Methods mailing list - see http://electorama.com/em for list info
Re: [EM] Generalizing manipulability
At 03:57 PM 1/18/2009, Kristofer Munsterhjelm wrote: Wouldn't it be stricter than this? Consider Range, for instance. One would guess that the best zero info strategy is to vote Approval style with the cutoff at some point (mean? not sure). Actually, that's a lousy strategy. The reason it's lousy is that the voter is a sample of the electorate. Depending on the voter's own understanding of the electorate, and the voter's own relationship with the electorate, the best strategy might be a bullet vote. Saari showed why mean cutoff is terrible Approval strategy. What if every voter agrees with you but one? The one good thing Saari shows is that this yields a mediocre outcome when /1 voters prefer a candidate, but also approve another above the mean. Essentially, the voter doesn't need to know anything specific about the electorate in a particular election, but only about how isolated the voter's position *generally* is. For most voters, zero-knowledge indicates a bullet vote unless there are additional candidates with only weak preference under the most-preferred one, such that the voter truly doesn't mind voting for one or more of them in addition. However, it would also be reasonable that a sincere ratings ballot would have the property that if the sincere ranked ballot of the person in question is A B, then the score of B is lower than that of A; that is, unless the rounding effect makes it impossible to give B a lower score than A, or makes it impossible to give B a sufficiently slightly lower score than A as the voter considers sincere (by whatever metric). Yes. Indeed, I've suggested that doing pairwise analysis on Range ballots, with a runoff when the Range winner is beaten by a candidate pairwise, would encourage maintenance of this preference order. Think of Range as a Borda ballot with equal ranking allowed and therefore with empty ranks. (Not the ridiculous suggestions that truncated ballots should be given less weight). If a voter really has weak preference between two candidates, the obvious and simple vote is to equal rank them. But then where does one put the empty rank? There are two approaches, and both of them are sincere, though one approach more accurately reflects relative preference strength. There are ways to encourage that expression. But here is the real problem: trying to think that a zero-knowledge ballot is somehow ideal is discounting the function of compromise in elections. That is, what we do in elections is *not only* to find some sort of supposed best candidate, but also to find compromises. That's what we do in deliberative process where repeated Yes/No voting is used to identify compromises, until a quorum is reached (usually a majority, but it can be supermajority). Deliberative process incorporates increasing knowledge by the electorate of itself. It extracts this with a series of elections in which sincerity is not only expected, it's generally good strategy. In that context, approval really is approval! If a majority agrees with your approval, the process is over. I consider election methods as shortcuts, attempts to discover quickly what the electorate would likely settle on in a deliberative environment. As such, it is actually essential that whatever knowledge the electorate has of itself be incorporated into how the voters vote. And that's what happens if, in a Range election, voters vote von Nuemann-Morganstern utilities. They have one full vote to bet. They put their vote where they think it will do the most good. They can put it all on one candidate, i.e., bullet vote. They can put it on a candidate set, thus voting a full vote for every member of the set over every nonmembe, i.e., they vote Approval style. They can split up their vote in more complex ways. What they can't do in this setup is to bet more than one vote. I.e., for example, one full vote for A over B, and one full vote for B over C. If we arrange their votes in sequence, from least preferred to most, the sum of votes in each sequential pairwise election must total to no more than one vote. Calling them VNM utilities sounds complex, but it's actually instinctive. If we understand Range, we aren't going to waste significant voting power expressing moot preferences. Suppose someone asks you what you want. But you understand that you might not get what you want. You prefer ABCD, lets say with equal preference steps. You think it likely that A or B might be acceptable to your questioner, but not C or D. You have so much time to convince your questioner to give you what you argue for. How much time are you going to spend trying to convince the person to give you C instead of D? You might mention it, but you wouldn't put the weight there unless you thought that the real possibilities were C or D. Voter knowledge of the electorate is how elections reach compromise, and it's very important. Of course, there is
