Decentralizability and responsibility under unequal skills Takashi Hayashi∗† September 2, 2014

Abstract We study how decentralizability constraints put restrictions on allocation rules in production economies with unequal skills, in particular what views about how people are responsible for their skills are feasible under such constraints. First we show that if we insist on efficiency strategy-proofness leaves us little variety of views about how people are responsible for their skills. If we do not insist on efficiency strategy-proofness allows for the role of equity, but it requires larger efficiency loss. We then consider Nash implementation, and show that informational constraint with regard to skill levels puts almost no restriction on allocation rules there. We then characterize a class of implementable solutions, which allow for the role of equity at smaller efficiency loss.

1

Introduction

People are born with different technological characteristics which are embodied and cannot be transferred. People are born with different levels of skill, in particular. Although many kinds of skill will be acquired rather than inherited, how people acquire skills is severely affected by how they are given opportunities to learn and how they are trained when they ∗ †

Adam Smith Business School, University of Glasgow. email: [email protected] I thank seminar participants at Hitotsubashi, Kobe, Kyoto and Osaka.

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are young, which are often not their choice. It is not an obvious question if they are responsible for such characteristics. We do not intend to give an immediate answer to this question. We view that it is a quantitative question, and the answer to it should be something that is calibrated through the process of pinning down a solution. Thus we aim to characterize which answers are feasible under natural restrictions, so that such calibration work is made operational. The primary restriction we consider is decentralizability. There are two kinds of potential manipulability here, one is about preferences and the other is about skill levels. In particular, if an individual is penalized because of having higher skill he would have an incentive to hide his skill. As the primary condition of decentralizability we adopt a version of strategy-proofness condition stating that one can never gain by misreporting his preference or by hiding his skill. In the second part of the paper we consider implementability in Nash equilibria. Although it is harder to play Nash equilibria, if we could do that it allows more possibility for social choice rules. In the current problem there exists a ”trivial” solution which is even strategy-proof, efficient and satisfies many other requirements: each individual simply optimizes on his own production function, and no transfer or taxation is made. This solution, however, can be very severe against unskilled people even when they are not responsible for that. Thus our question is whether it is the only option. Whether we insist on efficiency of allocation under the restriction of decentralizability may affect the range of possible views about how people are responsible for their skills. When it comes to the concept of fairness as absence of envy, Pazner and Schmeidler [14] consider an envy-freeness criterion based on the presumption that nobody is responsible for his skill at all, and demonstrate that it is in general incompatible with efficiency of allocation. On the other hand, Varian [18] proposes an envy-freeness criterion based on the presumption that everybody is perfectly responsible for it and shows that it is compatible with efficiency. It remains unclear, however, if efficiency is compatible only with the requirement of no envy based on the presumption of perfect responsibility. When it comes to the concept of fairness as a distributive requirement that nobody should be made worse off than certain lower bound, the choice of such lower bound reflects one’s position on responsibility, since some choices of lower bound are favorable for the skilled and some are favorable for the unskilled. As far as we see from the works by Fleur-

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baey and Maniquet [4, 5], taking the lower bound to be no-work-no-pay results in treating unskilled people severely, and taking the lower bound to be the outcome of optimizing under the lowest skill results in favoring the unskilled more. We should notice, however, that these are the implications obtained when the criterion is combined with the requirement of efficiency. In exchange economies the efficiency requirement itself leaves us relatively free to choose fairness criterion, and they are relatively orthogonal to each other. On the other hand, in production economies with unequal skills how a distributive criterion puts restriction depends severely on whether we impose efficiency as well or not. Thus what position on responsibility each axiom conveys is more subtle here. Since we aim to identify which answers are feasible under the restriction of decentralizability, we aim to identify a class of allocation rules meeting the restriction, so that we can tell where the above positions are located within the class and the above subtlety is made clear. In this paper we investigate two paths, one on which we insist on efficiency, the other on which we do not. We do this for each of the two decentralizability conditions. Under efficiency, everybody should receive his wage equal to his productivity and the argument reduces to how lump-sum transfers should be. We first demonstrate that choice of lower bound has to have a particular implication about how people should be responsible for skills, even under the weaker strategy-proofness requirement in which skill levels are known. This motivates us to characterize a class of lump-sum transfers without making a particular choice of lower bound. The characterized class is a family which includes two well-known rules as endpoints, one is the ”trivial” solution and the other is that the transfer is made so as to equalize potential income across individuals which they receive when they spend all their time on labor. We show, however, that under the stronger strategy-proofness requirement in which skill levels are unknown the only possibility is the first endpoint. This tells us that it is hard to escape from the ”trivial” solution if we insist on efficiency. This suggests that using distortion as a device to achieve certain kinds of fairness deserves consideration. If we do not insist on efficiency of allocation, the choice of lower bound still leaves us a variety about how people should be responsible for their skills. It

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characterizes a class of distortive taxation rules which includes two prominent rules as endpoints, one is the ”trivial” solution and the other is that everybody receives the same wage per hour which however cannot exceed the lowest productivity in the society. Note, however, that the second endpoint is Pareto-dominate by the first endpoint, the trivial solution. In the second part of the paper we investigate the consequence of allowing Nash implementation with potentially multiple equilibria. We provide a characterization of social choice correspondences which are implementable in Nash equilibria via mechanisms in which individuals report skill profiles in the society in addition to the standard kind of message such as price (wage) and quantity. Because of the nature that nobody can exaggerate his skill, it turns out that Nash implementability on this aspect puts no restriction on social choice correspondences, except that nobody should be forced to spend his whole time on labor without any pay. Under efficiency, allowing Nash implementation in potentially multiple equilibria does not have much to add, since it simple leaves indeterminacy of lump-sum transfers or it has the same implication as startegy-proofness does. On the other hand, if we do not insist on efficiency, Nash implementation if we could do it allows us to endogenously determine wages which are distorted in general but support equitable allocations with smaller efficiency loss. Thus such equitable solution is not Pareto dominate by the ”trivial” solution, in contrast to the implication of strategy-proofness.

Related literature The literature on decentralizability is huge, so we can review directly relevant ones only. The classic contributions on strategy-proofness are Gibbard [7] and Satterthwaite [15], which are written in abstract social choice settings. In economic environments, Hurwicz [8], and Zhou [22] and Serizawa [16] show that strategy-proofness and efficiency lead to impossibility or dictatorship in exchange economies. The classic paper on Nash implementation is Maskin [11], which is written in the abstract social choice setting. In the setting of exchange economy, Hurwicz [9] and Thomson [17] provide characterization of solutions implementable in Nash equilibria. Hurwicz, Maskin and Postlewaite [10] provide sufficient conditions for Nash implementability when the designer does not know initial endowments or production technolo4

gies. They consider implementability via a mechanism in which each individual submits his endowment or production technology, where he cannot exaggerate it while he can hide or withhold a part of it. Yamada and Yohihara [19, 20] provide conditions for allocation rules in production economies with unequal skills which are implementable in Nash equilibria in a different type of mechanism in which people submit how many hours to work. Pazner and Schmeidler [14] demonstrate that the requirement of absence of envy is incompatible with efficiency of allocation when nobody is taken to be responsible for his skill. On the other hand, Varian [18] proposes a notion of envy-freeness based on the presumption that everybody is responsible for his skill. Fleurbaey and Maniquet [4] characterize an efficient allocation rule which satisfies Maskin monotonicity, an implementability condition and also an informational efficiency condition, and a weaker version of envy-freeness based on the first presumption, stating that it holds only for some fixed reference preference. Despite of taking the weaker version of envy-freeness, they obtain that under additional axioms the set-valued solution must include an allocation rule that is almost immediate to the solution with perfect responsibility. Indeed, they show that under an additional axiom of individual rationality the set-valued solution must include the ”trivial” solution, which suggests that under efficiency escaping from it is hard. We show, too, that it is hard to escape from the ”trivial” solution under efficiency and decentralizability, but it is even more serious here, because our characterization is about what the solution must select rather than what it must contain, and any of the characterizing axioms says nothing about responsibility. Fleurbaey and Maniquet [5] propose an efficient resource allocation rule which satisfies Skill Monotonicity, an axiom stating that having higher skill hurts nobody. They provide a characterization of a rule satisfying the axiom, together with an distributional axiom stating that nobody should be made worse of than the outcome of optimizing based on the lowest skill in the society and some other ones. Note that decentralizability conditions are not considered there, since as it is true in other environment as well decentralizability conditions and solidarity requirements are often in trade-offs, and it is true in the current paper as well. In the framework of fair social orderings, Fleurbaey and Maniquet [6] characterize social orderings over allocations of labor and consumption based on the presumption that nobody

