The point of criticism is that employmen_Definition & Reality in the General Theory of Political Economy

separate issues. Our own analysis in this book shows that a return to

full employment is possible. The main instrument is to get rid of the

tax void. Employment does not depend upon growth per se but employment

depends upon a properly working system to allocate the work that is

being done in an economy. Growth comes only into the story when we

aspire at higher welfare by means of higher productivity. If we don’t

want growth, we can easily imagine a stagnant economy. That said, most

economies aspire at a growth in welfare. We can do this by designing

new products or by material investments or by creative ways to

reorganise production. [104] Then the problem returns of optimising

investments that define real savings. Since some sections of the

economy are devoted to investments, there is also the Keynesian

phenomenon that investments influence activity, income and nominal

savings.

The paradigm to ‘minimize’ the role of government in investment was

misguided since the relation between growth and employment was

misspecified. Now that we know that the tax void was the main cause of

stagflation we can reconsider the paradigm. The argument that remains

is that government meddling supposedly caused failed investments. The

answer to that argument is (i) that failures must be judged on a

case-by-case manner, by Cost Benefit Analysis, and (ii) that one should

include the concept of Keynesian recession and that some investments

might seem a failure but actually are beneficial. Note that there is no

need for a government deficit since the analysis on the dynamic

marginal rate shows that progressive taxes need not be a drawback for

the richer. If growth is the issue, then the true issue is its

optimality in terms of level and composition and effects.

The line of thought that I would suggest is that this optimum requires

competing investment banks that develop plans during the economic

upswing that can be implemented during the economic downswing. Who

worries about pensions and the EU Lissabon Strategy is advised to

consider this approach. Since the market is an anonymous beast that may

or may not generate such competition, it remains the challenge for

governments to mastermind and manage it all.

Book VII

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_. [105]

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, [106] 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. [107]

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

_A_

_B_

_ _

_B_

_C_

_ _

_C_

_A_

Red

_A_

_B_

_C_

Green

_C_

_A_

_B_

Blue

_B_

_C_

_A_

_Total_

_100_

_40_

_60_

_ _

_35_

_65_

_ _

_25_

_75_

_B_

_C_

_A_

It is, in all clarity, not that easy to aggregate votes on more than

two topics. [108] 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 _p_0_ _we have

_O(p_0_ & ~p_0_). _But this means _Op_0_ & O~p_0_, _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.

[109]

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

_A_

_B_

_C_

_ 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_ = {_x_1, …, _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_

_A_

_B_

_C_

_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_

_A_

_B_

_C_

_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.

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.

Book VIII Supportive notions

37. On the nature and significance of a free lunch

It has been a cause of wonder for the present author why other

economists are not more outspoken on the Tax Void, and why above

theorem on the possibility of returning to full employment meets such

disbelief as it apparently does. In the course of time, I found that

the following issue forms part of the explanation.

Many economists think that there are no free lunches. It may even be a

dogma or mantra to them. With this general attitude, they close their

eyes to the free lunch that presently exists in the inefficient labour

market. They adhere to their ‘no free lunch’ philosophy regardless of

what arguments other people forward. My diagnosis is that this is one

of the reasons why the debate on unemployment is rather stuck.

It actually can be shown that the economy is full of free lunches. We

will discuss two examples below, namely the examples of the consumers

surplus and economic growth. By regarding these examples we will better

appreciate the _nature and significance_ (as Robbins might say) of a

free lunch. When the possibility of a free lunch is accepted, then we

can discuss unemployment in more realistic terms.

Some quotes

The American science fiction writer Robert Heinlein once created a

rough Moon Colony where the rules of the free market are exploited to

their limits. In this colony the phrase “Your money or your life” is

not a criminal threat but a sound business proposal - and a bargain for

many as well. In the same vein all incidents in the novel are subject

to bets - and after some consideration, the reader of this novel may

well accept this as a useful system of rational contingent forward

markets. Then, properly, the slogan & law of this Moon Colony is

TANSTAAFL: “There Aint No Such Thing As A Free Lunch”.

