Arrow (1951, 1963) introduced an axiom “Independence of Irrelevant Alternatives”  (AIIA) that has caused much misunderstanding. That axiom here has been baptised the “Axiom of Pairwise Decision Making” (APDM). Thus the axiom remains the same, only the name is different. The new name is much clearer about what the axiom really means in normal English.

Since the name “IIA” is so entrenched in the literature, this change of name requires some explanation. The explanation is along the lines:

·         There is the distinction between voting and deciding.

·         Items that cause cycles cannot be called ‘irrelevant’ for decision making.

·         The criterion to separate the relevant items from the irrelevant ones is rather the budget and is not necessarily found in pairwise voting for all items.

Arrow's axioms on using the whole commodity domain and universal preferences introduce the possibility that we might also be obligated to consider farfetched items. Arrow introduced the APDM to limit this effect again, since it allows that a decision on our current issues can be taken independently from other farfetched possibilities. It is reasonable that people neglect farfetched possibilities. Thus Arrow on one hand opens the door wide for such farfetched possibilities, and on the other hand introduces a strict condition that kills the relevance of this. The whole looks reasonable, since people in fact neglect farfetched possibilities.

Yet, the whole does not conform with the practical situations in Parliaments, where the problem is defined for existing voters and where the issues on table are given by the budget set.

Thus, (a) the notion of ‘irrelevance’ is dealt with by considering the budget set, (b) the axiom can be named after what it properly does: pairwise decision making.

If we want to deal with possibly farfetched preferences of some citizens, which is the moral meaning of the axiom of universal preferences, then we should work towards practical procedures that work. Assuming inconsistent axioms is not a good way to deal with that moral question.

The following sections use formal logic.

Lemma A.I:  AF implies that a constitution p satisfies the property Op   p.

First proof: AF means that desires (Op) in conflict with reality (~p) are not entitled to consideration. But p ~(Op & (~p)) is equivalent to p Op   p.  Q.E.D.

Second proof: We already concluded that the most preferred point (Op) would also be the chosen point (p). Thus p Op   p. (If the point is not preferred, then the implication is true ex vacuoso.) Q.E.D.

Discussion: We have enlarged the commodity domain with constitutions, and hence the axiom of feasibility becomes a bit stronger. The extension itself is rather weak, since we only extend on consistency (and not empirical validity). Our criterion is as that a reasonable society would stick to its rules. The gap between Is and Ought still exists in principle, but can in practice be bridged by the human effort to attain one’s ends.

Theorem A.1: For a reasonable society, the AMC is invalid.

First proof by rationality & moral consistency (DA): Assume Oa. But a   ~a, and with DA we get O~a. But this gives a preference inconsistency Oa & O~a. Hence ~Oa. Q.E.D.

Second proof by rationality & moral consistency (DA): Assume Oa. Since a   falsum  we find Ofalsum. Thus for some p₀ we have O(p₀ & ~p₀). But this means Op₀ & O~p₀, and that is a preference inconsistency. Hence ~Oa. Q.E.D.

First proof by realism (AF): Assume Oa. By the lemma p Op   p we find a. But then we have ~a & a, which is an inconsistency. Hence ~Oa. Q.E.D.

Second proof by realism (AF): Since ~a  and above lemma ~a  ~Oa, hence ~Oa. Thus the axioms are not morally desirable either. Q.E.D. Note: q   p is equivalent to ~p   ~q, and we may take q = Op.

When the axioms would be morally desirable, then the derived contradiction would be morally desirable - but nobody can be asked to do the impossible. Hence the axioms are not morally desirable. This is a seemingly simple reasoning scheme, but destructive to the accepted view.

Theorem A.2: For a reasonable society, the ARC is invalid.

Proof: Given AF, infeasible choices are not considered. Since ~a, apparently a is not feasible, and the Arrow constitution is not reasonable. So it is invalid that the axioms would be reasonable. Q.E.D.

Discussion: As we stated above, we have enlarged the commodity domain with constitutions, and hence the axiom of feasibility becomes a bit stronger. The extension itself is rather weak, since we only extend on consistency (and not empirical validity). But the conclusion is strong. No reasonable society in its right mind would want to accept Arrow’s axioms as its constitution. Supposedly at a chaotic Boston Tea Party a constitution c = a might be tried, but pretty soon rational people would see that they should make another constitution, for otherwise the situation will remain chaotic, and the Tea Party will not go down into history as a notable event.