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is responsible for his skill level. In the framework of axiomatic bargaining, Yoshihara [21] characterize a class of efficient allocation rules which allow people to be partially responsible for their skill levels. Chambers and Hayashi [2] study allocation rules in the setting of pure exchange economy where people are responsible only partially for their initial endowments. They maintain imposing efficiency of allocation, since efficiency itself leaves us a variety of which positions about fairness to take in exchange economies. They characterize a class of codes of lump-sum transfers which are parametrized by how much people are responsible for their endowments. On the other hand, since efficiency may not be compatible with equity in the current setting, distortions as well as lump-sum transfers may be used as a device to meet certain fairness criteria.

2

The model

Consider a production economy with n individuals, in which one consumption good is produced only from labor input. Assume that nobody holds any amount of consumption good in the outset and everybody has 1 unit of time which can be used for labor or leisure. We normalize the minimal possible level of consumption equal to zero as it is taken to be an implicit fixed factor here. Let li denote individual i’s labor hour for each i = 1, · · · , n, then the total amount of consumption good produced is given by n ∑

a i li ,

i=1

where ai > 0 denotes i’s skill level. Let a = (a1 , · · · , an ) ∈ Rn++ denote a profile of skill levels, which is taken to be variable in this paper.

∑ One may think of a more general form of production function such as f ( ni=1 ai li ). We

view that most of ethical problems arising in the issue of unequal skills are already there in the simplest case of additive linear production, as far as the notion of individual skill remains clear, and thus we restrict attention to it.1 1

Further general form such as f (l1 , · · · , ln ) will allow complementarity between individuals’ labor in-

puts. Then it becomes a non-obvious question what we mean by an individual’s own skill, even without decentralizability arguments.

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Each individual’s consumption space is [0, 1] × R+ . When (li , ci ) ∈ [0, 1] × R+ is given to individual i it means he works for li units of time and consumes ci units of the consumption good, while he enjoys 1 − li units of time for leisure. Thus, an allocation (l, c) = (l1 , · · · , ln , c1 , · · · , cn ) ∈ [0, 1]n × Rn+ is said to be feasible if n ∑

ci ≤

n ∑

i=1

a i li .

i=1

We also use the notation (l, c) = ((l1 , c1 ), · · · , (ln , cn )) in an interchangeable manner. Given a = (a1 , · · · , an ) ∈ Rn++ , let F (a) denote the set of feasible allocations under a. Let R denote the set of preferences over [0, 1] × R+ , which are strictly convex, strongly decreasing in labor hours and strongly increasing in consumptions, and allow continuously differentiable representation over (0, 1) × R++ with the following properties: for any individual i, given his preference Ri ∈ R and its differentiable representation ui the marginal rate of substitution of consumption for labor at (li , ci ) ∈ (0, 1) × R++ defined by M RS(Ri , (li , ci )) = −

∂ui (li ,ci ) ∂li ∂ui (li ,ci ) ∂ci

satisfies lim M RS(Ri , (li , ci )) = 0

ci →0

lim M RS(Ri , (li , ci )) = ∞

li →1

lim M RS(Ri , (li , ci )) = 0

li →0

For any individual i, given his preference Ri ∈ R and a compact convex set B ⊂ [0, 1] × R+ , let m(Ri , B) denote the maximal element in B with respect to Ri , which is uniquely determined since Ri is strictly convex. A social choice function is a mapping φ : Rn++ × Rn → [0, 1]n × Rn+ such that φ(a, R) ∈ F (a) for all (a, R) ∈ Rn++ × Rn , where φi (a, R) denotes the allocation for i = 1, · · · , n. Also, given (a, R) and i, let li (a, R) denote the labor component of φi (a, R) and ci (a, R) denote the consumption component of φi (a, R). It will be helpful to note here what we meant by ”trivial” solution in the introduction. For each i, given wi > 0 and ti > −wi , let B(wi , ti ) denote i’s budget set with slope wi and intercept ti . 7

Definition 1 φ is said to be the trivial solution when it holds φi (a, R) = m(Ri , B(ai , 0)) for all (a, R) ∈ Rn++ × Rn and i = 1, · · · , n.

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Axioms

3.1

Axioms on efficiency, envy-freeness and welfare bounds

First we state the axiom of allocative efficiency. Pareto Efficiency (PE): For all (a, R) ∈ Rn++ × Rn , there is no there is no (l′ , c′ ) ∈ F (a) such that (li′ , c′i )Ri φi (a, R) for all i = 1, · · · , n and (li′ , c′i )Pi φi (a, R) for at least one i. In the current setting Pareto efficiency implies that we do not waste resources and allocations are in the interior, under a very mild condition (see Proposition 1 below). Resource Efficiency (RE): For all (a, R) ∈ Rn++ × Rn , it holds n ∑

ci (a, R) =

i=1

n ∑

ai li (a, R).

i=1

Interior Solution (IS): For all (a, R) ∈ Rn++ × Rn , it holds φi (a, R) ∈ (0, 1) × R++ for all i. The following is a very mild requirement that nobody should be forced to spend whole his time on labor without any pay. Non-Confiscatory (NC): For all (a, R) ∈ Rn++ × Rn , it holds φi (a, R) ̸= (1, 0) for all i. The claim below is immediate. Proposition 1 PE implies RE. PE and NC imply IS. Also, IS implies NC. Next we list axioms of absence of envy. What kind of envy is ”justifiable” depends on whether people are responsible for their skills, and because of this we consider two versions of envy-freeness. The first one presumes that nobody is responsible for his skill at all. 8

Envy-Freeness (EF): For all (a, R) ∈ Rn++ × Rn and for all i, j = 1, · · · , n, it holds φi (a, R)Ri φj (a, R). The second one, one other hand, presumes that everybody is perfectly responsible for his skill. n n ∗-Envy-Freeness For }all (a, R) ∈ ( (∗-EF): { ) R++ × R and for all i, j = 1, · · · , n it holds a l (s,R) φi (a, R)Ri min j jai , 1 , cj (a, R) .

