Book VII. Social Choice

Social Choice

33. Introduction

Kenneth Arrow (1950, 1951, 1963) presented an Impossibility Theorem in which he showed that decisions about ‘the general welfare’ are impossible in certain cases or have to be left to a dictator. Arrow presented some five axioms that each seemed reasonable when considered by itself, and he argued as well that these axioms are morally desirable and fitting to the concept of ‘general welfare’. He also formulated the problem in general terms so that it concerns choices on goods or people. Subsequently, he derived a contradiction. This result caused quite some consternation, but eventually the mathematical rigour caused acceptance, and since then the Theorem forms the core of many books, such as Sen (1970) and Mueller(1989). The Theorem was also one of the reasons to award Arrow the Nobel Prize in economics.

A voting example is given by the US Presidential election of 2000. Apart from the problems around the ballot process itself, there was a more basic problem: with main contenders Bush, Gore and Nader, Bush got elected, but in another system, such as a run-off between the two ‘major’ contenders, the Nader vote apparently would have switched largely to Gore, making him the US President. So the choice depends as much upon the system chosen as on the preferences. Can we find a generally good system ? Arrow’s Theorem suggests ‘No’.

Arrow’s Theorem has had a huge influence on scientific and political thought. Part of this influence is subtle, where skepsis arises about the concept of ‘democracy’. That shiny goal loses its appeal when we don’t know how representatives should be elected and when morally desirable rules would be impossible. Opting for the natural forces in the social process may be more pragmatic. The influence of the Theorem can sometimes be more explicit. Next to the model of the utility maximising individual, there is the model for society as a whole and then the maximisation of a Social Welfare Function (SWF). But when a morally acceptable SWF is impossible, what would be the use of research into such an inherently flawed concept ? Many nations co-ordinate their economic policy, and have created institutions for this, like the Council of Economic Advisors (US), the Commissariat du Plan (France), the Sachverständigenrat (Germany), and the Central Planning Bureau (Holland). Such an institution, given its role in the co-ordination of economic policy, could be expected to do reseach on the national SWF. However, those institutions tend to abstain from that kind of research, pointing to Arrow’s Theorem as one of the arguments, if not the major argument.

Over the years an ‘accepted view’ has grown in economics concerning the meaning of Arrow’s Theorem. This accepted view however has also implied a kind of moral stagnation.

There are two main reasons to reconsider the accepted wisdom on the meaning of the Theorem and to rekindle the debate on it. The first reason is destructive, since it rejects Arrow’s position; the second reason is constructive, since it provides an alternative.

These reasons are: (1) There is a distinction between the mathematical framework on one hand and its interpretation on the other hand. The Theorem holds, and the impossibility holds for Arrow’s axioms, but the questions of reasonableness and moral desirability are of a different kind. (2) The area of application of Arrow’s axioms seems rather static, while reality is dynamic. By considering the role of time, there is more scope for morality, and then one can identify a voting procedure that many would find attractive.

The two following chapters develop these arguments subsequently. Readers interested in more details are referred to Colignatus (2001), “Voting Theory for Democracy”. That book develops the theory of direct single seat elections from the bottom up while it also provides programs (in Mathematica) to eliminate the tedious work of the calculations of the various voting procedures.

34. The solution to Arrow’s difficulty in social choice

Summary

Arrow’s Theorem holds that no constitution can satisfy certain properties. In annex to that theorem, Arrow claims that those properties are reasonable and morally desirable. In Arrow’s view there thus is the difficulty that people desire a constitution that cannot exist. While the Theorem stands as a mathematical result, the additional claims concern some other matters, namely the domains of reasonableness and morality. It are these claims that have caused much confusion in the literature. It is shown here that the claims are unwarranted, since inconsistent properties are neither reasonable nor morally desirable. It is shown too that Arrow’s axiom of Pairwise Decision Making (formerly known as the Independence of Irrelevant Alternatives) is not realistic, and thus unattractive. We show the existence of some constitutions without that axiom that are consistent and might be optimal to many. The major error made by Arrow and his students is to mix up the context of scientific discovery and learning with the context of application to the real world by educated people.

Introduction

Arrow (1950, 1951, 1963) showed that if certain properties are postulated for a constitution, then such a constitution would not exist. This result has been checked by numerous scholars, is accepted by this author, and thus stands as a mathematical theorem. In fact, we will give a short proof below.