Note that Arrow adopts feasibility, but also wants to impose infeasible conditions.

When Arrow’s axioms would be reasonable, then they would have to be consistent as well. However, they are inconsistent. Thus they are not reasonable. This seems a rather simple scheme of reasoning, but it destroys the impact of the Theorem.

For the axioms, there is the subtle difference between ‘reasonable’ and ‘seemingly reasonable when considered by itself’. The following is a good analogy. For a bicycle we want round wheels for when it rides. For a bicycle we also want square wheels, so that it does not fall when it stands still. But there are no round squares ! Ergo, conditions that seem reasonable by themselves, create something impossible and decidedly unreasonable when combined. To conclude ‘there is no good bike’ would however be absurd. Admittedly, it is a good teaching method to first convince students that something would be reasonable, and then have them derive a contradiction. As with the buying of a bad second-hand car, the students learn to be careful, and they learn a respect for science and the value of modesty. This teaching method however overshoots when people remain believers of the reasonableness of the assumptions - as apparently happened with the assumptions of Arrow’s Theorem. A paradox is only a seeming contradiction. Thus there must exist a system that we are willing to accept as the optimal one.

Many mathematicians have been sensitive to the distinction between ‘reasonable’ and ‘seemingly reasonable when considered by itself’, but the literature also abounds with instances where this distinction is not applied with sufficient care. Part of the accepted view thus is a case of bad communication of the incrowd with the larger public. (Given above quotes, the incrowd however might be small. Quis custodet custodes ?)

The selection of the culprit axiom is straightforward. We order the axioms by preference, for example AD > AWP > AU > APDM. From ~a, we conclude that we have to drop one of the axioms. We drop the least preferred one. My discussion on Condorcet’s example should generate support for the rejection of APDM. Basically though, scientists can only advise on preferences, and the proper decision is up to the body politic.

Lemma A.II:  If all agents have a > APDM then, with AWP, society has [AU, AWP, AD] > APDM.  Note: here [x, y, z] means the unordered set.

Proof: obvious.

Discussion: When all people put AU, AWP and AD in any individual order, but all would have APDM below these, then society can reject APDM unanimously. In fact, the condition AU might as well be regarded as part of the definition of a SWF-GM, and similarly, AWP could as well be regarded as part of the definition of the notion of collective preference. So the real choice concerns AD and APDM, or between dictatorship or not.Here a selfish dictator and his associates would have ¬AD > APDM > AD. The Jorgenson quote suggests his preference for a benevolent and non-selfish dictatorship, but, also since such dictatorships tend to turn sour, my impression is that he would eventually be an associate of a real dictator. Most likely, he did not understand the situation when the quote was printed.

Note that ordering the axioms means that the deontic predicate O is not homogeneous. This means that deontic logic may be more related to preference theory than deontic theorists think.

Consistent constitutions violate one of the axioms of Arrow’s Theorem. Violating one of these axioms is to be considered useful for reasonableness and morality, rather than the reverse. (That is what we proved above.)

One general feature is a Status Quo that persists when there are ties.

One example already has been mentioned in the discussion of the Condorcet problem. With majority voting, a cycle means indifference, and there are various ways to solve ties. One possible solution is the persistence of the Status Quo.

Another example constitution is the “Pareto-Majority” rule. One first selects all Paretian improvements from the Status Quo. That is, those points where some advance while nobody loses. There may be more Paretian points, such as B > A and C > A, with the Status Quo as A. When there is no Paretian order between B and C, then it suffices to decide on these points by simple majority. Of course, with more than two points, majority voting can result into cycling, but that again means indifference, which could be settled by dice, by the chairperson, or by other creative ways.

See my home page and The Economics Pack for implementation of these rules in the program Mathematica. Little helps so much as a trying it out for yourself.

Our discussion arrives at a conclusion that differs from the literature, and thus warrants a reappraisal of that literature. This reappraisal is not the topic of this paper, but some examples are useful.

(1) Note that the Tobin quote above was misleading. The problem with ‘unborn generations’ should not be mixed up with the Arrow difficulty. The Tobin problem actually can have a rather simple solution. It are the preferences of the currently living that matter, and what they prefer for the future unborn (which can also be based on a forecast of such preferences). These future preferences cannot logically be included, since they don’t exist yet.

