Talk:Arrow's impossibility theorem

From Wikipedia, the free encyclopedia
Jump to: navigation, search
edit·history·watch·refresh Stock post message.svg To-do list for Arrow's impossibility theorem:

Here are some tasks awaiting attention:
  • Expand : *Mention local IIA in the interpretations section
    • Give more "real world" examples that are easy to visualize

first sentence[edit]

I don't get that first sentence at all: "demonstrates the non-existence a set of rules for social decision making that would meet all of a certain set of criteria." - eh? (Sorry if this was discussed already, don't have time to read it all now, nor rack my brains on trying to decipher that sentence.) --Kiwibird 3 July 2005 01:20 (UTC)

That was missing an "of", but maybe wasn't so clear even with that corrected. Is the new version clearer? Josh Cherry 3 July 2005 02:48 (UTC)

From Dr. I.D.A.MacIntyre[edit]

I make three comments. Firstly the statement of the theorem is careless. The set voters rank is NOT the set of outcomes. It is in fact the set of alternatives. Consider opposed preferences xPaPy for half the electorate and yPaPx for the other half. ('P' = 'is Preferred to'). The outcome is {x,a,y} under majority voting (MV) and Borda Count (BC). (BC allocates place scores, here 2, 1 and 0, to alternatives in each voter's list.) The voters precisely have not been asked their opinion of the OUTCOME {x,a,y} compared to, say, {x,y} and {a} - alternative outcomes for different voter preference patterns. All voters may prefer {a} to {x,a,y} because the result of the vote will be determined by a fair lottery on x, a and y. If all voters are risk averse they may find the certainty of a preferable to any prospect of their worst possibility being chosen. This difference is crucial for understanding why the theorem in its assumptions fails to represent properly the logic of voting. As I show in my Synthese article voters must vote strategically on the set of alternatives to secure the right indeed democratic outcome. Here aPxIy for all voters would do. ('I' = 'the voter is Indifferent between'). Indeed as I show in The MacIntyre Paradox (presently with Synthese) a singleton outcome evaluated from considering preferences can be beaten by another singleton when preferences on subsets (here the sets {x}, {x,y}, {a} etc) or preferences on orderings (here xPaPy, xIyPa, etc) are considered. Strategic voting is necessary because this difference between alternatives and outcomes returns for every given sort of alternative. (Subsets, subsets of the subsets etc). Another carelessness is in the symbolism. It is L(A) N times that F considers, not, as it is written, that L considers A N times. Brackests required. In a sense,and secondly, we could say then that the solution to the Arrow paradox is to allow strategic voting. It is the burden of Gibbard's theorem (for singleton outcomes - see Pattanaik for more complex cases) (reference below) that the Arrow assumptions are needed to PREVENT strategic voting. The solution to the Arrow problem is in effect shown in the paragraph above. For the given opposed preferences with {x,a,y} as outcome voters may instead all be risk loving prefering now {x,y} to {x,a,} and indeed {a}. This outcome is achieved by all voters voting xIyPa. But in terms of frameworks this is to say that for initial prferences xPaPy and yPaPx for half the electorate each, the outcome ought to be {a} or {x,y} depending on information the voting procedure doesn't have - voters' attitudes to risk. Thus Arrow's formalisation is a mistake in itself. The procedure here says aPxIy is the outcome sometime, sometimmes it is {a} and sometimes were voters all risk neutral it is{x,a,y}. These outcomes under given fixed procedures (BC and MV) voters achieve by strategic voting. We could say then that Gibbard and Satterthwaite show us the consequences of trying to prevent something we should allow whilst Arrow grieviously misrepresents the process he claims to analyse

Thirdly if you trace back the history of the uses that have been made of the Arrow - type ('Impossibility') theorems you will wonder at the effect their export to democracies the CIA disappoved of and dictatorships it approved of actually had. Meanwhile less technical paens of praise for democracy would have been directed to democracies the US approved of and dictatorships it didn't. All this not just in the US. I saw postgraduates from Iran in the year of the fall of Shah being taught the Arrow theorem without any resolution of it being offered. It must have been making a transition to majority voting in Iran just that bit more difficult. That the proper resolution of the paradox is not well known (and those offered above all on full analysis fail to resolve these Impossibilty Theorems and in fact take us away from the solution) allows unscrupulous governments to remain Janus faced on democracy. There certainly are countries that have been attacked for not implementing political systems that US academics and advisors have let them know are worthless.