Arrow also claimed, annex to the theorem, and this will be at issue here, that those properties would be reasonable and morally desirable. He recently repeated that claim in the Palgrave (1988:125). He writes:

“(...) conditions to be imposed on constitutions (...)”

“(...) there is no social choice mechanism which satisfies a number of reasonable conditions”.

For clarity it is useful to introduce the following abbreviations for the theorem and its companion claims, and their conjunction:

AT = the Arrow Theorem

ARC = the Arrow Reasonableness Claim = the properties are reasonable

AMC = the Arrow Moral Claim = that they are to be imposed

AGV = the Arrow General View = AT & ARC & AMC

Note that Arrow’s phrasing on ARC and AMC is a bit ambiguous. The “to be imposed” might not be moral but merely logical, in a sense that one needs at least some conditions to make a constitution. However, the topic of collective choice is distinctly a moral one. Secondly, Arrow emphasises what is to be imposed and what is reasonable, but he may not be in a position to impose his views and morals on us. The best interpretation of the situation likely is as follows. Presume that Arrow sees the Founding Fathers at work. He then retreats to his office, and conjectures: ‘If I interprete correctly what they want, then it are these properties.’ Thus the ARC and AMC are not quite Arrow’s personal ideas. Above quotes can best be interpreted as factual statements on what people apparently want and consider reasonable.

Arrow’s general view has been accepted in many places in the literature and textbooks, see Luce & Raiffa (1957), Johansen (1969), Sen (1986) or various other entries in that same Palgrave. For example, Tobin (1990):

“We know there is no way to aggregate individual preferences into social rankings (...). As if this were not obvious, Kenneth Arrow proved it rigorously years ago. The impossibility applies to aggregations across contemporaneous cohorts, a fortiori across generations living and unborn.”

In a much used book on Cost-Benefit Analysis (CBA), A.K. Dasgupta & D.W. Pearce (1980):

“(...) no escape route (...) seems yet to be available.”

Apparently feeling that Arrow's argument destroys the foundations of CBA, they find themselves forced, rather grudgingly, to reduce CBA to something like information gathering.

In an otherwise recommendable volume of Statistical Science, Gill & Gainous (2002) find:

“In fact, he proved that unless one is willing to violate one of a set of reasonable democratic norms, (…inconsisteny...) is an inevitability. (…) Therefore, collective social decisions cannot yield a truly democratic system in this sense.”

Jorgenson (1990), once president of the Econometric Society, concludes ‘more positively’ to dictatorship:

“The classic result of social choice theory is Arrow’s (...) impossibility theorem, which states that ordinal noncomparability of individual welfare orderings implies that a consistent social ordering must be dictatorial, corresponding to the preferences of a single individual.”

Not everybody falls for dictatorship. The impact of the AGV generally comes from the fact that people find themselves, either from moral obligation or from reasonableness, wanting the impossible. And many simply stay in that fixture.

Note the subtlety in that fixture. The impossibility is logical and not just empirical. An example may help. Let me confide that I want to found a new university on the island of Crete. However, I am not that rich, so I want something impossible. This however does not put me into a fixture, since I am used to the fact that I cannot afford some things that I want. However, the Arrow general view concerns a logical impossibility, which is something quite different.

We can usefully recognise:

reasonable = rational & realistic

Reasonableness is the intersection of rationality and empirical realism. Nonexistence may derive from empirical circumstances or from logical impossibility. Irrationality however is always unrealistic. Inconsistency cannot exist, in the true empirical sense. For example a round square cannot exist. The nonexistence of the Arrowian constitution similarly derives not from empirical reality but from logical necessity.

Given the AGV, the question arises what the reasonableness and moral presumptions of Arrow’s claims actually are. Are these claims as strong as conjectured ?

My position is as follows:

1. As has been said on ‘round tables’, it is not rational to postulate inconsistent properties. People involved in a learning process may indeed make inconsistent assumptions. However, once the inconsistency is discovered, it is no longer considered to be rational to adopt those assumptions. People may enjoy ‘roundness’ and ‘squareness’, but having both simultaneously is seen to be inconsistent, even inconceivable, and hence unreasonable. The Arrowian properties are unreasonable in the exactly same manner. Arrow’s pitfall is to confuse the learning process, his context of discovery, with real world applications by educated people.