(2) Arrow 1951 also stated:

“If consumers’ values can be represented by a wide range of individual orderings, the doctrine of voters’ sovereignty is incompatible with that of collective rationality.”

This is clearly inaccurate. The statement suggests that we have to adopt Arrow’s axioms, while the sensible thing is to reject these axioms and to adopt both voters’ sovereignty and collective rationality.

(3) One of the more interesting points made here is the distinction between the learning process and the end result. How should Arrow’s result be presented in the future ? Is it possible to maintain the teaching strategy to call the axioms ‘reasonable’, then have the students get into a fixture, and them let them find a way out ? It is good teaching practice ! However, in a Palgrave meant for a wider audience (or a general encyclopedia that even might be read by dictators), it might be improper to call Arrow’s axioms ‘reasonable’. It should be ‘seemingly reasonable’ at the least.

Note that the phrase then becomes less enchanting:

‘there is no social choice mechanism which satisfies a number of seemingly reasonable conditions’.

(4) I am a bit shocked by Mueller’s (1989, p406-407) discussion of Arrow’s general view. One would expect a more critical attitude, but finds instead:

“The Arrow and Sen theorems (...) raise fundamental questions about the possibility of establishing collective choice procedures satisfying minimally appealing normative properties (...) But the negative side should not be overemphasized. We have suggested that both sorts of paradoxes might be avoided with the use of cardinal, interpersonally comparable utility information. Arrow explicitly eschewed the use of such information, and the independence of irrelevant alternatives [thus Pairwise Decision Making / TC] axiom was imposed to rule out voting procedures that might make use of such information (... But it) is possible that the citizens may be trusted to make these comparisons in an ethically acceptable way.”

Well, interpersonal comparison of course occurs, minimally, when we assign votes to people, assign rights to put topics on ballot, and the like. So interpersonal comparison is not as bad as many economists seem to think. But my solution to Arrow’s difficulty does not rely on  cardinality and cardinal comparison. So, disappointingly, Mueller both accepts the idea that Arrow would cause ‘questions’ about the possibility of social choice, and he comes with a wildly wrong conclusion. This is supposed to be a modern textbook !

(5) What is important, is that the development of economic theory and the development of real economies have been hindered by the confusion generated by the standard explanation. Where decision makers were divided, some interested in social welfare and others not, the latter group was provided with decisive gunpowder - and beware of people who have an ideology and even wield a mathematical theorem to prove their lunacy. Generations of students have been taught by Nobel Prize laureats that research into social welfare would be subject to impossibilities. Creative energy has been directed to enlarging the impossibilities rather than to devising structures that might improve practical situations. Practical research into social choice functions and parameters has been aborted, all with reference to a misunderstood theorem !

Economic research also leads to a suggestion of a constitutional amendment, see Colignatus (1996b) and the appendix. I hope that this present chapter helps to clarify that this kind of research is a useful type of economics.

(6) This analysis also clarifies a confusion about the relation of constitutions to the SWF. While many economists argued that constitutions could not be reasonable or morally acceptable, they did accept the Bergson-Samuelson SWF, even though the latter was derived from the former - and nobody seems to care about this inconsistency. Which is now removed, since the properties of the constitution are projected into the SWF.

Arrow’s Theorem has given some problems in the literature, see the quotes above. We have achieved the following solution:

·         There is more clarity now, by the distinction between the theorem proper (a   falsum), the moral claim (Oa) and the claim on reasonableness (AF and I(~a)).

·         The arguments above on rationality and morality have a destructive character since they reject the accepted view. In another perspective they are constructive, since they allow the formalisation of (meta) notions, and bring these back into mathematics again (notably the voting on constitutions).

·         From a mathematical point of view, the Arrow axioms are incomplete for decision making in a reasonable society.

·         It has been shown that the APDM is undesirable. Dropping APDM is not a sad state of affairs, as is sometimes suggested in the literature, but a sign of understanding group decision making.

·         The Arrow axiomatisation does not capture the truly desirable properties required for a constitution, both by incompleteness and APDM.

·         There are detail results, such as the distinction between voting and deciding, the integration of preference theory and deontic logic, and a proof of Arrow’s Theorem that shows clearly the abuse by APDM.

·         We have given examples of consistent constitutions that many might regard as optimal.

