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Revealed Meta-Preferences: Axiomatic Foundations of Normative Assessments in the Capability Approach
“It is a mistake to set up physics as a model and pattern for economic research. But those committed to this fallacy should have learned one thing at least: that no physicist ever believed that the clarification of some of the assumptions and conditions of physical theorems is outside the scope of physical research." (Ludwig von Mises, 1949, p. 4)
George Stigler described Adam Smith’s The Wealth of Nations as a “stupendous palace erected on
the granite of self-interest” (Stigler, 1982, p. 136). Indeed, no assumption has been more central to
the development of modern economic theory than this one, and in the eyes of many of its
detractors, as well as its supporters, this assumption has come to define the discipline1. As Prof. A.K.
Sen has pointed out, however, there is nothing intrinsically “econonomic” about the assumption of
selfishness and there is no reason why the study of the allocation of scarce resrouces could not start
from a richer representation of the complexity of human nature (Sen A. K., Rational Fools: A Critique
of the Behavioural Foundations of Economic Theory, 1977; Sen A. K., Equality of What?, 1979; Sen A.
, Rationality and Freedom, 2002, p. 21; Sen A. K., Choice, Welfare and Measurement, 1997). Sen’s
point poses a fundamental challenge, not only for mainstream neoclassical economists, but also for
proponents of his own so-called “capability approach”, as it calls for the reconstruction, from
sounder foundations, of an edifice that could rival the internal coherence and solidity that has been
achieved by classical and neoclassical theory through more than two centuries of theoretical
refinement.
This paper focuses on a small piece of this vast problem, one that was pointed out by Martin
Ravallion in a recent piece in which he criticised what he perceived as the arbitrariness of various
“mashup” indices of dubious conceptual credentials that have sprung up in recent years (Ravallion,
2010) – particularly as part of the effort to operationalise the “capability approach”. Unlike
conventional economic measures, such as GDP, which are constructed on the basis of a body of
economic literature, he argues, these “mashup” indices contain “key decision variables that the
1 The most famous example of such a confusion of the discipline for its axioms is probably found in Becker’s exploration of the “economics” of issues pertaining to the fields of biology, criminology and even religion (Becker, 1976).
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analyst is free to choose, largely unconstrained by economic or other theories intended to inform
measurement practice” (Ravallion, 2010, p. 3). In particular, he argues that the weights chosen, for
instance, in UNDP’s Human Development Index and more recent Multidimensional Poverty Index, do
not rest on the sort of theoretical justification that allows neoclassical economists to claim that
“market prices are defensible weights on quantities in measuring national income” (Ravallion, 2010,
p. 3).
Indeed, much of the efforts of welfare economics over the past couple of centuries have been
concerned precisely with the problem of linking the common variables that furnish economic theory
(e.g. prices, markets, goods) with the normative concepts (e.g. utility, efficiency, maximisation) that
can infuse moral value into these otherwise inert terms. The problem, however, is that all these
technical innovations have aimed to link normative welfare economics back to one particular ethical
framework, namely the utilitarian school, of which Adam Smith was one of the main proponents (Sen
A. K., On ethics and economics, 1987). Hence, once all the technical artifices of the theory are
removed, one is left with an argument that in fine relies on the acceptance of the notion of utility as
an ultimate standard of value for assessing the “goodness” of individual or social outcomes. And this
is one of Sen’s main contentions against neoclassical welfare economics (Sen & Williams,
Utilitarianism and Beyond, 1982). Indeed, as Sen has shown, there is no particular reason why
normative assessments of economic phenomena could not be made in terms of, say, Kantian or
Aristotelian ethics. Why, for instance, should we consider that a market equilibrium is efficient when
it maximises aggregate utility or balances marginal utilities, rather than, say, when it ensures that
fundamental social and economic rights are fulfilled?
Section 1 of this paper offers a very succinct review of the trail of theoretical innovations that have
been brought into economic theory over the past two centuries to justify the use of prices as
normative weights. In section 2, we look at Sen’s critique of a key step in this chain, namely the
revealed preference theory, and its implications for the formalisation of the same. In section 3,
finally, we propose a response to Ravallion, in the form of two normative ranking rules that are
consistent with Sen’s axiomatic representation of human nature.
The Normative Value of Market Prices
Utility, Prices and Market Equilibria
The fundamental arguments that constitute the pillars of neoclassical welfare economics can be
traced back to the founder of classical economics in The Wealth of Nations. Chief among these is the
idea that the interaction of self-interested individuals in a free market place will, in time, generate a
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natural equilibrium, which is not only stable, but – more importantly from a normative point of view
– optimal in the sense that it is the most conducive to the common good:
“[Every individual] generally, indeed, neither intends to promote the public interest, nor knows how much he is promoting it. By preferring the support of domestic to that of foreign industry, in such a manner as its produce may be of the greatest value, he intends only his own gain, and is in this, as in many other cases, led by an invisible hand to promote an end which was no part of his intention. Nor is it always the worse for the society that it was no part of it. By pursuing his own interest he frequently promotes that of the society more effectually than when he really intends to promote it.”(Smith, Book IV: Of Systems of Political Economy, 1776, p. 400). 2
The notion of “common good” is here defined in utilitarian terms, that is, in Bentham’s definition
“the greatest happiness of the greatest number” (Bentham, 1789). In the original formulations
proposed by Bentham, the assessment of the amount of “common good” generated by various social
arrangements was to be made through a very specific so-called “felicific calculus”, involving a
cardinal comparison of the net amounts of pleasure and pain experienced by each individual.
(Bentham, 1789, p. chap. 4). Bentham’s proposal faltered, however, due to the unobservable nature
of the notion of utility and the consequent impossibility to measure and compare a unit of pleasure
and pain across individuals.
