Arrow: Criticism and Defence

A while ago I promised a reply to John Lawrence's Criticisms of Arrow. Here it is, point-by-point. (Hope you don't mind me copying them John - if you do leave a comment and I'll edit your bit out)

1) Arrow doesn’t allow ties. Example: For alternatives a, b and c, the only acceptable social choice for Arrow is one of the following: abc, acb, bac, bca, cab, cba. A tie would be of the form {abc, bca} where the parentheses indicate that the two social rankings abc and bca are tied.

I don’t think Arrow disallows ties. See below, but ‘indifference’ counts as a decision. Of course, trying to choose between policies x and y, knowing society is indifferent between them doesn’t much help. Perhaps we can resolve indifference by tossing a coin, as many have suggested for tied elections. Technically that doesn’t fit Arrow’s general project, as it only breaks deadlock, it doesn’t produce a social ranking – but then, I think he’s wrong to require such anyway.

2) Arrow assumes that a tie is the same as an indifference. Let’s assume 2 alternatives: x and y. Also that half of the individuals in society prefer x to y – xPiy for half – and half the individuals prefer y to x – yPix. Pi represents the preference ordering of the ith individual. Arrow would say that society is indifferent between the 2 alternatives – xIy. However, society is not indifferent between the 2 alternatives. Society is evenly divided between the 2 alternatives. Half prefer x to y and, let us assume, very passionately. Half prefer y to x and, let us assume, equally as passionately. Society would be indifferent if every individual was indifferent between the 2 alternatives.

Arrow doesn’t just ‘assume’ this. There’s an argument p.50 ff. intended to establish that if x and y tie, yet we take one to be preferred, we get inconsistent results. (The argument was too technical to follow easily, let alone recall or reproduce here, but believe me there is one)

Of course, this doesn’t quite answer the challenge, as the criticism isn’t that either x or y should be preferred – it’s that there’s a fourth possibility: xPy, yPx, xIy and xTy where the T relation stands for ‘ties with’.

I think there is something to this, but note there’s now a problem. Jonathan says “Society would be indifferent if every individual was indifferent between the 2 alternatives” but it isn’t clear if this is merely sufficient or also necessary. What if 99% were indifferent and the 1% evenly split each way? Surely that’s also social indifference rather than a tie. What if the 1% mostly support x? Is the social choice now xPy or xIy? The problem is that introducing this extra possibility means we’re now longer dealing with a binary x or y choice, because tying is (unlike indifference) a third option.

3) According to Arrow the only information an individual may specify is pairwise binary comparisons. Any other information is irrelevant. However, there is no reason why an individual shouldn’t specify as much information as possible. For instance, if x, y and z are the alternatives, Arrow would say that the only relevant information is in comparing x to y, x to z and y to z. It is easy to imagine a grid with values from 1 to 100, for example. An individual could place x, y and z on this grid in any position. This would convey more information than binary comparisons Why shouldn’t each individual have unlimited freedom of expression?

If you can’t compare interpersonal utility, the problem is this ‘extra information’ is meaningless. I may rank a, b and c as 1, 50 and 100 and you may rank them 1, 99, 100, but this doesn’t mean anything. We can’t conclude I prefer b more than you do, so why allow voters to express something that has no meaning?

4) There is no such thing as an irrelevant alternative. The number of alternatives determines the underlying grid. If that grid changes due to the death of one of the candidates, for instance, the individual’s preferences, if he is required to specify them within a grid determined by the number of candidates, may change. Preferences may become indifferences and vice versa.

This is to some extent a fundamental disagreement, too close to premises to resolve, and one where John essentially sides with Borda against Condorcet and Arrow. It doesn’t seem obviously wrong to say all that matters in choice between x and y is the relative ranking of x and y, however. Why compare them to non-option z, particularly in light of the above point that it doesn’t even allow some kind of interpersonal comparison?

The death of a candidate is actually a slightly different case. Arrow himself confuses independence of irrelevant alternatives with contraction consistency, which is related but can be distinguished. Still, suppose you rank aPbIcPd then c dies. It seems natural to suppose the remaining preferences simply carry over – aPbPd – unless some good reason is given why they might change (non-strategically).

5) Since there is no such thing as an irrelevant alternative, one of Arrow’s “rational and ethical” criteria is invalidated. Therefore, his entire analysis is invalidated.

This is rather hasty if 4) hasn’t been sufficiently proven…

6) We don’t know how an individual would vote if one candidate died, for example. If the number of candidates changes, the individuals must be repolled or else some probabilistic assumptions would have to be made about how they would have voted. You can’t just assume that because an individual voted aPbIc, if a dropped out, the individual would still be indifferent between b and c. For example, let us assume that there are 3 candidates, a, b, c and Hitler. A voter might vote aIbIcPHitler with the rationale that any candidate would be preferred to Hitler. Then, if Hitler dropped out, true preferences among a, b and c might emerge.