2. Similarly, one cannot be morally obligated to a logical impossibility. Hence Arrow’s properties are morally undesirable.

These points will be clarified below.

Note that people have in practice rejected some of Arrow’s properties. Even those scholars who seem to accept the general claim AGV, accept, a fortiori, the implied inconsistency, and thus in practice drop some assumptions to cope with the real world. Unfortunately, however, the literature has not converged to some agreement on which properties are best to drop. The position of this paper will be to forward the proposition that the Arrow axiom of Pairwise Decision Making (formerly known as the Independence of Irrelevant Alternatives) is the culprit to kill. It is a bad axiom for rational collective decision making, since it appears to be incongruent with that very notion itself.

In the following we develop the concepts, give a short proof and discussion of Arrow’s Theorem, construct the argument against the claims, reappraise the literature, and conclude.

Basic concepts

Please note that we will have to redefine some symbols for this chapter only.

Let X be the commodity domain. An element in the commodity domain can be called an item or a candidate. An agent is a compound of various properties such as utility, wealth etcetera. Let S be the set of possible compounds on X. With n agents, our interest concerns the function c: Sn S. which maps the society into an aggregate compound. This is generally called the ‘Arrow type of social welfare function’ or simply a constitution.

A constitution differs from the ‘Bergson-Samuelson type of social welfare function’ (SWF) - and the latter is defined directly over X as SWF: X [0, ).

Arrow’s Theorem concerns Social Welfare Function Generating Mechanisms (SWF-GMs) like the c above. Thus, a constitution can be seen as a mechanism that uses the population as input and generates a SWF that orders all elements in the commodity space. This can be compared to a Social Decision Function (SDF) that selects only one element, namely the best of a budget set. This can be weakened further by considering preference orderings instead of functions. Constitutions generally associate better with SDF-GMs since parliaments generally don’t care ordering all proposals. However, these concepts can be translated into each other via varying the budget set. Since the SWF is the conventional concept in economics, the word “constitution” can remain associated with a SWF-GM.

It suffices to restrict S to preference orderings. These orderings satisfy reflexivity, transitivity and completeness. It is important to add that there is no cheating. Let R denote normal preference, P strict preference, and I indifference. When there is no confusion, we can also use the symbols , < and =. A suffix denotes an individual preference, otherwise it is the aggregate. An element in Sn is called a profile, and R = c(R1, ...Rn).

There are the following Arrowian axioms:

AWP the weak Pareto principle

AU universal domain (wide ranging preferences)

AD no dictator

APDM pairwise decision making (the axiom

f.k.a. independence of irrelevant alternatives)

a AWP & AU & AD & APDM.

The Arrow Theorem can be expressed in various equivalent logical forms:

AT a falsum

AT’ a ~a

AT” ~a

AT”’ (AWP & AU & APDM) ~AD

with falsum a contradiction or falsehood and ~ the negation sign. If something leads to a contradiction, then we conclude to the falsehood of the assumptions themselves.

There is a Kantian distinction between technical, pragmatic and moral (categorical) imperatives. Utility, as commonly regarded by economists, likely is of the pragmatic kind. Interestingly, theorists on morality have developed something called ‘deontic logic’, which appears to give many similar results as economic theory. Deontic logic however applies to propositions and not to commodity domains. It is possible, though, to integrate all these kinds of preferences into an integral utility index, when we replace a point x in the commodity domain by a statement “The state of the world is x”. This integral utility index likely would be lexicographic, in that some moral and constitutional issues might dominate pragmatic results in the commodity domain. Thus, while we would use the same symbols R, P and I, we would need to look into the structure of the index to find the Kantian distinction as made by the particular agent. We conclude that we can usefully introduce and apply some terms from deontic logic. Define:

Ap (~p p) means that p is allowed (at least as good as ~p)

Op (~p < p) means that p is a moral obligation (one ought to p)

An exemplaric deontic result is:

Op ~(A(~p))

Deontic logic allow us to translate:

AMC = Oa

The use of deontic logic allows a forceful restatement of Arrow’s difficulty in social choice:

Oa & ~a

Let us consider some more properties of morality and deontic logic.