The marginalist revolution in economic theory, in particular the work of William Jevons (Jevons,
1871) and Alfred Marshall (Marshall, 1890) helped to circumvent this difficulty by shifting the
attention away from aggregate utility towards marginal utility. Although the total intensity of utility
generated by a given good could not be measured, the marginalist revolution made it possible to
make binary comparisons between goods of the sort: an additional unit of increases utility more
than an additional unit of . This opened the door for replacing Bentham’s demanding cardinal
felicific calculus with the significantly more modest ordinal principle of Pareto-optimality, whereby a
social state would be considered optimal not if it maximised the “greatest happiness of the greatest
number” but if it was such that “no individual could be made better off without making someone
else worse off” (Pareto, 1906).
Another marginalist, Leon Walras, used the analysis of marginal utilities and production costs to
study, from a positive rather than normative perspective, the way in which markets for goods and
services achieve general equilibrium across all markets in an economy by adjusting relative prices so
2 It is worth noting, in passing, that the “Invisible hand” is none other than the hand of God, who has the power to – somehow – transform the uncoordinated actions of self-interested individuals into an outcome that is both stable and beneficial: “The happiness of mankind, as well as of all other rational creatures, seems to have been the original purpose intended by the author of Nature, when he brought them into existence. No other end seems worthy of that supreme wisdom and divine benignity which we necessarily ascribe to him; and this opinion, which we are led to by the abstract consideration of his infinite perfection is still more confirmed by the examination of the works of nature, which seem all intended to promote happiness, and to guard against misery.” (Smith, The Theory of Moral Sentiments, 1761, p. 237).
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as to eliminate surplus demand or supply (Walras, 1874). In the so-called Walrasian equilibrium,
relative prices will reflect consumers’ marginal rates of (utility) substitution between goods, as well
as producers’ marginal rate of transformation (or opportunity cost).
The first fundamental theorem of welfare economics, then, simply states that the Walrasian
equilibrium is – by construction – Pareto-efficient in the sense that individuals cannot increase their
marginal utility further by exchanging goods that confer them low marginal utility for goods that
generate higher marginal utility (Arrow & Debreu, Existence of a Competitive Equilibrium for a
Competitive Economy, 1954). The first fundamental theorem of welfare economics has thus been
widely interpreted as a formalisation Adam Smith’s original insight – albeit, reformulated in terms of
ordinal rather than cardinal utility.
Revealed Preference Theory
There are numerous criticisms that have been formulated against this model for the simplistic and
idealised assumptions it makes about the functioning of complete and competitive markets (e.g.
(Greenwald & Stiglitz, 1986)). In this paper, however, we will choose to disregard those and focus,
instead, on problems associated with the second part of this normative defence of market prices as a
standard of relative welfare. First, of course, the notion of Pareto-efficiency falls far short, as a
normative standard, of the original utilitarian social objective of aggregate welfare maximisation.
However, in the absence of observable or cardinally measureable utility, Pareto efficiency has
prevailed as the closest possible approximation to Bentham’s initial proposition that may be
compatible with the minimal empiricist requirements of verifiability (Hicks, A Revision of Demnand
Theory, 1956, p. 6).
A more serious problem with this formulation, as Samuelson pointed out, was that, in the absence of
observable preferences, even the minimal normative claims of Pareto efficiency seemed tenuous and
liable to the same criticism that marginalists had formulated against classical economists for their
normative reliance on the notion of utility:
Consistently applied, however, the modern criticism [of classical utility theory] turns back on itself and cuts deeply. For just as we do not claim to know by introspection the behaviour of utility, many will argue we cannot know the behaviour of ratios of marginal utilities or of indifference directions. Why should one believe in the increasing rate of marginal substitution, except in so far as it leads to the type of demand functions in the market which seem plausible? (Samuelson, A Note on the Pure Theory of Consumer's Behaviour, 1938, p. 61)
In order to address this flaw, Samuelson proposed a key piece of theoretical engineering which has
come to be known under the term of revealed preference theory. Samuelson initially claimed that
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the purpose of his contribution was to rid economic theory “of the last vestiges of utility analysis”
(Samuelson, A Note on the Pure Theory of Consumer's Behaviour, 1938, p. 62) by creating a theory
that would be based on nothing else than observable behaviour. However, as Sen argued, if
successful, this theory would quickly empty consumer choices of the normative power they derived
from utilitarian postulates about the value of utility: if behaviour didn’t reveal anything about
underlying preferences, price ratios would not tell us anything about marginal rates of substitution
and market equilibria would thus tell us nothing about Pareto-optimality. From a normative
perspective, therefore “the rationale of the revealed preference approach” must lie in the
“assumption of revelation and not in doing away with the notion of underlying preference” (Sen A.
K., Behaviour and the Concept of Preference, 1973, p. 244).
As long as individuals behave consistently, Samuelson argued, we could infer from their choice of
over , when is available, that they prefer over (i.e. that gives them more utility than , if we
assume utility maximisation). Hence, while utility itself may be unobservable, sufficient fragments of
it can be indirectly observed to allow us to make the type of ordinal claims needed for normative
welfare assessments (Little, 1949; Houthakker, 1950). Later Samuelson, building on Little (Little,
1949), went on to show how indifference curves could be approximated through the iterative
comparison of binary preference relations over choice bundles (Samuelson, Consumption Theory in
Terms of Revealed Preference , 1948).
Here, we will build on the notation used by Sen (Sen A. K., Choice Functions and Revealed
Preference, 1971) and Arrow (Arrow, Rational Choice Functions and Orderings, 1959) to formalise
the argument. Let be the set of all possible options and the set of all non-empty and finite
subsets, , of . The non-empty “choice set” ( ) represents elements of any given subset of
that have been chosen by individual . Henceforth, we will write ’s choice set over simply as for
short. The decision maker ’s preferences are represented by transitive and binary relations, ,
defined over the elements of contained in . Let = {1, … , , … , } be the set of all individuals
holding preferences . Then Γ = { , … , } represents the set of all possible preference
relations over .