--86.128.143.185

Moved from the article. --Gwern (contribs) 19:43 11 April 2007 (GMT)


From Dr. I. D. A. MacIntyre.

I am at a loss to understand why other editors are erasing my comments. Anyone who wishes to do so can make a PROFESSIONAL approach to Professor Pattanaik at UCR. He will forward to me any comments you have and, if you give him your email address I will explain further to you. Alternatively I am in the Leicester, England, phone book.


I repeat: the statement of the theorem is careless. For a given set of alternatives, {x,a,y} the possible outcomes must allow ties. Thus the possible outcomes are the SET of RANKINGS of {x,a,y}. The other editors cannot hide behind the single valued case of which two things can be said. Firstly Arrow allowed orders like xIyPa (x ties with y and both beat a). Secondly if only strict orders (P throughout) can be outcomes how can the theorem conceivably claim to represent exactly divided, even in size, societies where for half each xPaPy and yPaPx.


Thus compared with {x,a,y} we see that the possible outcomes include xIyPa, xIyIa and xIaIy. In fact for the voter profile suggested in the previous paragraph under majority voting and Borda count (a positional voting system where,here 2, 1 and 0 can be allocated to each alternative for each voter) the outcome will be xIaIy. The problem that the Arrow theorem cannot cope with is that we would not expect the outcome to be the same all the time for the same voter profile. For for the given profile, and anyway, voters may be risk loving, risk averse or risk neutral. If all exhibit the same attitude to risk then respectively they will find xIyPa, aPxIy and xIaIy the best outcome. (Some of this is explained fully in my Pareto Rule paper in Theory and Decision). But the Arrow Theorem insists that voters orderings uniquely determine the outcome. Thus the Arrow Theorem fails adequately to represent adequate voting procedures in its very framework.



To repeat the set of orderings in order (ie not xIyPa compared with xIaIy, aPxIy etc). Thus all voters may find zPaPw > aPxIy > xIaIy > xIyPa if they are risk averse. ({z,w} = {x,y} for each voter in the divided profile above). The plausible outcome xIaIy is thus Pareto inferior here to aPxIy. In fact any outcome can be PAreto inoptimal for this profile. (For the outcomes aPxIy, xIyPa and xIaIy the result will be {a}, and a fair lottery on {x,y} and {x,a,y} repsectiively. The loving voter for whom zPaPw prefers the fair lottery {x,y} compared with {a} and hence xIyPa to aPxIy.


The solution is to allow strategic voting so that in effect voters can express their preferences on rankings of alternatives. Under majority voting such strategic voting need never disadvantage a majority in terms of outcomes, and as we see here, can benefit all voters. (Several of my Theory and Decison papers discuss this).


We are very close to seeing the reasonableness of cycles. For 5 voters each voting aPbPc, bPcPa and cPaPb the outcome {aPbPc, bPcPa, cPaPb, xIaIy} seems reasonable. This is not an Arrow outcome but one acknowleding 4 possible final results. But then the truth is, taking alternatives in pairs that with probability 2/3 aPb as well as bPc and cPa. What else can this mean except that we should choose x from {x,y} in every case where xPy with 2/3 probability.


(In the divided society case above if all voters are risk loving the outcome {aPxIy} is preferred by all voters to some putative {xPaPy, yPaPx}. The possible outcomes for voting cycles are to be found in my Synthese article.)


I go no further. Except to make five further comments. Firstly those who like Arrow's theorem can continue so to do, as a piece of abstract mathematics, but not as a piece of social science, as which it is appallingly bad. Arrow focuses on cyclcical preferences and later commentators like Saari have fallen into the trap of thinking opposed preferences not a problem for the Arrow frame. In fact both sorts of preferences are a problem for the erroneous Arrow frame. That is the way round things are. The Arrow frame presents the problems. The preferences are NOT problematic.


Secondly I reiterate strategic voting, which is a necessary part of democracy, need never allow any majority to suffer (see my Synthese article for the cyclical voting case). Indeed majorities and even all voters can benefit. Majority voting with strategic voting could, then, be called consequentialist majoritarian.