The gap between Is and Ought (Sein und Sollen) means the rejection of p p Op (‘If something is, then it should be like that’) and, in principle, p Op p (‘what ought to be, is achieved’).

Note what this actually means. A statement p has a truthvalue 1 (true) or 0 (false), depending upon the state of the world. A statement Op has a ‘truthvalue’ 1 (ought) or 0 (not-ought) depending upon one’s preferences. Applying the logical calculus for the propositional operators , ~, , & thus is a mental exercise, where empirical and preferential statements are first given the common denominator of ‘accepting as valid’. Also, it may be that in one case both p and Op are accepted, but the rejection of p p Op means that it is rejected as a rule.

Moral consistency is reflected in the Deontic Axiom:

DA p,q (Op & (p q)) Oq

There is some discussion between moral theorists whether DA really holds. It may be felt that the logic is not very compelling for empirical relations of dubious causality. However, if p q reflects a logial truth, then DA is commonly accepted.

On reasonableness, it seems a bit better to attach the properties to the agents rather than to the propositions or commodities. Useful axioms then are:

AF feasibility, X is the budget set (rather than the whole space)

ARe agents are realistic (they only consider feasible options, accept AF)

I thus agree with Arrow’s 1950 statement: “My own feeling is that tastes for unattainable alternatives should have nothing to do with the decision among the attainable ones; desires in conflict with reality are not entitled to consideration.” Thus, also, when one point is (socially) most preferred, it is the one consumed.

The most complex property seems to be good old rationality. It appears that we better introduce the information set or knowledge base I(.) and state the condition that it must contain the Arrow Theorem. Then:

ARa agents are rational (they accept logic, have a preference ordering, are morally consistent (DA), and are educated on Arrow’s Theorem (I(~a)))

The I(~a) condition is a novel aspect, that, however, should not come as a surprise, given what we said in the introduction. There is a difference between a learning process and a result. In a common classroom or used-car-salesman strategy, people are goaded into buying some axioms as reasonable and attractive, and then burn themselves, which teaches them. This may be called rational from the viewpoint of learning. This paper however concentrates on the after-learning-rationality, the kind of rationality that makes learning so worthwhile.

How does Arrow’s original approach relate to the inclusion of I(~a) ? Arrow (1950, 1951, 1963) has no incorporation of learning - though he later has written on ‘learning by doing’ - so it might be that he assumes standard economic rationality. If that would be perfect foresight, then I(~a) is implied. However, it is better to hold that Arrow in that period discussed constitutional choice for agents and not by agents. The choice for people then is made by some algorithm or calculating machine. His axioms do not describe educated people involved in constitutional choice. Alternatively put, another new result in this chapter is the widening of the scopes of utility and rationality to the inclusion of knowledge about the constitutional process itself. In that sense the original Arrowian axioms can be called incomplete. Alternatively, if the idea is that these axioms concern educated people, then there is a hidden inconsistency, in that reasonable agents are assumed to regard inconsistent axioms as reasonable.

Hence:

ARC = ARe & ARa

Restatement of Arrow’s Theorem

It appears very useful to discuss the example given by the Marquis de Condorcet 1785. Sen (1970) gives a simple example that appears to be presented first by Nanson 1882. A similar example is reproduced in Table 12, and I will refer to it as “the Condorcet case”. There are three parties and three topics A, B and C on ballot, and the numbers of seats and the preferences are such that, with pairwise voting and a majority rule, a cycle results: A < B < C < A.

Table 12: Condorcet 1785

Party

Seats

Topics ordered by preference

Pairwise vote

Low

Mid

High

Red

Green

Blue

Total

100

It is, in all clarity, not that easy to aggregate votes on more than two topics. For two topics one can indeed ask for pro and contra, and find a majority (and occasional ties, for which exist tie-breaking rules). For two topics one can indeed ask for pro and contra, and find a majority (and occasional ties). For more topics, votes will scatter across the topics, and there will often be no clear majority. Therefor, pairwise voting is a good strategy to get the required information on the preferences. However, pairwise voting apparently also causes problems. So, basically, the search is for a strategy without such problems. And that is, basically, also the suggested value of Arrow’s Theorem: that it states that there would be no such good strategy.