Re: [EM] Generalizing manipulability
Hi, Manipulability by voter strategy can be rigorously defined without problematic concepts like preferences or sincere votes or how a dictator would vote or or how a rational voter would vote given beliefs about others' votes. Let X denote the set of alternatives being voted on. Let N denote the set of voters. Let V(X,N) denote the set of all possible collections of admissible votes regarding X, such that each collection contains one vote for each voter i in N. For all collections v in V(X,N) and all voters i in N, let vi denote i's vote in v. Let C denote the vote-tallying function that chooses the winner given a collection of votes. That is, for all v in V(X,N), C(v) is some alternative in X. Call C manipulable by voter strategy if there exist two collections of votes v,v' in V(X,N) and some voter i in N such that both of the following conditions hold: 1. v'j = vj for all voters j in N-i. 2. vi ranks C(v') over C(v). The idea in condition 2 is that voter i prefers the winner given the strategic vote v'i over the winner given the sincere vote vi. That definition works assuming all possible orderings of X are admissible votes. I think it works for Range Voting too (and Range Voting can be shown to be manipulable). The following may be a reasonable way to generalize it to include methods like Approval (and if this is done then Approval can be shown to be manipulable): Call C manipulable by voter strategy if there exist two collections of votes v,v' in V(X,N) and some voter i in N and some ordering o of X such that all 3 of the following conditions hold: 1. v'j = vj for all j in N-i. 2. o ranks C(v') over C(v). 3. For all pairs of alternatives x,y in X, if vi ranks x over y then o ranks x over y. The idea in condition 3 is that vi is consistent with the voter's sincere order of preference. For example, approving x but not y or z is consistent with the 2 strict (linear) orderings x over y over z and x over z over y. It's also consistent with the weak (non-linear) ordering x over y,z. Approving x and y but not z is consistent with x over y over z and y over x over z and x,y over z. Interpreting o as the voter's sincere order of preference, condition 2 means the voter prefers the strategic winner over the sincere winner. Another kind of manipulability is much more important in the context of public elections. Call the voting method manipulable by irrelevant nominees if nominating an additional alternative z is likely to cause a significant number of voters to change their relative vote between two other alternatives x and y, thereby changing the winner from x to y. We observe the effects all the time given traditional voting methods. It explains why so many potential candidates drop out of contention before the general election (Duverger's Law). It explains why the elites tend not to propose competing ballot propositions when asking the voters to change from the status quo using Yes/No Approval. I expect this kind of manipulability to be a big problem given Approval or Range Voting or plain Instant Runoff or Borda, but not given a good Condorcet method. The reason manipulability by irrelevant nominees is more important than manipulability by voter strategy is that it takes only a tiny number of people to affect the menu of nominees, whereas voters in public elections tend not to be strategically minded--see the research of Mike Alvarez of Caltech. Regards, Steve -- On 1/17/2009 10:38 PM, Juho Laatu wrote: --- On Sun, 18/1/09, Jonathan Lundell jlund...@pobox.com wrote: On Jan 17, 2009, at 4:31 PM, Juho Laatu wrote: The mail contained quite good definitions. I didn't however agree with the referenced part below. I think sincere and zero-knowledge best strategic ballot need not be the same. For example in Range(0,99) my sincere ballot could be A=50 B=51 but my best strategic vote would be A=0 B=99. Also other methods may have similarly small differences between sincere and zero-knowledge best strategic ballots. My argument is that the Range values (as well as the Approval cutoff point) have meaning only within the method. We know from your example how you rank A vs B, but the actual values are uninterpreted except within the count. The term sincere is metaphorical at best, even with linear ballots. What I'm arguing is that that metaphor breaks down with non-linear methods, and the appropriate generalization/abstraction of a sincere ballot is a zero-knowledge ballot. I don't quite see why ranking based methods (Range, Approval) would not follow the same principles/definitions as rating based methods. The sincere message of the voter was above that she only slightly prefers B over A but the strategic vote indicated that she finds B to be
