Arrow's Theorem is a Lie

This system is a reinvention of range voting.

Like taw says, for causally rational voters this procedure collapses into approval voting. (Though Warren D. Smith of the Center for Range Voting argues that range voting is still better. In computer simulations (technical report), under many conditions, voters who (causally irrationally) choose non-extreme scores collectively reduce aggregated Bayesian regret. Non-extreme scores may help voters simulate non-causal rationality by a "nursery effect".)

I've thought about another voting system that might not collapse. In this system, each voter has the same power as in approval voting. Each voter can choose their approvals themselves, if they have the resources to choose tactically good votes themselves. The system also has a "public voting tactician", which is to voting what a public defender is to legal defense. Voters can give their preferences to the public voting tactician in the form of a utility function, and the tactician finds an equilibrium approval voting strategy for all the voters given their preferences.

(The real reason I am interested in this system is that it might get around problems of incommensurability and scaling of utility functions while still using full utility information.)

The tactician should be chosen to minimize the advantage which voters or groups of voters can get by using any other procedure to choose their votes, or by reporting their preferences non-honestly.

Parts of this system which that description doesn't define are:

• What information does the tactician have? Does it have the preferences of the other voters? Does it have the votes of the voters who did not give preferences to the tactician? If yes, then this may not be a secret ballot, and non-public-tactician voting tactics may be at an unfair disadvantage, preventing competitive pressure to change the public tactician. If no, then where does its beliefs about other voters' votes come from?
• What kind of equilibrium? A voter only has a causal incentive to vote sincerely and admissibly (i.e. monotonic non-decreasing and non-constant in their preferences) when their vote can make a difference. What prevents "all the voters vote for a least collectively preferred candidate" from being an equilibrium, if, in this equilibrium, no voter has a causal incentive to change their vote? Are mixed strategies or uncertainty needed to define the equilibrium? I think the system reduces the voters' utility functions to simple preference orderings unless it models more than infinitesimal uncertainty about other voters' strategies.
• Will there be voters or groups of voters who can do better by using a non-deterministic method to choose their input to the voting system? If so, should the tactician sacrifice determinism?
• Is the best public voting tactician computationally intractable?

The procedure you're proposing collapses into approval voting immediately.

Nobody has any reason to vote anything else than 10 (possibly minus epsilon) or 0 (possibly plus epsilon). If you like A slightly more than B, and estimate A will get 81 points, while B will get 90 points, the optimal behaviour is to vote 10 for A (and everything better than A), and 0 for B (and everything worse than B), any other vote is strictly worse for you. There is no scenario under which voting anything else than 0 and 10 is better than extreme votes.

Approval voting is prone to tactical voting (this should added to requirements, Arrow's Theorem talks about preferences, as it assumes voting is uniquely determined by them) - you never need to order candidates in a way that reverses your preferences, but approval threshold depends on what you think others will vote like. If you think A > B > C, and think C is leading, you vote 10,10,0. If you think B is leading, you vote 10,0,0. It also fails at determinism.

If we give people predefined allowance of points (like n alternatives need scores 1 to n, or get n points for arbitrary distribution) it fails independence immediately, and is prone to preference reversal in tactical voting, which is even worse (if A > B > C, and C is leading, but B has some chances, you vote 0, 10, 0, with A < B).

Even if you can only state a preference ordering and nothing else, there's no reason to keep IIA, as the ranking of irrelevant alternatives is evidence of preference strength.

Apparently Arrow's theorem and the Gibbard-Satterthwaite theorem are the same thing, so there's a real impossibility theorem in there, with the real lesson being that you can't avoid tactical voting.

Great post, voted up. I would add Amartya Sen's Liberal Paradox to the list, particularly relevant because it's regarded as a corollary of Arrow's Theorem.

It purports to show a problem with liberalism (actually, what most people would now call libertarianism). However, it does so by equating rights with obligations -- in other words, preventing people from waiving or otherwise trading away their rights, which is pretty much the essence of (what its proponents consider) the chief mechanism by which libertarianism achieves Pareto-optimality.

So, to Sen, a property right to an apple means you can never transfer the right to the apple to another person, which is of questionable applicability to real scenarios. He uses this constraint to show that libertarianism isn't Pareto-optimal because pesky rights will get in the way of Pareto-improvements. But if a move is really Pareto-optimal, then the relevant parties will waive the relevant rights! (Neat tongue-twister there, by the way.)

Strangely, in developing the alleged problem, Sen passes right by a much stronger argument against libertarianism, that property rights can hinder the well-being of disproportionately many people, even if it wouldn't be Pareto-optimal to make the move (e.g. the one hold-out is made worse off).

I can only guess that his purpose in presenting the Liberal Paradox was to get that argument across to readers' minds, but without having to step out as a defender of that view. That would actually have been a clever move.

