Praesidium

Tuesday, January 31, 2006

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.

8 Comments:

At 4:35 am, Anonymous John Lawrence said...

Thanks, Ben, to giving serious considerations to my comments. It's true there is a fair degree of repetition in a number of the points I made.

I'll just consider one of your responses tonight. Hopefully, I'll get to some of the others later.

You say:
"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?"

I say you don't need to compare interpersonal utility. I don't think there's a way to do it or a need to do it. Using the modified Borda count (with apology I call the Lawrence count: http://www.socialchoiceandbeyond.com - click on Lawrence count), for instance, you would just be giving 49 more points to b than I would thereby increasing b's chances of election somewhat. By my ranking b 99 and c 100, I am using my vote strategically to do everything I can to get "a" elected or these may represent my true feelings among the candidates. It doesn't make any difference as long as neither one of us has any relative advantage in the voting process.

You have a more favorable view of b than I do and would much prefer b to c while I am almost indifferent between b and c. Whether or not you like "a" more than I do or dislike c more than I do is irrelevant since neither you nor I can increase the point spread between any 2 candidates by more than 99 points. Therefore, we have an equal vote irregardless of whether or not you feel more strongly than I do.

 
At 6:50 pm, Anonymous John Lawrence said...

Well, today, Ben, I'll respond to your response to my point #1. You say "I don’t think Arrow disallows ties." I beg to differ with you and offer the following argument.

Let's say there are 6 candidates standing for election: X1, X2,...,X6. The voters cast one vote for their favorite candidate and the candidate with the most votes wins. Of course, there are other methods one could use depending on whether a candidate gets a majority etc., but let's say in this simple method the one with the most votes wins. A legitimate voting method, no?

Now let's consider the case in which 2 or more of the candidates get the same number of votes. Then we have a tie among 2 or more candidates. In the extreme case, if all candidates receive the same number of votes, we could have a six way tie. What to do about such a situation is not our concern for the present. We merely note that this simple election method could indicate clearly whether (a) a candidate is a winner or (b) there is a tie among 2 or more candidates.

Now consider that each candidate is actually a preference list. We identify X1 with aRbRc; X2 with aRcRb; and so on: X6 with cRbRa.

 
At 7:16 pm, Anonymous John Lawrence said...

I guess I'm constrained by the number of words per comment. Oh well, I'll continue on. Hope you don't mind. So each candidate is a preference ordering among 3 alternatives and each voter votes for one of those preference orderings. By Arrow's Definition 4 (p. 23, "Social Choice and Individual Values") the social welfare function (SWF) must provide ONE social ordering of the form xRyRz for each possible set of individual orderings.

Therefore, in Arrow's formulation of the problem, there is no way to account for a tie among 2 or more of the candidates where the candidates are indeed preference orderings. What Arrow does offer is indifferences among alternatives (by virtue of the "R" operator) not ties among orderings.

#2) (I guess it is going to allow me a comment of any length after all.)

Consider the case of 3 alternatives: a, b, c. Let us say there are 3 voters and they vote, respectively: abc, bca, cab. Now if we identify these orderings with candidates X1, X2 and X3, respectively, we have a 3 way tie among X1, X2 and X3. Well, no problem for our simple voting system. We could notate a 3 way tie as: {abc, bca, cab}. Big problem for Arrow. He cannot accomodate a simple tie situation due to his Definition 4!!

Isn't it rational that the above situation should lead to a 3 way tie? Let's call this "Rational Fact #1. now let's say c drops out of the race, and we're left with the votes ab, ba and ab. The winning social ordering would be ab, right? Let's call this "Rational Fact #2."

These two facts taken together invalidate Arrow's Condition #3: Independence of Irrelevant Alternatives. In fact, if you accept the rationality of Facts #1 and #2, you must consider Arrow's Condition #3 to be irrational. Remember his is a general theory which must apply to every case including this one.

Some of my papers like a A General Theory of Social Choice attempt to prove that a solution involving tie orderings is always possible, and Arrow's Impossibility Theorem is only true if you accept his rather limited formulation of the general problem.

More later...

 
At 7:36 pm, Anonymous John Lawrence said...

This is so much fun, I think I'll continue a bit longer. In response to my point #2, you say: "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."

Yes, Arrow attempts to prove that, for 2 individuals, if one prefers y to x and one prefers x to y then society is indifferent between y and x. I would say that in the second line of his proof his assumptions are wrong. He says "we would have xP1y and yP2x, but not xIy." I would say we have a tie between x and y. There's no necessity for society to be indifferent just because the electorate is evenly divided!

 
At 7:56 pm, Anonymous John Lawrence said...

In the second part of you response to #2, you say: "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 no longer dealing with a binary x or y choice, because tying is (unlike indifference) a third option."

A SWF in general doesn't have to deal with the various cases you mention in a totally rational manner. One SWF might map the 99% indifference case into a social indifference. Another SWF might map it into xPy if 99% were indifferent and 1% prefered x to y.

For a SWF to exist it only has to provide a consistent mapping from domain to range. Many such SWFs could exists, some better than others according to different criteria.

I grant you that we're no longer dealing with a binary choice. But the individuals have to decide among 3 possibilities: xPy, yPx and xIy. Society actually has to decide among 7: xPy, yPx, xIy, xPy ties yPx, xPy ties xIy, yPx ties xIy and all three (xPy, yPx and xIy) are tied.

Part of Arrow's problem, I think, is that he oversimplifies the analysis leading to consistent results only if you accept his formulation and conceptualization of the problem. Basically he just proves the voter's paradox, a fact that's been known for 200 years. He could just as well have shown that the voter's paradox can be interpreted as a tie which is broken if one of the candidates drops out. Then you could use the methods of your PhD thesis to break the tie!