“The Arrow Theorem does not in fact show what the popular interpretation frequently takes it to show. It establishes, in effect, not the impossibility of rational choice, but the impossibility that arises when we try to base social choice on a limited class of information.”

This is not correct. Using the information provided by pairwise voting results, we can decide to a tie (deadlock, indifference) when such might arise. It is the adoption of the APDM axiom that, wickedly, turns this indifference into an inconsistency. The APDM does not mean lack of information, it only corrupts the information that exists.

“At the risk of oversimplification, let me briefly consider one way of seeing the Arrow theorem. Take the old example of the “voting paradox,” with which eighteenth-century French mathematicians such as Condorcet and Jean-Charles de Borda were much concerned. If person 1 prefers option x to option y and y to z, while person 2 prefers y to z and z to x, and person 3 prefers z to x and x to y, then we do know that the majority rule would lead to inconsistencies. In particular, x has a majority over y, which has a majority over z, which in turn enjoys a majority over x. Arrow’s theorem shows, among other insights it offers, that not just the majority rule, but all mechanisms of decision making that rely on the same informational base (to wit, only indi­vidual orderings of the relevant alternatives) would lead to some inconsistency or infelicity, unless we simply go for the dictatorial solution of making one person’s preference ranking rule the roost.”

Locating the problem in the informational base is erroneous. Clearly, majority decision does not lead to inconsistencies, for it is the use of the APDM axiom that does so - and we don’t need it for majority decisions. The Arrow Theorem does not show that there are inconsistencies for all mechanisms - we namely can use mechanisms without APDM.

“This is an extraordinarily impressive and elegant theorem — one of the most beautiful analytical results in the field of social science. But it does not at all rule out decision mechanisms that use more — or different — informational bases than voting rules do. In taking a social decision on economic matters, it would be natural for us to consider other types of information.”

I don’t know about “extraordinarily impressive and elegant”. Condorcet came up with his paradox, as earlier people came up with paradoxes when dividing by zero, as Bertrand Russell had his set-paradox, and as the Cretian Epimenides said “All Cretians are liars.” Arrow’s Theorem solves the Condorcet paradox by showing that we must not use APDM - though Arrow apparently did not realise that. The theorem is basic, and we must be glad that we have it, as APDM apparently can cause a lot of confusion, as the last 50 years have shown.

“Indeed, a majority rule — whether or not consistent — would be a nonstarter as a mechanism for resolving economic disputes. Consider the case of dividing a cake among three persons, called (not very imaginatively) 1, 2, and 3, with the assumption that each person votes to maximize only her own share of the cake. (This assumption simplifies the example, but nothing fundamental depends on it, and it can be replaced by other types of preferences.) Take any division of the cake among the three. We can always bring about a “majority improvement” by taking a part of any one person’s share (let us say, person 1’s share), and then dividing it between the other two (viz., 2 and 3). This way of “improving” the social outcome would work — given that the social judgment is by majority rule — even if the person thus victimized (viz., 1) happens to be the poorest of the three. Indeed, we can continue taking away more and more of the share of the poorest person and dividing the loot between the richer two—all the time making a majority improvement. This process of “improve­ment” can go on until the poorest has no cake left to be taken away. What a wonderful chain, in the majoritarian perspective, of social betterment!”

Remember that Sen writes this book for a general audience of economists who will not have gone deeper in social choice theory. Though Sen now relates basic truisms, his reasoning nevertheless is a bit off. Indeed, Western democracies tend to have property rights and a “status quo” rule, and a Madisonian philosophy that democracy actually exists to protect the minorities. We use all kinds of additional information, in order to settle problems of fairness and equity. Thus the majority rule is not suggested for the raw form that Sen uses as an example. Then, crucially, when Sen suggests that this example clarifies that we must use more information to solve the Arrow paradox, then this is a non-sequitur. His argument becomes seductive, since the reader is seduced into thinking that, indeed, we use more information. But the truth is that we use this additional information to solve equity matters, and not to solve the Arrow inconsistency.

“Rules of this kind build on an informational base consisting only of the preference rankings of the persons, without any notice being taken of who is poorer than whom, or who gains (and who loses) how much from shifts in income, or any other information (such as how the respective persons happened to earn the particular shares they have). The informational base for this class of rules, of which the majority decision procedure is a prominent example, is thus extremely limited, and it is clearly quite inadequate for making informed judgments about welfare economic problems. This is not primarily because it leads to inconsistency (as generalized in the Arrow theorem), but because we cannot really make social judg­ments with so little information.

