Rev. Econ. Design (2014) 18:3–10DOI 10.1007/s10058-014-0158-1

LEAD ARTICLE

Intransitive social indifference and the Arrow dilemma

Allan F. Gibbard

Published online: 4 February 2014© Springer-Verlag Berlin Heidelberg 2014

Abstract This paper deals with a proposal by Frederic Schick for resolving Arrow’sdilemma of social choice. Schick challenges Arrow’s assumption that rational socialindifference is transitive. It turns out, however, that even if we drop this assumption, westill face a dilemma as embarrassing as Arrow’s. Any constitution which satisfies theremaining conditions is a liberum veto oligarchy. This means a unique set of individualscarries any issue on which it is unanimous, and engenders social indifference on anyissue which divides it. A formal statement and proof of this theorem follows an informalpresentation of the result.

Keywords Arrow’s theorem · Oligarchy · Quasitransitivity

JEL Classification D71

Arrow’s well-known book (Arrow 1963) bears an unwelcome message. Apparentlythere is no rational way to amalgamate individual preferences into a scheme of socialpriorities. Call such an amalgamation of individual preferences a constitution. To

This paper was written in the 1969–1970 academic year when I was an Assistant Professor of Philosophyat the University of Chicago. It is based on a term paper (Gibbard 2014) that I wrote for theArrow–Rawls–Sen seminar at Harvard University in the Autumn term of 1968–1969. For a discussion andhistory of both papers, see Weymark (2014). Numbered footnotes are from the original manuscript.Symbolic footnotes have been added by the Associate Editor, John Weymark. Minor errors have beensilently corrected. I am very grateful to John Weymark for arranging for this publication and for hiscareful editorial work, including his wonderful explanatory notes. I am also grateful to Nicholas Baigentfor identifying some of the minor errors alluded to above.

A. F. Gibbard (B)Department of Philosophy, University of Michigan, Ann Arbor, MI 48109, USAe-mail: gibbard@umich.edu

123

4 A. F. Gibbard

show that there can be no rational constitution, Arrow suggests several conditions anyrational constitution should satisfy, and then proves that no possible constitution doessatisfy them all. The result seems paradoxical, for each condition looks desirable andinnocuous. Arrow’s proof shows definitively, though, that if each of the Arrow condi-tions indeed is necessary for constitutional rationality, then no rational constitution ispossible.

To resolve Arrow’s dilemma, then we would have to show at least one of theArrow conditions unnecessary for rationality in constitutions. Frederic Schick (1969)suggests the hitch is in a condition Arrow calls “Collective Rationality”.1 I shall statethe condition later; at this point I keep to informal exposition. According to Arrow, ifsocial choice is rational, it must be possible to reconstruct it in the following way. On thebasis of the individual preference ordering, society first orders all the alternatives whichmight be available to it, without regard to which alternatives actually are available toit. Call this ordering the social preference ordering. Then given the set S of availablealternatives, a social choice is acceptable if out of all alternatives in S, it comes outtop, or tied for top, in the social preference ordering.

Schick accepts that rational social choice must be derivable from a social preferencerelation among all the alternatives which might be available. He accepts also thatthe social preference relation between any two alternatives which might be availableshould not depend on which alternatives are actually available. He denies, however,that a rational social preference relation must always be an ordering. An ordering ofa set V is a two place relation which for all x, y, and z ∈ V satisfies the followingconditions:

(asymmetry of preference) x Py ⊃ ∼y Px

(transitivity of preference) x Py · y Pz ⊃ x Pz

(transitivity of indifference) x I y · y I z ⊃ x I z

where ‘x I y’ means ∼x Py∼y Px .∗ Schick bogles at the third condition. He acceptsthe first two: he agrees that rational social preference is asymmetrical and transitive. Ofthe requirements in Arrow’s Collective Rationality condition, Schick rejects only tran-sitivity of social indifference.2,† He denies that social indifference must be transitiveif it is to be rational.