 
At 2:09 am, Anonymous John Lawrence said...

Ben, I can't believe you watched the Superbowl. I'm one American that didn't watch it. Actually it's a good time to go someplace as the freeways are clear of traffic!

Today I'll deal with point #4. You say: "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)."

See my paper Social Choice, Information Theory and the Borda Count p.6 and especially Figure 4 for all kinds of examples of how a person's individual preference orderings can change as a function of the number of candidates or alternatives being considered.

Quoting: "Figure 4 shows, in general, how a rational individual will project his or her “true” preferences by expressing them in various situations in which a differing number of slots are available. It can be seen that, if one or more alternatives are removed from the original ordering (due to the death of one or more candidates, for instance), the rational individual will project his or her “true” preferences onto the number of slots appropriate to the number of remaining candidates in order to come up with his or her new preference ordering."

I reiterate that having more "slots" than candidates does not necessarily mean that you're dealing with "interpersonal comparisons of utilities." It just means that you're collecting more information regarding the individual's preference orderings. There would not be a need to ioncrease the number of slots beyond an individuals's "sensitivity level" that is the finest level in which an individual can discriminate between 2 alternatives. Also the same number of slots would have to be mandated for each individual in order for each vote to have the same power. Otherwise, one person's vote could count more than another's.

I'll try to explain it briefly by the following:

If there are a number of candidates and 1 dies, now you're comparing the candidates on a coarser grid, that is, there are fewer slots. Therefore, a very small preference of one candidate over another on the finer grid can turn into an indifference between them on the coarser grid.

By the way what is "contraction consistency?"

 
At 2:38 am, Anonymous John Lawrence said...

Point#5. I think I prove this sufficiently in Social Choice, Information Theory and the Borda Count

Point#6. You say: "This rather repeats 4). If there are true preferences between a, b and c, why shouldn’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”.

Because the "anyone but Hitler strategy" would want to add as many points to everyone else's point totals as it could. Ranking Hitler as 0 and a, b and c as 100, for example, would give a , b or c the greatest chance of beating Hitler. I maintain that a person should be able to use his vote strategically. The literature that is strategy-phobic is barking up the wrong tree to my way of thinking.

Point #8: You say: "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."

I'll take it you agree with me on this "so long as all have the same opportunities."

Point#8: Saari is saying that, when more information is provided, Arrow's Impossibility Theorem is moot. More information could be, for example, more slots than the number of candidates or more than just binary comparisons.

Point#9: You say: "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)."

What is expressed is more information down to the sensitivity level of the voter. More slots. More information. Less slots. Less Information. See Shannon, "The Mathematical Theory of Communication." Ice cream wouldn't be relevant unless it was directly comparable with the other alternatives all of which would be foods I presume.

Point#10. I respectfully disagree. Restricting information is restricting freedom. Requiring an individual to express a preference coarser than his sensitivity level is restricting information hence freeedom of expression in some sense.

Point #11. How you resolve ties is essentially another subject. The Borda Count does it by turning ties into indifferences, but, in general, a tie is not an indifference.

That's it for today. No sense in straining my brain!

 
At 10:05 pm, Anonymous John Lawrence said...

I'll try to finish this up today.

Point#12 and #16 - You say: "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)." and "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)."

By Definition 9 on p. 46 of "Social Choice and Individual Values," Arrow states: "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."

Condider the case when the number of individuals such that xRiy is as great as the number of individuals such that yRix. According to the above definition, xRy. That is x is prefered or indifferent to y. Def. 9 does not say x is indifferent to y in that case. It is prefered or indifferent. We don't know in the individual cases whether xPiy, yPix or xIiy. All we know is that, for this specific case, the number who vote xRiy equals the number who vote yRix. Yet according to the definition, xRy. Therefore, x is getting special treatment in this case.

Point#13. I'm not sure what point I'm making here either except that by Axiom 1 (p. 13) Arrow defines a tie as an indifference. Arrow says that you can have both xRy and yRx. That would be a tie. He implies that that is the same as an indifference. He proves on p. 50 that if 2 individuals have opposing interests, society will treat this as an indifference not a tie because his analysis won't allow ties.

Point#14 - You say: "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."

I thought a strong ordering was one in which no indifferences were possible. That is the individuals could express only xPiy or yPix but not xIiy or xRiy or yRix. It is clearly possible if the number of voters is even that the number who have xPiy could be equal to the number who have yPix. That would be a tie, no? Society would have to have xPy or yPx or a tie between the two since no indifferences are allowed at the societal level either. It seems clear that a tie is the only reasonable solution.

Point#15 - You say: "nothing is lost by using R notation." Let's say the individuals submit their votes in terms of P and I. Then the social amalgamating mechanism converts these to the R notation before amalgamating them to come up with the social choice in terms of the R notation. I think information has been lost since any particular individual could change Ps to Is and vice versa and the SWF would not know the difference. It would come up with the same social choice.

Point #17: In your example, 997 have xRiy and 996 have yRix so, according to Arrow, society has xRy. However, if you don't restrict the analysis as Arrow does and let society have xPy, yPx or xIy, and treat each possibility equally, then clearly xIy.

That concludes my response to your response. Clearly, my 17 points are somewhat redundant. However, I think my analysis is essentially correct. It's all in how you formulate and conceptualize the problem. Arrow's formulation is quite narrow, and a more general formulation can lead to a different result than the one he obtained. If you're interested, I'd be willing to pursue this. If not, nice chatting and good luck!

 

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