Strategy-proof fuzzy aggregation rules

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Abstract

We investigate the structure of fuzzy aggregation rules which, for every permissible profile of fuzzy individual preferences, specify a fuzzy social preference. We show that all fuzzy aggregation rules which are strategy-proof and satisfy a minimal range condition are dictatorial. In other words, there is an individual whose fuzzy preferences determine the entire fuzzy social ranking at every profile in the domain of the aggregation rule. To prove this theorem, we show that all fuzzy aggregation rules which are strategy-proof and satisfy the minimal range condition must also satisfy counterparts of independence of irrelevant alternatives and the Pareto criterion. There has been hardly any treatment of the manipulability problem in the literature on social choice with fuzzy preferences.

Introduction

According to philosophers, natural language is infected with vague predicates. For example, predicates such as “thin”, “red” and “tall” are commonly taken to be vague. These predicates are “vague” because they admit borderline cases. Some people are borderline thin, for example. Some reddish-orange patches of colour are borderline red, and so on. The existence of vagueness implies that many of the predicates we commonly use lack sharply defined boundaries. For example, on a scale of height, there is no sharp boundary between people who are tall and people who are not tall. The boundaries of vague predicates are blurred, not sharp.1

According to the standard view in philosophy, vagueness creates problems for classical logic.2 For example, if Jim is borderline thin then the sentence “Jim is thin” is neither true nor false, violating the principle of bivalence. The law of excluded middle, that “Jim is thin or Jim is not thin” is true, seems suspect too.

Several theories of vagueness have been proposed. The one most commonly used by economists is fuzzy set theory.3 But why should economists be interested in vagueness? One argument put forward in the social choice literature is that preference relations are vague.4 This is an interestingly claim since most philosophical studies of vagueness assume that predicates are the semantic concepts infected with vagueness, not relations. But what do these economists mean when they say that preference relations are vague? Here are two examples of what they have in mind.

Imagine that you are comparing two possible jobs, and what you care about is how they fare with respect to salary and excitement. Imagine too that one job offers a higher salary than the other, but is less exciting. Which job would you prefer? Often it is hard to say, but not always. For example, one job could offer a much higher salary than the other, and yet only be marginally less exciting. In such cases, it seems plausible that you would prefer the better paid job to the more exciting one. The reason for this is that you are probably willing to trade-off slightly less excitement in order to obtain a much higher income. Unfortunately, things are not always as straightforward as this. For example, one job could offer a much higher salary than the other, and yet be much less exciting too.

In cases like this, it is extremely difficult to put the jobs in a clear order. This may be due to the existence of unresolved conflict in the mind of the decision maker. How does this conflict manifest itself? One way is through feeling that your preferences are “divided” in some sense. For instance, in this example, you might feel that to some extent the better paid job is at least as good as the more exciting job. However, at the same time, you might also feel that to some extent the more exciting job is at least as good as the better paid job.

Here is another example, inspired by an argument made by Sen (1970). Imagine that a social planner is attempting to rank two social states, A and B, in which only two people live. Suppose that person i is better-off in A than in B, but that person j is better-off in B than in A. In other words, A and B cannot be ranked by the traditional Pareto principle. Does this mean that the planner cannot put the states in order? Quite rightly, Sen argues no. For example, it might be the case that state A is much better for person i than state B, and yet only slightly worse for person j. If so, it would be reasonable for the planner to place A above B in the social ordering. In other words, the planner could judge that the gains to person i in state A outweigh the losses to person j.

However, as Sen points out, this does not mean that it is always possible to rank Pareto incomparable states. For example, if the gains to person i in A are “similar” to the losses to person j, then the planner may conclude that it is extremely difficult to put the states in a clear order.5 In cases like this, the planner may feel that his preferences are “divided” in some sense. In other words, the planner may feel that to some extent state A is at least as good as state B. However, at the same time, the planner may also feel that to some extent state B is at least as good as state A.

These examples appeal to the fact that preferences are often “conflicting” or “ambiguous” in some sense.6 Put more simply, we might just say that in these examples preferences are vague. Fuzzy set theory has been used by economists to represent preferences like these.

