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∗Comment on “The Veil of Public Ignorance”†Geoffroy de ClippelFebruary 2010Nehring(2004)proposesaninterestingmethodologytoextendtheutilitariancriteriondefinedundercompleteinformationtoaninterimsocialwelfareorderingallowingtocom-pare acts. The first axiom defining his approach, called “State Independence,” requiresthe interim social welfare ordering to be consistent with ex-post utilitarian comparisons:if it is commonly known that the act f selects in each state an outcome that is sociallyprefererred according to the utilitarian criterion to the lottery selected by an alternativeact g, then f should be interim socially preferred to g. The second axiom is a classicalcondition of consistency with respect to interim Pareto comparisons: if an act f interimPareto dominates and act g, then f should be interim socially preferred to g. Nehringproves that 1) these two axioms are incompatible if there is no common prior, and 2)that the unique interim social welfare ordering that satisfies these two axioms when thereis a common prior is the ex-ante utilitarian criterion.The purpose of this comment is to show that Nehring’s methodology does not provehelpful in finding ways of extending other classical social welfare orderings. I show in-deed that the corresponding state-independence property becomes incompatible with theinterim Pareto criterion for a very large class of common priors, as soon as the socialwelfare ordering satisfies the strict Pigou-Dalton transfer ...
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∗ Comment on “The Veil of Public Ignorance”
† Geoffroy de Clippel
February 2010
Nehring (2004) proposes an interesting methodology to extend the utilitarian criterion defined under complete information to an interim social welfare ordering allowing to com-pare acts. The first axiom defining his approach, called “State Independence,” requires the interim social welfare ordering to be consistent with ex-post utilitarian comparisons: if it is commonly known that the actfselects in each state an outcome that is socially prefererred according to the utilitarian criterion to the lottery selected by an alternative actg, thenfshould be interim socially preferred togsecond axiom is a classical. The condition of consistency with respect to interim Pareto comparisons: if an actfinterim Pareto dominates and actg, thenfshould be interim socially preferred tog. Nehring proves that 1) these two axioms are incompatible if there is no common prior, and 2) that the unique interim social welfare ordering that satisfies these two axioms when there is a common prior is the ex-ante utilitarian criterion. The purpose of this comment is to show that Nehring’s methodology does not prove helpful in finding ways of extending other classical social welfare orderings. I show in-deed that the corresponding state-independence property becomes incompatible with the interim Pareto criterion for a very large class of common priors, as soon as the social welfare ordering satisfies the strict Pigou-Dalton transfer principle (strict PD for short). I also show that his impossibility result in the absence of a common prior extends to any social welfare ordering that satisfies PD. The Pigou-Dalton principle states that transfer-ring utility so as to reduce inequality should never hurt from a social perspective. Strict PD requires that the resulting utility profile is socially strictly preferred. PD is often viewed as a very appealing axiom in social choice theory, and indeed all the classical social welfare orderings (e.g. utilitarian sum, egalitarian minimum, and Nash product)
∗ K. Nehring, 2004,Journal of Economic Theory119, 247-270 † Department of Economics, Brown University, Providence, Rhode Island - declippel@brown.edu. Fi-nancial support from the National Science Foundation (grant SES-0851210) is gratefully acknowledged.