Re: [EM] Generalizing manipulability
On Jan 17, 2009, at 10:38 PM, Juho Laatu wrote: --- On Sun, 18/1/09, Jonathan Lundell jlund...@pobox.com wrote: On Jan 17, 2009, at 4:31 PM, Juho Laatu wrote: The mail contained quite good definitions. I didn't however agree with the referenced part below. I think sincere and zero-knowledge best strategic ballot need not be the same. For example in Range(0,99) my sincere ballot could be A=50 B=51 but my best strategic vote would be A=0 B=99. Also other methods may have similarly small differences between sincere and zero-knowledge best strategic ballots. My argument is that the Range values (as well as the Approval cutoff point) have meaning only within the method. We know from your example how you rank A vs B, but the actual values are uninterpreted except within the count. The term sincere is metaphorical at best, even with linear ballots. What I'm arguing is that that metaphor breaks down with non-linear methods, and the appropriate generalization/abstraction of a sincere ballot is a zero-knowledge ballot. I don't quite see why ranking based methods (Range, Approval) would not follow the same principles/definitions as rating based methods. The sincere message of the voter was above that she only slightly prefers B over A but the strategic vote indicated that she finds B to be maximally better than A (or that in order to make B win she better vote this way). (I'd use rating/ranking opposite to that. No?) I was making a smaller point, that the actual values in Range and the approval cutoff point in Approval are hard to interpret as sincere or not. On the other hand, we need a voter's sincere linear ordering of the candidates (ranking?) in order to be able to say whether an *outcome* is better or worse. Election-Methods mailing list - see http://electorama.com/em for list info
Re: [EM] Generalizing manipulability
Jonathan Lundell wrote: On Jan 17, 2009, at 4:31 PM, Juho Laatu wrote: The mail contained quite good definitions. I didn't however agree with the referenced part below. I think sincere and zero-knowledge best strategic ballot need not be the same. For example in Range(0,99) my sincere ballot could be A=50 B=51 but my best strategic vote would be A=0 B=99. Also other methods may have similarly small differences between sincere and zero-knowledge best strategic ballots. My argument is that the Range values (as well as the Approval cutoff point) have meaning only within the method. We know from your example how you rank A vs B, but the actual values are uninterpreted except within the count. The term sincere is metaphorical at best, even with linear ballots. What I'm arguing is that that metaphor breaks down with non-linear methods, and the appropriate generalization/abstraction of a sincere ballot is a zero-knowledge ballot. Wouldn't it be stricter than this? Consider Range, for instance. One would guess that the best zero info strategy is to vote Approval style with the cutoff at some point (mean? not sure). However, it would also be reasonable that a sincere ratings ballot would have the property that if the sincere ranked ballot of the person in question is A B, then the score of B is lower than that of A; that is, unless the rounding effect makes it impossible to give B a lower score than A, or makes it impossible to give B a sufficiently slightly lower score than A as the voter considers sincere (by whatever metric). Election-Methods mailing list - see http://electorama.com/em for list info
Re: [EM] Generalizing manipulability
--- On Sun, 18/1/09, Jonathan Lundell jlund...@pobox.com wrote: On Jan 17, 2009, at 10:38 PM, Juho Laatu wrote: --- On Sun, 18/1/09, Jonathan Lundell jlund...@pobox.com wrote: On Jan 17, 2009, at 4:31 PM, Juho Laatu wrote: The mail contained quite good definitions. I didn't however agree with the referenced part below. I think sincere and zero-knowledge best strategic ballot need not be the same. For example in Range(0,99) my sincere ballot could be A=50 B=51 but my best strategic vote would be A=0 B=99. Also other methods may have similarly small differences between sincere and zero-knowledge best strategic ballots. My argument is that the Range values (as well as the Approval cutoff point) have meaning only within the method. We know from your example how you rank A vs B, but the actual values are uninterpreted except within the count. The term sincere is metaphorical at best, even with linear ballots. What I'm arguing is that that metaphor breaks down with non-linear methods, and the appropriate generalization/abstraction of a sincere ballot is a