Thirdly, and if this is what is getting up the other editos noses then leave just this out because it is most important that everyone stops being fooled about MAJORITY VOTING by Arrow's theorem and his Nobel driven prestige, anyone who thinks Arrow has a point has been led astray. If US academics and advisors believe he has then why do we bomb countries for not being demcracies? And if no one does then why was the theorem taught unanswered to Iranian students here in the UK during the year of the fall of the Shah? If Iran is not a demcracy to your liking, I am speaking to the other editors, a good part of the reason is the theorems you are protecting, I can assure you. No one can be Janus faced about this. paricualrly not by suppressing solutions to the Theorem in a dictatorial way.


Fourthly to restrict the theorem to linear orderings which Arrow does not do is pointlessly deceptive. For it hides the route to the solution (keeping 'experts' in pointless but lucrative employment?). For even in that case the set of strict orders on the set of alternatives is NOT what voters are invited to rank.


Lastly the hieroglyths above are wrong too. The function F acts on L(A) N times. L does not operate on A N times as the text above claims. Brackets required!


From Dr. I. MacIntyre : Of course any account of the Arrow Theorem and its ramifications is going to please some and displease others so I add this comment without criticism.

It seems to me that strategic behaviour in voting (and more generally) is such an important part of human behaviour that how various voting procedures cope with it will turn out to be the most useful way of distinguishing between them.

Indeed one could go so far as to say that strategic behaviour, properly understood and interpreted, also provides the key to resolving the Arrow 'Paradox'.

To that end, and anyway because of its importance I think it would be useful in this Wikipedia article to indicate, at least, the tight connection between the constraints Arrow imposes on voters in order to derive his theorem and what must be imposed on them to avoid the logical possibility of 'misrepresentation' or strategic behaviour. That is, the role of Arrow's assumptions in Gibbard's Theorem should, I think, be spelt out at least informally.

Many writers have suggested resolutions to the Theorem without paying any real attention to strategic voting. As a result they have missed what is certainly majority decision making's best (and I think decisive) defence. For under majority decision making strategic voting can benefit majorities, even all voters (sic!) (see my Pareto Rule paper in Theory and Decision) and no majority ever need suffer. No other rule (eg the Borda Count rule) defends its own constitutive principle in this way.

As a result of these omissions (of any acknowledgement of the ubiquity of strategic behaviour and of the Arrow - Gibbard connection) the technical literature in recent years has lost realism in its accounts of democratic behaviour and leaves its readers with the impression that democracy is best saved by abandoning majority voting. (As Borda Count does). Such an odd view of best voting practice is likely to encourage dictators and discourage even the strongest of democrats. Perhaps that is the intended effect. For one could argue that the way majority mandates have been de - legitimised is the worst legacy of the Arrow Theorem so that just redistribution has been thwarted in South Africa, Northern Ireland and elsewhere in localities better known by you readers than I.

I. MacIntyre n_mcntyr@yahoo.co.uk 27th April 2007

IIA Range Voting Counter-example[edit]

The footnote concerning IIA versus range voting (with the 9 & 1 comparison) seems irrelevant. What is it trying to demonstrate? I recommend it be removed. --Osndok (talk) 00:17, 20 October 2010 (UTC)