“Acceptable social rules would tend to take notice of a variety of other relevant facts in judging the division of the cake: who is poorer than whom, who gains how much in terms of welfare or of the basic ingre­dients of living, how is the cake being “earned” or “looted” and so on. The insistence that no other information is needed (and that other information, if available, could not influence the decisions to be taken) makes these rules not very interesting for economic decision making. Given this recognition, the fact that there is also a problem of inconsistency—in dividing a cake through votes — may well be seen not so much as a problem, but as a welcome relief from the unswerving consistency of brutal and informationally obtuse procedures.”

Sen is aware that his reasoning is not strict (vide his use of “primarily” and “also”) but, still, he makes the suggestion, which is erroneous.

Indeed, the spirit of “impossibility” is not, I believe, the right way of seeing Arrow’s “impossibility theorem.” [footnote] Arrow provides a gen­eral approach to thinking about social decisions based on individual conditions, and his theorem—and a class of other results established after his pioneering work — show that what is possible and what is not may turn crucially on what information is taken into effective account in making social decisions. Indeed, through informational broadening, it is possible to have coherent and consistent criteria for social and economic assessment. The “social choice” literature (as this field of analytical exploration is called), which has resulted from Arrow’s pioneering move, is as much a world of possibility as of con­ditional impossibilities. [footnote]”

This quote just repeats the error - and adds a string of perceptions to sweeten the cake. The  footnotes are references to his “Collective choice and social welfare”, his Handbook contribution and the Nobel lecture, Sen (1999b), and add no news, for us, to the essence discussed here. Indeed, the obviously relevant Nobel lecture just repeats the error.

Hence, Sen basically does not understand the problem. I do value his work on social choice since it was a useful guide to me in making Arrow’s result accessible, and in seeing the various perspectives of it. As Newton is reported to have said: “Standing on the shoulders of giants, we can look further.” I cannot wait till Sen writes me that he enjoys my solution !

Andreu Mas-colell, Michael Whinston and Jerry Green ’s 1995  “Microeconomic Theory” is just wonderful. A great book. Generally speaking, though, since they erroneously write: “Either we must give up the hope that social preferences could be rational in the sense introduced in Chapter 1 (i.e. that society behaves as an individual would) or we must accept dictatorship.” (p780). And the subsequent discussion indeed leads the student in the bogs and misdirections so typical of 20^th century ‘social choice theory’. The math is OK, but concerns something like the question of how many angels can dance on a pin’s head - and the whole induces the student to become wary of social decision making. (To be sure: I appreciate the other qualities, and have used the book for sections of my Economics Pack.)

Summary

Theory shows that voting is subject to paradoxes, while it also appears that a voting result is caused as much by the procedure as by the voters’ preferences. From a moral point of view, the choice of the procedure then is the major issue. A key insight is that morality presumes time. In a static world everything is given and there is no place for individuals who have to ponder their moral choices. The real world is dynamic however and the most challenging voting paradoxes concern budget changes. The paper develops a new “Borda Fixed Point” mechanism that provides a better protection to surprises by such budget changes. Under dynamics, Donald Saari’s argument on symmetry is less convincing.

The currently accepted view is sometimes expressed as that ‘there is no ideal voting scheme’. The former chapter destroyed that view. There is no mathematical reason to think that such an ideal cannot exist. Since Arrow’s axioms must be rejected, they do not form an ideal. An ideal still can exist, but apparently it is different than originally thought. Perhaps people have different ideals, but then the non-existence of a common ideal derives from empirically different opinions and not from mathematical reasons. Since people can benefit from co-operation, they can still aspire at a scheme that all can agree upon.

Above analysis does not answer the positive question yet what would be a generally good system. The main point here is that everyone should determine this for oneself. Theory can only help to remain consistent. The following is a suggestion for a scheme that is consistent and that could appeal to many.

One important idea is that time plays a role. The basis for this idea is that, abstractly, morality presupposes time. Without time there would be no morality. In a static world everything is given, and there is no place for an individual who has to ponder his or her moral choices. As economists, we can draw static utility functions and isoquants, but those are abstractions, and they might distract from the real moral problem. The moral problem is that now a decision has to be made while the consequences appear later. Afterwards, everything can be explained deterministically (which is the meaning of ‘explanation’), and by hypothesis, determinism will also hold for the future. Yet, in the mean time forecasts are imperfect, there is fundamental uncertainty, and that creates the possibility of morality (or the illusion of morality).