TANSTAAFL is rather “accepted wisdom” in the economics profession, and

not something that is subject to critical discussion. There are only

few explicit statements on the supposed absence of a free lunch. A

recent statement is by Cnossen & Van Ewijk (1995):

“No society limited in resources can for a moment proceed from the

premise [sic] that there is such a thing as a free lunch. Dispassionate

analysis of the problem and hard-headed calculation of the costs of

alternative courses of action are called for. This applies especially

to the economics discipline, which gives center stage to the concept of

opportunity costs.”

So, evidently, in the views of these authors, people disagreeing to

their views on this issue are emotional or soft-headed !

Coase (1994:200) has a fine anecdote:

“Charles Walgreen in 1936 withdrew his niece from the University of

Chicago because he had been informed that the university taught free

love and communism. I know nothing about the university’s teaching on

communism but presumably Mr. Walgreen would not have been mollified to

learn that the true Chicago view is that there is no such thing as a

free love. Eventually, however, Mr. Walgreen was convinced that he had

been misinformed (...)”

The British newspaper The Economist (1994b) and the Dutch economist Van

Bergeijk (1994) state, in reaction to proposals by Snower, that there

would be no free lunch on the labour market. Even with current

unemployment, it would not be possible to change taxes, contributions

and benefits in such manner that this would raise employment

opportunities for the unemployed without other agents having to pay

some bill.

These latter authors use arguments for their views. So their judgement

does not seem dogmatic. However, their arguments have been refuted.

Authors like Snower and myself, and many others, have also pointed to

the possibilities for improvement in the labour market, and these

arguments have not met with convincing rejections. So it may well be

that TANSTAAFL works its ways in the back of the minds and hinders

proper balancing of arguments.

We somehow might welcome the Cnossen & Van Ewijk statement, since it

makes explicit what often is only implicit. In the following I shall

deal with the problem in general. I hope to banish TANSTAAFL to the

domain of science fiction, so that thereafter we can discuss the labour

market in more useful terms.

Consumers surplus

The more innocent examples of free lunches happen around us every day.

For example, in a free country, a transaction occurs only when both

parties get something out of it. TANSTAAFL adepts will hold that when

there is a transaction, and people pay for their lunch, then there

clearly is no free lunch. However, the theory of the consumers surplus

reminds us that you may pay for your lunch, but likely not as much as

you might be willing to pay. If you would not get more out of it, there

would be little point is actually doing the transaction. In everyday

life, we see few people exchanging dollars for dollars, just for the

fun of it. So if _p _is what you pay for your lunch, and if _wtp _is

your willingness to pay, then _wtp - p _is your free lunch.

One might argue that the TANSTAAFL conjecture properly reads that _p

_0. Thus TANSTAAFL-ists accept that _wtp > p, _but the point would be

that you have to invest a nonzero amount before you can reap greater

benefits. It would seem to me that the following is the proper reaction

to this:

1. We might accept a definition that ‘no free lunch’ means _p

_0.

2. However, that definition does not warrant universal truth.

Some goods have _p _= 0, notably endowments, ideas and, in a sense,

public goods.

3. So, please then, do not use this mal-definition to kill

arguments on the labour market that concern new ideas.

4. And, please see the point that it may be advisable to define

‘_p _0’ ‘there are some costs’, and ‘_wtp > p’ _‘there_ _is a free

lunch’_._

In a sense, the discussion might only be about words. But there are

also emotional connotations involved, that should cause us to be rather

careful in that choice.

Economic growth

Economic growth is another instance of manna from heaven, and also a

phenomenon that has been with us since the dawn of mankind.

An invention in one industry will generally have consequences for the

entire economy. The industry of origin can seldom claim all proceeds.

When the optimal ratio of production factors changes, then prices

change. E.g. just by mentioning the possibility of other prices, one

signals to the other parties that there is room for discussion. The

other parties will use that room, and their knowledge and possessions,

to claim part of the economic value of any innovation. Other parties

have had no effort in bringing about the innovation, but they consider

themselves partners in the industry, they know their leverage, and,

thus, exploit it. Their advantage not only concerns the consequences of

a better product, but also an improvement of their income position.

