Talk:Independence of irrelevant alternatives

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Contents 1 Approval and range voting 2 LIIA 3 Preferences set 4 LIIAC 5 Possible new section in article: IIA in Arrow's social choice theory

[edit] Approval and range voting

Is this finally a demand were Approval Voting works better than any Condorcet method (besides simplicity)?

Yeah, Approval does satisfy IIA. Of course, it doesn't satisfy the Condorcet criterion -- Arrow showed that no method can satisfy both the Condorcet criterion and IIA. This is easy enough to see. If you have a race that includes a cycle of A > B > C > A, and your method currently selects A, then if B drops out, you'll now have C > A, and C will win. So the race is between A and C, under the control of B (an "irrelevant alternative" -- he is not the winner in either case). This is why many analysts consider the IIA criterion flawed, and instead say that meeting Condorcet is good, and meeting the local IIA criterion is good, but total IIA is not important as long as those other two are met. I'm not clear why another editor considered the example of how the existence of cycles trivially implies IIA failure a "special pleading" for Condorcet. It seems, to me, critical to understanding Arrow's theorem and IIA's role in it. Rmharman 17:45, 14 June 2006 (UTC)

Local IIA is clearly special pleading since it is designed to say "Condorcet methods satisfy IIA (except when they do not in which case it doesn't really matter)". The role I see of IIA in Arrow's theorem is to show that democracy is complicated, and is vulnerable to tactical behaviour, and that there is no objectively optimal solution to the questions which result. I see nothing special about the Condorcet criterion compared with many other criteria, and if you do then that is your point of view. --Henrygb 22:39, 14 June 2006 (UTC)

The point of LIIA is that there are some sets of preferences (cyclic ones) under which no method can satisfy IIA. Satisfying IIA under all possible scenarios in which it is possible to satisfy it is not meaningless. Saying that it's "special pleading" is like saying that building a "divide by 0" exception into a programming language is special pleading -- "your math program can solve all divisions, except the ones it can't, which don't really matter." Well, yes, exactly. We don't need to provide an "answer" to a divide by 0 request, other than, "that's meaningless." In the context of cyclic preferences, IIA is meaningless. Saying that "democracy is complicated" and all systems are vulnerable to tactical manipulation under some circumstances, doesn't mean it's impossible to make meaningful comparisons about how vulnerable a given electoral system is -- under what conditions odd behaviors might emerge, and how likely those conditions are in the real world. As for whether Condorcet is special -- that depends on whether you think that majoritarianism is, in general, a good thing; Condorcet/Smith is the clearest majoritarian criterion, though note that even the most basic version, "a candidate with a majority of first choice votes must win," is contradictory to IIA, as I showed in the example in the article. The only way to satisfy IIA is to have an "election" that doesn't actually consider the will of the voters -- either have it be dictatorial, or random, or something silly like that. In any case, I don't privilege Condorcet above all other concerns -- I'm fine with the variety of Condorcet failure that can occur under Approval Voting, for example, as long as voters have access to good polling information and can make choices with some understanding of the consequences -- but in general, a Condorcet-winner seems to have a pretty strong argument that they're the majority choice. Rmharman 21:13, 23 June 2006 (UTC)

There are plenty of methods that satisfy IIA. Not only are there dictatorial methods, random methods, and others, but the cardinal methods (such as Approval voting and Range voting) have a particular attractiveness. But they have other issues. Condorcet methods and the criterion tend to presume one-dimensional politics and often lead to the election of inoffensive candidates rather than contraversial ones. You may like that, but it is still a POV. --Henrygb 01:37, 24 June 2006 (UTC)