But without transitivity of social indifference, Arrow’s proof does not go through.Hence to deny transitivity of rational social indifference is to deny that Arrow’s proofshows a rational constitution impossible. If Schick is right, Arrow’s proof is of nogreat significance.

I shall not examine Schick’s grounds for denying that social rationality requiressocial indifference to be transitive. My thesis here is that intransitive indifference will

1 I use the terminology of Arrow (1967, p. 225) since in Arrow (1963) this condition is not labelled.∗ The symbols ⊃, ∼, and · mean ‘implies’, ‘not’, and ‘and’, respectively.2 I was introduced to the idea of dropping transitivity of social indifference from the Arrow conditions byAmartya K. Sen. I originally develeloped this paper in connection with unpublished work by Sen.† The unpublished work referred to in footnote 2 was published as Sen (1969). See also Sen (1970).

123

Intransitive social indifference 5

not free us from the Arrow dilemma in any helpful sense. True, not only does Arrow’sparticular proof require transitivity of social indifference, but the theorem with transi-tivity of social indifference deleted is false.3,∗ There is a constitution which satisfiesthe remaining conditions. This, however, is no cause for jubilation. The constitutionswhich satisfy the Arrow conditions without transitivity of social indifference are of apeculiarly unpalatable kind, as I shall show. Hence even without transitivity of socialindifference the Arrow conditions have consequences which are hard to swallow.

The only constitutions which satisfy the Arrow conditions minus transitivity ofsocial indifference are what I shall call liberum veto oligrachies. A constitution is aliberum veto oligarchy if there is a set A of individuals to whom it gives the followingpowers. (1) Whenever everyone in A prefers x to y, society prefers x to y. (2) Ifsomeone in A prefers x to y, then society does not prefer y to x . As a consequence, ifsomeone in A prefers x to y and someone else in A prefers y to x , then society prefersneither, and is indifferent. The oligarchs control society, then, and any one of themcan veto a social preference. The purpose of this paper is to prove my assertion thatliberum veto oligarchies are the only constitutions which satisfy the Arrow conditionswith transitivity of social preference deleted.

A liberum veto oligarchy is a terrible way to govern a large group. If the set ofoligarchs is small, the system is as undemocratic as my name for it suggests, for a smallgroup can get its way whatever anyone outside the group wants. If the set of oligarchsis large, the system is more democratic but likely to be paralyzed. Numerous oligarchswill rarely be unanimous, and when they are not, society will have no preference. Inthe egalitarian case when everyone is an “oligarch”, no one’s preference can ever beover-ridden in forming the social preference, and social decision is paralyzed by theslightest controversy. Thus, however large or small the set of oligarchs, a liberum vetooligarchy is a poor system.

In short, a liberum veto oligarchy is not a rational constitution, but only a liberumveto oligarchy satisfies the Arrow conditions minus transitivity of social indifference.This leaves us where we started. Either we must deny that a consitution can be rational,or we must reject one of the Arrow conditions Schick accepts. Schick’s suggestionleaves us in a constitutional quandry as great as Arrow’s.

This ends the informal exposition of the result I shall prove and its significance. Ishall now expound the problem, state the theorem, and offer a proof.

Take a society σ of n individuals, whom we designate with numbers 1, . . . , n. LetU be the set of all alternatives which might be available to the society σ . Suppose, now,that for each individual 1 to n we have an ordering Pi of alternatives in U accordingto his preferences. Call such an ordering a ballot, and call the n-tuple 〈P1, . . . , Pn〉 ofballots cast by individuals 1, . . . , n an election. (I shall use ‘P’ for 〈P1, . . . , Pn〉, ‘P′’for 〈P ′

1, . . . , P ′n〉, and so forth.) A constitution is a rule for choosing among alternatives

available to society on the basis of an election. For any election and any non-empty setS of available alternatives, the constitution designates at least one member of S as anacceptable social choice. In other words, a constitution is a function c(S, P) defined

3 Sen has proved this independently, in unpublished work.∗ Sen’s result referred to in footnote 3 appeared in Sen (1969, 1970).