However, in this Condorcet example, we may clearly conclude that the cycle primarily means that there is a tie. The situation is in a deadlock, and the group, as a collectivity, is indifferent. That there are indifferences or ties, is nothing special. Standard economic analysis allows agents to be indifferent (we even draw indifference curves), so groups should be allowed to be indifferent too. In Condorcet’s example, indifference is even a logical choice, since when we assume something else, then we quickly run into difficulties.

There is the famous case of Buridan’s Ass (AD 1358). A donkey stands between two equal stacks of hay, at equal distances. He cannot decide which stack to take, and dies of starvation. The upshot of this parable is that rational beings can devise a decision. Constitutions generally state what happens when there are ties. Commonly the Status Quo persists. (This may happen even if it was one of the topics under ballot, and apparently was rejected at that stage.) Alternatives are that the chairman decides, or points are (re-) negotiated, and one can use dice.

It is important to see the difference between voting and deciding. In two stages, the chairperson first lists the votes, and then only secondly gives the decision with a tick of the hammer. Table 12 essentially gives a voting field, and no decision yet. There is no inconsistency as long as we record these results as voting scores, for example “B has more votes than A in a pairwise comparison”. There only arises an inconsistency when we change this into a preference, i.e. decide that “B is better than A”. There are additional rules that translate the field into a unique decision. Part of paradoxical element in voting derives from confusing voting and deciding.

We can use Condorcet’s example to give a short proof of Arrow’s Theorem, restricting our attention to majority voting.

Proof: The group decision in the Condorcet case is indifference, so that B = C. Under the axiom of universality we can look at various preference profiles, of which Condorcet’s example is only one. Now regard the adjusted profile such that the preferences on B and C remain the same, but the preference on A drops to the lowest position. The new profile thus is {A < B < C, A < C < B, A < B < C}. Since the preferences on B and C have not changed, the APDM outcome on B and C should be the same. Majority voting now however results into B < C which differs from B = C. Contradiction. Thus there is a counterexample to the axioms. So the axioms are inconsistent. Q.E.D.

The merit of this short proof is that it clearly shows the awkwardness of the APDM. In the case of Condorcet’s example the conclusion B = C is a sound decision, and in the case of the adjusted example the conclusion B < C is sound too. That preferences outside of the pair B and C have changed is vital to the group decision, since the shift helps a change from clear indifference to clear preference. The preferences on other topics are quite relevant, and not ‘irrelevant’. APDM excludes vital information about the preferences - to be precise: it destroys information that exists - and it should come as no surprise that paradoxes and inconsistencies arise. The APDM is incongruent with the notion of group decision making. Perhaps an individual can exclude information about other topics, but a group cannot. (Or a brain that works as a group cannot.) It is a surprise that APDM has not been killed right in 1951.

A note on the name of APDM

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.

A lemma

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.

Rejection of the Arrow Moral Claim (AMC)

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.

Rejection of the Arrow Reasonableness Claim (ARC)

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 ?)

Selection of the culprit axiom.

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.

Examples of consistent constitutions

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.

A reappraisal of the literature

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.

(7) It is relevant to note that I gave this analysis earlier, in Colignatus (1990c, 1992a). This chapter is almost 99% the same as 1997b, and a a rephrasing of the main principles. I have had no success so far in getting a publication, neither at the CPB nor in a journal.

Conclusion

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.

Addendum: Sen’s restatement in “Development as freedom”

Sen (1999a:250-253) contains a short summary discussion on his view on the Theorem. First I quote him and then give my comment. Sen states:

“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 !

Addendum: Mas-colell, Whinston and Green, “Microeconomic Theory”

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 20th 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.)

35. Without time, no morality

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.

Introduction

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.

Control of natural forces in the social process

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.

Three traditional methods

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 following example is taken from Saari (2001ab). Consider a budget of three candidates A, B and C, and let there be 114 voters. When we neglect indifference and use strict preference only, then with 3 candidates there are 3! = 6 possible ways of ranking them. Table 13 contains an arbitrary allocation of those voters over such preferences. The highest ranking candidate gets rankorder weight 3, the second gets weight 2, and the least preferred candidate gets weight 1. In the table we can read for example that there are 33 candidates with preference A > B > C. > > Table 13: Voting example

Number of voters

Candidates and their rank order weight

Sum 114

Results of the procedures

Mostly preferred

33+0 = 33

14+25 = 39

25+17 = 42

Borda

230

242

212

Pairs: A vs B

A vs C

B vs C

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.