To illustrate, suppose that ai = 2 and aj = 6, and consider an allocation (li , ci ) = (0.4, 1) and (lj , cj ) = (0.2, 1.2). When nobody is responsible for his skill since (0.2, 1.2) dominates (0.4, 1) it is legitimate for i to envy j. However, when everybody is responsible for his skill we need to take it into account how much one has to work if he is to make the same output as another one does. In the current example in order to make j’s production 6 × 0.2 = 1.2 individual i has to work 1.2/2 = 0.6 units of time. Hence the right comparison here is between (0.4, 1) and (0.6, 1.2), not between (0.4, 1) and (0.2, 1.2). The meaning of the min operator is as follows. Suppose that ai = 2 and aj = 20, and consider an allocation (li , ci ) = (0.4, 1) and (lj , cj ) = (0.2, 5). Then, in order to make j’s production 20 × 0.2 = 4 individual i has to work 4/2 = 2 units of time, that is impossible. In such case we truncate i’s time of labor and compare between (0.4, 1) and (1, 5). The following result is known (originally due to Pazner and Schmeidler [14]), but needs to be shown for the current domain of preferences. Proposition 2 There is no SCF which satisfies PE and EF. Proof. See Figure 1, where 1 has higher skill and 2 has lower skill. Without loss of generality, assume the series of indifference curves are such that 1’s MRS is constant equal to a1 along the vertical dotted line in the right, and 2’s MRS is constant equal to a2 along the vertical dotted line in the left, down to sufficiently low points close to the horizontal axis. Let (l1 , a1 l1 ) denote 1’s preference maximization point under his own production possibility set, and (l2 , a2 l2 ) denote 2’s preference maximization point under his own production possibility set. By PE, 1’s labor hours have to be l1 and 2’s labor hours have to be l2 , while there may be lump-sum transfer of consumption. Let t∗ be such that (l1 , a1 l1 − t∗ )I2 (l2 , a2 l2 + t∗ ). In order that 2 does not envy 1 he should receive (l2 , a2 l2 + t) 9

a1 l1 − t∗ a2 l2 + t∗
















r




a1

R2 AKA A


R2
AKA 6
A R1
 a2  
  
r 


l2

l1

Figure 1: Incompatibility between Efficiency and Envy-freeness

with t∗ ≤ t ≤ a1 l1 . That is, 2’s labor-consumption pair must lie on the thick vertical segment in the left, and 1’s labor-consumption pair must lie on the thick vertical segment on the right. However, since the thick vertical segment in the left is above 1’s indifference curve passing through (l1 , a1 l1 ), we have (l2 , a2 l2 +t)P1 (l1 , a1 l1 ), implying (l2 , a2 l2 +t)P1 (l1 , a1 l1 −t), hence 1 envies 2. Let us list some axioms about bounds on welfare. First one says that nobody should be made worse off than status quo, where the status quo is taken to be the point of nowork-no-pay. Individual Rationality (IR): For all (a, R) ∈ Rn++ × Rn and for all i = 1, · · · , n, it holds φi (a, R)Ri (0, 0). Second one says that nobody should be made worse off than status quo, where the status quo point is such that everybody spends all his time on labor and gets paid according to the average skill. Average Skill Lower Bound (ASLB): For all (a, R) ∈ Rn++ × Rn and for all i = 10

1, · · · , n, it holds φi (a, R)Ri (1, a), where a =

1 n

∑n i=1

ai .

Third one says nobody should receive surplus more than the largest possible amount. No Excess Surplus (NES): For all (a, R) ∈ Rn++ × Rn , it holds (0, M RS(Ri , φi (a, R))li (a, R))Ri φi (a, R) for all i = 1, · · · , n. Since marginal cost of labor is increasing, M RS(Ri , φi (a, R))li (a, R) is the largest possible measure of an individual i’s labor cost at φi (a, R). NES states that he should not be compensated more than that. The following claim is immediate. Proposition 3 Each of IR and ASLB implies NC.

3.2

Axioms on decentralizability and comparative properties

Here we list axioms on decentralizability and related comparative properties. First one says that nobody can gain by misreporting his preference or hiding his skill. n−1 Strategy-Proofness (SP): For all i, for all (a−i , R−i ) ∈ R++ × Rn−1 and for all

(ai , Ri ), (a′i , Ri′ ) ∈ R++ × R with a′i ≤ ai , it holds φi ((ai , a−i ), (Ri , R−i ))Ri φi ((a′i , a−i ), (Ri′ , R−i )). When skill levels are known we face a weaker constraint that nobody can gain by misreporting his preference. In the literature it is called ”strategy-proofness,” but in order to emphasize that informational incompleteness is limited to preferences we call it Preference Strategy-Proofness. Preference Strategy-Proofness (PSP): For all a ∈ Rn++ , for all i and R−i ∈ ×Rn−1 and for all Ri , Ri′ ∈ R, it holds φi (a, (Ri , R−i ))Ri φi (a, (Ri′ , R−i )). 11

The following result is helpful. Lemma 1 Assume PE and NC. Then, φ satisfies PSP if and only if for all a ∈ Rn++ , for all i and R−i ∈ Rn−1 and for all Ri , Ri′ ∈ R with M RS(Ri′ , φi (a, (Ri , R−i ))) = M RS(Ri , φi (a, (Ri , R−i ))) it holds φi (a, (Ri′ , R−i )) = φi (a, (Ri , R−i )). Proof. Note that PE and NC imply IS. ”If” part: Suppose φi (a, (Ri′ , R−i ))Pi φi (a, (Ri , R−i )). Then we can take Ri′′ ∈ R such that M RS(Ri′′ , φi (a, (Ri , R−i ))) = M RS(Ri , φi (a, (Ri , R−i ))) = ai , M RS(Ri′′ , φi (a, (Ri′ , R−i ))) = M RS(Ri′ , φi (a, (Ri′ , R−i ))) = ai and and φi (a, (Ri′ , R−i ))Pi′′ φi (a, (Ri , R−i )). Then the condition implies φi (a, (Ri′′ , R−i )) = φi (a, (Ri , R−i )) and φi (a, (Ri′′ , R−i )) = φi (a, (Ri′ , R−i )), which is a contradiction. ”Only if” part: Suppose M RS(Ri′ , φi (a, (Ri , R−i ))) = M RS(Ri , φi (a, (Ri , R−i ))) and φi (a, (Ri′ , R−i )) ̸= φi (a, (Ri , R−i )). By PE, IS and the assumption, φi (a, (Ri , R−i )) is supported by a line with slope ai and intercept ti , and φi (a, (Ri′ , R−i )) is supported by a line with slope ai and intercept t′i , where t′i ̸= ti . Without loss of generality, assume t′i > ti . Then we can take Ri′′ ∈ R such that its corresponding wealth-expansion path (li (t), ci (t)) = m(Ri′′ , B(ai , t)), t > −ai exhibits that li (t) is non-increasing in t and ci (t) is increasing in t, and (li (ti ), ci (ti )) = φi (a, (Ri , R−i )), and φi (a, (Ri′ , R−i ))Pi′′ φi (a, (Ri , R−i )). Then we have φi (a, (Ri′′ , R−i )) = φi (a, (Ri , R−i )) because otherwise either of the two dominates the other, which leads to a violation of PSP. Hence we obtain φi (a, (Ri′ , R−i ))Pi′′ φi (a, (Ri′′ , R−i )), which is a violation of PSP. The following is a technical but mild condition. Convexity (CON): For all a ∈ Rn++ , for all i and R−i ∈ Rn−1 , the set Oi (a, R−i ) = {φi (a, (Ri , R−i )) : Ri ∈ R} is convex. 12