This rather repeats 4). If there are true preferences between a, b and c, why hsouldn’t they emerge before Hitler’s death? Thus if the individual would want to vote aPbPc without Hitler in the frame, they should want to vote aPbPcPHitler with Hitler, preserving “the rationale that any candidate would be preferred to Hitler”.

7) An individual shouldn’t be prevented or constrained from using his vote in a strategic way. In any rational voting system there is what Arrow calls the “Positive Association of Social and Individual Values.” Therefore, an individual will express preferences based on the candidate set in order to prevent a particular individual from being elected or to help guarantee that another individual will be elected if he feels that strongly about some particular candidate. The individual shouldn’t be required, as Arrow does, to vote consistently when the candidate set changes. For example, let’s assume that the candidate set consists of candidates a, b, c and Jesus. It would be an entirely rational vote to specify (Jesus)PaIbIc. Compared to Jesus, the individual is indifferent among a, b and c. Now let us suppose that Jesus drops out of the race. Then it would be entirely rational for the individual to vote aPbPc. In other words, the voter voted strategically to get Jesus elected and should be allowed to do so. If Jesus is not in the race, the individual’s true preferences among a, b and c emerge. Arrow’s condition, “Independence of Irrelevant Alternatives” is not rational after all.

Again, this criticises 4), but explicitly claiming the individual should be allowed to vote strategically. I’m not sure about that. Sometimes I think strategic voting is defensible, so long as all have the same opportunities to use their votes, it’s up to people how they do so. Certainly no argument for strategic voting is given here, however, so nothing to convince anyone who already believes it’s wrong.

8) “Adopting an informational perspective, then, [Arrow’s theorem] just state[s] that procedures for three or more candidates require more information than just the relative rankings of pairs.” Saari, DG, (1995), Basic Geometry of Voting, Springer-Verlag, Berlin.

I don’t see the point being made here.

9) Rankings have been considered to be cardinal or ordinal where ordinal represents a simple ranking and cardinal allows more information. Cardinal rankings supposedly allow “preference intensity” to be represented. It’s not about preference intensity; it’s about freedom of expression. Let’s add a third type of ranking: digital. An ordinal comparison for 3 candidates can be specified by 3 bits since there are 6 possibilities. If we allow more bits, then more information and relevant information can be gleaned from each individual. Allowing each individual to specify his or her preferences using the same number of bits eliminates the “interpersonal comparisons of utility,” another of Arrow’s bugaboos. There is no preference given to one individual over another because of supposed greater need. Therefore, allowing more than just ordinal information is just as impersonal as allowing only ordinal information.

This claims to be like 3) but without making interpersonal comparisons, however I don’t see what is supposedly expressed. How about another box on the ballot allowing voters to express their favourite flavour of ice cream? Surely that’s also more expression and therefore better (though I don’t see how it should influence the decision in question).

10) Arrow constrains freedom of expression.

See 3) and 9). I don’t think Arrow constrains ‘free expression’ in any troubling sense – he wouldn’t repeal the First amendment, for example – all he restricts is what influences social choice.

11) Arrow confuses ties and indifference in the binary case in which there are 2 alternatives and n voters. See my paper, "Neutrality and the Possibility of Social Choice" He says majority rule when there are only 2 alternatives is the only case where social choice actually works. But according to his analysis, if done correctly, social choice isn’t even possible with 2 alternatives. The key point is that when the number of voters who prefer x to y, N(x,y), equals the number of voters who prefer y to x, N(y,x), you have a tie between the solutions xPy and yPx which I indicate {xPy, yPx}. This is not the same as xIy, x is indifferent to y.

This is basically point 2) repeated. (Again, I suggest tossing a coin to resolve ties).

12) Arrow violates the Principle of Neutrality in his analysis of binary majority rule. The Principle of Neutrality states that each alternative (in this case x and y) must be treated in the same way. No alternative may be given preferential treatment.

How does Arrow violate Neutrality then? Neither x nor y is given preferential treatment (unless ties favour status quo, which is slightly non-neutral). But see 16).

13) In the binary case, Arrow assumes that a tie in the domain of individual votes implies a social indifference. The domain consists of all possible combinations of votes by the individual voters. The range consists of all the possible choices by society as a whole i.e. social choices.

I’m not sure what point is being made here, but since it concerns ties again I suggest it’s much the same as 2) and 11).

14) Arrow states (p. 12, 13 of “Social Choice and Individual Values”): “A strong ordering…is a ranking in which no ties are possible.” WRONG! If n/2 voters prefer y to x and n/2 voters prefer x to y (n being even), this clearly is a tie! This section clearly shows Arrow’s confusion between the concept of a tie and the concept of indifference. He thinks that both xRy and yRx imply a tie. Wrong again. They imply an indifference.