Approval and Range do not meaningfully "satisfy" IIA, they simply fail to provide the information that would allow you to determine that it has been violated. Given a description of voters' ordinal preferences, one can make guesses at how they will vote in Approval or Range, but the two systems don't map transparently to each other. If you define the right sort of strategy for mapping ordinal preferences to cardinal ratings, you can recreate the example, where A wins with B in, but C wins with B out. Dictatorial or random methods select a winner, but it's hard to justify calling either an election method. As for your last line, it seems to be generally understood that Condorcet favors centrism somewhat less than Approval or Range, which, at least in theory, favor a "broad" strategy (aiming to provide middling utility for many people) over a "deep" one (providing high utility for few people). Condorcet doesn't clearly favor either variety of strategy. The notion that centrism is automatically equivalent to being "inoffensive" is simplistic, and probably wrong in many cases -- if one imagines a re-run of the 2000 US presidential election using AV amongst Bush, Gore, and McCain, it seems highly likely that McCain would win, precisely because he was (at the time) seen as being more bold than either candidate. "Centrist champions" tend to get driven out by first-past-the-post and other Duvergerian methods. Rmharman 22:16, 22 August 2006 (UTC)

What a strange way of looking at the world. The key point about range voting (for honest voters) is that it not only allows voters to say whether they prefer one candidate over another, but also allows them to say by how much they care. So it allows more information (assuming the range is big enough) than ordinal preference voting. There is no failure to provide information. --Henrygb 23:27, 22 August 2006 (UTC)

I'm sure you can see why Approval doesn't provide full ranking info, but my apologies, I should've been clearer on why I don't consider Range to provide the full ranking info either. If we can safely assume honesty ratings, it would provide rank info. But there is a very strong incentive for voters whose real utilities (on a scale from 0 to 99) are something like "Minor Party = 99, Major Party A = 51, Major Party B = 0" to make an exaggerated claim of utility for Major Party A. They do not expect Minor Party to win, and if they want to maximize their expected utility from the election, they need to maximize the difference in their ballot between the two likely winners. Thus, they will actually cast an Approval-style ballot, with Major Party A given the same maximum score as Minor Party. You follow? In any case, I have, in recent months, started to come around to the view that rating-based systems may, nonetheless, have advantages over ranking-based systems, even if the ranking system collects approval info. The Center for Range Voting has some interesting work from KVenzke suggesting that even Condorcet does not fully escape "duopoly problems". I haven't decided yet whether I think their cases where betraying a favored third-party in favor of a major party is encouraged are things that a strategic voter could realistically anticipate. Rmharman 17:53, 22 November 2006 (UTC)

I accept the point that for some people there is an incentive for tactical voting (I wrote the final paragraph of Range_voting#Example), but even then most of them might give Major Party A 98 rather than 99. And if they saw preventing Major Candidate B from winning as their overwhelming priority then it suggests they see no significant difference between Minor Party and Major Party A. --Henrygb 22:57, 22 November 2006 (UTC)

[edit] LIIA

LIIA implies Condorcet (and Smith), but Condorcet (and even Smith) doesn't imply LIIA. KVenzke 03:21, 20 June 2006 (UTC)

Is there a name for the subset of Condorcet methods that satisfies LIIA? Rmharman 21:13, 23 June 2006 (UTC)

I don't know of a name for this subset of methods. I would have to just call them "methods which satisfy LIIA." KVenzke 19:41, 25 June 2006 (UTC)

I edited the beginning of the LIIA section, because it seemed to give the impression that LIIA is practically original research, and that it doesn't even have an agreed-upon name. Also, it seemed to be POV to mention Condorcet methods here, as more specifically it is Smith methods that may satisfy it. I'm not sure about the rest of the section: It seems to be a general argument against the utility of IIA rather than an argument in favor of LIIA. KVenzke 20:14, 25 June 2006 (UTC)

The point is that IIA is incompatible with some very basic ideas about democracy and social choice (like "if there are only two options, the one preferred by a majority should win") and as such is a very silly criterion to be applying to election methods. Thus, the only way the concept of "irrelevant alternatives" can have any meaning at all is under LIIA, where candidates that are members of a Smith set of more than one element are not mis-defined into the "irrelevant" class. IIA is badly named! Rmharman 22:16, 22 August 2006 (UTC)

It is fine to say that the majority criterion and independence of irrelevant alternatives conflict. That does not make either good, bad or silly. But you are displaying your POV when you say "the only way the concept of irrelevant alternatives can have any meaning at all is under LIIA" - it simply is not true. --Henrygb 23:27, 22 August 2006 (UTC)