It is indeed implied by reasonable conditions. Proposition 4 PSP, PE and NC imply CON. Proof. It follows from Theorem 1. The following result is helpful. Lemma 2 Assume IS and CON. Then, φ satisfies PSP if and only if for all a ∈ Rn++ , for all i and R−i ∈ Rn−1 and for all Ri , Ri′ ∈ R with M RS(Ri′ , φi (a, (Ri , R−i ))) = M RS(Ri , φi (a, (Ri , R−i ))) it holds φi (a, (Ri′ , R−i )) = φi (a, (Ri , R−i )). Proof. ”If” part: Suppose φi (a, (Ri′ , R−i ))Pi φi (a, (Ri , R−i )). Then we can take Ri′′ ∈ R such that M RS(Ri′′ , φi (a, (Ri , R−i ))) = M RS(Ri , φi (a, (Ri , R−i ))), M RS(Ri′′ , φi (a, (Ri′ , R−i ))) = M RS(Ri′ , φi (a, (Ri′ , R−i ))) and and φi (a, (Ri′ , R−i ))Pi′′ φi (a, (Ri , R−i )). Then the condition implies φi (a, (Ri′′ , R−i )) = φi (a, (Ri , R−i )) and φi (a, (Ri′′ , R−i )) = φi (a, (Ri′ , R−i )), which is a contradiction. ”Only if” part: Note that φ satisfies SP if and only if it holds φi (a, (Ri , R−i )) = m(Ri , Oi (a, R−i )) for all i, for all Ri ∈ R and R−i ∈ Rn−1 . Since φi (a, (Ri , R−i )) = m(Ri , Oi (a, R−i )), the strict upper contour set over φi (a, (Ri , R−i )) with respect to Ri and Oi (a, R−i ) are separated by the line with slope M RS(Ri , φi (a, (Ri , R−i ))). Under the assumption that M RS(Ri′ , φi (a, (Ri , R−i ))) = M RS(Ri , φi (a, (Ri , R−i ))) the strict upper contour set over φi (a, (Ri , R−i )) with respect to Ri′ and Oi (a, R−i ) are separated by the line with slope M RS(Ri , φi (a, (Ri , R−i ))). Therefore m(Ri′ , Oi (a, R−i )) = φi (a, (Ri , R−i )), which implies φi (a, (Ri′ , R−i )) = φi (a, (Ri , R−i )). Next axiom states that if one’s allocation is unchanged when his preference to report changes it should not affect the others, otherwise it will create an incentive for the others to bribe him. Non-Bossiness (NB): For all a ∈ Rn++ , for all i and R−i ∈ Rn−1 and Ri , Ri′ ∈ R, if φi (a, (Ri′ , R−i )) = φi (a, (Ri , R−i )), then φj (a, (Ri′ , R−i )) = φj (a, (Ri , R−i )) holds for all j ̸= i. Last axiom we introduce basically says that when an individual’s skill is higher it should not hurt anybody else.

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Weak Skill Monotonicity (WSM): For all i, for all ai , a′i ∈ R++ with a′i ≤ ai , for all a−i ∈ Rn−1 ++ , and for all R ∈ R, it holds φj ((ai , a−i ), R)Rj φi ((a′i , a−i ), R) for all j ̸= i. By ”weak” we mean the axiom does not cover the condition that the individual himself should not get hurt when his skill is higher, since it is covered by SP.

4

Characterization: Efficient rules

Here we characterize the class of efficient allocation rules. Under efficiency, each individual’s wage has to be equal to his productivity, and arguments reduce to how lump-sum transfers should be. The first result below states basically that under Preference Strategy-Proofness any transfer to or from each individual has to be independent of his preference. Also, under Strategy-Proofness such transfer to each individual has to be non-decreasing in his skill. For each i, given wi > 0 and ti > −wi , let B(wi , ti ) denote i’s budget set with slope wi and intercept ti . Theorem 1 φ satisfies PSP, PE ans NC if and only if there exists a list of functions t = ∑ (t1 , · · · , tn ) with ti : Rn++ ×Rn−1 → R and ti (a, R−i )+ai > 0 for all i and ni=1 ti (a, R−i ) = 0 for all (a, R) ∈ Rn++ × Rn , such that φi (a, R) = m(Ri , B(ai , ti (a, R−i ))) holds for all (a, R) ∈ Rn++ × Rn and i = 1, · · · , n. Moreover, such φ satisfies SP instead of PSP if and only if ti is non-decreasing in ai for each i = 1, · · · , n. Proof. It suffices to show the sufficiency of the axioms. Pick any (a, R) ∈ Rn++ × Rn , then by PE and NC for each i the allocation φi (a, R) is supported by the straight line with slope ai and intercept denoted by ti (a, R). In order to show that ti (a, R) is independent of Ri , suppose ti (a, (Ri′ , R−i )) > ti (a, (Ri , R−i )). Then we can take Ri′′ ∈ R such that M RS(Ri′′ , φi (a, (Ri , R−i ))) = M RS(Ri , φi (a, (Ri , R−i ))) and φi (a, (Ri′ , R−i ))Pi′′ φi (a, (Ri , R−i )). 14

By Lemma 1 we have φi (a, (Ri′′ , R−i )) = φi (a, (Ri , R−i )). Hence we obtain φi (a, (Ri′ , R−i ))Pi′′ φi (a, (Ri′′ , R−i )), which is a violation of PSP. Now we show ti is non-decreasing in ai under SP. Suppose ti (a′i , a−i , R−i ) < ti (ai , a−i , R−i )

for a′i > ai . Since ti is independent of i’s preference, we can take Ri such that φi (ai , a−i , Ri , R−i )Pi φi (a′i , a which is a violation of SP. Next result states that in the above class allocation rules adding the individual rationality axiom implies that the solution has to be trivial, in which everybody has to be responsible for his skill. Theorem 2 φ satisfies PSP, PE and IR if and only if it holds φi (a, R) = m(Ri , B(ai , 0)) for all (a, R) ∈ Rn++ × Rn and i = 1, · · · , n. Moreover, the same assertion holds when we replace IR by NES. Proof. It suffices to show the sufficiency of the axioms. Note that IR implies NC. Pick any i, pick any a ∈ Rn++ , i and R−i ∈ Rn−1 . Suppose ti (a, R−i ) < 0. Since ti (a, R−i ) is independent of Ri we can take Ri ∈ R such that (0, 0)Pi φi (a, (Ri , R−i )), which is a violation of IR. ∑ Hence ti (a, R−i ) ≥ 0 holds for all i. Since ni=1 ti (a, R−i ) = 0 follows from feasibility and PE we have ti (a, R−i ) = 0 for all i. To see that the same assertion holds when we replace IR by NES, we show ti (a, R−i ) ≤ 0. Suppose ti (a, R−i ) > 0. Since ti (a, R−i ) is independent of Ri we can take Ri ∈ R such that (0, M RS(Ri , φi (a, (Ri , R−i )))li (a, (Ri , R−i )))Pi φi (a, (Ri , R−i )), which is a violation of NES. ∑ Hence ti (a, R−i ) ≤ 0 holds for all i. Since ni=1 ti (a, R−i ) = 0 follows from feasibility and PE we have ti (a, R−i ) = 0 for all i.

PSP is purely an incentive constraint or an informational efficiency condition, and by itself says nothing about how people are responsible for their skills, although it puts a restriction on what options of views about responsibility are ”feasible”.

15

IR is not very explicit about the notion of responsibility in this setting since apparently all people have equal amounts of time, but actually it does since taking no-work-no-pay as the lower bound is favorable for the skilled. We can see this by replacing IR by ASLD, which is viewed as the ”dual” of IR. The replacement results in selecting a solution which equalizes potential incomes to be earned when they spend all their time on labor. This solution severely punishes the skilled people, though. Theorem 3 φ satisfies PSP, PE and ASLB if and only if it holds φi (a, R) = m(Ri , B(ai , a − ai )) for all (a, R) ∈ Rn++ × Rn and i = 1, · · · , n, where a =

1 n

∑n i=1

ai .