Model

In a general equilibrium framework we consider an economy with 400

units of labour and 600 units of capital. The economy produces food and

clothing, and a social welfare function (SWF) determines the optimal

combination. Here, our SWF will be a Cobb-Douglas function that

neglects the distribution of income:

(SWF)

Labour _a _en capital _k _are allocated to the food (v) and clothing

(k) industries via _av + ak = 400 _and _kv + kk = 600. _Industrial

output is determined by the production functions. Here we take

CES-functions, that have a constant elasticity of substitution between

capital and labour:

Equilibrium and the optimum are found at 278 units of food and 253

units of clothing, with a distribution of the factors of production of

_av/ak_ = 299/101 and _kv/kk_ = 210/390.

The allocation can be shown using two figures. Figure 36 confronts the

social welfare function with the Production Possibility Curve (PPC).

Figure 36: Social Welfare and the Production Possibility Curve

The PPC gives those combinations of food and clothing that can be

produced with the scarce resources. The choice of the highest possible

value of the SWF generates a tangent of a contour of the SWF with the

PPC. The tangent gives the optimal price ratio (thus trading ratio) of

food and clothing.

Figure 37 confronts the production functions of the separate industries

in an Edgeworth-Bowley diagram. The food industry has its origin in the

lower left-hand corner, and the clothing industry has its origin in the

top right-hand corner. The amounts of capital and labour that are not

allocated to the food industry are allocated to the clothing industry.

The drawn contour for the food industry gives those combinations of

capital and labour that produce the same amount of food. That contour

is touched in a tangent by a contour of the clothing industry. The

collection of all tangency points is called the contract curve. The

tangent drawn here passes through the optimum selected by the SWF. This

tangent thus also determines the price ratio of wages and capital rent.

Now we assume that there is an innovation in the clothing industry.

This innovation can be of technical or organisational origin, and it

causes that the same garment can be produced with a little less labour

but a little more capital. To be concrete: the production possibility

is discovered that can be stated in the production function _clothing =

CES_[0.2, 0.5]_._ Is this innovation useful ? The answer appears to be

that labour is the factor that is relatively scarce and that this

innovation allows its better use, so that welfare can rise to 282 units

of food and 269 units of clothing. The allocation of factors of

production becomes _av/ak_ = 309/91 and _kv/kk_ = 202/398.

Figure 37: Edgeworth-Bowley diagram for the factors of production

Figure 38 and Figure 39 present the same plots as before so that one

may see how the economy changes. The figures speak for themselves. It

will be clear that our analysis is comparative statics. How quickly the

prices change, and how quickly the agents react, will be a question of

dynamics.

Figure 38: SWF and PPC of two situations

Figure 39: Edgeworth-Bowley of two situations

The free lunch

Above model was not perfect but helps us to understand how a free lunch

percolates through the economy. It helps us to understand what a free

lunch actually is.

In above model, the innovation falls from heaven like manna. The

innovation is the free lunch. One may see the tautology: If you accept

the model, then there is a free lunch; and you accept the model if you

see innovation as a free lunch.

One may hold that above model is incomplete. One would want to

introduce a separate R&D sector, and then there will be a balancing of

R&D costs and the expected increase in national income. As an

economist, I’m very much in favour of developing such models. However,

actually doing this only moves the question one station further, and

does not answer the proper question. For, it is possible that an

economy spends 99% of its resources to R&D, and still does not come up

with innovations. Good ideas remain like manna from heaven.

You may hold the view that agents already expect economic growth, so

that they will not regard it as a free lunch. This reminds of the

attitude of some children of rich parents who expect a rich inheritance

and who don’t show gratitude for their daily bread. The point to note,

though, is that the concept of a free lunch is not an expectational

variable, but one of circumstance. There is a free lunch or not,

whatever one expects. Indeed, as another example, our wealth is a

cumulation of free lunches in the past. That we don’t experience this

as a free lunch anymore, is more a sign that we are spoiled, rather

than a sign of our dynastic rationality.

And even if we would design a revised expectational concept of a free

lunch: then perfect foresight or rational expectations are only

assumptions. There is always the possibility of a surprise idea. The

future is uncertain (though predictable) - even though our scientific

predisposition is deterministic.