Borda Fixed point

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, …, xn} 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.

Relation to Saari’s work

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.

Table 14: Start with 48 voters: Borda B, Condorcet B

Candidates and their rank order weight

Number of voters

Borda weighted total

116

124

A versus B

A versus C

B versus C

Table 15: Add 27 ‘neutral’ others: Borda B, Condorcet A

Candidates and their rank order weight

Number of voters

Borda weighted total

170

178

102

A versus B

A versus C

B versus C

Secondly, however, my problem remains that there is the phenomenon of budget changes. Note that Saari’s example uses a changing electorate rather than a changing budget. My suggestion is that a change in the electorate would require a new vote, while we would want to avoid that in case of a change in the budget. The Borda method would be best, only when the budget would be really given. When it might change, the application of cancellation to all subsets becomes doubtful, since subsets change. There is a fundamental uncertainty with respect to the future. Consider the following example. At a specific point in time, the population of a nation is given, and thus the vote for a President has a specified budget: the population. But, uncertainty sets in again, when people may withdraw from the race. Only a few actually run. Hence, we might well want a rule to deal with possible changes in the budget. Hence, it is not logically required that we cancel votes for all possible subcycles (also for candidates who are not in the race). Saari is very strong on the argument that when we accept cancellation in one case, then we should do so in all cases. I am more sensitive to the exception: when ‘if one, then all’ does not hold.

Concerning Table 14 and Table 15, my reasoning is - contrary to Saari - that the added votes cannot be neglected. The argument of rotational symmetry breaks down when we compare a winner with the alternative winner - which is a pair - while rotational symmetry requires a third candidate or more. For the pair, the addition has an effect. When we consider unrefined winner B and its alternative winner A, then the added votes are in favour of A and no longer ‘neutral’. While C is important since it shows a cycle for a subgroup of voters, another view is that C could be neglected since it is not a fixed point. Canditate C is a typical example of an irrelevant candidate that can cause a preference reversal in Borda voting. Namely, let us consider Table 15 under Borda voting, and let C decide to drop from the race: then A becomes the winner. The Borda Fixed Point method has been developed precisely to deal with that kind of preference reversal.

Thus, when you select your voting method then you must choose between the properties exemplified by this case. (1) Borda is subject to preference reversal. In the example of Table 15, when C drops out, then there would be switch from B to A. (2) The Borda Fixed Point method still depends upon the voting field. In this example, when 27 voters drop out, then there is a switch from A to B.

The choice basically is whether we attach more importance either to the voters or to the candidates. Saari suggests that the candidates are more important, since he cancels the votes of 27 voters and keeps C in the race. I would say that the voters are important and that candidate C is less relevant. The proper question would be whether the winner is a convincing winner. Of course, C can become an important candidate when we add other voters. But then the argument is that those voters count, rather than C.

Consider the impact of semantics. While it has been a long standing notion that cycles may also be taken as indifference, so that the votes cancel, Saari now rephrases this as rotational symmetry, and he suggests that acceptance of rotational symmetry implies acceptance of it for all cases and subsets. The label might be a common mathematical label, but I have a problem with that label in the realm of morality (and the implied universality). Human beings seem to have biological preference for symmetry, and by labelling something as ‘symmetry’, it becomes more attractive. When discussing the different voting schemes, we should be aware of such effects, and try to focus on what the properties really mean, and we should make a proper distinction between a property that is universal and a property that is dependent upon the situation. Perhaps it might be analysed as the ‘mathematical frame of mind’ that acceptance of a property for one set also implies acceptance for all other (sub-) sets, but my conclusion is that when we look closer, that there is room for more subtlety. Indeed, it might well be that considerations of symmetry apply to the static situation, but that we need other considerations for dynamics.

Another example for this need for subtlety is that the ‘rotational symmetry’ argument breaks down on the status quo (see below).