123

6 A. F. Gibbard

for all non-empty subsets S of U and all elections P. Its value is always a non-emptyset of available alternatives. Formally, for every S, P in the domain of c,

c(S, P) = ∅. (1)

c(S, P) ⊆ S. (2)

The Arrow conditions on c are as follows:

I. Collective Rationality. This condition contains three parts, which I shall state. Itsays that the social choice is generated by a social ordering of U which is a function ofthe election P alone. The social ordering P , or f (P), is thus independent of S, the setof available alternatives. Part (a) asserts the existence of a social preference relationdependent on P alone, and says that the social choice set c(S, P) is the set of elementsin S to which nothing in S is socially preferred. Parts (b) and (c) state that the socialpreference relation f (P) is an ordering. Hence parts (b) and (c) presuppose part (a).In this paper, I discuss the consequences of denying part (c).

(a) Generation (by a Social Preference Relation). For every election P, there isa social preference relation P = f (P) such that∗

x ∈ c(S, P) iff x ∈ S · y ∈ S ⊃y ∼y Px . (3)

It follows that f (P) is asymmetrical. Otherwise, let P = f (P) and suppose x Py ·y Px . Then by (3), c({x, y}), P) is empty, contrary to (1). Hence f (P) satisfies thefirst of the three defining conditions of an ordering. There remain

(b) Transitivity of Social Preference. For every P, f (P) is transitive.

(c) Transitivity of Social Indifference. Let P = f (P) and let x I y mean∼x Py∼y Px . Then I is transitive.

The remaining Arrow conditions can be stated without assuming the Generationcondition I(a). The exposition is simplified, however, by placing conditions on thefunction f posited by I(a), rather than placing them directly on c. For that reason, Ishall follow Arrow’s example in Arrow (1963), assume I(a), and place the conditionson f . I shall use P to mean f (P), P ′ to mean f (P′), and so forth.

II. Independence (of irrelevant alternatives). If in elections P and P′, x Pi y ≡ x P ′i y

and y Pi x ≡ y P ′i x for each individual i , then x Py ≡ x P ′y. In other words, whether

x Py depends soley on individual preferences between x and y.

III. Unanimity. If for every individual i , x Pi y, then x Py.

IV. Nondictatorship. There is no individual i such that for every election P and everyx and y, x Pi y ⊃ x Py.

Arrow proved that no constitution satisfies conditions I–IV. The proof, however,depends on condition I(c), Transitivity of Social Indifference. There are constitutionswhich satisfy all the remaining Arrow conditions. Take, for example, the constitution

∗ The subscript y on ⊃ indicates that the implication holds for all such y.

123

Intransitive social indifference 7

which makes the social choice set, c(S, P), the set of Pareto optimal alternatives inS,∗

x(x ∈ S · ∼(∃y) : y ∈ S · (i) y Pi x).

It is generated by the relation “is unanimously preferred to”, (i) x Pi y, and that rela-tion clearly satisfies every Arrow condition but I(c). Dropping I(c) makes possible aconstitution.

The constitutions dropping I(c) makes possible, however, are thoroughly unsatis-factory, as the theorem I shall now prove shows. Note, by the way, that the proof at nopoint uses transitivity of individual indifference. Hence while I have defined a ballotas an ordering of U , the proof would go through if a ballot were defined more broadly,as any relation satisfying asymmetry and transitivity of preference on U .

Definition A constitution c which satisfies generation condition I(a) is a liberum vetooligarchy if there is a unique set λ of individuals such that for every election P, x , andy,

(i) [i ∈ λ ⊃i x Pi y] ⊃ x Py(ii) [(∃i) : i ∈ λ · x Pi y] ⊃ ∼y Px .

It follows that in a liberum veto oligarchy, if one member of the oligarchy λ prefers xto y and one prefers y to x , then both ∼y Px and ∼x Py, and hence x I y.

Theorem A constitution which satisifies conditions I(a), I(b), II, and III is a liberumveto oligarchy.†

Notation A bar indicates complementation in society σ , so that μ, for example, isi(i ∈ σ · i ∈ μ). For a given P , we write ‘Pλxy’ for

i ∈ λ ⊃i x Pi y.