Proof. It suffices to show the sufficiency of the axioms. Note that ASLB implies NC. Pick any i, pick any a ∈ Rn++ , i and R−i ∈ Rn−1 . Suppose ti (a, R−i ) < a − ai . Since ti (a, R−i ) is independent of Ri we can take Ri ∈ R such that (1, a − ε)Ii φi (a, (Ri , R−i )) for some ε > 0, which is a violation of ASLD. ∑ Hence ti (a, R−i ) ≥ a−ai holds for all i. Since ni=1 ti (a, R−i ) = 0 follows from feasibility we have ti (a, R−i ) = a − ai for all i. Example 1 Here is a subclass of the class characterized above in the first statement: given β ∈ [0, 1], the allocation rule is defined by φi (a, R) = m(Ri , B(ai , β(a − ai ))) for all (a, R) ∈ Rn++ × Rn and i = 1, · · · , n, where a =

1 n

∑n i=1

ai . Notice that when β = 0

it reduces to the trivial solution and when β = 1 it reduces to the solution to equalize potential incomes. Note, however, that it does not satisfy SP except when β = 0, since it in general hurts an individual when he has higher skill. Now let’s not insist on either of IR (or NES) or ASLB and return to the class of allocation rules as characterized in Theorem 1. It is already a stringent property, however, that transfer to an individual have to be nondecreasing in his skill under SP, since it should be typically less skilled people who will need transfers if we insist on certain kind of fairness. This suggests that it is hard to save unskilled people even in the presence of skilled people. 16

Weak Skill Monotonicity is an axiom stating that if one has more skill it should not hurt anybody else, while the requirement that it should not hurt himself is covered by SP. The result below states that when we strengthen PSP to SP and add WSM the lump-sum transfers must be zero, which is the case of the trivial solution. Theorem 4 φ satisfies SP, PE, NC and WSM if and only if it holds φi (a, R) = m(Ri , B(ai , 0)) for all (a, R) ∈ Rn++ × Rn and i = 1, · · · , n. Proof. It suffices to show the sufficiency of the axioms. Pick any a, a′ ∈ Rn++ with a′ ≥ a and pick arbitrary R ∈ Rn . Suppose ti (a′i , a−i , R−i ) < ti (a, R−i ). Since R ∈ Rn was arbitrary and ti is independent of Ri we can take it so that m(Ri , B(ai , ti (ai , a−i , R−i )))Pi m(Ri , B(a′i , ti (a′i , a−i , R−i ))), that is, φi (ai , a−i , R)Pi φi (a′i , a−i , R), which is a violation of SP. Hence ti (a′i , a−i , R−i ) ≥ ti (a, R−i ). Suppose tj (a′i , a−i , R−j ) < tj (a, R−j ) for some j ̸= i. Since R ∈ Rn was arbitrary and tj is independent of Rj we can take it so that m(Rj , B(aj , tj (ai , a−i , R−j )))Pj m(Rj , B(a′i , tj (a′i , a−i , R−j ))), that is, φj (ai , a−i , R)Pj φj (a′i , a−i , R), which is a violation of WSM. Hence tj (a′i , a−i , R−j ) ≥ tj (a, R−j ) for all j ̸= i. ∑n ∑n ′ ′ Since t (a , a , R ) = j −i −j i j=1 j=1 tj (ai , a−i , R−j ) = 0, we obtain tj (ai , a−i , R−j ) = tj (a, R−j ) for all j = 1, · · · , n. By repeating the above argument we obtain tj (a′ , R−j ) = tj (a, R−j ) for all j = 1, · · · , n. Pick any a, a′ ∈ Rn++ . Then, since tj (a, R−j ) = tj (a ∧ a′ , R−j ) and tj (a′ , R−j ) = tj (a ∧ a′ , R−j ) we obtain tj (a′ , R−j ) = tj (a, R−j ) for all j = 1, · · · , n. Hence tj is independent of a for all j = 1, · · · , n, and we write it by tj (R−j ). Now we show tj (R−j ) ≥ 0. Suppose tj (R−j ) < 0 for some j. Then there exists a ∈ Rn++ such that tj (R−j ) + aj < 0, which violates feasibility of allocation. Hence tj (R−j ) ≥ 0 for all j. Since

∑n

j=1 tj (R−j )

= 0, we obtain tj (R−j ) = 0 for all j.

None of the above axioms says anything explicit about how much people are responsible for their skills, but the conjunction of them allows only the trivial case that everybody has to be perfectly responsible for his skill. 17

5

Characterization: How much efficiency loss should we bear?

Here we investigate what we can do when we do not insist on efficiency. Although we can use both wage distortions and lump-sum transfers, from the previous arguments we already know how restrictive lump-sum transfers can be. Hence we would focus on what we can do with wage distortions. The following result characterizes when we can focus on tax distortion. Notice that imposing the axioms on welfare bounds still leave us flexibilities for considering how much people are responsible for their skills, while they leave no such flexibility under efficiency. Theorem 5 φ satisfies PSP, CON, IS, IR and NES if and only if there exists a list of functions w = (w1 , · · · , wn ) with wi : Rn++ × Rn−1 → R++ for each i such that φi (a, R) = m(Ri , B(wi (a, R−i ), 0)) holds for for all (a, R) ∈ Rn++ × Rn and i = 1, · · · , n. Moreover, such φ satisfies SP instead of PSP if and only if wi is non-decreasing in ai for each i = 1, · · · , n. Moreover, such φ satisfies SP instead of PSP and WSM additionally if and only if wi is non-decreasing in a for each i = 1, · · · , n.

Proof. It suffices to show the sufficiency of the axioms. Pick any (a, R) ∈ Rn++ × Rn , then for each i the allocation φi (a, R) is supported by the straight line with slope denoted by wi (a, R) and intercept denoted by ti (a, R). Suppose ti (a, R) < 0. Take Ri′ such that M RS(Ri′ , φi (a, R)) = M RS(Ri , φi (a, R)) = wi (a, R) and (ε, 0)Ii′ φi (a, R) for some ε > 0. By Lemma 2 we have φi (a, (Ri′ , R−i )) = φi (a, R). But this violates IR in the economy (a, (Ri′ , R−i ))). Suppose ti (a, R) > 0. Take Ri′ such that M RS(Ri′ , φi (a, R)) = M RS(Ri , φi (a, R)) = wi (a, R) and (0, wi (a, R)li (a, R) + ε)Ii′ φi (a, R) for some ε > 0. By Lemma 2 we have φi (a, (Ri′ , R−i )) = φi (a, R). But this violates NES in the economy (a, (Ri′ , R−i ))). In order to show that wi (a, R) is independent of Ri , suppose wi (a, (Ri′ , R−i )) > wi (a, (Ri , R−i )). Then we can take Ri′′ ∈ R such that M RS(Ri′′ , φi (a, (Ri , R−i ))) = M RS(Ri , φi (a, (Ri , R−i ))), M RS(Ri′′ , φi (a, (Ri′ , R−i ))) = M RS(Ri′ , φi (a, (Ri′ , R−i ))), and φi (a, (Ri′ , R−i ))Pi′′ φi (a, (Ri , R−i )). 18

By Lemma 2 we have φi (a, (Ri′′ , R−i )) = φi (a, (Ri , R−i )) and φi (a, (Ri′′ , R−i )) = φi (a, (Ri′ , R−i )), which is a contradiction. The following result says that if we impose Envy-Freeness additionally everybody should be paid the same wage which cannot exceed the lowest productivity in the society. That is, the Pareto dominant solution within class of solutions is that the common wage is equal to the lowest productivity. Note that this is Pareto-dominated by the trivial solution. Theorem 6 φ satisfies PSP, CON, IS, NB, IR, NES and EF if and only if there exists a function w0 : Rn++ → R++ with w0 (a) ≤ mini ai for all a ∈ Rn++ , such that φi (a, R) = m(Ri , B(w0 (a), 0)) holds for for all (a, R) ∈ Rn++ × Rn and i = 1, · · · , n. Moreover, such φ satisfies SP instead of PSP if and only if w0 is non-decreasing in a. Proof. It suffices to show the sufficiency of the axioms. Pick any (a, R) ∈ Rn++ × Rn , then for each i the allocation φi (a, R) is supported by the line with slope wi (a, R−i ) which passes the origin. Note that it follows from Lemma 3 that wi (a, R−i ) is independent of Ri . We show that wi (a, R−i ) = wj (a, R−j ) for all i, j. Suppose wi (a, R−i ) < wj (a, R−j ), then we can take Ri′ ∈ R such that M RS(Ri′ , φi (a, R)) = M RS(Ri , φi (a, R)) and φj (a, R)Pi′ φi (a, R). By Lemma 2 it holds φi (a, (Ri′ , R−i )) = φi (a, (Ri , R−i )). By NB it holds φj (a, (Ri′ , R−i )) = φj (a, (Ri , R−i )). Then we obtain φj (a, (Ri′ , R−i ))Pi′ φi (a, (Ri′ , R−i )), which is a violation of EF. Hence we have wi (a, R−i ) = wj (a, R−j ) for all i, j. Then wj (a, R−j ) has to be independent of Ri , and wi (a, R−i ) has to be independent of Rj . Hence both wi (a, R−i ) and wj (a, R−j ) are independent of Ri , Rj . By repeating this argument wi for all i is independent of R, and rewrite it by wi (a). Since all wi (a) are equal again rewrite it by w0 (a). Suppose w0 (a) > ai for some i. Then take Ri such that m(Ri , B(w0 (a), 0)) yields sufficiently large li and Rj for all j ̸= i such that m(Rj , B(w0 (a), 0)) yields sufficiently ∑ ∑ small lj , so that w0 (a) j=1 lj > nj=1 aj lj , which is a violation of feasibility. On the other hand, if we impose ∗-EF instead of EF wage vectors must be proportional to skills. In contrast to the rule with common wage above, this class of solutions is compatible with RE, the stronger but still mild requirement of efficiency, and we go back to the trivial solution once we impose it. 19