The preference for over is designated by , meaning that “ is at least as good as
according to the preference relation ”. The superscript indicates that preference relations are
based on utility maximising calculus. Hence, is equivalent to saying that confers with at
least as much utility as , and can be interpreted in marginalist terms, if and represent units of
and for given quantities. The symmetric and asymmetric elements of will be designated,
respectively, by and .
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The axiom of revealed preference (ARP) then states in its original, weak, form:
∀( , ) ∈ ∀ ∈ : ∈ ∈ \ ⇒ ¬ (1)
Plainly put, if both and are part of ’s opportunity set, but only is chosen, we can conclude, at
the least, that does not prefer to .
Sen’s Critique
Plural Motivations
Implicit in the revealed preference axiom is the assumption that choice is motivated by one thing and
one thing only, namely utility maximisation. Consequently, if an option is observed to have been
chosen, it must be because it was deemed by the chooser to provide more utility than any of the
available alternatives. It is this assumption that allows the revealed preference axiom to reconnect
price ratios with marginal rates of substitution, and it is this assumption that Sen questions. In
practice, he argues, though self-interest maximisation is undoubtedly one of the most powerful
driving forces of human conduct, it is but one of the factors that motivate action, alongside things
like commitment, compliance with moral rules, religious principles or social norms. It is this capacity
to disregard one’s own narrow self-interest and act in the interest of the common good that allows
individuals to avoid suboptimal outcomes in problems such as the prisoners’ dilemma, as well as
numerous other social problems, such as taxation, provision of public goods, etc. (Sen A. K.,
Behaviour and the Concept of Preference, 1973, p. 250).
Importantly, Sen is adamant to point out that he is not interested in the sympathy or altruism that
the prisoners may feel for each other, which may lead them to incorporate the other’s welfare into
their own maximisation function. Instead, he insists, it is their capacity to “suspen[d] calculations
geared towards individual rationality” (i.e. self-interest maximisation) and to “behave as if they are
maximizing a different welfare function from the one that they actually have” (i.e. to see a problem
from someone else’s perspective) that makes the solution interesting (Sen A. K., Behaviour and the
Concept of Preference, 1973, p. 252)3.
For the purpose of the present argument, make a distinction between two fundamental types of
motivations described by Sen, namely (1) self-centred motivations, which can include altruism,
3 This ability of individuals to dissociate themselves from their own preferences and interests, in order to pursue a higher common objective, is what makes the difference between what Rousseau calls “the General Will” and “the will of all” (Sen A. K., Behaviour and the Concept of Preference, 1973, p. 250).
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sympathy, etc., and (2) reasoned motivations, which we denote by ∈ Γ = { , … , }. The
former type of motivations are idiosyncratic in the sense that they are defined by a person’s
particular interests, inclinations, desires, fears, etc. – more broadly captured by the term utility. The
latter type of motivations is here defined as those that allow us to disregard our own particular
circumstances and act, as Sen calls it, “as if” we are maximising someone else’s utility. The latter is
derived from our capacity for universalisation, that is, the faculty of reason to abstract from our own
circumstances in order to act in accordance with a law or a moral principle4. It is also reason that
allows us to generalise from the observation of our own behaviour that if we are able to distance
ourselves from our own self-interest, others should be able to do the same. It is thus reason that
allows the prisoners to resolve their dilemma by abstracting from their particular circumstances in
order to reach an agreement that transcend the individual interests of each prisoner. As Sen points
out, it is also this capacity that allows us to reach mutually beneficial agreements through politics and
even in many market transactions (Sen A. K., Markets and freedoms, 1991).
While we do not exclude the possibility that different individuals’ idiosyncratic preferences may
coincide, due to sympathy or convergence of interests, for instance, we must recognise the
possibility that they may diverge due to divergence of interests and tastes. Therefore, we define
as :
Definition 1: Γ = | ∃( , ) ∈ , ≠ , . . ≠ .
In addition to utility-based preferences, Sen argues, individuals have reasoned preferences, also
called “preferences over preferences” or “ ‘metarankings’, reflecting what [an individual] would like
his preference to be” (Sen A. K., The Journal of Philosophy, 1983, p. 25). Without going as far as
claiming that the latter motivations must concord with a universal law of reason, we will define them
as the ability of individuals, who are part of a group, to take a bird’s eye view on a situation, and
recognise that there is a common interest that transcends their individual interests. We thus define
by their convergence across individuals in a given group. Let be the set of all non-empty
subsets of and let ∈ designate a group of individuals = (1, … , ):
Definition 2: Γ = | ∀ ∈ : ( , ) ∈ ⇒ = .
4 Formally, Kant defines pure reason as the “faculty of the unity of the rules of understanding under principles” (Kant, Kritik der reinen Vernuft, 1781, p. A302/B359), that is the capacity for abstraction or universalisation. Judgement, then is the capacity “of thinking the particular as contained under the universal” (Kant, Critique of the Power of Judgement, 1790, p. 5:179), that is to link a particular situation to a universal rule.
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Once we admit the possibility of plural motivations, the technical trick of revealed preferences, which
allowed us to draw inference about underlying utility from observable behaviour, ceases to be
effective, since observed behaviour could equally well have been cause by moral commitment, norm
following or even self-sacrifice, neither of which tell us anything about a person’s level of utility5. The
table below indicates various possible outcomes that can result from a combination of preferences
over the outcomes and :
Table 1: Possible choices resulting from combinations of utility and meta-preferences
Utility
Reason
∈ ,∈ \
∈ ,∈ \
( , ) ∈ ,( , ) ∈
∈ ,∈ \
∈ ,∈ \
( , ) ∉ ,( , ) ∈ \
∈ ,∈ \
( , ) ∉ ,( , ) ∈ \
( , ) ∈ ,( , ) ∈
∈ ,∈ \
∈ ,∈ \
∈ ,∈ \
∈ ,∈ \
( , ) ∉ ,( , ) ∈ \
∈ ,∈ \
( , ) ∉ ,( , ) ∈ \
In this example, we are assuming that strict preferences will prevail over weak or indifferent
preferences in cases where there is a conflict between utility and reason. In cases where the
individual has strict preferences for both and simultaneously, he will choose both. In cases where
the individual is indifferent both in terms of utility and reasoned preferences, he chooses neither.