A fuzzy set is the extension of a vague predicate. So if “red” is vague then the set of red objects is a fuzzy set. To be precise, let X denote the universal set and let W denote a subset of X in the classical sense, WX. The set W is characterized by a function fW:X{0,1} where fW(x)=1 for xW, and fW(x)=0 for xW. Given xX, fW(x) is the degree to which x belongs to W. The generalization to a fuzzy set occurs by permitting this degree to take more than two values, typically by allowing any value in [0,1].7 So a fuzzy subset G of X is characterized by a function fG:X[0,1]. If fG(x)=1 then x “clearly” belongs to G, and if fG(x)=0 then x “clearly does not” belong to G. In between there are varying degrees of belonging. So if G is a vague property and if x is borderline G, then x’s degree of G ness is some number between 0 and 1.

A fuzzy (binary) relation F defined on X is characterized by a function fF:X×X[0,1]. If the semantic concept this fuzzy relation is meant to represent is (weak) preference, then fF(x,y) can be interpreted as the degree of confidence that “x is at least as good as y”. This is not the only possible interpretation of fF(x,y) but it is the one we prefer. Fuzzy relations can be used to represent vague preferences, since the degree to which you are confident that one alternative is at least as good as another is represented by a number in [0,1].8

It is interesting to know whether economic theory can be reconstructed on the basis of fuzzy preferences. The literature on this topic is now quite extensive. For instance, if individuals have fuzzy preferences then how do they make choices?9 Can we demonstrate that a competitive equilibrium exists and prove counterparts to the first and second welfare theorems?10 Does Arrow’s impossibility theorem hold in the context of fuzzy preferences?11 This paper makes a contribution to this growing body of work.

Like much of the literature, our focus is on social choice theory.

We investigate the structure of fuzzy aggregation rules which, for every permissible profile of fuzzy individual preferences, specify a fuzzy social preference. We show that all fuzzy aggregation rules which are strategy-proof and satisfy a minimal range condition are dictatorial. In other words, there is an individual whose fuzzy preferences determine the entire fuzzy social ranking at every profile in the domain of the aggregation rule. To prove this theorem, we show that all fuzzy aggregation rules which are strategy-proof and satisfy the minimal range condition must also satisfy counterparts of independence of irrelevant alternatives and the Pareto criterion. There has been hardly any treatment of the manipulability problem in the literature on social choice with fuzzy preferences.12

Of course, this result is the celebrated Gibbard–Satterthwaite theorem but in the context of fuzzy preferences and fuzzy aggregation rules.13

What is responsible for the impossibility theorem in this paper? The minimal range condition is mild. It stipulates that, for each pair of social alternatives, two profiles exist in the domain of the fuzzy aggregation rule which produce different social values in {0,1} for this pair. In other words, for any pair of social alternatives, a and b say, a profile exists in the domain of the fuzzy aggregation rule which ensures that society is confident to degree one that “a is at least as good as b”. In addition, a profile exists in the domain of the fuzzy aggregation rule which ensures that society is confident to degree zero that “a is at least as good as b”.

This condition rules out fuzzy aggregation rules which assign constant values to pairs of alternatives, irrespective of individual preferences.

Given this, the real culprits behind the theorem are probably our strategy-proofness condition and the assumption we make about the transitivity of fuzzy preferences.

Let us deal with strategy-proofness first. On first impressions, it does seem desirable for fuzzy aggregation rules to be immune to preference misrepresentation. Indeed, this normative position seems to be implicitly accepted in the large literature on strategy-proof social choice.14 If we accept this, then obviously we need some way of prohibiting profitable misrepresentation within the framework of fuzzy aggregation. How do we accomplish this?

The condition we employ can be described as follows. Take any pair of alternatives (a,b) and any fuzzy preference profile in which you truthfully report your preferences. At this profile you are confident to some degree that “a is at least as good as b”. However, imagine that at this profile the fuzzy aggregation rule assigns a larger social degree of confidence to (a,b) than the one you happen to hold. Then, if the fuzzy aggregation rule is strategy-proof, whenever you misrepresent your preferences, the social degree assigned to (a,b) will either rise, or remain constant. Conversely, if the social degree assigned to (a,b) is smaller than your individual (a,b) value, whenever you misrepresent your preferences, the social degree assigned to (a,b) will either fall, or remain constant.

Furthermore, this condition holds for all pairs of alternatives, and not just (a,b). It also holds for all individuals, and not just you.