I write "Whether such a claim is correct depends on how each condition is reformulated." As I wrote in User talk:Osndok, the footnote demonstrates that Rv violates Arrow's IIA. Of course, Rv satisfies a weakened IIA. So, it depends on how IIA is formulated.--Theorist2 (talk) 00:54, 20 October 2010 (UTC)
As discussed, I see that the example you have given is translating a range-voting example into rank-order results. I think we agree that range voting does not apply to Arrow's theorem, as the example could just as easily be: one voter scores all the candidates the same, therefore range voting does not produce an ordered result, therefore it "violates" a precondition. It still does not seem relevant to me. --Osndok (talk) 02:59, 20 October 2010 (UTC)
If you define a social welfare function so that it excludes indifference, then it means such a case is ignored because it is deemed uninteresting, not the point, excluded for simplification, etc. Why do you consider the uninteresting case in which all alternatives have the same score? Of course, it does violate the domain condition. But you can redefine the domain of a swf so that indifference is allowed. Then, the same score alternatives are treated as indifferent. If you think the same score case is important, then just use the latter definition. Don't use the former uninteresting definition. There, we regard IIA as important. We care if Rv satisfies it. The footnote is obviously relevant since it clarifies that the assertion that Rv satisfies all conditions depends on how those conditions are defined. Without the footnote, the reader might think Rv satisfies Arrow's original IIA.--Theorist2 (talk) 05:19, 20 October 2010 (UTC)
What is at issue here is that range voting (by-definition) satisfies "general IIA". The only reason it does not satisfy "Arrow's IIA" is because you must translate it into ranks. Range voting does not fall under the scope of Arrow's theorem, so why should this be included? As best I can see, you are adding a new statement; that, "range-voting-when-translated-into-ranks does not satisfy part of arrow's theorem" (it is original research, and [IMO] not relevant). --Osndok (talk) 16:28, 20 October 2010 (UTC)
Why should this be included?---Because, a "solution" to Arrow's impossibility is already included. As you say, Range voting does not fall under the scope of Arrow's theorem. But it is already cited as an example of a rule that "can be considered to satisfy the spirit of" Arrow's conditions. I think it is best to delete the citation (after all, very few professional works mention Rv). But doing so would not be very effective, since someone will add the same thing later anyway. A compromise solution is to retain the citation to Range voting, but clarify what it means for Rv to satisfy the conditions. For most readers, it is enough to know that "Whether such a claim is correct depends on how each condition is reformulated." For someone who cares about Rv, (supposing academic sincerity) it is important to know which formulation of IIA Rv violates. (If the footnote is deleted, then they will request the statement "Whether such a claim is correct depends on how each condition is reformulated" be removed, because it is unfounded. The result is that most reader will incorrectly think Rv satisfies all of Arrow's conditions. I think that is not a desirable situation.)
Let's not hide the fact that proposed solutions like Rv do not actually satisfy all of Arrow's conditions. By being academically sincere, I think more professionals will begin to link to the article and contribute to it. That should be good news to the supporters of Rv in the long run.--Theorist2 (talk) 13:47, 23 October 2010 (UTC)
I have replaced the footnote that Osndok complained about. There was nothing original in it, but it seems easier just to mention the well known fact from the well known source (Sen) than to find an exact source supporting the particular example in the footnote. The problem resolved.--Theorist2 (talk) 09:55, 24 October 2010 (UTC)
Can I take this issue up again? Range voting does not satisfy Arrow's IIA or Samuelson's cardinal version of IIA - and nor does any mechanism using cardinal utility. Kalai and Schmeidler (1979) demonstrate this rather clearly. Every cardinal preference is also an ordinal preference - since it expresses a ranking over outcomes. Hence, mechanisms that make use of cardinality are still subject to Arrow's theorem. (There is nothing in Arrow's theorem that requires preferences to be ordinal-but-not-cardinal.) Here is a counter-example: Suppose there are 3 outcomes - A,B,C and 3 agents (with utilities u,v,w, respectively). The cardinal preferences are in profile 1 are given by: {u1(A,B,C,D)=(9,3,0,5), v1(A,B,C,D)=(0,2,1,3) and w1(A,B,C,D)=(8,3,2,1)} whilst preferences in profile 2 are given by: {u2=(3,1,0,9), v2=(0,10,5,6) and w2=(8,3,2,1)]. By Range voting, the social preference according to the first profile is A>D>B>C, whilst the social preference by the second profile is D>B>A>C. Note that w is the same in both profiles, and u and v are cardinally equivalent over the subset {A,B,C} in both profiles - i.e. u1(x)=3*u2(x) and v1(x)=0.2*v2(x). Then by IIA, Range voting should rank A,B and C in the same way - but it doesn't. Finally, the "weakened" notion of IIA in footnote 28 is surely not helpful. Consider two preference profiles in which all but agent 1 have the exact same utility and agent 1's utility is different only in that his utility in the second profile is twice his utility in the first profile. Clearly the two profiles are identical - even agent 1's preferences are exactly the same. You would hope that the social choice axioms would say that the social choice function must choose the same social ranking under both profiles. But the weakened notion of IIA does not require this. It is weak indeed! --Gparames (talk) 21:15, 25 March 2013 (UTC)
Every cardinal utility expresses an ordinal preference, but in translating a utility function into a preference, you lose certain information. A social welfare function in Arrow's sense cannot use that lost information. In other words, Arrow's theorem does require preferences to be ordinal-but-not-cardinal. The definition of a social welfare function does that. Mechanisms (like Range voting) that make use of cardinality are generally not a social welfare function in Arrow's sense. So to deal with such mechanisms, you need to redefine IIA. Provided that IIA is defined so that only ordinal information is taken into account (as you like), it is correct to say Range voting violates IIA. However, many people prefer defining IIA in the weaker sense, where cardinal information is also taken into account. That way, the redefined IIA can reflect the strength of preference, treating (u(x), u(y), v(x), v(y)) = (1,0,0,10) and (10,0,0,1) differently.Theorist2 (talk) 00:09, 26 March 2013 (UTC)