While it is possible to conceive of other plausible combinations of preference relations to outcomes,
the precise form of the relation will not matter here. Instead, we are interested in showing that, in
the presence of dual motivations it will no longer be possible to infer underlying preference relations
with certainty from observed choices, as was done in ARP, since each choice pattern will be
associated with, not one, but several plausible combinations of underlying preference relations.
Depending on the assumptions we choose about the form of underlying decision making mechanism,
5 There have been attempts to circumvent the problem of inference by expanding the definition of preferences to include anything sought by the individual, be it for religious reasons, moral reasons, pathological desires or some other form of “want” (Hicks, Value and Capita, 1939). Such an expansion of the definition of preference, however, would quickly empty the term of its normative content by equating it with the descriptive notion of “behaviour”(Harsanyi, 1997). If preferences are defined as “behaviour” the problem of inference simply gets pushed up by one level from preferences to underlying motivations, without enabling us to reconnect observed behaviour with the underlying notion of utility from which any utilitarian justification ultimately derives its normative force.
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we will still be able to make certain weaker claims based on observed choices. Table 1, for instance,
allows us to make the following claim, which is a close but weaker relative to the ARP:
∀( , ) ∈ ∀ ∈ : ∈ ∈ \ ⇒ ¬ Ψ ∪ Ψ(2)
where Ψ and Ψ describe the events and , respectively. The axiom now states that if
is observed to choose over , he cannot possibly have a strong preference for over . Given the
uncertainty we now face about underlying preference relations, it may be more appropriate to
represent these claims as probabilities. Axiom (2), for instance, would thus become:
∀( , ) ∈ ∀ ∈ : Ψ ∪ Ψ | ∈ ∈ \ = 0(3)
Henceforth, we will write the probability of an event having occurred, given that we observe that
has chosen over (i.e. given ∈ ∈ \ ), simply as ( ∙ | ).
Let χ and χ designate the events and , respectively. We will assume that ∀( , ) ∈and ∀( , ) ∈ , ( | ) and ( | ) satisfy the following intuitive properties, which are
consistent with definitions 1 and 2:
1. Existence: | > 02. Completeness: ⋃ | = 13. Uncertainty: | = |4. Anonymity: | = |5. Independence : ∙⋂ ∙ | ⋂ = ( ∙| ) × ∙ |
Loosely put, property 1 simply states that the probability that an observed choice, , is motivated by
selfish motives is strictly greater than 0 (i.e. there is a possibility that even a seemingly virtuous act is
motivated by selfish motives). Property 2 states that a given act, , must have been motivated either
by utility based motives or by reasoned meta-preferences. Property 3 captures the fact that, with no
additional information about underlying preferences, we cannot know whether a given behaviour is
motivated by self-interested utility-based considerations or by moral concerns. Property 4 states that
the probability that a given act is motivated by selfish motives is equal across all individuals (i.e. there
are no intrinsically more virtuous individuals). Property 5, finally, simply makes the point that the
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probability of any event occurring is independent across individuals (i.e. what motivates me is not
dependent on what motivates my neighbour).
Meta-Preferences and Consensus
In the “rationalist”6 perspective advocated by Prof. Sen, it is not underlying levels of utility that
constitute a legitimate standard of value, but rather the reasoned assessment that individuals make
of their own preferences, i.e. their metarankings (Sen A. , Rationality and Freedom, 2002). Indeed,
utility-based preferences may be determined by culture, upbringing, genetics or other factors over
which the individual has no control (Nussbaum, 2003) and they may be adaptive, in the sense of
being moulded by prejudice, fear or simply resignation to an unjust fate (Sen A. K., Resources, Values
and Development, 1984). The question that is of interest for the purpose of operationalisation of the
capability approach is thus whether meta-preferences, , can be, if not observed, at least inferred
from looking at choices . in the same way as Samuelson’s revealed preference theory had allowed
us to infer unobservable utility-preferences from observed choices.
At first sight, the answer to this question would appear to be negative: in the same way as it was
impossible to infer from the observation of choice, whether was caused by rather than , we
cannot infer whether was caused by rather than . Importantly, we do not exclude the
possibility that = , such that idiosyncratic preferences , may coincide with moral
preferences, , either due to chance or sympathy (e.g. deriving utility from someone else’s
achievement). A same behaviour , e.g. charitable giving, could therefore be caused by either self-
interested motives (e.g. attempting to secure loyalties or relieving one’s own displeasure at the sight
of poverty) or moral motivations (e.g. a social obligation towards other less fortunate members of
the group). There is, however, a peculiarity in Sen’s concept of meta-preferences, which is captured
in definition 2, that may allow us to, if not resolve, at least manage this problem. To understand how,
we shall return to the predicament of our prisoners.
In the case presented by Sen, it would be easy to imagine that the two prisoners, having worked
closely together in crime and endured hardship jointly as a result, may have developed feelings of
sympathy for each other, and that these feelings may lead them to naturally include the other
prisoner’s utility in their own maximisation problem and thus choose the optimal solution, even
though it may not be the most rational from a narrow perspective of self-interest. But imagine now
6 Sen’s alternative conception of human nature links up to a non-utilitarian tradition, which he has referred to elsewhere as the “rationalist” tradition, including non-utilitarian thinkers, such as Kant, Rousseau among others (Sen 1999, p.255).
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that instead of two prisoners, the District Attorney is trying an organised gang of ten criminals, each
of whom have participated in the crime. As before, each prisoner has an overwhelming incentive to
denounce the other prisoners, but now the likelihood of there being a close bond of friendship
between each of the 10 prisoners is significantly diminished. Furthermore, the risk faced by each of
the prisoners that one of his co-inmates will defect has now increased ninefold.
