--- Abd ul-Rahman Lomax <[EMAIL PROTECTED]> wrote: > At 08:59 PM 11/7/2006, David Cary wrote: > >Both CRV and Wikipedia claim that range voting meets the Independence > >of Irrelevant Alternatives (IIA) criterion, but neither gives much > >justification for such claims. > > > >Electorama succinctly but informally describes IIA as: > > "if one option (X) wins the election, and a new alternative (Y) > >is added, only X or Y will win the election." > > Election criteria are supposed to be objective. One of the aspect of > this, and I think it is -- or should be -- standard in applying the > criteria, is that the mental state of the voters is not relevant. > What is relevant is what is expressed on the ballots. (It was a > failure to consider this that resulted in the error of considering > Approval to have failed the Majority Criterion.)
The comparison of individual voter's mental preferences to the group choice is at the core of many criteria, IIA and Majority Criterion among them. In many cases, these criteria have roots in the broader theory of social choice that allows consideration of a wide variety of possibilities, each associating some specified individual mental preferences to a collective decision or set of preferences without having to consider the mechanism, such as voting, for achieving that association. In an important sense, the involvement of voter mental preferences does not make the criteria any less objective. The criteria are not required to function as empirical criteria. They can work quite well as theoretical criteria. In this context, the voter mental preferences are not discovered but given, as in the example I gave, or even treated as mathematical variables. More precisely, in this context, human mental preferences are modeled by some mathematical object, for example some ordering of the candidates, a scoring function, etc. This approach facilitates careful thinking about voting systems by separating various issues and allowing consideration of a broad range of possibilities. A criterion expressed in terms of individual mental preferences is often uninteresting when applied to an election method. The application becomes interesting when there are restrictions on how the individual mental preferences are converted into votes, for example voting "sincerely", but other restrictions are possible. The restrictions convert the election method into a non-trivial social choice function. It possible to embed the restrictions into the criterion, but since the restrictions may not apply to all election methods, it can make more sense to invoke them as needed. > One can certainly claim that an election method does not allow voters > to express something, such as more than two ranks, or preference > strengths. But that's not relevant here. Formulation of a criterion often achieves precision and clarity at the expense of generality, for example by making it particular to a specific model of individual preferences. This is done even though the principle of the criterion could easily apply to other situations, or because there are different ways of generalizing the criterion. > > What does it mean to "add" a new alternative? In methods that are > purely ranked, we presumably assume that the new candidate fits > somewhere into the ranking system, and then the votes would be > changed accordingly, with no other complications. We would assume > that all relative ranks remain the same, for the candidates already > in the election, and that the newcomer is simply assigned one of > these ranks. If ranks must be exclusive, it is reasonable to allow > that a new rank is inserted at some position for the candidate, but, > again, except for that additional rank, nothing else changes. > > The Range equivalent could only mean that the new alternative does > not disturb the existing range ratings. And if this is the meaning, > Range satisfies IIA, quite clearly. For IIA there is clearly an implication that some things, but not necessarily others, are held constant when an additional alternative is added. The informal definitions fail to address exactly what is held constant. It is certainly reasonable to specify that the mental preferences and any rules or restrictions on how those preferences are converted into votes are held constant. Call that IIA-P. An alternative is to specify that the actual votes remain constant. Call that IIA-V. It is pointless to argue that one is right and the other is wrong. It is important to acknowledge the two possibilities and be specific about which one is being considered. Each version of IIA has something different to say. Wikipedia too often is deficient by ignoring those distinctions. > What Mr. Cary did was to assume that the voters were following a > strategy where one considers and adjusts Range votes based on the > "candidate space." I.e., with this strategy, one ranks the best at > max and the worst at zero, and then assigns values in between for the > others, based on expectation of value. I gave an example that is consistent with such a restriction, but does not require such a restriction. I did assume IIA-P and that the voter's preferences could be specified by a scoring function that was not bounded by the voting range. > However, this is internal process, and there are other reasonable > strategies for the voter to follow. Moving toward Approval style > voting is one of them. Approval satisfies IIA. Only if we think that > the voter is following some strategy which causes the voter to change > the approval votes already cast based on this newcomer could we say > that it does not satisfy it. > > But this is a ridiculous interpretation of IIA. Or at least it is one > that becomes very hard to satisfy. I do not find IIA to be crisply > defined. It seems to me that it is a criterion designed for ranked > methods, because one may make ready assumptions about the effect of > an added candidate on voting, by making presumptions about the effect of > rank. That the voters could have used other rules to convert their preferences into votes is not directly relevant to the validity of the counterexample. Making IIA general enough to apply to cardinal preferences as well as ordinal preferences is not really the problem either. The issue with getting Approval Voting with "sincere" voting to satisfy IIA-P is the issue of whether a "sincere" approval vote exists, is unique, or can change as alternatives are added. But those are recurring issues for Approval voting regardless of whether IIA-P is being considered. > The nomination of Genghis Khan would probably not cause me to adjust > my Range Votes for any candidate.... I'd simply rank him at zero. The > nomination of a candidate that was obviously better to me than anyone > else on the list, however, might indeed cause me to change my range > ratings, and such adjustments might indeed, if other voters did not > agree with me, cause a slippage for one of these such that he loses > and the newcomer also loses. But this is hardly an "irrelevant > alternative." I'd think of an irrelevant alternative as someone > outside the reasonable candidate space, or, alternatively, a clone of > a candidate already in that space. These would not ordinarily affect > the winner, except possibly to substitute the newcomer for the > previous winner, thus satisfying IIA. > There are two ways to make Range Voting satisfy IIA. One is to say that it satisfies IIA-V. IIA-V essentially only considers how a voting method tabulates votes. It ignores any consideration of how the voting method otherwise responds to voter preferences. For example, IIA-V restricts FPTP voting so that when a new alternative gets added only previous abstaining votes are allowed to vote for it; no voter is allowed to switch votes from another alternative. Range Voting can also satisfy IIA-P to a limited degree if the the model of voter preferences is limited to a scoring function that has a bounded range and the conversion of votes is restricted the positive linear function from the overall scoring range to the voting range, independent of the alternatives being considered. The ruminations about Ghengis Khan illustrate the limitations of both approaches to getting Range Voting to satisfy IIA. While there is value in recognizing preference strength in modeling people's preferences, in not being limited by an ordinal preference model, it is also important to not gloss over the issues in defining what preference strength is and how it might be represented, nor to avoid a careful analysis of its involvement in an election method. -- David Cary ____________________________________________________________________________________ Sponsored Link $200,000 mortgage for $660/mo - 30/15 yr fixed, reduce debt, home equity - Click now for info http://yahoo.ratemarketplace.com ---- election-methods mailing list - see http://electorama.com/em for list info