Saari has also developed an ingenious way to depict voting schemes geometrically. For 3 candidates, this becomes a triangle, and the different procedures can be calculated from that. It appears that these triangles are a good educational tool. However, my experience is that the computer programs (Colignatus (2001) uses Mathematica) are easier to use, since they take away the need for calculations, while they are available for more dimensions and also allow for indifference and not just strict preference. A complex scheme like the Borda Fixed Point also requires more work with the triangle, while in Mathematica it is a simple procedure call. It may be noted that above discussion of the Borda Fixed Point method has been simplified by assuming single winners. In practice, there can be ties, complicating the search, and requiring tie-breaking rules.

Pareto

Another consequence of the switch of attention from statics to dynamics is the recognition of a status quo.

There appears to exist another wide-spread confusion about ‘majority voting’. This idea is that a majority result would still be democratically valid, even if the winning decision implies a real loss for the opposition. The counter-example is when the majority decides that the minority pays $1 to the majority: this is not necessarily a morally acceptable situation, even though there is a majority. From a moral point of view, each voting scheme should have two rounds: a first round to select the Pareto improving points compared to the status quo, and then a second round to select the winner from those Paretian improvements. The majority rule thus can be regarded as only a tie-breaking rule, namely for the deadlock when there are more Pareto improving points. In elections of persons, the status quo can be a vacancy, and in that respect all candidates could be taken as Paretian. But the Paretian pre-condition cannot be skipped in general.

The Paretian condition may require some subtlety. Consider the family choice for a holiday to Greece or Spain, discussed above. If little Robby considers the holiday to Spain to be a deterioration from the status quo of not having a holiday at all, then there is moral argument to say that Spain is not a valid option to take a vote on. However, if it can be established in a first round that going on a holiday is unanimously a good idea, then Robby has to accept a possible majority decision in favour of Spain and against Greece.

One argument against the selection of Pareto improving points is that people might also cheat about these points. This argument is not convincing, since Pareto improvement is in one’s own interest. Indeed, little Robby might try to veto Spain by saying that he does not want a holiday, and thus he might be trying to bargain to get everybody to accept Greece. However, this ploy can be prevented by having that first round on having a holiday, since if he really wants a holiday anyhow, then he has to show this then. Careful construction of the voting process thus remains an issue.

A note on cheating

One of the key problems in voting theory is strategic voting behaviour, better known as cheating. In a scheme like Borda, cardinal utility has already been reduced to ordinal utility, so perhaps we should be lenient and allow voters to maximize their utility from the final outcome by manipulating their vote. But our opinion on this does not matter, since the ballot generally is secret and we cannot stop people from voting strategically anyway. In fact, my Mathematica programs, Colignatus (2001), contain routines for cheating. These are simple routines that assume both full information and that others don’t cheat, since the mathematics of cheating while assuming that others cheat too is rather complex, especially when nobody has full information about the true preferences. Given all this, one surmises that election results do not reflect the true state.

Thinking about these issues gave me an idea that might be helpful to elicit the true state. Suppose that each voter is informed in advance that there is a probability p that the ranking order that is submitted will be used by the election computer for strategic voting. If the voter submits his or her true ranking, then this is rewarded with probability p to improve the election result for that voter, and much better than the voter can, since the computer knows all submitted rankings. If the voter submits a strategically adapted ranking, then this is punished with probability p namely to improve the election result for that false ranking. Likely there is a specific value of p that would generate the most truthful election result. Unfortunately, I haven’t had time to develop this idea.

Conclusion

An election result is ‘as much’ the result of the procedure as of the preferences. Arrow’s Impossibility Theorem is complex and full with paradoxes, but the dependence of morality upon time provides a way towards solution.

There are two key conclusions:

(1) The Pareto condition for the candidates under ballot should not be neglected - i.e. that only those candidates are voted on that are an improvement compared to the status quo.

(2) The Borda Fixed Point can be seen as a compromise between the Borda and Condorcet procedures (on Paretian points), and provides a degree of protection against budget changes.

There is also another conclusion. Voting is complex, and becomes increasingly complex when the numbers of candidates and voters rise (especially when we also include indifference and not just strict preference). Direct election of a President becomes quickly infeasible for the more advanced voting procedures. From this observation we can conclude that it is better to have a proportional parlementary system, so that the elected professionals can use the advanced voting procedures to select the President. This approach of representation also prevents that there is a different electoral mandate for President versus Parliament. Note that the discussion above, on Arrow’s Theorem and the Borda Fixed Point method, considers single seat elections, and not multi-seat elections. But the complexity of direct single seat elections tends to support this conclusion on the overall system of proportional representation and indirect election of the chief executives.