The likelihood that sympathy alone will be sufficient to generate the desired outcome will now, as
neoclassical choice theory rightly predicts, be small. But we can also look at the problem from the
other angle and say that in the event that the ten prisoners are able to reach agreement, then it is
likely that something other than pure self-interest was at work. Of course, we can never know with
certainty by observing a given outcome whether the motivation behind it was mutual sympathy, cold
calculus of self-interest or reason’s recognition of duty under the moral law. But, the same logic that
tells us that ten individuals are less likely than two to reach agreement through self-regarding
motives alone, also implies that, as the consensus broadens, the likelihood that sympathy alone is at
work decreases and, consequently, the likelihood that the observed consensus reflects some form of
reasoned meta-preferences or common good increases.
This idea is captured in the following lemma, which follows from definitions 1-2 and properties 1-5.
We will represent the consensus between a group ∈ of individuals, = (1, … , ), as , which
we define as follows: = ⋂ ≠ ∅. We designate by the set of all possible non-empty and
finite consensus choice sets, , held by groups ∈ . The cardinality of a group of individuals
= (1, … , ),. is designated by | | = .
Lemma 1: ∀( , ) ∈ , 1 < | |, | | < ∞, with individual preference defined as 1-2, and satisfying
properties 1-5: ⊃ ⇒ ( ) ≥ ( ).
See Annex 1 for a proof of Lemma 1.
In the limit, as tends to ∞, we can think of a hypothetical consensus involving an infinite number of
individuals with all possible combinations of preference relations. Such a consensus would, by
construction, need to coincide with a universal law of reason since that is the only law that any
reasonable individual could voluntarily be expected to subject themselves to, regardless of their
personal circumstances, inclinations and interests. In other words, the probability of utility-based
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interests converging over an infinite number of individuals is 0. This idea is represented by the
following corollary, which follows from Lemma 1:
Corollary 1: ∀ ∈ , = { | = (1, … . , )}, 1 ≤ < ∞, with individual preference defined as 1-2,
and satisfying properties 1-5: lim→ ( ∪ | ) = ( ) = 1.
See Annex 2 for a proof of Corollary 1.
Normative Meta-Rankings
Probability-Based Ranking
In an ideal world, we would have liked to be able to reach categorical (yes/no) conclusions regarding
reasonableness and, why not, make cardinal comparisons of value (e.g. decision is three time more
reasonable than decision ), in the same way as utilitarians would have wished to make cardinal
comparisons of utility. While this may not be possible, the transient and contingent truths generated
by the process of inter-rational validation should at least allow us to make ordinal binary
comparisons between outcomes based on the nature and extent of the consensus on collective
choices. Based on Lemma 1, we propose the following general re-formulation of Axiom (3), which we
will call the Axiom of Revealed Meta-preferences (ARM):
∀( , ) ∈ ∀ ∈ , . . = { | = (1, … , )}: ( | ) = (∙, )(4)
Where (∙, ) is a monotonically non-decreasing function of . The axiom of revealed meta-
preferences simply states that the probability that an observed consensus, , has been generated by
a meta-preferences will be a function of the number of individuals reaching consensus. An outcome
over which there is a consensus among members of a group will thus be said to be revealed
meta-preferred to outcome with probability ( | ). In this perspective, the original revealed
axiom becomes a special case of this general axiom, in which = 1. In this case, we know from
property 3 that | = | , meaning that the observed choice has equal probabilities of
having been caused by either motive. Other special cases that have been discussed above, include
the case in which → ∞, in which case corollary 1 tells us that ( | ) = 1. A final special case is
that in which there is no consensus, that is = 0. By definition 2, an outcome over which there is no
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consensus, such as the public provision of amusement parks in society , for instance, is one in which
we know that individuals have been unable to overcome their divergence of interests in order to
agree on a common objective. Hence, in this case, we have ( ) = 0. All other cases, such that
1 < < ∞ are covered by Lemma 1.
At the moment, the axiom does not make any normative claim. Hence, except in the hypothetical
case of a universal consensus, the fact that a group, , reveals a meta-preference for, say, healthcare
( ) over amusement parks ( ) does not necessarily mean that healthcare is absolutely and
universally a more valid social objective than amusement parks. Indeed, another society, ,may
reach the opposite consensus to support amusement parks instead of than healthcare. In order to
compare the choices of societies and , we need to turn the probabilistic logic of Lemma 1 into a
normative rule. If we consider that value stems from reason rather than from utility, it would be
logical to make the normative value of a given choice directly proportional on the probability that it is
caused by rather than , using the following axiom:
Probability-Based Ranking (PBR): ∀( , ) ∈ ∀( , ) ∈ ∶1. ≻ ⇔ ( | ) > |2. ∼ ⇔ ( | ) = |
(5)
Where ≽ , meaning “is at least as good as”, describes the normative ranking of social states– not to
be confused with the positive description of individuals’ actual preferences over those states,
denoted and . Hence, whereas indicated that a group meta-preferred to , ≽indicates that it is also, and for that reason, preferable from a normative point of view. We will
denote the asymmetric and symmetric elements of ≽ by ≻ and ∽, respectively.
When combined with definitions 1-2 and propositions 1-5, this yields the following proposition:
Proposition 1: If ≽ is a binary transitive ranking over satisfying PBR and ARM, and if properties 1-5
hold, then ∀ , ∈ ∀( , ) ∈ , . . 1 < | |, | | < ∞ ∶ | | > | | ⇒ ≽ .
See appendix 3 for a proof of proposition 1.
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It is important to note that we are ranking choice sets, rather than the options that may be contained
in the choices that have been made. Hence, if groups and both agree on the public provision of
healthcare, we could have ≻ if | | > | |, which means that the moral force of ’s preference
for healthcare is greater than that of ’s preferences for healthcare. The ranking will therefore be
transitive over but may not be transitive over .