Another problem here is the article claims that the Gibbard–Satterthwaite theorem applies to range (score) voting. But the Gibbard–Satterthwaite page says it only applies to voting systems “where each voter ranks all candidates in order of preference”. In other words, G-S only applies to ordinal ranking systems, which means it does not apply to cardinal rating systems like score voting. Qaanol (talk) 02:23, 2 February 2014 (UTC)

which Pareto criterion[edit]

Pareto principle was recently linked; I doubt that's what the author of the phrase intended. On another hand, I'm not sure that Pareto efficiency is meant either, so I won't change it there. —Tamfang (talk) 04:18, 25 June 2011 (UTC)

Yes, it's Pareto efficiency, in particular the weak version. CRGreathouse (t | c) 16:35, 25 June 2011 (UTC)

voting vs. swf[edit]

The article presents the theorem as a voting result. But isn't AIT more general then that? In the sense that it says that no social welfare function which satisfies the stated criteria exists - in other words, it's not just that a voting system can't do something, it's that there is no way to aggregate individual preferences into a social preference.Volunteer Marek (talk) 20:23, 1 January 2012 (UTC)

This is a good point, Volunteer Marek. I think the issue is that Arrow's theorem is a theorem of mathematics that can be interpreted in terms of voting, or in terms of preference aggregation. i.e. the determination of a social preference ordering from individual preference orderings. I would call the latter interpretation "social choice", subject to a relatively insubstantial caveat mentioned below. I would not say it is more general, just different. I think the difference between these two interpretations should be made clear. The main issue is whether you take each individual preference ordering to be a ranked list of alternatives submitted by that individual, which is then used in a voting procedure, or to be that individual's actual preferences over the alternatives. Different mathematical requirements on the choice rule/ voting system may be intuitively appealing on these two different interpretations (this is indeed mentioned in the article).

There is perhaps a minor caveat: It may be, though, that the "social choice" language is usually applied to a slightly different formulation, in which there is a "choice function" that for each subset X of alternatives, says which subset of X consists of acceptable choices for society. I think there is likely to be an equivalence between this formulation and the one where the output is social preference ordering (the choice function will return, on input X, the subset of X consisting of alternatives maximal with respect to the social preference ordering restricted to X), and there will be a way of recovering a ranking from a choice function defined on all subsets of alternatives as well....). This strikes me as a fine point to be avoided if possible, but perhaps discussed under the heading of Social Choice Theory or Preference Aggregation, and just mentioned in passing (and cited) in this article.

I may do some editing based on the above observations, though probably not before taking some more time to think about it, and doing some more research. I also have other minor issues regarding whether the individual and/or social preference orderings are assumed to be strict or not, which I think should be made crystal clear in the article. MorphismOfDoom (talk) 17:31, 2 November 2012 (UTC)

Volunteer Marek is right: Arrow's theorem shows that "there is no way to aggregate individual preferences [orderings] into a social preference [ordering]". However, as my added brackets and emphasis suggest, that is less restrictive than it sounds like. For instance, there are multiple ways to aggregate individual utilities into social utilities. Because this result only talks about preferences, its main application is to voting. Thus it would be nice to have a passage explaining that it could be applicable in other situations, but I find the focus on voting, especially in the intro, to be perfectly appropriate. Homunq () 11:10, 5 November 2012 (UTC)

On the name "impossibility theorem"[edit]

EDIT: So this got deleted last time, but I never got a reply out of the person who did it, so I'm simply re-posting it again.

Where does the term "impossibility theorem" come from? In both Arrow's 1950 paper "A difficulty in the concept of social welfare" and the 1951 book "Social choice and individual values" the only terminology I see is "The General Possibility Theorem for Social Welfare Functions". Could someone direct me to the cause of why everywhere this is called the "impossibility" theorem? Drozdyuk (talk) 20:45, 3 December 2012 (UTC) (originally posted 25 April 2012)

Rank-order voting and the definition of Arrow's theorem[edit]

     Arrow did not define his theorem in terms of rank-order voting. The theorem is about choosing an aggregate preference hierarchy from the collection of preferences that exist among the members of a social group. It is not necessary for preferences to be expressed as ordinal rankings for the theorem to apply. You cannot get around the Impossibility Theorem by replacing individuals' rank orderings with something else. None of Arrow's four criteria require that information about individual preferences come in the form of ranks, and it is not possible for any system, based on ranks or not, to satisfy all four of Arrow's criteria. For example, contrary to claims in the current draft of this article, Arrow's theorem applies to range voting and approval voting.