[edit] Preferences set

I deleted the piggybacking addition on my own parenthetical comment on the Morgenbesser anecdote because if it is correct (and I'm not convinced it is -- it needs at the least more exposition) it should be a separate "technical" paragraph on its own, not overloading what is meant to be an illustrative non-technical comment on an illustrative anecdote. The meaning of the addition is also not clear to me -- that the analogy is imperfect because the preference rankings could shift non-transitively with a new option even *after* the choices have been frozen? This seems meaningless to me -- how can one take account of the new option at all if the choices are already frozen? 142.103.168.33 04:19, 12 July 2006 (UTC)

I think that the comment was useful, although you're probably right in that it didn't belong in your parenthetical. The comment seemed to discuss the fact that preferences are taken as given for voting systems---anything that acts to change that is, strictly speaking, outside of the scope of the field. This is why there is a difference between runoff elections and IRV: preferences can only be stated once, frozen, for IRV, while they can change for various reasons in a true runoff election. See, for example, Corks' "Evaluating Voting Methods" (p. 3) CRGreathouse 03:12, 14 July 2006 (UTC)

I deleted the addition "not exactly rational" added by Noe to my own parenthetical comment (originally entered from another IP address) on the Morgenbesser anecdote because it misses the whole point of my comment -- that the apparently irrational non-transitive nature of Morgenbesser's choice could be a perfectly sensible and rational consequence of the "irrelevant" alternative changing the internal state of the decision-maker. There is nothing "irrational" in human cognition in something reminding you of something else -- the "else" thing then being used as the basis for a decision where the original reminder plays no part. 137.82.82.145 20:03, 22 November 2006 (UTC)

[edit] LIIAC

I don't understand why LIIAC is considered a weaker version of IIAC. IIAC doesn't imply LIIAC, does it? We should mention this if that's the case, or include a short proof if it isn't. Also, does someone have a reference to the Young & Levenglick paper introducing LIIAC? CRGreathouse 22:33, 13 July 2006 (UTC)

You are correct. It is easy enough to produce a range voting example (so meeting IIA) where the new candidate wins despite not being in the Smith set. --Henrygb 09:45, 14 July 2006 (UTC)

Are there any methods that adhere to both IIAC and LIIAC? Range voting has IIAC but not LIIAC; the Schulze method has LIIAC but not IIAC; the Borda count has neither IIAC nor LIIAC. I'm just trying to see if there's one with both -- they seems mutually exclusive, even though related. CRGreathouse 17:36, 16 July 2006 (UTC)

Silly me -- LIIAC → Smith criterion → majority criterion → not IIAC. CRGreathouse 16:06, 19 July 2006 (UTC)

[edit] Possible new section in article: IIA in Arrow's social choice theory

A recent edit deleted references in the article to Arrow (1951), b/c his usage of 'IIA' is different from the usage of 'IIA' in the rest of the article. One way of handling the ambiguity is with a Wikipedia:Disambiguation page and separate articles for each page. Another way is to define and discuss each uaage within the article itself (as was done in the article Qualitative economics). (This has already been done with the section IIA in econometrics.) Then the fact of different usages should be noted in the lead section. Arrow's usage of the same term is spelled out in the link to Arrow in the article. There's also a nice intuitive statement in Kenneth Arrow. The current intro to the article is

Independence of irrelevant alternatives (IIA) is an axiom often adopted by social scientists as a basic condition of rationality. The axiom states: If A is preferred to B out of the choice set {A,B}, then introducing a third, irrelevant, alternative X (thus expanding the choice set to {A,B,X} ) should not make B preferred to A. In other words, whether A or B is better should not be changed by the availability of X.

In Arrow's usage, the second sentence could be rephrased as:

IIA states: If A is preferred to B out of the choice set {A,B}, then no matter what preferences are for an unavailable third alternative X, B should not selected by the voting rule over A.