As formulated here, the normative rule now allows us to resolve cases in which there is a
disagreement between different normative standards, such as, say a tribal or customary law and a
national law or international convention. Although we can never reach certainty that one is
universally and unconditionally superior to the other, the logic developed in this paper implies that
the international consensus reached about fundamental human rights, for instance, will be more
likely to approximate some version of a universal law of reason than, say, the norms that govern a
small isolated community or tribe. This is because the former involves agreement between a large
number of different individuals with widely divergent views, cultures and interests, whereas in the
latter case it is more likely that agreement may have been facilitated by a strong convergence of
interests and experiences. Consequently, international human rights norms will carry more
normative weight than customary or tribal law. By the same logic, a national constitution should also
carry more normative weight than national legislation, insofar as it is protected by additional
procedural guarantees (e.g. two thirds majority or double majority) that mean that a broader
consensus would be needed to reach agreement on the former than on the latter.
This need not mean that lower order norms are unfit for normative assessments. When judging each
others’ actions, members of the tribe or the village may be content with, and indeed prefer, doing so
in accordance with their own set of rules and norms that they have all consented to. However, when
comparing members of different tribes or communities, or when asked to judge the validity of a
tribal norm or custom such as female circumcision, we will need to appeal to higher order consensus,
such as a national legislation or international human rights.
Consensus-Based Ranking
Finally, we propose an alternative normative ranking rule that captures the rationale of the above
discussion, but presents the advantage of being based entirely on the external and thus observable
properties of the consensus reached between different agents. Needless to say that the normative
power of this ranking ultimately is derived from the assumed link of consensuses to underlying meta-
preferences, much in the same way as the normative power of choice in the neoclassical framework
was derived from its assumed link to underlying utility. By avoiding references to unobservable
Draft: 27/05/2011 -15- © Sebastian Silva-Leander
properties of the individual decision-making mechanism, however, we make it possible to centre the
axiomatic discussion on the desirable political properties of a consensus-based ranking. We propose
the four following simple axioms to characterise our ranking:
Equality (EQU):
∀( , ) ∈ : ∽(6)
Monotonicity (MON):
∀( , ) ∈ , . . ( , ) ≠ ∅: ⊃ ⇒ ≻(7)
Independence (IND):
∀( , ) ∈ , ∀ℎ ∈ \( ⋃ ), . . ⋂ ≠ ∅, ⋂ ≠ ∅: ≽ ⇒ ⋂ ≽ ⋂(8)
Irrelevance of non-consensual preferences (INC):
∀( , ) ∈ , . . ⊃ ∀ℎ ∈ \ : ⋂ = ∅ ⇒ ⋂ ∼ ⋂(9)
The EQU axiom states that the choice of any given individual has the same normative force as that of
any other individual. The MON axiom states that the consensus reached by any given group of
individuals has more normative force than the consensus reached by any of its subgroups. IND states
that a ranking between two consensuses will not be affected by the inclusion of an additional
individual who agrees with both. These axioms echo fairly closely the axioms proposed by Pattanaik
and Xu (Pattanaik & Xu, 1990) to characterise a cardinal ranking rule. However, they exclude cases in
which individuals are unable to reach consensus. To complete the ranking we therefore need to add
a fourth axiom, to deal with those cases. We do this by using the INC axiom, which states that the
expansion of a group that has been unable to reach consensus will not increase the group’s
normative power.
Based on these four axioms, we are able to construct a ranking rule that is dependent solely on the
number of individuals reaching consensus.
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Theorem 1: If ≽ is a binary and transitive ranking over , satisfying EQU, MON, IND and INC, then
∀ ( , ) ∈ ∀( , ) ∈ ∶ ≽ ⇔ | | ≥ | |.
See Annex 4 for a proof of theorem 1.
It is easiest to think of and as sets of individuals (e.g. electorates). However, since the consensus-
based ranking is not directly dependent on the properties of individual motivational structures, it is
also possible to think of a different unit, such that and representing sets of countries, for instance.
This would, for instance, be relevant for international human rights legislation, where currently
countries have equal weights in the decision-making process regardless of their population. Based on
this logic, we could for instance state that the Universal Declaration of Human Rights’ endorsement
of the right to healthcare carries more normative force than the recognition of the same right in the
European Convention on Human Rights, since the group having agreed on the latter (all European
countries) is a strict subset of the group having agreed on the former (all countries in the World).
From a strict perspective of revealed meta-preferences, however, as well as from the perspective of
other normative frameworks, such decision-making mechanisms that does not take the individual as
its starting point may violate the principle of equality of individuals.
Finally, we should note that by taking the existing consensus as the starting point of the assessment,
and thus disregarding the decision-making process, we are implicitly imposing a number of
assumptions that will need to be clarified. In particular, we have assumed that the decisions reached
reflect a genuine consensus in the sense that no individual has been subjected to any form of
coercion, intimidation or deceit. We may also need to add assumptions regarding the nature of the
decision makers. In particular, we need to assume that no individual has more power than other and
can influence the decisions of his followers. Finally, we are, of course, assuming that everyone has
access to the same level of information and has sufficient information to make free and informed
decisions. All of these assumptions may be optimistic and none may in fact ever be met in reality.
However, they are not fundamentally less plausible than the basic set of assumptions needed to
prove the fundamental theorems of welfare economics. In fact these assumptions echo quite well
the ones that characterise the Weberian general equilibrium, such as perfect and competitive
markets with perfect information and no monopolies. As such, they may provide an adequate basis
for exploring the conditions under which these ideals may be realised and the reasons for which they
fail to do so.
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Conclusion
Let us conclude this discussion by attempting an answer to the valid query that had been raised
about measures, such as the Human Development Index or Multidimensional Poverty Index, which
attempt to capture key insights of Sen’s so-called “capability approach”. What, Ravallion asked,
would be acceptable normative standards for selecting and weighting indicators in such an index?