Let me rephrase the point that I want to make here. There are data

(exogenes or endowments such as soil, sun, technical relations and the

like), the economy depends on the use of these, and the development of

the economy can be described in terms of the developments in these

data. The data are for free. Ideas are part of these data, and the

(major) source of uncertainty. In this terminology, there are free

lunches_ by definition_. That is the crux. When economists better deal

with their definitions, we get better economics.

Conclusion

Our discussion on the consumers surplus showed that much may be a

matter of words. However, using an abstract argument and a concrete

small general equilibrium model, we showed that innovation and economic

growth are an example of a free lunch for the whole economy. Our

intention was to refute the attitude of “there aint no such thing as a

free lunch”. Hopefully, this refutation creates more room for

discussion of proposals concerning the present immense inefficiency on

the labour market. The latter discussion is especially important, since

the major proposals for solving the inefficiency concern _ideas_ by

impartial economists.

Note 1999: I was afraid that I would clash with Paul Krugman on this

issue, since he has a Fortune column ‘No Free Lunch’. To my great

relief, Krugman (1999:167) however writes: “And this brings us to the

deepest sense in which depression economics has returned. The

quitessential economic sentence is supposed to be “There is no free

lunch.”; it says that there are limited resources, that to have more of

one thing you must accept less of another, that there is no gain

without pain. Depression economics, however, is the study of situations

where there _is_ a free lunch, if we can only figure out how to get our

hands on it, because there are unemployed resources that could be put

to work. In 1930 John Maynard Keynes wrote that “we have involved

ourselves in a colossal muddle, having blundered in the control of a

delicate machine, the working of which we do not understand.” The true

scarcity in his world - and ours - was therefor not of resources, or

even of virtue, but of understanding.” Hurray!

38. Proper definitions for uncertainty and risk

This discussion will present proper definitions for uncertainty and

risk. Such definitions are required since the current definitions in

common use are rather erroneous and generate conceptual problems.

Uncertainty

The new definitions are - see also Figure 40:

(1) First there is the distinction between _certainty_ and

_uncertainty_.

(2) Uncertainty forks into _known categories_ and _unknown categories_.

(3) Known categories forks into _known_ and _unknown probabilities_.

(4) Unknown probabilities forks into_ assuming a uniform

distribution_ (Laplace) or use_ _non-probabilistic _techniques like

minimax _or _neglect._

Note that these definitions only use certainty, knowledge and the

distinction about categories (category-uncertainty), and that they do

not use the term ‘risk’. Thus an independent definition of ‘risk’ is

possible.

A.S. Hornby (1985) “Oxford Advanced Learner’s Dictionary of Current

English” defines ‘uncertain’ as: “1 changeable; not reliable: _~

weather; a man with an ~ temper_. 2 not certainly knowing or known:

_be/feel ~ (about) what to do next; a woman of ~ age_, one whose age

cannot be guessed”. The above fits this.

Figure 40: A diagram of the new definitions

Risk

Hornby (1985) defines ‘risk’ as: “(instance of) possibility or chance

of meeting danger, suffering loss, injury, etc.” Also: “at the ~ risk

of / at ~ of, with the possibility of (loss etc.)”.

Thus, if there are possible outcomes _O_ = {_o1, o2, ..., on_}, then

the situation is risky if at least one of the _o_’s represents a loss.

The risks are the _oi_ that are losses, thus _Risks_[_O_] = {_oi_ _O

| oi_ is a loss}. The risk factors are the positions or index numbers

of the risky outcomes, the i’s, or the dimensions (the causes that make

such positions to be filled).

We will use the term ‘valued risk’ when a risk is valued with money or

utility. When all risks have been made comparable by valuing them, then

we can add them, and we will use the term _expected risk value_ for the

_expected value of the ‘valued risks’_. Then, crucially, once these

definitions are well understood, then we may also use_ ‘the risk’_ for

the expected risk value. [110]

With such understanding, risk will be _r _ = -E_x_<0[_x_] [111] or for

short _r_ = -E[_x_ < 0]. [112]

Valued risk deals with the cases when probabilities are known _or_ when

unknowns are assumed to be uniformly distributed over known categories.