36. Some notes on ethics

The following notes on ethics are not well developed but the points are useful to observe.

(a) I was struck by Keynes’s quote: “along the line of origin at least, economics - more properly called political economy - is a side of ethics” (Skidelsky (2000:264)). This is a point that is commonly not seen by the general public who associate economics with money, and neither by many economists who don’t appreciate the subject of political economy.

(b) Ethics focusses on survival and the good life (“flourishing”). That is, just like laboratory animals require an optimal environment, humans have their own conditions for flourishing. Csikszentmihalyi (1997), “Living well. The psychology of everyday life”, clarifies the required balance between challenge and competence: too much challenge causes stress while too little challenge causes boredom. The Rasch model, also known in psychology as the item-response model, or the Elo model used for Elo rating in chess, seems to fit the situation.

(c) Colignatus (2003), “On the value of life”, essentially focusses on survival: the lifeyears saved and the allocation over individuals. On the quality of life, the “flourishing”, I only have a rough outline “On the price of health”.

(d) The chapter “Without time, no morality” of course links with the discussion in chapter 19 on determinism and free will, and the general importance of ‘dynamics’ for this book.

(e) There was a seminar by McCloskey on virtue ethics that was illuminating and that I can advise to who has a chance to attend. Smith (1759, 1984), “The theory of moral sentiments”, featured strongly.

(f) A general point in ethical theory is that people aren’t really ‘souvereign consumers’. They grow from dependent children to mature adults to dependent seniors, so that there is always a degree of dependency. Political economy takes this into account. The standard economic approach that assumes souvereign consumers however can still be useful for analysis even while being limited in this respect.

(g) Another point concerns the distinction between ‘rules’ and ‘rhetorics’. In ethics, it does not suffice to have rules only, since these must be applied to practical situations – where rhetorics apply. In law, there are not only laws but also courts. Current literature in economics tends to emphasize rules. If economics had courts too then there might be less imbalance. The suggestion that there be economic courts links with the idea of an Economic Supreme Court.

(h) There are some other advisable books that enrich our understanding of humanity, (social) behaviour, ethics and its biological roots, which form the input for and target of political economy. Tiger (1992), “The pursuit of pleasure”, mollifies the economistic calculus of utility, which at the same time clarifies that it still can be useful to use small abstract (simplistic) models to develop arguments that can improve the lifes of many. Damasio (2003), “Looking for Spinoza”, delves into the brain to understand human emotion and feeling. Though many dimensions exist, there still is the pain and pleasure dichotomy that links to ethics. Damasio also notes that biological ‘emotions’ (generally) arise split-seconds before being reflected in ‘feeling’ in the mind. This phenomenon raises the question of ‘free will’ and the reader is referred to that section in chapter 19 above. De Waal (2001), “Tree of origin”, discusses whether primate behavior can tell us something about human social behaviour, and the same themes arise. Cavalli-Sforza (2000), “Genes, peoples and languages”, focusses on recent human evolution. Diamond (1997), “Guns, germs, and steel”, makes us aware of the impact of mere geography. All these books clarify that political economy can be of value for humanity by keeping an open eye for the study of humanity itself.

(i) Cavalli-Sforza (2000:207) concludes with this statement: “It will be necessary, for example, to be more successful in spreading the necessary moral values to the whole world. Is the amount of deception, hatred, exploitation, and unrestrained selfishness we observe in almost every society inevitable ? We need not be too pessimistic and should admit that people do not always display their worst qualities. But it would be valuable to learn exactly the conditions that elicit these destructive tendencies, in order to systematically prevent them. Overpopulation and extreme competition for valuable resources undoubtedly contribute. Our aptitute for social engineering is limited, although we must become more serious about work in this area, so as to end - or at least reduce - major social ills such as poverty, ignorance, population growth, racism, drug addiction, crime, and other social epidemic and endemic diseases that afflict us. Our efforts in this regard can be helped by studying cultural transmission and the forces of conservatism that hinder useful innovations, as well as the danger posed by promoting and accepting great changes too soon.” I can only agree with this, and the current book fits this objective.