The argument developed in this paper allows us, first of all, to discard the use of market prices as
acceptable normative weights in Sen’s framework. Market prices are the product of an iterative
process involving a very large number of individual decisions taken by consumers, individually, based
on tastes, needs and perhaps even moral concerns (and/or producers individually, based on
production costs, etc.). At no point does this process require individuals to reach out and seek
compromise with other individuals who have different interests and inclinations. Consequently,
market prices do not tell us anything about people’s reasoned assessments of the various options
they face, and may, at best, provide some fragments of information about individuals’ marginal rates
of utility substitutions between different options. From the perspective of Sen’s theory, this thus
undermines the normative justification for using a measure, such as GDP, which effectively rely on
the market price to weigh different goods or dimensions of wellbeing.
Let us now consider another measure that assigns relative monetary values to different goods,
namely a public budget that has been voted by a democratically elected parliament. Unlike market
prices, a budget will required lengthy discussions and a process of inter-rational validation in order to
agree on which priorities the community should invest in an in what proportions. The weights
provided y public spending ratios in a national budget could therefore, unlike market prices,
constitute a valid normative standard in the construction of an aggregate index.
It is possible to think of other standards that may carry information about weights. The Millennium
Declaration, for instance, describes an implicit weighting system, as it defines eight equally important
goals, each of which is composed of a number of targets, which in turn are measured by various
indicators. The Universal Declaration of Human Rights also makes reference to weights and even to
elasticities of substitution when it states that all rights are indivisible and inalienable. More
importantly, it also implicitly assigns a weight of zero to a large number of rights, namely those that
are not included in the declaration.
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Which of these standards should be used in constructing a normative index of welfare will depend on
the purpose and the scope of the assessment. A national budget may, for instance, provide an
adequate standard for a punctual assessment of national policies, whereas a more demanding
standard, such as the Universal Declaration, may be required when making a comparison between
various countries and over several years. Finally, an assessment geared specifically towards
development issues, may be content with a more focused normative standard, such as the
Millennium Declaration.
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Annex 1: Proof of Lemma 1
Let ( , ) ∈ be two groups of individuals, such that ⊂ and 1 < | |, | | < ∞. Further, let
= \ .
Definition 1 combined with property 1 imply 0 < ( | ) < 1 for any > 1 since we
know that exists (property 1) and that there is at least one pair of individuals ( , ) ∈ for which ≠ .
Similarly, we know from property 1 that 0 < ( | ) ≤ 1 , since ⊂ , which implies
| | > | |, and hence, | | − | | > 0.
Consequently, and assuming independence across individuals (property 5), we have :
( | ) = | × ( | ) ≤ |(10)
Let us now look at the implication for ( | ). We know that:
( ⋃ | ) = ⋃ ⋂( ⋃ )| ⋂(11)
Applying the second distributive law,(11) is equivalent to:
( ⋃ | ) = ⋂ ⋃( ⋂ )| ⋂(12)
By the general property of additivity, (12) is equivalent to:
( ⋃ | ) = ⋂ | ⋂ + ( ⋂ )| ⋂ − ⋂ ⋂( ⋂ )| ⋂(13)
By the property of commutativity, (13) is equivalent to:
( ⋃ | ) = ⋂ | ⋂ + ( ⋂ )| ⋂ − ⋂ ⋂( ⋂ )| ⋂(14)
By the property of independence (property 5), (14) becomes:
Draft: 27/05/2011 -20- © Sebastian Silva-Leander
( ⋃ | ) = | ( | ) + ( | ) ( | ) − ⋂ | ( ⋂ | )(15)
By the property of completeness (property 2), we then have:
( ⋃ | ) = | ( | ) + ( | ) ( | ) − ⋂ | ( ⋂ | ) = 1(16)
We can now solve for ( | ) ( | ):
( | ) ( | ) = 1 − | ( | ) + ⋂ | ( ⋂ | )We know from (10) that | × ( | ) ≤ | . Furthermore, by definition of an
intersection, we know that ⋂ | ≤ | < 1 and ( ⋂ | ) ≤ ( | ) ≤ 1.
Consequently, we have:
( | ) ( | ) = 1 − | ( | ) + ⋂ | ( ⋂ | )≥ 1 − | + ⋂ | = ( | )
By property 5, finally, we have:
( | ) × ( | ) = ( ⋂ | ⋂ ) = ( | )And consequently,
( | ) ≥ ( | )
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Annex 2: Proof of corollary 1
Let ⋂ denote the consensus between two individuals, and , on the preferable choice in ,
such that ( , ) ∈ , ≠ and ⋂ ≠ ∅. For ⋂ to be non-empty, it must be the case that the
preferences of and overlap such that both and would choose over , either because they
derive utility from the same outcomes, hence and have both occurred, or because they are
using their meta-rankings, hence has occurred for both and , or a combination of these two.
By the general property of additivity of probabilities, we can write the probability of any of these
events having occurred as:
( ∪ | ) = | + ( | ) − ( ∩ | )(17)
Using the second distributive law, we can write the probability of this event having occurred jointly
for and as:
( ⋃ )⋂( ⋃ )| ⋂ = ( ⋂ )⋃ ⋂ | ⋂(18)
By the general property of additivity of probabilities (18) is equivalent to:
( ⋂ )⋃ ⋂ | ⋂= ⋂ | ⋂ + ⋂ | ⋂ − ⋂ ⋂ ⋂ | ⋂
(19)
By the property of commutativity, (19) is equivalent to:
( ⋂ )⋃ ⋂ | ⋂= ⋂ | ⋂ + ⋂ | ⋂ − ⋂ ⋂ ⋂ | ⋂
(20)
By applying property 5, we can re-write (20) as:
( ⋂ )⋃ ⋂ | ⋂= | | + | | − ⋂ | ⋂ |
(21)
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This can be generalised for the case of a group of individuals as:
( ∪ | ) = ( | ) + ( | ) − ( ⋂ | )
(22)
Where describes the event that simultaneously for all ∈ . In other words it describes the
case in which there is a concordance of interests, sympathy etc. describes the event that
simultaneously for all ∈ . The limit of equation (22) as tends towards ∞, is:
lim→ ( ∪ | ) = lim→ ( | ) + lim→ ( | ) − lim→ ( ⋂ | )
(23)
From definition 2, we know that is equal for all ∈ , which means that ⋂ = = .