Economic science is intended to help explain reality. In this reality, we see an evolution of human beings in a social process of natural forces. The basic concept is power, in a continuous process, so that the basic approach uses ratio scales and cardinal utility and not ordinal scales. Other assumptions than cardinality enter the discussion only when the group wants to control power, and for example introduce democracy. A common notion is that economists reject cardinality and interpersonal comparison of utility. However, the concept of ‘one person, one vote’ actually imposes some interpersonal comparison of utilities. Also comparing orderings of preferences implies some comparison of utilities. The proper perspective is rather that cardinality is deficient since people can cheat about their preferences (at least in the current state of technology). The major argument for ordinality is that it limits the room for cheating. If people could not cheat, interpersonal comparison likely would be much more popular amongst economists. The point that ordinality reduces interpersonal comparison thus seems less relevant than the point that cardinal comparisons are unreliable since people can cheat.

For example, when a family goes on holiday and has the choice between Spain or Greece, then little Robby might exaggerate his preference for Greece and say that he might as well die when Spain is selected. When the aggregation of preferences would be cardinal, such a huge negative weight for one option would certainly block it. Imposing ordinality limits the impact of cheating however. In common textbooks on voting theory, cheating comes in relatively late, but it is more adequate to start right away with that notion. The crucial insight is: Arrow’s Theorem and the voting paradoxes are the price that we have to pay in order to limit that impact of ‘stategic’ voting behaviour.

Arrow’s orginal question whether there could not exist a generally good voting mechanism remains a valid question, though. As history has shown, mathematicians are proficient in identifying paradoxes and in deriving new impossibilities, and one will not quickly find a suggestion for a generally good system. But it appears that when we consider the issue of time, then a solution tends to suggest itself. To understand this solution, it is useful to first consider three main contenders, i.e. the ‘traditional’ solutions provided by Plurality, Borda and Condorcet. There are other methods, but their properties are such that they need no consideration here.

In Plurality, all voters have one vote, and the candidate with the highest number is selected. Note the problems with this method. The criterion of ‘highest number’ does not imply that the winner must also have more than 50% of the vote. If this is additionally imposed, then this may require more rounds of voting, and then there is the difficult issue whether candidates have to drop out, and if so, how.

Borda’s method is to let each voter rank the candidates by importance, then assign weights given by the rank position, to add the weights per candidate for all voters, and then select the candidate with the highest value. Note that the method appears sensitive to preference reversal, see below.

Condorcet’s method is to vote on all pairs of candidates, and to select the one who wins from all alternatives. Note that such a “Condorcet winner” does not need to exist. In that case the margins of winning can be used to solve the deadlock - but this increases the sensitivity to who participates.

The different voting schemes result into different decisions:

1)      Plurality: Voters give one single vote to the candidate of their highest preference. For candidate A we consider its column, select the rows with the score 3, and add the associated numbers of voters 33 + 0 = 33. And so on. Candidate C gets most votes, namely 42.

2)      Borda: The votes are weighted with the rank order weight. De column for A is multiplied row by row with the number of voters 3 * 33 + 3 * 0 + 2 * 25 + … = 230. Candidate B gets most votes, namely 242. (Scores -1, 0, 1 might calculate easier.)

3)      Condorcet: Voting pairwise over A versus B, there are 33 + 0 + 25 = 58 voters who give A a higher rankorder than B. Etcetera. Candidate A appears to win from both B and C, and then is the “Condorcet winner”.

This example shows that A, B and C can all be winners, depending upon the method selected. The properties of the methods then are the true issue.

Above still neglects strategic voting. This could be represented by a change in apparent position. How do we evaluate this ? It appears that the Condorcet approach is least sensitive to cheating since in a pairwise vote there is an incentive to express one’s true preferences. Pairwise voting however can be unattractive since there need not be a Condorcet winner, or, when one exists, it may conflict with the preference rankings. One way to solve the complexity of choosing between these methods is to compromise by having a run-off election. The two top outcomes of Plurality or Borda are taken and then subjected to a pairwise vote as in Condorcet. There is one final consideration. Simply taking the two ‘top outcomes’ seems unduly simple, we should consider what these actually are. In France, the election between Chirac, Jospin, Le Pen and others caused Jospin’s votes to scatter over all kinds of smaller parties so that he dropped from the race while he was the Condorcet winner of both Chirac and Le Pen. When we are compromising, we should focus on determining the two main contenders.