We then write ‘Dλxy’ to mean

Pλxy Pλyx ⊃P x Py,

λ is weakly decisive on the issue xy. We write ‘D∗λxy’ to mean

Pλxy ⊃P x Py,

λ is decisive on the issue xy.

Decisiveness is transitive, in the sense that for all λ, x , y, and z, D∗λxy D∗λyz ⊃D∗λxz. For suppose D∗λxy D∗λyz, and suppose that Pλxz. If x Pz follows, thenD∗λxy D∗λyz implies Pλxz ⊃P x Pz, and this last is just D∗λxz. Proof of x Pz on

∗ The following expression reads as: the set of all x such that (a) x ∈ S and (b) there does not exist a ysuch that y ∈ S and y Pi x for all i . In general, (i) means ‘for all i’.† It is being implicitly assumed that n is finite with n ≥ 2 and that there are at least three alternatives in U .

123

8 A. F. Gibbard

the above assumptions: Let P′ be the same as P except that for individuals in λ, yis placed between x and z in P′, so that P ′λxy P ′λyz. By decisiveness, x P ′y andy P ′z, and hence by Transitivity of Social Preference, x P ′z. By Independence, sincethe individual orderings of x and z in P and P′ are the same, x P ′z iff x Pz. Thereforex Pz, and the transitivity of D∗ is proved.

Lemma I Dλxy ⊃ D∗λwz. If a set λ of individuals is weakly decisive on some issue,it is decisive on every issue.

The proof is essentially Arrow’s and Blau’s (Arrow 1963, pp. 98–100).∗ We provefirst

Dλxy ⊃ D∗λxz. (4)

Suppose Dλxy, and let P be an election such that Pλxz. We need to prove that x Pzfollows. Let P′ preserve the ballots in P, except that (1) for individuals in λ, y is placedbetween x and z, so that P ′λxy P ′λyz, and (2) for individuals not in λ, y is placedfirst, so that P ′λyx P ′λyz. Then since Dλxy and P ′λxy P ′λyz, we have x P ′y. AlsoP ′σ yz, and hence by Unanimity, y P ′z. By Transitivity of Social Preference, it followsthat x P ′z. Now P and P′ agree in the individual orderings of x and z alone, and henceby Independence, x P ′z iff x Pz, and we have shown x Pz.

By a similar argument, we show

Dλxy ⊃ D∗λwy. (5)

Suppose again that Dλxy, and this time let P be such that Pλwy. Then let P′ differfrom P only by placing x between w and y for individuals in λ and placing x last forindividuals not in λ. Then P ′λwx P ′λxy and P ′λwx P ′λyx . By Unanimity, wP ′x , byweak decisiveness, x P ′y, and by Transitivity of Social Preference, wP ′y. Hence byIndependence, wPy, and (5) is established.

Now putting z, y for y, w respectively in (5) we get

Dλxz ⊃ D∗λyz. (6)

Putting y, z, x for x, y, z respectively in (4), we get

Dλyz ⊃ D∗λyx . (7)

Since D∗ is stronger than D, we can claim implications (4), (6), and (7) to get

Dλxy ⊃ D∗λyx . (8)

We now have that Dλxy implies D∗λwy (5), D∗λyx (8), and D∗λxz (4). Hence bytransitivity of decisiveness,

Dλxy ⊃ D∗λwz (9)

∗ Blau’s proof appears in Blau (1956).

123

Intransitive social indifference 9

for all x, y, w, and z, proving the lemma.∗ ��Lemma II Dλxy Dμwz ⊃ D∗(λ ∩ μ)uv. In other words, if λ is weakly decisive onsome issue and μ is weakly decisive on some issue, their intersection is decisive onevery issue.