Definition 1 tells us that ⋂ = ∅ , since ≠ for some ( , ) ∈ . Hence, as tends to
infinity, lim→ ∏ ( | ) = (∅) = 0 and lim→ ∏ ( ⋂ | ) = (∅⋂ | ) =(∅) = 0. Consequently, in the limit, equation (23) will boil down to:
lim→ ( ∪ | ) = ( | )(24)
Since must have been motivated either by utility-based preferences or by reasoned meta-
preferences (property 2), we have lim→ ( ∪ | ) = ( ) = 1.
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Appendix 3: Proof of Proposition 1
Let ( , ) ∈ be two groups of individuals, such that | | > | | and let ⊂ such that | | = | |.By Lemma 1, we have:
( | ) ≥ ( | )By definition of ≽ and ≥, we have ≻⇒≽ and >⇒≥.
We can thus apply PBR1 to get ( | ) ≥ ( | ) ⇒ ≽ .
Further, we know that | | = | | > 1 . We can therefore choose two pairs of individuals
((ℎ, ), ( , )) ∈ , such that (ℎ, ) ∈ and ( , ) ∈ .
By property 4, we have ∀( , ) ∈ : | = | and ∀(ℎ, ) ∈ : | =| .
By property 5, we can combine these two pairs to obtain: ⋂ | ⋂ = | ×| = | × | = ⋂ | ⋂ .
Similarly, by using properties 4 and 5, we can add all remaining elements in and without
affecting the equality, to obtain: | = | .
By PBR2, we then have | = | ⇒ ∼ .
Since ≽ is transitive over : ≽ ∼ ⇒ ≽ .
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Appendix 4: Proof of Theorem 1.
It is easy to see that a ranking of consensus sets in terms of the number of individuals reaching
consensus would be transitive and would satisfy (EQU), (POC), (IND) and (INC).
In order to prove the theorem, we need to prove that for any binary transitive ranking ≽ satisfying
the four axioms:
∀( , ) ∈ , , ≠ ∅: | | = | | ⇒ ∼(25)
And
∀( , ) ∈ , , ≠ ∅: | | > | | ⇒ ≻(26)
Let us start with a proof of (25) by induction:
Axiom EQU gives us the basis of the induction for the case where = 1, as it tells us that for any
individuals ({ }, { }) ∈ : ∽ . Our inductive hypothesis is that this holds for all . In order to
prove the inductive step, we need to show that this holds true also for + 1. Consider the sets
( , ) ∈ , such that ≠ ∅ and ≠ ∅.
Let | | = | | = + 1 , with 1 ≤ ≤ | |. Further, let { } ∈ and let ⊂ , such that = \{ }.
Since | | = + 1 and since |{ }| = 1 by definition of a singleton, it follows that | | = .
Logically, { } must either be a member of or not be a member of . If { } ∈ , we know, by
construction of that ⋂ ≠ ∅. However, if { } ∉ it can either be the case that ⋂ ≠ ∅ or
that ⋂ = ∅. This gives rise to 3 possible cases:
Case 1 ({ } ∈ ):
Let ⊂ such that = \{ }.
By construction of and we know that ≠ ∅ and ≠ ∅.
Since | | = + 1 and |{ }| = 1, it follows that | | = .
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Since we know that | | = = | |, we can apply the inductive hypothesis to get ∽ .
By applying the IND axiom, we then get: ⋂ ∽ ⋂ , which,
By construction of and is equivalent to ∽ .
Case 2 ({ } ∉ ⋂ ≠ ∅):
If { } ∉ and | | = | | = + 1, then there must be a { } ∈ , such that { } ∉ .
Let ⊂ such that = \{ }.
By construction we know that ≠ ∅.
Since| | = + 1 and |{ }| = 1 by definition of a singleton, it follows that | | = .
We can thus apply the inductive hypothesis to get: ∽ , and then
Apply IND to obtain: ⋂ ∽ ⋂ .
By construction of , this is equivalent to ∽ ⋂ .
Let { } ∈ .
Since ⊂ it follows from { } ∉ that { } ∉ , and from { } ∈ that { } ∈ .
Consequently, we have: |( \{ })⋃{ }| = and | \{ }| = .
By applying the induction hypothesis, we get: \ ⋂ ∽ \ .
We can then apply IND to obtain: ⋂ ∽ .
We have now shown that ∽ ⋂ and that ⋂ ∽ .
Since ≽ is transitive, this implies that: ∽ .
Case 3 ({ } ∉ ⋂ = ∅):
This is a special case in which, by definition of a non-consensus, we have = 0 for ⋂ . So we need
to show that the indifference holds for all = 0 and + 1 = 1.
If it is the case that ⋂ = ∅, then there must be at least one element, { } ∈ , such that
⋂ = ∅.
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By INC, we have: ⋂ ∼ ⋂Since we know that ∈ , and that ⋂ = ∅ we can also use INC to expand around to
obtain: ⋂ ∼ ⋂ .
Since ≽ is transitive on , we have: ⋂ ∼ ⋂ ⋂ ∼ ⋂ ⇒ ⋂ ∼ ⋂The choice sets of individuals and can be compared using the EQU axiom, which yields: ∼ . By
definition of a singleton consensus set, we have: ⋂ ≠ ∅ and ⋂ ≠ ∅ , which covers the case
when = 1.
Proof for (26):
Let | | > | |. Now, let ⊂ , such that | | = | |.
By construction of , we know that ≠ ∅.
By (25), we have: ∼ .
Further, by MON, we have: ≻Finally, we can apply transitivity of ≽ to obtain: ≻ .
QED.
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