Example (adapted from Sen, Collective Choice and Social Welfare, 1970, p. 37): the choice between Hillary and McCain, given preferences of voters for each relative to the other in 2008, is not to be affected by their preferences for say George Allen (who by hypothesis is not on the ballot) in 2008. Thomasmeeks 15:59, 2 December 2006 (UTC) (spelling typo fixed) Thomasmeeks 12:29, 3 December 2006 (UTC)

I don't think they're different. There are arguments about minor differences in IIA (see the classic 1973 article of Paramesh Ray in Econometrica), but the concept of IIA in social science is literally based on that of Arrow. If there is a difference between the two uses you list, it's entirely because the one you list for Arrow is phrased wrongly. That's not the ordinary way of phrasing it. I don't have his seminal 1951 book (I've read it, though), but in the original 1950 paper he talks about removing an alternative, which can be clearly seen as equivalent to adding one. CRGreathouse (t | c) 21:08, 2 December 2006 (UTC)

I'm assuming that the wording in the lead correctly states IIA. If so, that wording requires changing to produce Arrow's IIA. The Kenneth Arrow cite above and my example above are dissimilar to those in the IIA article. The Econometrica cite in the IIA article (which I have not read) may bring out the same difference that I am asserting. Thomasmeeks 02:24, 3 December 2006 (UTC)

To be more explicit, in the article addition of X to those that are available could flip the observed choice of A or B (the spoiler effect of X not being chosen but affecting the choice of A or B), but in Arrow, availability of X removes tha applicability of his IIA condition to the choice of A & B. In his usage, what makes a variable 'irrelevant' is not that it is not chosen but that it is unavailable. Thomasmeeks 12:29, 3 December 2006 (UTC)

(New left margin to respond directly to CRG's points above). I thank CRG for prompting a more careful response. I've had a chance to look at Ray (1973) (thx JSTOR), cited above and in IIA. Arrow (1950 JPE, 1951) does use an example that equates removing X or adding X back as affecting the choice between A & B (the spoiler effect), just as CRG suggests. But, Ray (1973, pp. 989-90), cited above, demonstrates that that Arrow's own formal statement of IIA (called IIA(A) by Ray), is inapplicable to the example he cites ("Clearly, this [Arrow's example] is a violation of IIA(R[adner]-M[arshak]], not IIA(A)." p. 990). So, the distinction of Arrow's IIA condition from other uses does matter. Thomasmeeks 15:48, 6 December 2006 (UTC) (p. numbers above corrected Thomasmeeks 02:09, 7 December 2006 (UTC))

P.S. Concerning the last sentence, a couple more results from the Ray (1973) article supporting it: Ray proves that the IIA(R-M) condition (congruent with the current IIA-article usage) is consistent with Arrow's other conditions. That is, Arrow's impossibility theorem cannot be derived with IIA(R-M) instead of IIA(A) (p. 989, Theorem 2). Second, as between IIA(R-M) & IIA(A), neither can be derived from the other. Still, there is overlap in the conditions, and certain kinds of information restriction can reduce IIA(R-M) to IIA(A) in which case the impossibility result recurs, though in that case IIA(R-M) adds nothing to the information restriction (pp. 990-91). Thomasmeeks 02:09, 7 December 2006 (UTC)

Just state Arrow's definition here, which from Social Choice and Individual Values seems to be:

• 2. Independence of irrelevant alternatives I: Let S be a subset of hypothetically available (not merely conceivable) social states, say x and y, from the set of social states. Let R₁, ..., R_n and R₁' , ..., R_n' be any 2 sets of orderings with the following property: For every voter i , the ranking of x and y in S is (by construction) the same. (Different voters could still have different rankings of the 2 social states.) Then the subset S for each of the 2 sets of orderings maps to an identical social choice.

--Henrygb 00:02, 3 December 2006 (UTC)

Works for me (if any background info deemed necessary to make that exposition self-contained and clear to those who had not read the source article were worked in). Thomasmeeks 11:43, 3 December 2006 (UTC)

I'll work on this. As the above suggests, there are different ways IIA could go, from new sections to new articles to spinning off sections to a disambiguation page. In my edits, I'll try to respect the integrity of the whole article and make edits as plain as I can. Improvements are always welcome as is rational criticism. (minor edit) Thomasmeeks 11:16, 7 December 2006 (UTC)