zero-knowledge ballot. I don't quite see why ranking based methods (Range, Approval) would not follow the same principles/definitions as rating based methods. The sincere message of the voter was above that she only slightly prefers B over A but the strategic vote indicated that she finds B to be maximally better than A (or that in order to make B win she better vote this way). (I'd use rating/ranking opposite to that. No?) Yes, sorry about the confusion. I was making a smaller point, that the actual values in Range and the approval cutoff point in Approval are hard to interpret as sincere or not. On the other hand, we need a voter's sincere linear ordering of the candidates (ranking?) in order to be able to say whether an *outcome* is better or worse. OK. I think people are most often (e.g. on this list) expected to have an internal preference order of the candidates, also when the ballots of the method does not express it. I also think that most often people on this list assume that Approval votes are expected to be strategic while Range votes are expected to be sincere (except that many assume votes to be normalized, and that can already be seen to be a strategy). Juho Election-Methods mailing list - see http://electorama.com/em for list info
[EM] Generalizing manipulability
On Jan 8, 2009, at 4:45 PM, Abd ul-Rahman Lomax wrote: The whole concept of strategic voting is flawed when applied to Range. Voters place vote strength where they think it will do the most good -- if they think. Some don't. Approval is essentially, as Brams claimed, strategy-free, in the old meaning, and the only way that it was at all possible to call it vulnerable was that critics claimed that there was some absolute approval relation between a voter and a candidate. It would be useful to generalize the concept of strategic voting (and the related concepts of manipulation and sincerity) to other than linear ballots (that is, a ballot with an ordinal ranking of the voter's preferences). With linear ballots (and so Borda, IRV and various Condorcet methods) we define a sincere ballot as the one a voter would cast if the voter were a dictator, and manipulability as the ability of a voter to achieve a better result (where better means the election of a candidate ranked higher on that voter's sincere ballot) by voting insincerely or strategically--that is, by casting a ballot different from their sincere ballot. An election method that is not manipulable in this sense is defined to be strategy-free. A two-candidate plurality election is strategy-free. Most interesting elections are not. With any practical election method using linear ballots, manipulation cannot succeed unless the voter has knowledge of how the other voters are voting. This knowledge need not be perfect. I propose (and I don't claim that this is original, though I don't recall seeing the definition) that we use this observation to generalize the idea of manipulability to election methods, such as Range and Approval, that do not use linear ballots, thus: An election method is manipulable if a voter has a rational motivation to cast different ballots depending on the voter's knowledge (or belief) of the ballots of other voters. In such an election, a voter should vote strategically when the ballot that will produce the best outcome (for that voter) depends on the behavior of the other voters, the strategy consisting of determining, by some means depending on the method, which ballot that is. For example, in an Approval election, with a preference of ABC, we will always vote for A, but whether we vote for B depends on how well we believe B and C are doing with other voters. If we believe that C cannot win, then we vote for A only, to improve our chance of electing A over B. If we believe that C is a serious threat, then we vote for A and B, to improve our chance of rejecting C. The generalization of a sincere ballot then becomes the zero- knowledge (of other voters' behavior) ballot, although we might still want to talk about a sincere ordering (that is, the sincere linear ballot) in trying to determine a best possible outcome. It seems to me that it's clearly desirable to be able to optimize the outcome by casting a sincere linear ballot. Such a ballot is reasonably expressive (that is, it contains more information about my preferences than, say, a plurality or approval ballot) without (in itself) requiring me to strategize. Unfortunately, no such election method exists, and many (most?) of the arguments on this list are over the tradeoffs implied by that sad fact. The best we can do is to find a method in which it's very unlikely that we can improve our outcome by voting other than sincerely. Election-Methods mailing list - see http://electorama.com/em for list info