Let us reconsider the dynamic process that occurs within an economy. We see that under the influence of time, the budget changes continuously. A voting scheme naturally requires that there is a list of candidates, but one cause for paradoxes is that that list is not fixed. For example, in the Borda vote above, B is selected, but if C decides to withdraw (or gets a heart attack), then we would expect B to remain the winner, but suddenly it is A (see the Condorcet vote A versus B). Remember also the Bush, Gore and Nader case. We could consider a procedure to be better when the choice is less dependent upon changes in the budget.

A way to achieve this is to use the notion of a ‘fixed point’. For a function f: D R, for some domain D and range R, the point p is a fixed point iff f(p) = p. Let us consider this concept for voting.

Let P be the voting procedure, and let X = {x₁, …, x_n} be the budget with all the candidates. Let the unrefined winner be w = P(X). Let Y be the budget when w does not participate, Y = X \ {w}. Let the ‘alternative winner’ be v = P(Y) = v(w), i.e. the candidate who wins when the first winner w does not participate. This is not simply the run-off between the winner and the common runner-up, since the selection of the alternative winner requires the recalculation of the preference weights. This alternative winner can be seen as a ‘summary’ of the opposition to w. The scheme is a compromise since the Condorcet pairwise condition holds for the winner and the alternative winner. While these notions are defined with respect to the unrefined winner, we can generalise this to any winner, and in particular to our optimal winner.

An alternative condition for winning in general is the ability to win from one’s strongest opponent. This gives the fixed point condition. Define f(x) = P(x, P(X \ {x})), which is the general function ‘the vote result of x and its alternative winner’. Then w* is the solution to the fixed point condition x = f(x):

w* = P(w*, v(w*)) = P(w*, P(X \ {w*})) = f(w*)

When the unrefined winner w is not a fixed point, i.e. when the unrefined winner w = P(X) appears to lose from v, so that w P(w, v), then the search process can start again from v.

It appears that this fixed point voting procedure reduces the dependence upon budget changes. There can still be a dependence, but it is not as large as without the condition.

In Table 13, the Borda Fixed Point winner is A. With B the Borda winner, A is the alternative winner when B does not participate, and B loses from A in a pairwise match; starting the search from A, its alternative winner is B, and A wins from B.

More on this can be found in Colignatus (2001). That book has also been intended as a textbook and it developed Mathematica programs for the various voting schemes and data manipulations. Given the complexity of the matter, this working environment has appeared a great advantage.

Donald Saari (2001ab) showed that Borda’s method is the only method that satisfies certain symmetries. His suggestion is that the Borda rule ‘therefor is best’. This argument does not convince by itself since ‘symmetry’ is not by itself a moral category. Dynamics is linked to morality, by the notion that morality presumes time, and thus seems a better angle.

Consider direct symmetry first. Suppose that your preference is A > B > C and that my preference is C > B > A. The direct symmetry consideration is that we might both abstain from a vote and stay home, since our preferences strictly oppose each other. Saari noted too that voting cycles can be catalogued under the mathematical concept of rotational symmetry. His subsequent suggestion is that cancellation should hold for all symmetries for all subsets of voters.

What happens when cancellation of ‘rotational symmetry’ is applied to subsets ? The following is an example by Saari that cancellation isn’t trivial then. In Table 14 there are 48 voters, and B is selected by both Borda and Condorcet. In Table 15, 27 voters have been added who have the mentioned rotational symmetry, with 9 for each subgroup. Now Borda still selects B, but Condorcet, and the Borda Fixed Point, select A. In Saari’s view, Borda satisfies symmetry, and ‘hence’ is the better method.

My reasoning is a bit different. First of all, note that I myself have used an argument similar to that of Saari. In my view, the typical Condorcet situation of three preferences A > B > C, B > C > A and C > A > B results into indifference rather than an inconsistency, and I use this against Arrow’s analysis. So I agree with Saari’s view that such votes cancel. I applaud Saari’s insight that if you apply cancellation for all cycles in all subsets, then the logic is to get rid of Condorcet’s method and to use Borda’s method.