It is not customary to use the term ‘risk’ for unknown categories. For

example, it is uncommon to say, or write economics papers about this,

that “all our lives are _at risk_ of a suddenly imploding universe, or

black hole hitting Earth, or waking up as a cockroaches”. Such real

‘Acts of God’ are commonly neglected. Note though that it still remains

possible to say that a situation is risky even though one cannot put a

number to it. Above expectation may be indeterminate since one may lack

knowledge about the probability distribution or even the categories.

_ _

_Relative_ risk is defined as _r(t)_ = _t_ - E[_x < t_] for some target

level _t_. Risk (or _absolute_ risk) takes _t_ = 0, and relative risk

would allow for a different target level. [113]

An interesting application is when _x_ is a stochastic rate of return

and _r _the certain rate, so that there is relative risk _r(r)_ _=_ _r_

- E[_x < r_]. This relative risk answers the question: What is the

probable loss with respect to a target return of _r _? Here, _r_ -

_r(r) = _E[_x < r_] gives the weight of underperformance in the total

target return (which weight has to be compensated by probable profits

to achieve the target).

_ _

_Conditional _(relative) risk is defined as _k(t)_ = _t_ - E[_x_

| _x < t_] for some target level _t_. With respect to rates of

return, conditional risk _k(r)_ answers the question: What would one

expect to lose with respect to _r_, if earnings actually underperform

and fall below _r_. Indeed, _r_ - _k(r)_ would give your expected

return when actually underperforming.

Conditional risk is related to relative risk by the property that E[_x_

| _x < t_] = E[_x < t_] / Pr[_x < t_]. The probable loss thus is

corrected for the probability of the loss. Or, the probability measure

in the expectation is corrected so that a density is taken that sums to

1. [114]

Example

In everyday parlance, profit and loss are nonnegative concepts. For

example, if the difference between revenue and costs is $-10, then your

loss is $10. It is only in mathematical economics that profits are

defined as a general profit function such that ‘negative profits’ are

possible. To understand risk, we however return to the everyday

parlance convention.

Let us have a prospect that can give _profit_ with probability _p_, and

_loss_ with probability 1 - _p_. We denote this as Prospect[_profit,

-loss, p_]. We call _profit * p_ ‘probable profit’ and _loss * (1 - p)_

‘probable loss’. Then the following definitions apply:

· Expected Value = _ _= _p profit + (1 - p) (-loss)_ = probable profit

- probable loss

· Risk = risk value = expected value of the risks = probable loss = _(1

- p) loss_

· Risk Ratio = Risk / (ExpectedValue + Risk) = _(1 - p) loss / (p

profit)_

· Thus: Expected Value = _p profit _(1 - Risk Ratio)

· Risk Probability = cumulative probability of all losses (in this case

1-_p_)

Risk is the (absolute value of the) down side of a bet. A venture is

judged to be risky if the probable loss is large. Note that this notion

still is somewhat vague. A probable loss can be large because of the

probability or because of the sum of money involved. This vagueness is

unfortunate, in some respects, but here is little to be done about it,

since this vagueness is inherent in working with probabilities. In

fact, this vagueness is an essentially positive aspect of working with

probabilities. For, when we have different prospects, then we can order

and evaluate them on risk, neglecting differences in losses and

probabilities.

Colignatus (1999, 1999a) further develops these notions for simple

binary prospects, multidimensional prospects, joint prospects, and

continuous probability densities. An interesting application is the

‘Markowitz efficiency frontier’, but now with risk rather than the

spread.

Wrong use in economics 1921-2005

The above definitions are proper in the sense that they conform to

every day parlance and the definitions provided by Hornby’s dictionary

op. cit.. The definitions provided here however differ from the use

within the economics literature. First there are the definitions of

Knight (1921) that have been adopted widely in economics, as for

example in The New Palgrave (1998:III:358). Or it has become custom in

finance to associate risk with the standard deviation. And some

mathematical statisticians use another concept of risk. Let us discuss

these in turn.

