I've explained it this way.

He who sets the agenda for the meeting controls the outcome.

Or the shorter version,

It's good to be king.

I've heard this explained to me by a certain individual in a number of ways, but the question I never seem to figure out the answer to is...what does this actually mean in practical terms? If a little bit of the gaming of the system is going to have to happen, I guess I can be satisfied if, for the *most* part, a fair allocation is achieved.

Isn't it Gibbard-Sattherwaite that says that the only strategy-proof (truth-revealing dominant strategy implementable) social choice function must have a dictator?

the question I never seem to figure out the answer to is...what does this actually mean in practical terms?

I'm a member of a Graduate Student Organization, and we're having a vote on whether to endorse, oppose, or take no position on an issue. What Arrow's Impossibility Theorem told us was that since there was no "perfect" solution for this -- and because there were some very bad solutions -- we needed to be careful to come up with something that worked decently but not constrained to find something which was/is perfect. Which was pretty useful, all told.

Actually, I think Arrow assumed honesty. The Condorcet problem (gaming the system) is another reason why voting leads to strange outcomes.
Oh one thing, Arrow assumed people gave only a preference ordering. The complicated gamable system is to ask people how much they prefer A to B. The optimal dishonest strategy is to say one cares a whole lot even if one barely cares.

My reading of Arrow is that, when our voting system leads to a silly outcome, we should not assume that it can easily be improved by changing it so that it would not have given that outcome. There is always some risk of a silly outcome.

An example (and not just according to the preference orderings of Brad, Kenneth Arrow and me) is 2000, when Nader won the election for Bush because people whose preferences were Nader>Gore>Bush voted for Nader. Notice that strategic voting would have given an outcome preferred by most Americans (Gore rather than Bush).

One might think that with plural voting such problems would be resolved (each voter ranks candidates, the candidate with fewest 1st place votes is eliminated and the 2nd place vote of those voters counts as their vote etc till only one candidate left). In fact, as Arrow showed, this complicated scheme can lead to silly results (even if Palm beach voters could have handled this much more complicated ballot).

So my reading is don't get upset about Bush. No system is perfect. Sad to say no meer proof can reconcile me to the world's misfortune.

Your recasting of Arrow's theorem in terms of information channels makes it start to sound like Ashby's Law, but googling them doesn't seem to show any research connecting the two. Ashby's claim is that a feedback channel has to be able to carry enough information to control all of a system's degrees of freedom, and simple pairwise voting can only stably control a system with one degree of freedom. Since few interesting systems are so limited, voting is inevitably an impossible control mechanism for anything complicated enough to meet Arrow's requirements for a reasonable system.

My dim recollection is that the Impossibility Theorem holds that there does not exist a satisfactory decision function for a population which exhibits arbitrary preferences (where "satisfactory" is defined by conditions including "non-dictatorship"). One interesting thing to look into is whether the conditions of "satisfactoriness" make sense...I remember being somewhat sceptical about non-dictatorship in particular. (IIRC, non-dictatorship turns out to mean that the result of the social decision function MUSTN'T match exactly the preferences of any member of the population. I'm not sure that that condition really matches the intuition we associate with "dictatorship.")

But the really interesting result was the (necessary? sufficient? memory fails...) condition for the existence of a "satisfactory" social decision function -- namely that individual preferences must have some structure, and in particular that they should be consistent with a distance function for some common "policy space."

And, since observed political preferences DO seem to be highly structured (there really are such things as liberals and conservatives) there's good reason to suspect that a satisfactory social decision function does exist under real-world preferences.

At least that's what I remember.... I'm sure someone will throw a rock at my head if I'm wildly off-base.

One of the criteria in Arrow theorem is too strong--the "Independence to Irrelevant Alternatives Criterion", or the IIAC.

Essentially, the IIAC says that if you remove any losing candidate from the election, the winner should not change. This sounds reasonable, but it doesn't deal with "preference loops" very well.

Assume that the voters, in a pairwise election, prefer candidate A over B, B over C, and C over A. Assume that the A/B election has a margin of 100,000 votes, the B/C election has a margin of 100,000 votes, and the C/A election has a margin of 5 votes. (Why the voters would hold these preferences, I couldn't tell you.)

Now under a Condorcet system, we'd resolve this loop by throwing out the weakest link--the C/A election with the 5-vote margin. This gives us A over B over C, with the victory going to A.

But what if we apply the IIAC? According to the IIAC, we should be able to remove candidate B from this election without affecting the victor. This leaves us with only the C/A victory by a 5-vote margin, and makes C the victor. Thus, Condorcet voting fails the IIAC.

Basically, the Arrow theorem denies you any flexibility in resolving preference loops, and then shows that--without this flexibility--no voting system can handle an election with preference loops. That's rather obvious.

If you weaken the IIAC to say, "Removing any candidate outside the Smith set (roughly, 'the set of pairwise loops') should not change the outcome of the election," then you can easily satisfy all the Arrow criteria.