A strong ordering is one where no ties are possible. That’s definitional. If voters are evenly split, you get a tie, but that’s because the result isn’t a strong ordering! As for the tie/indifference distinction, see 2) and 11) – this increasingly seems John’s main, perhaps only, complaint.

15) Arrow’s R notation. On p. 12 of “Social Choice and Individual Values,” Arrow states: “Preference and indifference are relations between alternatives. Instead of working with two relations, it will be slightly more convenient to use a single relation, ‘preferred or indifferent.’ The statement ‘x is preferred or indifferent to y’ will be symbolized by xRy.” Emphasis added. Slightly more convenient? Ridiculous. What voter votes in such a way that they are “preferred or indifferent” between x and y. What would be the meaning? Maybe I prefer x to y or maybe I’m indifferent? I don’t know which? The net result is that the voter is constrained to make choices of this nature when he damn well knows he prefers x to y or he damn well knows he is indifferent between x and y. Heuristically, the R notation is nonsense. If I’m going to list my preferences, I can do so unambiguously using P and I. For example, aPbIcPd would indicate that I prefer a to b, c and d; I’m indifferent between b and c; and I prefer a, b and c to d. See "Arrow's Consideration of Ties and Indifference"

It is nowhere implied that voters don’t know whether xPy or xIy. The R notation is simply a representational device, equivalent to ‘equal to or great than’. It’s true all Rs can be cached out in terms of P and/or I, but perhaps sometimes we don’t know which so to write ‘xPy or xIy’ is more troublesome than simply ‘xRy’. Conversely, nothing is lost by using R notation. We can express xPy by ‘xRy and not yRx’ and xIy by ‘xRy and yRx’ (the ties/indifference problem not withstanding). The R relation allows us to represent everything we want – great than, equal to, and equal-or-great – using a single formula. In that respect, it’s more flexible and economical. Granted it doesn’t seem particularly necessary, but it’s far from absurd, and doesn’t have the implications John tries to draw.

16) Definition 9 (p. 46) The case of two alternatives. “By the method of majority decision is meant the social welfare function in which xRy holds if and only if the number of individuals such that xRiy is at least as great as the number of individuals such that yRix.” This is totally ridiculous. First of all it violates one of Arrow’s five “rational and ethical principles” which all social welfare functions must comply with: the principle of neutrality. When the number of individuals such that xRiy equals the number of individuals such that yRix, why is the solution xRy? Why not yRx which is equally as valid? In fact it is a tie between xRy and yRx, or according to Arrow’s own terminology, when xRy and yRx, then xIy not xRy! But wait, there is more. If half the individuals prefer x to y and half prefer y to x, we have a tie between x and y: {xPy, yPx}. If all the individuals are indifferent between x and y, we have a societal indifference: xIy. These are not the same thing! If more prefer x to y than prefer y to x, we have xPy and vice versa. If some individuals are indifferent between x and y, but more prefer x to y than prefer y to x, we have xPy and vice versa. This pretty well covers all the cases. Arrow is determined to ignore the significance of a tie and to turn a societal indifference into a tie. See "Arrow's Consideration of Ties and Indifference"

This seems to explain what was meant by 12), by repeating and elaborating the claim majority rule is non-neutral. It is, however, mistaken. Jonathan asks why xRy and not yRx. It is also yRx. As states in the previous section, xRy and yRx can hold together and imply xIy. Thus Jonathan either has no point, or it’s merely the ties/indifference thing again (see 2), 11), etc).

17) Arrow defines an indifference as a tie.

Yup, see comments on 2), 11) and 16).

It seems John’s only serious criticism is to treating ties and indifference as equivalent. It’s true, this does worry me as well. I wondered why, for May too, a split 4/993/3 between xPy, xIy and yPx results in a social choice xPy. It hardly seems intuitive but, as I said above, to introduce indifference as if it was a third option takes us away from the binary choices being dealt with.

There does seem to be a difference between everyone’s indifference and an even split between x and y – though in both cases I might suggest tossing a coin for decision-making, albeit for slightly different reasons (in the first simply to make a decision, as an individual might do in ‘Buridan’s ass’ like cases; in the second to be fair to everyone, i.e. give them an equal chance of getting what they want).

I don’t know what else to say about this issue. There’s certainly room for many whole papers on the topic. I think, however, it’s all these criticisms boil down to. (Plus some unsubstantiated assertions of rights to ‘free expression’ and ‘strategic voting’) – So it’s far from clear to me that Arrow is refuted, and the presentation of ‘17 criticisms’ significantly overstates the objection to make the case look stronger than it is.