Uncertainty and risk

The New Palgrave, Eatwell c.s. (1998:III:358), gives the current common

view:

“The most fundamental distinction in this branch of economic theory,

due tot Knight (1921), is that of risk versus uncertainty. A situation

is said to involve _risk_ if the randomness facing an economic agent

can be expressed in terms of specific numerical probabilities (these

probabilities may either be objectively specified as with lottery

tickets, or else reflect the individual’s own subjective beliefs). On

the other hand, situations where the agent cannot (or does not) assign

actual probabilities to the alternative possible occurences are said to

involve _uncertainty_.”

Indeed, most economic texts use this distinction in this manner (at

least, up to now). However, I cannot disagree more. The objections to

Knight’s concept are:

(a) Certainty and uncertainty are _binary_. So, if a situation is

not uncertain, then we have certainty, and there is no assigning of

probabilities.

(b) If I am uncertain about a situation and assign equal

probabilities to all cases - the Laplace suggestion - then according to

Knight this no longer is uncertainty!

unknown probabilities, but the distinction is between _events_ and

_human thought_.

Figure 41 contains a diagram of the objectionable use of terms

1921-2005.

Figure 41: A diagram of the current but objectionable use of terms

The diagram clarifies the inconsistency with the binary character of

certainty/uncertainty, the curious treatment of “Laplace”, and the

over-use of terms by introducing the term ‘risk’ where there already is

the qualification that the probabilities are known.

Risk is not the variance

The finance literature often uses the term ‘risk’ for the variance or

spread (standard deviation) of the distribution of the rates of return

of investments. This would be an improper use of the term. Suppose that

one has a very profitable venture without the possibility of a loss.

Suppose that the rate of return of this venture has a large variance,

from mildly profitable to highly profitable. Is this a risky venture ?

No, not in the usual understanding of the term.

Risk is not the negative of expected revenue

In mathematical statistics, some authors, like Ferguson (1967), define

‘risk’ as ‘expected loss’. However, it appears that they actually

regard ‘loss’ as the negative of total returns (i.e. - revenue), so the

definition used is -(_p profit + (1-p) (-loss))_, which is the negated

expected value. This use of the term ‘risk’ is inappropriate. My

proposal is to use the word “due” to stand for the negative of expected

value, so that the standard statistical decision theory (with the game

against nature) can be described as minimising _due_.

Note on Bernstein’s “Against the gods”

I came across Bernstein (1996) “Against the gods”, and found it equally

entertaining as his “Capital Ideas”. One comment is that Bernstein

indeed emphasises Knight’s and Keynes’s statements on “uncertainty”. My

answer to that is, again, that unknown probabilities or even unknown

categories indeed are serious cases of uncertainty, so that earlier

writers on the subject were right in emphasising that seriousness.

However, we should not be tempted to reserve the word “uncertainty” to

only those cases. So with all due respect to Knight and Keynes, the

definitions provided here are the proper ones.

Note on Wilson & Crouch (2001)

Wilson & Crouch (2001), “Risk-benefit analysis”, adopt the same

definition of “risk” as discussed here. I saw this only after the first

edition of this book. Since professor Wilson has been teaching on the

subject for decades and his book only collects his teaching material I

apparently only rediscovered what was already clear to him. Perhaps my

presentation is a bit clearer since I use the formal E[.] notation.

This chapter remains useful since it clarifies the confusions from the

other definitions. Where risk is the product of probability and

severity, this book also benefits from the emphasis on this definition,

since, where I started to develop this argument after the Fall of the

Berlin Wall in 1989, we have to deal with a future where there are huge

dangers: though with only a small probability but on balance a relevant

risk.

Book IX

Reduced form

39. The possibility of full employment in the welfare state

Introduction

Above we noted that the structural form of western welfare states is

quite complicated. We would like to have a more enduring result than

awareness of complexity, and therefor we adopt the Definition & Reality

methodology. As said, a_ proposition - _as a statement on reality - can

be regarded as a mathematical theorem about/within a model of stylized

facts. When there is a tautology, we attain truth by definition. So we

now (a) restate what we consider to be the stylized facts, (b) define

our concepts, (c) develop theorems and proofs, (d) link back to

conclusions about reality.