     I am concerned that individuals striving to promote certain voting systems have badly damaged this article. I think it was less flawed three or more years ago, before the unsupported claims about rank-order voting were added. Redefining the Theorem in terms of rank-order voting was a serious error committed by someone who presented no evidence that the previous definition was wrong or that his/her definition in terms of rank-order voting was right.

     Furthermore, as currently written, the article might lead readers to believe, falsely, that rank-order voting methods have a special flaw, revealed by Arrow, and because of that, we ought to avoid such methods. — Preceding unsigned comment added by Mbmiller (talkcontribs) 20:36, 22 April 2015 (UTC)

@Mbmiller: Can you explain how it applies to range voting or approval voting? This also says that it doesn't: http://rangevoting.org/ArrowThm.html 71.167.64.83 (talk) 04:48, 2 September 2016 (UTC)

External links modified[edit]

Hello fellow Wikipedians,

I have just added archive links to one external link on Arrow's impossibility theorem. Please take a moment to review my edit. If necessary, add {{cbignore}} after the link to keep me from modifying it. Alternatively, you can add {{nobots|deny=InternetArchiveBot}} to keep me off the page altogether. I made the following changes:

When you have finished reviewing my changes, please set the checked parameter below to true to let others know.

YesY An editor has reviewed this edit and fixed any errors that were found.

  • If you have discovered URLs which were erroneously considered dead by the bot, you can report them with this tool.
  • If you found an error with any archives or the URLs themselves, you can fix them with this tool.

If you are unable to use these tools, you may set |needhelp=<your help request> on this template to request help from an experienced user. Please include details about your problem, to help other editors.

Cheers.—cyberbot IITalk to my owner:Online 21:03, 27 February 2016 (UTC)

Dr. Hillinger's comment on this article[edit]

Dr. Hillinger has reviewed this Wikipedia page, and provided us with the following comments to improve its quality:


Arrow's theorem is about the agggregation of orderings. No voting procedure in actual use is of this form. They are all in one way or other cardinal. The importance of Arrow's theorem is therefore in my opinion much overrated.


We hope Wikipedians on this talk page can take advantage of these comments and improve the quality of the article accordingly.

We believe Dr. Hillinger has expertise on the topic of this article, since he has published relevant scholarly research:


  • Reference : Hillinger, Claude, 2004. "Utilitarian Collective Choice and Voting," Discussion Papers in Economics 473, University of Munich, Department of Economics.

ExpertIdeasBot (talk) 17:35, 26 July 2016 (UTC)


In what sense are ordinal voting systems actually cardinal? 71.167.64.83 (talk) 04:46, 2 September 2016 (UTC)

Dubious[edit]

This article states that Arrow doesn't apply to range systems, but that Gibbard–Satterthwaite still does.

Yet Gibbard–Satterthwaite theorem says right in the first sentence that it only applies to ranked systems, and the range voting website also says that Gibbard–Satterthwaite doesn't apply to it. http://rangevoting.org/GibbSat.html

"Wait a minute. Doesn't Range voting (in the ≤3-candidate case) satisfy all GS criteria, accomplishing the "impossible"?! Huh? ... How can this be? The explanation is simple. The Gibbard-Satterthwaite theorem only applies to rank-order-ballot voting systems." 71.167.64.83 (talk) 04:43, 2 September 2016 (UTC)

Both of those are based on using Satterthwaite's version of the theorem. Gibbard's version is more general, and applies to any simple (one-step) multiplayer game type where the number of players, possible player actions, and possible outcomes are all finite and more than 2. (I say "game type" rather than "game", because each player's payoffs for each outcome must be allowed to vary.) It states that if there is a single optimum "honest" move for each player independent of what the other players do, then the game is a dictatorship. Thus, this version applies to any voting rule, whether rated, ranked, or other. Homunq () 18:23, 18 October 2016 (UTC)