Loosely speaking, what this means is as follows. Whenever someone unilaterally switches from telling the truth to lying, the fuzzy social ranking moves “at least as far away” from their truthful ranking as was initially the case. In other words, whenever someone misrepresents their preferences, the “distance” between their truthful ranking and the social ranking (weakly) increases. In such circumstances, individuals do not gain by misrepresenting their preferences. We say that a fuzzy aggregation rule is strategy-proof if and only if it satisfies this property.

Obviously there may be other ways of formulating a strategy-proofness condition within the framework of fuzzy aggregation, but this one strikes us as a natural place to start. As we remarked earlier, the study of misrepresentation within the fuzzy framework is in its infancy, and so we leave to future work the question of whether weakening this condition permits a possibility result.

We now deal with transitivity. As with all papers that deal with fuzzy preferences, the way the transitivity condition is formulated is crucial. In this paper, our fuzzy preferences (both individual and social) are assumed to satisfy “max-min” transitivity. This is probably the most widely used transitivity assumption in the literature on fuzzy relations, although it is somewhat controversial.15 It is certainly the case that weaker transitivity conditions exist. However, we are aware of no experimental evidence which shows that people with fuzzy preferences do not satisfy max-min transitivity. In the absence of such evidence, we feel that this condition cannot simply be dismissed out of hand.16 This fact, combined with its status as the most widely used condition in the literature, means that max-min transitivity is a natural place to start.

As with our strategy-proofness condition, we leave the question of weakening transitivity to future research. This approach keeps us firmly in the tradition of social choice theory, in which impossibility theorems are established in particular axiomatic frameworks. These frameworks are subsequently revised to see if possibility theorems can emerge.

The paper is organized as follows. Section 2 contains notation and definitions, including the important definition of an “Arrovian” fuzzy aggregation rule. In Section 3, we prove that all Arrovian fuzzy aggregation rules are dictatorial. In Section 4, we introduce our definition of a strategy-proof fuzzy aggregation rule with minimal range, and prove that all such rules are Arrovian. Our theorem follows immediately.

Section snippets

Preliminaries

Let A be a set of social alternatives with #A3. Let N={1,,n},n2, be a finite set of individuals. Ā={(a,b)A2|ab} is the set of ordered pairs of distinct social alternatives.

Definition 1

A fuzzy binary relation (FBR) over Ā is a function f:Ā[0,1]. An exact binary relation over Ā is an FBR g such that g(Ā){0,1}.

Let T denote the set of all FBRs over Ā.

You will notice that our fuzzy binary relations are defined over Ā and not A2. This is rather non-standard. However, we have done this in order to avoid

Arrovian FARs are dictatorial

In what follows wn denotes the vector with w[0,1] in all n places.

Lemma 1

An Arrovian FAR Ψ is neutral.

Proof

  • Case 1:

    If (a,b)=(c,d) then the result follows immediately from the fact that Ψ is Arrovian.

  • Case 2:

    (a,b),(a,d)Ā. Take (r1,,rn)Hn such that rN(b,d)=1n. Then PC implies that r(b,d)=1. Since r is max-min transitive, we have r(a,d)min{r(a,b),r(b,d)}. Therefore, r(a,d)min{r(a,b),1} and so r(a,d)r(a,b).

    In addition, since rN(b,d)=1n and individual preferences are max-min transitive, it follows that rN(a,d)rN(a,b).

Strategy-proof FARs

We denote by (r1,,ri,,rn)Hn the profile obtained from (r1,,ri,,rn) when individual i replaces riH with riH.

We write riri=Ψ(r1,,ri,,rn) and riri{a,b} denotes the restriction of riri to (a,b). Similarly, rijrirj=Ψ(r1,,ri,,rj,rn) and rijrirj{a,b} denotes the restriction of rijrirj to (a,b).

Definition 7

An FAR Ψ is strategy-proof if and only if it satisfies the following property.

  • (SP)

    For all (a,b)Ā, all (r1,,rn)Hn, all iN and all riH, both (i) and (ii) hold:

  • (i)

    r(a,b)<r

Acknowledgement

We are grateful to seminar participants at the University of Bath, University of Birmingham, University of East Anglia, Universidad de Murcia, Universität Osnabrück, University of St. Andrews and the Université de Caen for helpful comments and suggestions. We would also like to thank a referee of this journal. Financial support from MEC/FEDER Grant SEJ2004-07552 and the NUI Galway Millennium Fund is gratefully acknowledged.

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