The reduced form that is most relevant concerns the (long run)

comparative statics of the regimes of full employment (1950-1970;

Japan/Sweden) and unemployment (1970-2005).

This kind of comparative statics should not induce us to think that we

abolish dynamics, though. Stagflation has both a dynamic (inflation)

and a static or stationary (unemployment) aspect. When we skip proper

dynamics and discuss regime switches in which unemployment features as

an important switch variable, then Phillipscurve processes are included

in the switching process, even though they don’t feature explicitly in

the reduced form.

To attain the necessary level of generality, we use a reduced form

where the economy is mapped into a model with three types of agents.

One type is the net receiver; and two types are net tax payers. Since

the latter two points give a line, that single line represents the

state of the economy. The regime switch depends upon the choice of tax

parameters.

Stylized facts

There are regimes of full employment (1950-1970; Japan/Sweden) and

unemployment (1970-2005).

In the welfare state, it is more efficient to have full employment.

Unemployment causes lower income - not only directly as in

old-fashioned capitalism but also, more noteworthy, by the additional

benefit burden. Unemployment can have an adverse effect on inflation

when it causes a shift of the Phillipscurve.

It turns out that the propositions that are most interesting, from the

viewpoint of political economy, do not require continuity, and can be

formulated by assuming dichotomous High and Low productivity labour,

combined with one class of Benefit recipients. This assumption allows

for a reduced form formulation that allows for generality. For

expository reasons we can take social subsistence and productivities as

purely constant. In the simple mathematical model the dichotomy gives

fixed numbers, in actual observation they are subgroup averages which

depend upon general equilibrium processes. The benefit level is rather

not an average but a threshold, like the surface of the sea at

Scheveningen beach. The words _Benefit_, _High_, and _Low_ give letters

BHL, and this abbreviation may be pronounced - converged upon after

many walks - as ‘beachly’.

It is a stylized fact that welfare states are BHL. Checking this

requires next definitions.

Concepts

Here we will redefine variables such as _H, Z, b, n etcetera. _Also the

reduced tax function will be _T_(.) as opposed to structural _T_[.].

These redefinitions hold for this chapter 39 and chapter 40 - that

together form a reduced form unity.

Definition: Biological subsistence, for survival, is _S._

Definition: An economy is a welfare state iff people without

income are not left to charity, stealing or death, but get a benefit

_B._ The benefit _B_ has the following properties:

i. the net benefit has the social subsistence level _B S_,

ii. people on benefit may not work, [115]

iii. eligible are:

iii-a. permanent benefit recipients (e.g. ‘the elderly’)

iii-b. people able to work but currently unable to earn at least

net_ B _(these people are called ‘the unemployed’).

Remark: it is useful to have category (iii-a) in the model. It

introduces a degree of sufficient complexity. When there are levies

even under full employment, then it is easier to understand that wrong

co-ordination may cause a switch to unemployment. But (iii-a) might

count zero people.

Remark: Property (iii-b) has the effect of a legal minimum wage. It

sets a floor in the market. We might introduce a benefit threshold (for

workers) _XB_ such that _S XB < B,_ but for expository reasons, we

take _XB = B._

Remark: The reservation wage effect is as follows. When vacancies with

net income higher than _B_ are registered, then the relevant

unemployment benefits are simply scratched. This mimics the array of

measures needed for continuous reality.

Remark: This definition implies that people working with subsidies in

the Swedish/Japanese case are not on ‘benefit’. Such subsidies thus

must be accounted differently, basically as part of taxes.

Remark: The black economy (another form of working while on welfare) is

neglected. We neglect also the case that some people hate being on

welfare, and thus continue working even when their net earnings are

below the benefit threshold (_S < net earnings < XB _).

Definition: A welfare state is _bhl_ iff it remains meaningful to

trisect its membership into the economic classes of Low and High

productivity workers and permanent Benefit recipients.

Definition: A welfare state is nonrevolutionary, iff its economic

classes and their data are stable across the change of employment

regime.

Definition & Reality in the General Theory of Political Economy · 第 13 / 17 章

The point of criticism is that employment and growth are rather