Theorem 7 If φ satisfies PSP, CON, IS, NB, IR, NES and ∗-EF, then there exists a function λ : Rn++ → (0, 1] such that it holds φi (a, R) = m(Ri , B(λ(a)ai , 0)) for all (a, R) ∈ Rn++ × Rn and i = 1, · · · , n. Moreover, IS is replaced by RE in the above if and only if it holds φi (a, R) = m(Ri , B(ai , 0)) for all (a, R) ∈ Rn++ × Rn and i = 1, · · · , n. The converse statement is true for PSP, CON, IS (resp. RE), NB, IR and NES, and it is true for ∗-EF whenever m(Ri , B(ai , 0))Ri (1, aj ) for all i, j. Proof. It suffices to show the sufficiency of the axioms. Pick any (a, R) ∈ Rn++ × Rn , then for each i the allocation φi (a, R) is supported by the line with slope wi (a, R−i ) which passes the origin. Note that it follows from Lemma 3 that wi (a, R−i ) is independent of Ri . We first show that

wj (a,R−j ) wi (a,R−i )

=

aj ai

for all i, j. Suppose

wj (a,R−j ) wi (a,R−i )

>

aj . ai

Let (l, c) = φ(a, R), then there are two cases. Case 1: Suppose

aj l j ai

≤ 1. Then, since

wj (a,R−j ) wi (a,R−i )

>

aj ai

we have

wj (a, R−j )lj > wi (a, R−i ) (

aj l j , wj (a, R−j )lj ai take Ri′ ∈ R ( Then we can ) aj l j , wj (a, R−j )lj Pi′ φi (a, R). ai

which implies

)

a j lj , ai

∈ / B(wi (a, R−i ), 0).

such that M RS(Ri′ , φi (a, R)) = M RS(Ri , φi (a, R)) and

By Lemma 2 it holds φi (a, (R(i′ , R−i )) = φi (a, (Ri , R)−i )). By NB it holds φj (a, (Ri′ , R−i )) = φj (a, (Ri , R−i )) and we obtain

aj l j , wj (a, Ri′ , R−j,i )lj ai

Pi′ φi (a, (Ri′ , R−i )), which is a viola-

tion of ∗-EF under (a, Ri′ , R−i ). Case 2: Suppose

aj l j ai

> 1. Then, since

wj (a,R−j ) wi (a,R−i )

>

aj ai

wj (a, R−j )lj > wi (a, R−i )

we have a j lj > wi , ai

which implies (1, wj (a, R−j )lj ) ∈ / B(wi (a, R−i ), 0). Thus by the similar argument as in Case 1 we obtain a contradiction. 20

Hence we have

wj (a,R−j ) wi (a,R−i )

=

aj ai

for all i, j. Then wj (a, R−j ) has to be independent of

Ri , and wi (a, R−i ) has to be independent of Rj . Hence both wi (a, R−i ) and wj (a, R−j ) are independent of Ri , Rj . By repeating this argument wi for all i is independent of R, and rewrite it by wi (a). Because

wj (a) wi (a)

=

aj ai

for all i, j, we have wi (a) = λ(a)ai for some common λ(a) > 0. By

feasibility we have λ(a) ≤ 1. It is straightforward that under RE we obtain λ(a) = 1. Example 2 Here is an example of a subclass which includes the trivial solution and the solution with common wage being equal to the lowest skill as ”endpoints”. Given λ ∈ [0, 1], the solution φ : Rn++ × Rn ↠ [0, 1]n × Rn+ is defined by φi (a, R) = m(Ri , B(λai + (1 − λ) min aj , 0)) j

for all (a, R) ∈ Rn++ × Rn and i. Note that when λ = 1 it reduces to the trivial solution and when λ = 0 it reduces to the solution with common wage being equal to the lowest skill.

6

Nash implementation of social choice correspondences

6.1

Nash implementation

Here we investigate which options are feasible if we consider Nash implementation in potentially multiple equilibria. A social choice correspondence (SCC) is a correspondence φ : Rn++ × Rn ↠ [0, 1]n × Rn+ such that φ(a, R) ⊂ F (a) for all (a, R) ∈ Rn++ × Rn . Local Independence (Nagahisa [12]) is an axiom stating that only marginal rates of substitution should matter. It basically says that allocations are coming from some at least sub-optimal individual choices under ”market-like” mechanisms. It is known to be a necessary and sufficient condition for Nash implementation via ”market-like” mechanisms (see Dutta, Sen and Vohra [3]). Given a social choice correspondence φ, let φ◦ denote its interior given by φ◦ (a, R) = φ(a, R) ∩ ((0, 1)n × Rn++ ) for all (a, R) ∈ Rn++ × Rn . 21

Local Independence (LI): For all (a, R), (a, R′ ) ∈ Rn++ × Rn and (l, c) ∈ φ◦ (a, R), if M RS(Ri , (li , ci )) = M RS(Ri′ , (li , ci )) for all i = 1, · · · , n, then (l, c) ∈ φ◦ (a, R′ ). The key feature with regard to manipulation in reporting technological characteristics is that one cannot exaggerate his technology while he can hide a part of it. In the setting of production economy as formulated by Arrow and Debreu [1], Hurwicz, Maskin and Postlewaite [10] show that Nash implementability under such condition puts no restriction on social choice correspondences, except that nobody should receive zero consumption, the worst possible allocation. The translation of their condition to the current setting is that nobody should be forced to spend whole his time on labor without any pay. Non-Confiscatory (NC): For all (a, R) ∈ Rn++ × Rn and (l, c) ∈ φ(a, R), it holds (li , ci ) ̸= (1, 0) for all i. The following result is shown in the appendix, where the definition of implementability is stated precisely. Theorem 8 Suppose φ satisfies NC and n ≥ 3. Then it satisfies LI if and only if it is Nash-implementable via an economic mechanism with skill-reporting.

6.2

Axioms

Below are translations of the previously discussed axioms to the setting of social choice correspondence. Pareto Efficiency (PE): For all (a, R) ∈ Rn++ ×Rn and (l, c) ∈ φ(a, R), there is no there is no (l′ , c′ ) ∈ F (a) such that (li′ , c′i )Ri (li , ci ) for all i = 1, · · · , n and (li′ , c′i )Pi (li , ci ) for at least one i. Resource Efficiency (RE): For all (a, R) ∈ Rn++ × Rn and (l, c) ∈ φ(a, R), , it holds n ∑

ci =

i=1

n ∑

ai li .

i=1

Interior Solution (IS): For all (a, R) ∈ Rn++ × Rn , it holds φi (a, R) ⊂ (0, 1) × R++ for all i. 22

Again the following claim is immediate. Proposition 5 PE implies RE. PE and NC imply IS. Also, IS implies NC.