It's not clear to me how even a perfect dictator avoids Arrow's theorem, unless he's Robinson Crusoe. In any other circumstance, the dictator's decisions are bound to create externalities (both positive and negative). Unless we discount to zero the social welfare implications of those externalities, how can we conclude that the dictator's decision/preference unambiguously promotes social welfare?

Expanding on the "thin information channel":
voters in plurality (vote for one) voting use log_2 n bits; approval n bits; Borda or STV, which give the full preference order, n log n bits; Condorcet voting, maybe n^2 log n bits. Warren Smith analyzed a voting scheme where each voter gives each candidate a real number between zero and one; that's obviously a lot of bits.

johnchx:

"(IIRC, non-dictatorship turns out to mean that the result of the social decision function MUSTN'T match exactly the preferences of any member of the population. I'm not sure that that condition really matches the intuition we associate with "dictatorship.")"

Not so. That f(x1,x2,...,xn)=x1 for some set of arguments is not the same as defining f to be f(x1,x2,...,xn):=x1.

That social decision happens to exactly match my preference in a particular outcome does not make me a dictator. I am a dictator if the social decision matches my preference no matter what I prefer and no matter what anyone else prefers.

enfant terrible:

That social decision happens to exactly match my preference in a particular outcome does not make me a dictator.

That was exactly my point. IIRC, the definition of "dictatorship" used in laying out the Impossibility Theorem does not match our idea of "dictatorship."

I'm noticing that a lot of people seem confused about what Arrow's theorem is about. It has NOTHING to do with procedures. It is about the (non-)existence of an abstract mathematical function taking a set of individual preference functions to a "social" preference function with certain specified properties. Of course, it seems fair to conclude that, if a function F does not exist, then there does not exist an algorithm A which implements F. But the non-existence of the algorithm isn't really the point...the point is that there is "right answer" for the algorithm to yield.

Correction: the last sentence should read "...the point is that there is NO 'right answer' for the algorithm to yield."

johnchx: e.t. is right. Dictatorship does not mean that your preferences coincide with the social decision. It means that your preferences coincide with the social outcome, regardless of all other individuals' rankings. (Ie, that your prefs always/must coincide with the social decision.) Which does sound like the usual sense of "dictatorship".

Your morning smile: Joe Frazier beats Muhammad Ali, George Foreman toasts Joe, Muhammad beats George.

I think it's another mapping of the n-body problem. Across all the sciences, most multi-compartment, reticulated models remain indeterminate, for a varying set of reasons that sometimes include the nature of the compartments, and the computations between them (unless special conditions are applied for the solution).

This does not matter for real politics and government, though, because we learned eons ago to form one-to-many structures, or hierarchies, of many different depths and widths. Here, a center makes the decision. And with representative democracy, we continually dissolve and re-form the hierarchies, as the needs and conditions change.

The needs and conditions can be outside or inside the system; for example, external forcings such as new environmental limitations or war, or internal re-routings, such as new communication avenues like the internet.

Direct democracy will tend only to work in small groups where everybody knows everybody--so if somebody objects, or changes his mind to go along, this crucial fact is directly and personally noted by everyone else: an important function for future cohesion.

Arrow's Theorem is a fascinating abstraction, but the real world diverted far from it, a short time after the first single-cell organisms started to subdivide. Probability and statistics are a separate and somewhat mysterious reality, but they seem to be only valid from the coarse-grained side, and we should not fall into a fallacy of decomposition, or suppose that impossibilities are not routinely administered.

The thing about "assuming honesty" isn't quite right. Arrow is saying, "even if honesty" it won't work.

I think Arrow's impossibility theorem means that we should not call the winner of the academy award for acting the "best" actor. There is no sense in which that is true. We can't say that the outcome in any particular voting system is the "most preferred." It's just not true. That is what Arrow is telling us. It's no stronger than that--but it's wrong, as Lee Arnold says, that has no ramifications for actual democracy. A proper reading of Arrow should undermine one's faith in democracy; for even democracy with complete honesty and no manipulation and perfect information cannot aggregate people's opinions in a way that forms a consistent set of meta-opinions.

When I looked at a proof of Arrow's Theorem I was pretty much underwhelmed.

It says that every voting system can have a pathological case where things don't turn out the way you'd naturally want. For that particular proof they accepted "dictator" as one person who can change the vote by changing his mind, one time. And when it's possible to have a vote that wins by one vote, then everybody who's needed for that one-vote majority could change the result by changing their votes, so they're all dictators. QED.

This isn't something I'd think should never be allowed to happen.

Maybe there are better proofs that use less of a tricksy definition of "dictator". Arrow got a Nobel prize for his result, so it ought to have more meat on it than that particular proof suggests.

J Thomas: That's not quite right. Suppose x P[i] y means "i prefers x to y", and x P y means "society prefers x to y". Dictatorship is defined in Arrow as 'There exists an i such that x P[i] y implies x P y for arbitrary preferences P[j], j!=i' (paraphrase, not an exact quote -- the quote's too darn long). Obviously this is not true of majority voting -- one person cannot change the outcome unless other persons' choices are just so.

Re: your example, remember that Arrow proves that majority voting IS consistent with his axioms when there are only 2 choices. (He even comments that this is related to the 2 party system, as basis or justification, which seems very arguable.) The impossibility theorem only applies when the number of choices is greater than 3.