I've never understood why this issue is taken seriously.

I'm not sure how critical it is, but your method fails the IIA test described in Formal Statement of the Theorem on the wiki page. Changing the strengths assigned to outcomes can alter the social choice without changing individual rankings.

"There are numerous law of economics which appear to fall into this category; however, so far as I can tell, few of them have been mathematically formalized."

For better or worse, just about everything in economics has been mathematically formalized. If you are wondering about a specific "law" or concept I can try to dig up examples.

I don't see how this can possibly be correct. Suppose we require that no person can give the same score to 2 different options. Your reasoning would indicate that we can then combine these scores in a way that produces a total score for every option that satisfies the four conditions. Conditions 1,2, and 4 are stated in ways that don't depend on whether we use rankings or scores; condition 3 explicitly assumes the use of a score. So if we take the output scores, and list them in order, we get an output ranking, which must still meet those conditions, hence violating Arrow's theorem.

[...] - Independence of irrelevant alternatives: If we prefer A to B as a group, and we all assign the same scores to A and B when C is introduced, then the total score of A must still be higher than that of B, even after C is brought in. Hence, our decision between A and B does not depend on C.

Wrong : this assumes we assign scores to alternatives in a way that doesn't depend of he others. This is wrong for two reasons:

• Tactical voting, as taw pointed out

• We value thing as compared to alternatives : if A is good and B is so-so, you'll rate B differently if there was a third alternative C that's downright awful.

You didn't "beat" arrow's theorem by finding an alternate way to express preferences, but by making the fourth constraint less strict. Once you release that constraint, I'm sure you can find an algorithm that uses preference ordering instead of preference rating that fulfills the constraints just as well (i.e. not very well).

(Still, I agree with the general point, that we should be wary of over-interpreting what mathematical models tell us)

I believe that Arrow's Theorem has been widely misinterpreted, and with a slightly different interpretation it makes more sense.

Here is my argument:

For every collection of preferences, there must be some outcome that satisfies Arrow's Theorem. Like, for voting there must be some winner or there is some sort of tie. Either there is a winner or there is not. Arrow's Theorem can't demand that there has to be a winner and there cannot be a winner.

Since for any collection of preferences there must be some outcome that satisfies Arrow's Theorem, we can make an acceptable voting system by simply looking at each collection of preferences and going through the finite list of outcomes until we find one that satisfies Arrow's Theorem.

The reason people say that no voting system can always work, is that they imagine that voting systems cannot have ties. Obviously, no voting system that never results in a tie can be completely fair, nor will it satisfy Arrow's Theorem.

When there are more than two candidates and more than one voter, you can have a tie in new ways, it isn't necessary to have two candidates with exactly the same number of votes. You can also have a tie when -- one way or another -- the voters collectively prefer A to B, B to C, and C to A. Then of course there is no valid way to say which they prefer most. If the voting system prefers A then it's wrong because they prefer C to A, etc.

So a good voting system should not declare a winner in that case.

Here's a further conclusion. A voting system which does not collect enough information, cannot detect that A > B > C > A and so it will not know to declare a tie. So voting systems which satisfy Arrow's Theorem must somehow ask the right questions.

This voting system fails Pareto efficiency. Given three alternatives A, B and C, which I love, like, and hate respectively, I would vote 10-10-0. Get a second person like me, and we have options A and B equal even though we both prefer A.

And yes, this is a rational way of voting, if my dislike of C trumps any difference between A and B, and if I didn't know the preferences of the other voter.

Actually, it similarly fails the Independence of irrelevant alternatives. In the example where there are three options, A, B and C which I love, like and hate respectively, I would asign 10-10-0 to the three options, but would assign 10-0 to if A and B were the only two options.

an other example: cox's theorem.

Which laws of economics are you thinking of?

Your system fails the first two criteria:

Dictatorship: (ETA: My point here was wrong. I missed that the voters could only select numbers from 0 to 10.) Suppose that one of the voters can write numbers much, much larger than any number that that the other voters can write. (Maybe this voter can write faster and knows clever notations for writing very large numbers.) Then this voter can be dictator by writing a number next to its choice that exceeds the sums of the largest numbers that the other voters can write.

Determinism: The same preference orders, but expressed with different numbers, can lead to different outcomes. (Examples available upon request.)

Eg., if you thought that Q was the best, D was the second best, and A was the worst, you would write down Q > B > (...) > A.

Did you mean to write "B was the second best"?

I may have misunderstood something, but I don't think your system excludes pivotal voters. For example, with the aforementioned ten people, it is possible that everyone gave every alternative except B score 0, and alternative B got score 1 from every member of the group. If one of the members then changes her score for A to 10 and for B to 0, we have A(10) > B(9).

This does appear fixable with a scale tailored to the number of people and alternatives.