Envy-Freeness (EF): For all (a, R) ∈ Rn++ × Rn , for all (l, c) ∈ φ(a, R), and for all i, j = 1, · · · , n, it holds (li , ci )Ri (lj , cj ). ∗-Envy-Freeness (∗-EF): For all (a, R) ∈ Rn++ × Rn , for all (l, c) ∈ φ(a, R), and for all i, j = 1, · · · , n, aj lj ai

≤ 1 implies (li , ci )Ri

(

aj l j , cj ai

) .

Individual Rationality (IR): For all (a, R) ∈ Rn++ × Rn and (l, c) ∈ φ(a, R), and for all i = 1, · · · , n, it holds (li , ci )Ri (0, 0). Average Skill Lower Bound (ASLB): For all (a, R) ∈ Rn++ × Rn and (l, c) ∈ φ(a, R), and for all i = 1, · · · , n, it holds (li , ci )Ri (1, a), where a =

1 n

∑n i=1

ai .

Again the following claim is immediate. Proposition 6 Each of IR and ASLB implies NC. No Excess Surplus (NES): For all (a, R) ∈ Rn++ × Rn and (l, c) ∈ φ(a, R), it holds (0, M RS(Ri , (li , ci ))li )Ri (li , ci ) for all i = 1, · · · , n.

23

6.3

Efficient rules

Under efficiency, considering Nash implementation in potentially multiple equilibria does not have much to add, since it simple leaves indeterminacy of lump-sum transfers or it has the same implication as startegy-proofness does. We would include the results on this direction for completeness, however. Definition 2 A correspondence T : Rn++ ↠ Rn with ti + ai > 0 for all a ∈ Rn++ and t ∈ T (a) is called lump-sum transfer structure. Given a lump-sum transfer structure T , say that φT is the lump-sum transfer solution generated by T if for all (a, R) ∈ Rn++ × Rn and (l, c) ∈ F (a) it holds (l, c) ∈ φT (a, R) if and only if there exists t ∈ T (a) such that (ci , li ) = m(Ri , B(ai , ti )) holds for all i. The following says that Local Independence puts no restriction on T (a), while any lump-sum solution satisfies it. Theorem 9 If φ satisfies PE and NC, then φ = φT for some some lump-sum transfer structure T . Moreover, φT with any lump-sum transfer structure T satisfies LI, PE and NC. Proof. The converse statement is straightforward, hence we just show the first claim. Also, by IS (which is implied by PE and NC) we have φ◦ = φ. Pick any (a, R) ∈ Rn++ × Rn and any (l, c) ∈ φ◦ (a, R). Then for each i there is a unique vector (wi , bi ) with wi > 0 such that (li , ci ) is the solution to preference maximization under the constraint c′i = wi li′ + bi . By PE, wi = ai for every i. Now for each a ∈ Rn++ , let T (a) be the set of lump-sum transfers which support some allocation in φ(a, R). Then φ(a, R) = φT (a, R) for every (a, R) ∈ Rn++ × Rn . The following results can be shown in the same way as before, which confirms that LI has the same implication as PSP does when we impose efficiency and an axiom of bounds on welfare. Theorem 10 φ satisfies LI, PE and IR if and only if it holds φi (a, R) = {m(Ri , B(ai , 0))} 24

for all (a, R) ∈ Rn++ × Rn and i = 1, · · · , n. Moreover, the same assertion holds when we replace IR by NES. Theorem 11 φ satisfies LI, PE and ASLB if and only if it holds φi (a, R) = {m(Ri , B(ai , a − ai ))} for all (a, R) ∈ Rn++ × Rn and i = 1, · · · , n, where a =

6.4

1 n

∑n i=1

ai .

Distortions

Nash implementation in potentially multiple equilibria if we could do it allows us to endogenously determine wages which are distorted in general but support allocations with smaller efficiency loss. Like before, although we can use both wage distortions and lump-sum transfers, we would focus on what we can do with wage distortions, since from the previous arguments we already know what we can do with lump-sum transfers. Definition 3 A correspondence W : Rn++ ↠ Rn++ with is called wage structure. Given a wage structure W , say that φW is the wage solution generated by W if for all (a, R) ∈ Rn++ × Rn and (l, c) ∈ F (a) it holds (l, c) ∈ φW (a, R) if and only if there exists w ∈ W (a) such that (ci , li ) = m(Ri , B(wi , 0)) holds for all i, and

n ∑ i=1

ci =

n ∑

ai li .

i=1

Theorem 12 φ satisfies LI, RE, IS, IR and NES if and only if φ = φW for some wage structure W . Proof. The converse statement is straightforward, hence we just show the first claim. Also, by IS we have φ◦ = φ. Pick any (a, R) ∈ Rn++ × Rn and any (l, c) ∈ φ◦ (a, R). Then for each i there is a unique vector (wi , bi ) with wi > 0 such that (li , ci ) is the solution to preference maximization under the constraint c′i = wi li′ + bi .

25

Suppose bi < 0. Take Ri′ such that M RS(Ri , (li , ci )) = M RS(Ri′ , (li , ci )) = wi and (0, 0)Pi′ (−bi /wi − ε, 0)Ii′ (li , ci ) for some ε > 0 with −bi /wi − ε > 0. By LI we have (l, c) ∈ φ(a, (Ri′ , R−i )). But this violates IR in the economy (a, (Ri′ , R−i ))). Suppose bi > 0. Take Ri′ such that M RS(Ri , (li , ci )) = M RS(Ri′ , (li , ci )) = wi and (li , ci )Pi′ (0, M RS(Ri , (li , ci ))li ) By LI we have (l, c) ∈ φ◦ (a, (Ri′ , R−i )). But this violates NES in the economy (a, (Ri′ , R−i ))). Now for each a ∈ Rn++ , let W (a) be the set of wages which support some allocation in φ(a, R). Then φ(a, R) = φW (a, R) for every (a, R) ∈ Rn++ × Rn . What we gain by allowing Nash implementation is that there is an allocation rule satisfying Envy-Freeness with smaller efficiency loss. Theorem 13 φ satisfies LI, RE, IS, IR, NES and EF if and only if for all (a, R) ∈ Rn++ ×Rn and (l, c) ∈ φ(a, R) there exists w0 ∈ R++ such that (ci , li ) = m(Ri , B(w0 , 0)) holds for all i, and

n ∑ i=1

ci =

n ∑

ai li .

i=1

Proof. It suffices to show the sufficiency of the axioms. Also, by IS we have φ◦ = φ. Pick any (a, R) ∈ Rn++ × Rn and (l, c) ∈ φ(a, R), then for each i the allocation (li , ci ) is supported by the line with slope wi which passes the origin. We show that wi = wj for all i, j. Suppose wi < wj , then we can take R′ ∈ R such that M RS(Ri′ , (li , ci )) = M RS(Ri , (li , ci )) for all i and (lj , cj )Pi′ (li , ci ). By LI we have (l, c) ∈ φ(a, R′ ), but this violates EF. Note that in the previous argument based on strategy-proofness Envy-freeness can be achieved only by paying everybody the same wage which is no greater than the lowest skill. Here it is possible to determine the common wage endogenously in order to match demand and supply for consumption. To complete the argument, we state the ”dual” of the above result in which EF is replaced by ∗-EF.