IIAC is definitely the most important feature in Arrow's Theorem.

The thing is, it's hard to imagine any electoral system that, under realistic conditions, cannot be affected by a "strong spoiler" -- a candidate who if he enters the race, will be very popular with not only a fringe, but a fairly significant number of people, while still not commanding a majority of the "first choice" preferences.

Such a candidate is likely to skew results under any system, even ones that are plainly not subject to Arrow (e.g. Approval or Cardinal Ratings).

I'm with Eric. I always understood Arrow's Theorem to just mean that there are too many criteria to apply for every situation. Handling preference loops shouldn't be a high priority in a voting system anyway. I'd say monotonicity is much more important.

I run into too many people who think Arrow's Theorem just means all voting system are equally bad, so we should just stick with plurality. <sigh>

dan Arrow did not say that dictatorship was optimal. Only that it avoided silly outcomes. The IIAC means in this case that it shouldn't matter whether everyone agrees that Nader is a selfish twit, since he isn't going to win with 4% anyway. It mattered. Arrow's interest was only in how one could avoid such things. His answer, roughly, was that you can't in a democracy. Now if a dictator prefered Gore to Bush, Bush could not have been elected no matter what the dictator thought of Nader.

fling93. I think you just ran into another. I mean certainly not all voting systems are equally bad, but all existing voting systems seem to be the worst possible once you get to know them.
Take plural voting. In Cambridge MA you had a choice between populist crooks and otherwise sensible rent controllors. This led Larry Katz,a passionately political progressive economist to abstain in local elections. Hard to do worse than a system which can convince Larry Katz not to vote.

"all existing voting systems seem to be the worst possible once you get to know them."

I dunno about that. I'm pretty fond of Approval Voting and the Condorcet Method myself. Most of the criticisms of Condorcet either hinge on the plethora of ways to break ties. But this only happens when there are preference loops (as Eric described). Those shouldn't occur too often. And anyway, it's equivalent to the electorate saying that all of the candidates in the Smith set are about equally good, so the choice of tie-breaking method probably doesn't really matter that much.

And then there are those who harp on the fact that the Condorcet winner might not be on very many first-place ballots. I think that only happens if the electorate is polarized, in which case the moderate in the middle (that nobody loves but everybody likes) is highly preferable than picking somebody at one of the ends. So I'd say this a strength of the Condorcet Method.

And then there are those who decry its complexity, and that's when I pitch Approval Voting.

One way of putting Arrow's theorem is similar to pithy saying about MIT (and no doubt any high-stress university) - "Sleep, study, friends: pick two"

Thus, Arrow says that there exists at least one pathological set of individual preferences for any given voting method. If you are relaxed about the fact that you don't need a voting system that works for all possible preference sets but only most of them you can also find some satisfactory voting systems.

fling: "I'm pretty fond of Approval Voting and the Condorcet Method myself."

Condorcet Versus Approval:

Ballots consist of rank-order, with a "none of the below" mark, also called an "approval bar". A ballot can be written "BA|{EF}CD". (The | indicates the approval bar, braces indicate equally-ranked candidates. A truncated ballot, on which not all candidates are marked explicitly, is assumed to indicate that all unranked candidates are disapproved and equally ranked.)

We determine the highest-approved Smith Set winner, or unique Condorcet winner; call him CW. If CW is also the Approval Winner, or AW, then he wins.

If not: Determine the fraction of the vote CW receives in the pairwise match between AW and CW; vall this M, the pure-majoritarian victory margin.

In order for AW to win, AW must receive more than M approval in two sub-groups of ballots: those that do not rank AW over CW (i.e. do rank CW over AW or rank them equally), and those that do not rank CW over AW.

The idea behind this "override criterion" is that AW is a proposed compromise between those who prefer the pure-majority winner, and those who wanted somebody else; in order for the compromise to make sense, the candidate must be "highly" approved (where highly is defined by M) by members of both sides.

A typical race in which CVA chooses the compromise is:

16 A|BC
35 AB|C
34 CB|A
15 C|BA

The "centrists" here, who approved B, may be honest voters, or it may be that the electorate is polarized, and voters only approved B because they felt unsure of a victory for their partisan preference due to the near-perfect split. Either way, it's likely that "everyone's second choice" will be more acceptable to the general populace than either more-extreme candidate.

Addendum: CVA, while a fantastically robust system, is of course impractical because it's hard to explain to the average (read: innumerate) voter. I'm a fan of Approval for that reason -- simple to explain, cheap to implement. Approve Approval now! :-)

"CVA, while a fantastically robust system, is of course impractical because it's hard to explain to the average (read: innumerate) voter."

Yeah, I was about to say that. Fascinating system, though. I'll be sure to remember it.

Robert Waldman: I know Arrow did not say that dictatorship was optimal. My argument was about the social welfare implications of Arrow's theorem, namely that the theorem demonstrates that the social welfare implications of voting are indeterminate. That remains just as true for a dictator, as lone voter, as for democracy.

"Somewhere, someone, always gets screwed"