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satisfy it. The utilitarian criterion has the distinctive property of satisfying PD, but not its strict variant. I start the formal analysis by quickly reminding the content of Nehring’s (2004) model. LetIbe the finite set of individuals, letXbe the set of deterministicsocial alternatives, and letLbe the set of probability distribution overXIndivid-(with finite support). uals are expected utility maximizers. Letui:X→Rbe individuali’s von Neumann-Morgenstern utility function. There is a finite set Ω ofstatesthat determine the agents’ beliefspi: Ω→Δ(Ω), and hence their preferences overacts: X X X X ω ω ω fgif and only ifp(α)f(α)u(x)≥p(α) i i x i igx(α)ui(x), α∈Ωx∈X α∈Ωx∈X
for eachi∈I, eachf: Ω→ L, and eachg: Ω→ Lbelief functions are assumed to. The satisfy the following assumptions:
ω α ω Assumption 1(Introspection)p({α∈Ω|p=p}) = 1, for allω∈Ω and alli∈I. i i i
ω Assumption 2(Truth)p({ω})>0, for allω∈Ω and alli∈I. i
Introspection says that agents are always (at any stateω) certain of their own belief ω p. The Truth assumption requires that individuals put positive probability on the true i state; agents therefore can never be wrong in their probability-one beliefs. Nehring finally assumes that the individuals’ utility functions and the set of social alternatives are such that any real number can be derived as the utility of some lottery overX:
I Assumption 3(Rich Domain) For eachν∈R, there existsl P l(x)ui(x), for eachi∈I. x∈X
∈ Lsuch thatν=
α For anyα∈Ω,Ti(α) ={ω∈Ω|p(ω)>0}is the set of states that individualithinks i possible. Introspection and Truth implies that{Ti(ω)|ω∈Ω}forms a partition of Ω. Individualiknowsan eventE⊆Ω atαifTi(α)⊆E.Eiscommon knowledgeif everybody knowsE, everybody knows that everybody knowsEFormally,, and so forth. ifTIis the finest common coarsening of the individuals’ knowledge partitions, thenEis common knowledge atαifTI(α)⊆Eprobability distribution. A µ∈Δ(Ω) is acommon ω (A) =µ), for alli∈I,A⊆Ω, priorifpi(A|Ti(ω) andω∈Ω such thatµ(ω)>0. Asocial welfare ordering(under complete information) is a complete and transitive I U binary relationRdefined onR. Classical examples include the utilitarian criterion,R P P U E E withuR vif and only ifui≥vi, the egalitarian criterion,RwithuR vif i∈I i∈I
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1N N and only if mini∈Iui≥mini∈Ivi, and the Nash criterion,RwithuR vif and only if Πi∈Iui≥Πi∈Ivi.Pwill denote the strict component ofR, i.e.uP vifuRvand notvRu. Rsatisfies thePigou-Dalton transfer principle(PD) ifvRufor anyu, vsuch that there exist two individualsiandjsuch thatui< uj,vi≤vj,ui+uj=vi+vj,ui< vi, and u−ij=v−ij. Inequality is thus reduced when moving fromutov, sincevis obtained from uby “transferring” some utility fromjtoi, whileiwas and remains after the transfer with less utility compared toj.Rsatisfies thestrict Pigou-Dalton transfer principle (strict PD) ifvRuis replaced byvP u. Aninterim social welfare orderingis a transitive orderingI(even possibly incom-plete) defined on the set of acts. Two axioms will be imposed. The first is directly reproduced from Nehring (2004). Interim Pareto Dominance(IPD)fIg(resp.fig) whenever it is commonly α α known thatfg(resp.fg) for alli∈I. i i The second axiom is the direct analogue of Nehring’s second axiom, where the ex-post utilitarian criterion is replaced by a generic social welfare orderingR. State Independence GivenR(SI-R)fIgwhenever it is commonly known that f(ω)Rg(ω). Following Nehring’s terminology, say that a functionφ: Ω→Risacceptableif there P exists a collection (φi)i∈Ifrom Ω toRsuch thatφ=φiand such that it is common i∈I α knowledge that Eiφi>0, for alli∈I. Thinking ofφas determining an aggregate level of transferable utility in every state, being acceptable then means that there is a way to share this total amount of utility in each state so that it is common knowledge that the resulting contingent allocation of utilities is strictly interim individually rational. Nehring’s impossibility result follows from a classical characterization of the non-existence of a common prior (see Nehring (2004, Theorem A.1.(i)) who traces the result back to Morris (1994)): a common prior exists if and only ifφ= 0 is not acceptable. It is then straightforward to adapt Nehring’s (2004, Theorem 2) argument to show that his impossibility result in the absence of a common prior extends to any social welfare ordering that satisfies PD.
Proposition 1LetRbe a social welfare ordering that satisfies PD. If there is no common prior, then there is no interim social welfare ordering that satisfies both IPD and SI-R.
Proof:φ= 0 is acceptable, since there is no common prior, and hence there exists (φi)i∈I 1I The Nash criterion is defined only overR, and all the results presented in this comment apply ++ also to that more restrictive domain.