Proof Suppose Dλxy Dμwz. Then by Lemma I, D∗μyz. Let P be an election in whichx , y, and z are ranked by individuals in the following sets in the following orders:

λ ∩ μ xyz

λ ∩ μ zxy

λ ∩ μ yzx

λ ∩ μ zyx

Then

P(λ ∩ μ)xz P(λ ∩ μ)zx (10)

completely describes orderings of x and z, so that if x Pz holds in this case, then byIndpendence, x Pz for any election P for which (9) holds, and hence D(λ ∩ μ)xz.And indeed x Pz does hold in this case. For we have Pλxy, and since Dλxy, x Py.Similarly, Pμyz, and since Dμyz, y Pz. By Transitivity of Social Preference, x Pz.This proves the lemma. ��

With these two lemmas we prove the theorem. Take two alternatives x and y in U .There is a set decisive on issue xy, for by Unanimity, society σ is such a set. Let λ bea minimal set such that Dλxy, so that if μ ∈ λ and μ = λ, then ∼Dμxy. We showthat λ and only λ satisfies (i) and (ii) in the statement of the theorem.†

Part I. λ satisfies (i) by Lemma I, for since Dλxy, for all x and y D∗λxy.Part II. λ satisfies (ii). The proof is Arrow-like. Suppose on the contrary that for

some election P and individual j in λ, x Pj y but y Px . Partition society into three setsas follows.

μx = i(x Pi y) (hence j ∈ μx )

μxy = i(x Ii y)

μy = i(y Pi x)

Let P′ be an election in which x , y, and z are ranked as follows.

i ∈ μx ⊃i x Pi z · z Pi y

i ∈ μxy ⊃i x Ii y · x Pi z · y Pi z

i ∈ μy ⊃i y Pi x · x Pi z

∗ It may appear that this proof requires U to contain at least four alternatives, but this is not the case. Thealternatives x, y, w, and z need not all be distinct.† Here, (i) and (ii) refer to the two parts of the definition of a liberum veto oligarchy.

123

10 A. F. Gibbard

Then P′ preserves individual orderings of x and y, and thus by Independence, sincey Px , we have y P ′x . Everyone prefers x to z, and hence by Unanimity, x P ′z. Thereforeby Transitivity of Social Preference, y P ′z. Now since P ′μx yz P ′μx zy and y P ′z, byIndependence,

Pμx yz Pμx zy ⊃P y Pz,

or, in other words, Dμx yz.Now since also Dλxy, by Lemma II we have D∗(μx ∩λ)xy. But μx ∩λ is a proper

subset of λ since j ∈ λ but j ∈ μx . Since λ was defined as a minimal set such thatDλxy, this is a contradiction, and the supposition that (ii) is violated by λ is false.

Part III. λ is unique. For let μ satisfy (i) and (ii). Then from (i), Dμxy and byLemma II, D∗(λ∩μ)xy. Since λ is minimal, λ∩μ must equal λ, and λ ⊆ μ. Supposeμ = λ. Then let j ∈ μ, j ∈ λ and take an election P such that P{ j}xy Pλyx . From(ii) applied to μ, ∼y Px , but from (i) applied to λ, y Px . This contradicts Asymmetryof Social Preference, and hence the supposition that μ = λ is false. This proves thetheorem. ��

References

Arrow KJ (1963) Social choice and individual values, 2nd edn. Wiley, New YorkArrow KJ (1967) Values and collective decision-making. In: Laslett P, Runciman WG (eds) Philosophy,

politics and society, third series. Basil Blackwell, Oxford, pp 215–232Schick F (1969) Arrow’s proof and the logic of preference. Philos Sci 36:127–144

References added in 2013

Blau JH (1956) The existence of social welfare functions. Econometrica 25:302–313Gibbard AF (2014) Social choice and the Arrow conditions. Econ Philos 30, forthcoming (Originally written

in December 1968 or January 1969.)Sen AK (1969) Quasi-transitivity, rational choice and collective decision. Rev Econ Stud 36:381–393Sen AK (1970) Collective choice and social welfare. Holden-Day, San FranciscoWeymark JA (2014) An introduction to Allan Gibbard’s Harvard seminar paper. Econ Philos 30, forthcoming

123