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P such that (a)φi(ω) = 0, for eachω∈Ω, and (b) it is common knowledge that i∈I α 0, for all e two acts such thatu(f(ω)) =φ(ω) and Eiφi> i∈Inow. Let fandgbi i ui(g(ω)) = 0, for alli∈Iandω∈Ω (existence guaranteed under the assumption of Rich Domain). SinceRsatisfies PD, (a) implies thatg(ω)Rf(ω), for eachω, and hencegIf, ω by SI-Rthe other hand, (b) implies that is it common knowledge that. On fg, and i hencefIg, by IPD, which contradicts the previous comparison. Nehring proves that the converse is true whenRis the utilitarian criterion: if there is a common prior, then there exists an interim social welfare ordering that satisfies both U IPD and SI-R(in addition, it is unique, and it coincides with the ex-ante utilitarian criterion). I will show that this possibility result does not extend to other classical social welfare orderings. I will say that a functionφ: Ω→Risweakly acceptableif there P exists a collection (φi)i∈Ifrom Ω toRsuch thatφ=φi, and such that it is common i∈I α knowledg e thatEiφi≥0, for alli∈I. If there is no common prior, thenφ= 0 is acceptable, and hence a fortiori weakly acceptable. If there is a common priorµ, then φ= 0 is not acceptable, but it might be weakly acceptable in a non trivial sense meaning that there existsωin the support ofµandi∈Isuch thatφi(ω)6The next lemma= 0. offers a characterization of those common priors.
Lemma 1Suppose that there exists a common priorµ. Thenφ= 0is weakly acceptable K K in a non trivial sense if and only if there exist a sequence(ω)and a sequen of k k=1ce(ik)k=1 individuals such thatωk+16=ωk,ik+16=ik, andωk−1∈Tik(ωk), for eachk∈ {1, . . . , K}, with the convention0 =K.
0 ω ω φ=φ Proof:⇒) Let (φi)i∈Ibe a non trivial decomposition ofφNotice that= 0. Ei iEi i 0Tiω ifω∈T(ω). So, f∈Ω},E φwill de ior anyTi∈ {Ti(ω)|ωi inoteE φi, for some (or all) i ω∈Ti. Notice then that X X X X X Ti µ(T)E ω)φ(ω) = i iφi=µ(iµ(ω)φ(ω) = 0. i∈I Ti∈{Ti(ω)|ω∈Ω}ω∈Ωi∈I ω∈Ω
ω it must be thatE φ Hencei i= 0, for alli∈Iand allω∈Ω. Since the decomposition of φis non trivial, one can find aniand anωfor whichφi(ω)6Call him= 0. i2, call itω1, and let’s say to fix our ideas thatφi2(ω1)<0 (a similar reasoning applies if the inequality ω1 is reversed). SinceE φ= 0, there must existω∈T(ω) such thatφ(ω)>0. Call i2i21i2 i2 P o have thatω∈T( eφ(ω) = 0, there itω2. Sinceω2∈Ti2(ω1), we als1i2ω2). Sinci∈I i2 ω2 0. SinceE0, must exists another individual, call himi3, for whomφi3(ω2)<i3φi3= there must existω3∈Ti3(ω2) such thatφi3(ω3)>0. Sinceω3∈Ti3(ω2), we also have
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thatω2∈Ti3(ω3the argument, one of the new states, let’s say ¯). Iterating ω, will have already appeared previously, since Ω is finite. The subsequence starting atω¯, and ending at the state right before its reappearance, combined with the associated individuals (take i1as the individual that led to the reappearance ofω¯ - it must be that this individual i1is different fromi2, sinceφi2(ω¯)<0< φi1(¯ω)), satisfies the necessary condition, as desired. K be a sequen e statement, with the additional property ⇐) Let (ωk)k=1ce of states as in th that there is no shorter sequence of states with that property. It implies that
(∀k∈ {1, . . . , K}) :Tik(ωk)∩ {ωl|1≤l≤K}={ωk−1, ωk}.
(1)
I prove this by contradiction. Suppose thus, on the contrary, that there existk∈ {1, . . . , K}andl∈ {1, . . . , K} \ {k−1, k}such thatωl∈Tik(ωk). Henceωk−1∈Tik(ωl). Suppose first thatl < k−1. Ifil+1=ik, then one reaches a contradiction since the subse-quence that starts withl+ 1 and ends withk−1 is shorter than the original sequence and satisfies all the properties of the statement (ωl∈Tik(ωl+1) andωk−1∈Tik(ωl) imply that ωk−1∈Tik(ωl+1)). Ifil+16=ik, then again one reaches a contradiction, since the subse-quence that starts withland ends withk−1, changing onlyilintoik, is shorter than the original sequence while satisfying all the properties of the statement. Suppose now that l > k. Ifil=ik, then the subsequence that starts withkand ends withl−1 is shorter than the original sequence while satisfying the properties of the statement (ωl−1∈Tik(ωl) andωl∈Tik(ωk) imply thatωk∈Tik(ωl−1If)). This is not possible. il6=ik, then the subsequence that starts withkand ends withlis shorter than the original sequence (notice that it cannot be thatk= 1 andl=K, sincel6=k−1), while satisfying all the properties of the statement. Again, this is impossible, and we can conclude that (1) is indeed correct. Given anyα >0, construct the collection (φi)i∈Iby the following recursive formula:
φi1(ω1) =α,φi2(ω1) =−α, and (∀i∈I\ {i1, i2}) :φi(ω1) = 0 µ(ωk−1) φik(ωk) =−φik(ωk−1)µ(ωk), µ(ω) k−1 φik+1(ω (∀2≤k≤K) :k) =φik(ωk−1)µ(ωk), and (∀i∈I\ {ik, ik+1}) :φi(ωk) = 0
(∀ω∈Ω\ {ωk|1≤k≤K})(∀i∈I) :φi(ω) = 0. P ω Notice that, by construction,φi(ω) = 0, for alli∈I, andE φi= 0, for all i∈I i
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(ω, i)∈Ω×Ifor which there does not exist 1≤k≤Ksuch thati=ikandω∈Tik(ω). Consider now a pair (ω, i) and aksuch thati=ikandω∈Tik(ωproperty proved). The ω in the previous paragraph implies thatµ(T(ω))E φ)φ(ω) +µ(ω) i i i=µ(ωk−1ikk−1kφik(ωk). ω Ifk6t= 1, then it is straightf E φ= 0, by defini orward to check thai ition ofφik(ωk). If k= 1, then
µ(ω1)µ(ω2)µ(ωK−1) ω µ(T(ω))E(ω)α. . . +µ(ω)α= 0. i iφi=−µK1 µ(ω2)µ(ω3)µ(ωK)
ω HenceE φi= 0, for alli∈Iand allω∈Ω, and it is thus also common knowledge that i ω E iφi= 0, for alli∈I, as desired.
Notice that the condition characterizing priors for whichφ= 0 is weakly acceptable in a non trivial sense, is very weak. For instance, it is satisfied if there exist two individualsi andj, and a stateωsuch thatTi(ω)∩Tj(ωIndeed, suppose) contains at least two states. 0 that the intersection containsωin addition toωcondition in the Lemma is satisfied. The 0 withω1=ω,i1=i,ω2=ω, andi2=jparticular, of course, it is satisfied when,. In while facing uncertainty, all the individuals have the same information (Ti(ω) = Ω, for all i∈Iand allω∈condition in the Lemma is also satisfied when the informationΩ). The structure is derived from types with a joint probability distribution that has full support: a set of typesTiwith at least two elements is associated to each individuali, Ω =×i∈ITi, andµhas full support over Ω. Indeed, the condition of the Lemma is satisfied withi1= 2, 0 0 0 t , t),ω= (t , t , t i2= 1,i3= 2,i4= 1,ω1= (t1, t2, t−12),ω2= (t1,2−12 3 1 2−12), andω4= 0 example where the condition of the Lemma is satisfied, (t1, t2, t−12). Here is yet another while not falling in the two previous cases. Suppose that Ω ={ω1, ω2, ω3},I={1,2,3}, the first individual’s information partition is{{ω1, ω2},{ω3}}, the second individual’s information partition is{{ω1, ω3},{ω2}}, and the third individual’s information partition 3 is{{ω1},{ω2, ω3}}. The the condition of the Lemma is satisfies for (ωkchoosing) by k=1 i1= 2,i2= 1, andi3= 3. So finally here are two examples where the condition does not apply, and hence where a weakly acceptable decomposition ofφ= 0 is necessarily trivial. As a first example, consider the case where all the agents but one are fully informed. As a second example, consider the case whereI={1,2}, Ω ={ω1, ω2, ω3}, the first individual’s information partition is{{ω1, ω2},{ω3}}, and the second individual’s information partition is{{ω1},{ω2, ω3}}. Lemma 1 allows to show that IPD and SI-Rare essentially incompatible whenR satisfies strong PD. I will slightly strengthen SI-Rby requiring that the resulting interim
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