26

Theorem 14 If φ satisfies LI, RE, IS, IR, NES and ∗-EF then for all (a, R) ∈ Rn++ × Rn , (l, c) ∈ φ(a, R) it holds (ci , li ) = m(Ri , B(ai , 0)) holds for all i, and

n ∑

ci =

i=1

n ∑

ai li .

i=1

The converse statement is true for LI, RE, IR and NES, and it is true for ∗-EF whenever m(Ri , B(ai , 0))Ri (1, aj ) for all i, j. Proof. It suffices to show the sufficiency of the axioms. Also, by IS we have φ◦ = φ. Pick any (a, R) ∈ Rn++ × Rn and (l, c) ∈ φ(a, R), and let w be the supporting wage vector. We first show that Case 1: Suppose

aj l j ai

wj wi

=

aj ai

for all i, j. Suppose

≤ 1. Then, since

wj wi

>

aj ai

wj lj > w i ( which implies

aj l j , wj lj ai

)

wj wi

>

aj . ai

Then there are two cases.

we have aj lj , ai

∈ / B(wi , 0).

Then we can take R′ ∈ Rn such that M RS(Ri′ , (li , ci )) = M RS(Ri , (li , ci )) and

(

aj lj , wj lj ai

By LI we have (l, c) ∈ φ(a, R′ ), which is a violation of ∗-EF. Case 2: Suppose

aj l j ai

> 1. Then, since

wj wi

>

wj lj > w i

aj ai

we have

a j lj > wi , ai

which implies (1, wj lj ) ∈ / B(wi , 0). Thus by the similar argument as in Case 1 we obtain a contradiction. Hence there exists γ > 0 such that wi = γai for all i. By RE, we obtain γ = 1. Example 3 Here is an example of a subclass which includes the trivial solution and the solution with common wage as ”endpoints”. Given λ ∈ [0, 1], the solution φ : Rn++ × Rn ↠ [0, 1]n × Rn+ is a correspondence defined as follows: for all (a, R) ∈ Rn++ × Rn it holds (l, c) ∈ φ(a, R) if and only if there exists w0 ∈ R++ such that (ci , li ) = m(Ri , B(λai + (1 − λ)w0 , 0)) 27

)

Pi′ (li , ci ).

holds for all i, and

n ∑ i=1

ci =

n ∑

ai li .

i=1

Note that when λ = 1 it reduces to the trivial solution and when λ = 0 it reduces to the solution with common wage.

7

Concluding remarks

We have investigated how decentralizability requirements put restrictions on allocation rules in production economies with unequal skills. First we show that if we insist on efficiency strategy-proofness leaves us little variety of views about how people are responsible for their skills. If we do not insist on efficiency strategy-proofness allows for the role of equity, but it requires larger efficiency loss. We then consider Nash implementation, and show that informational constraint with regard to skill levels puts almost no restriction on allocation rules there. We then characterize a class of implementable solutions, which allow for the role of equity at smaller efficiency loss.

References [1] Arrow, Kenneth J., and Gerard Debreu. ”Existence of an equilibrium for a competitive economy.” Econometrica (1954): 265-290. [2] Chambers, Christopher P., and Takashi Hayashi. ”Resource allocation with partial responsibilities for initial endowments,” working paper, 2013. [3] Dutta, Bhaskar, Arunava Sen, and Rajiv Vohra. ”Nash implementation through elementary mechanisms in economic environments.” Economic Design 1.1 (1994): 173203. [4] Fleurbaey, Marc, and Francois Maniquet. ”Fair allocation with unequal production skills: The no envy approach to compensation.” Mathematical Social Sciences 32.1 (1996): 71-93.

28

[5] Fleurbaey, Marc, and Francois Maniquet. ”Cooperative production with unequal skills: The solidarity approach to compensation.” Social Choice and Welfare 16.4 (1999): 569-583. [6] Fleurbaey, Marc, and Francois Maniquet. ”Fair income tax.” Review of Economic Studies 73.1 (2006): 55-83. [7] Gibbard, Allan. ”Manipulation of voting schemes: a general result.” Econometrica (1973): 587-601. [8] Hurwicz, Leonid. ”On informationally decentralized systems.” Decision and organization (1972). [9] Hurwicz, Leonid. ”On allocations attainable through Nash equilibria.” Journal of Economic Theory 21.1 (1979): 140-165. [10] Hurwicz, Leonid, Eric Maskin, and Andrew Postlewaite. ”Feasible Nash implementation of social choice rules when the designer does not know endowments or production sets.” The Economics of Informational Decentralization: Complexity, Efficiency, and Stability. Springer US, 1994. 367-433. [11] Maskin, Eric. ”Nash equilibrium and welfare optimality*.” The Review of Economic Studies 66.1 (1999): 23-38. [12] Nagahisa, Ryo-ichi. ”A local independence condition for characterization of Walrasian allocations rule.” Journal of Economic Theory 54.1 (1991): 106-123. [13] Nagahisa, Ryo-Ichi, and Sang-Chul Suh. ”A characterization of the Walras rule.” Social Choice and Welfare 12.4 (1995): 335-352. [14] Pazner, Elisha A., and David Schmeidler. ”A difficulty in the concept of fairness.” Review of Economic Studies (1974): 441-443. [15] Satterthwaite, Mark Allen. ”Strategy-proofness and Arrow’s conditions: Existence and correspondence theorems for voting procedures and social welfare functions.” Journal of Economic Theory 10.2 (1975): 187-217. [16] Serizawa, Shigehiro. ”Inefficiency of strategy-proof rules for pure exchange economies.” Journal of Economic Theory 106.2 (2002): 219-241. 29

[17] Thomson, William. ”On allocations attainable through Nash equilibria, a comment.” [18] Varian, Hal R. ”Equity, envy, and efficiency.” Journal of Economic Theory 9.1 (1974): 63-91. [19] Yamada, Akira, and Naoki Yoshihara. ”Triple implementation by sharing mechanisms in production economies with unequal labor skills.” International Journal of Game Theory 36.1 (2007): 85-106. [20] Yoshihara, Naoki, and Akira Yamada. Nash Implementation in Production Economies with Unequal Skills: A Complete Characterization. No. a536. 2010. [21] Yoshihara, Naoki. ”Characterizations of bargaining solutions in production economies with unequal skills.” Journal of Economic Theory 108.2 (2003): 256-285. [22] Zhou, Lin. ”Inefficiency of strategy-proof allocation mechanisms in pure exchange economies.” Social Choice and Welfare 8.3 (1991): 247-254.

A

Nash implementation via an economic mechanism with skill-reporting

Given some m, let Di ⊂ Rm denote the message space for each i. Let D =

∏n i=1

Di . Given

a ∈ Rn++ , consider an outcome function g[a] : D → [0, 1]n × Rn+ such that g[a](d) ∈ F (a) and comp {gi [a](di , d−i ) : di ∈ Di } is convex for all d−i ∈ D−i , where compA denotes the ”comprehensive hull” of A ⊂ [0, 1] × R+ defined by compA = {(x, y) ∈ [0, 1] × R+ : x ≥ l, y ≤ c, (l, c) ∈ A}. Let g denote the function which maps each a ∈ Rn++ into g[a]. We call (g, D) = (g[a], D)a∈Rn++ an economic mechanism Definition 4 φ is said to be R-implementable via an economic mechanism if there exists an economic mechanism (g, D) such that g[a] (N E((g[a], D), R) = φ(a, R) for all a ∈ Rn++ and R ∈ Rn . 30

Proposition 7 Suppose φ satisfies IS and n ≥ 3. Then it satisfies LI if and only if it is R-implementable via an economic mechanism. Proof. Proof is similar to the one for Theorem 3.2 in Dutta, Sen and Vohra. [3]. Necessity of LI is straightforward. We prove its sufficiency. For each i, let Di = Rn++ × (0, 1)n × Rn++ × (0, 1) where its element di = (wi , xi , yi , qi , ri ) consists of wi = (wi1 , · · · , win ) ∈ Rn++ , xi = (x1i , · · · , xni ) ∈ (0, 1)n , yi = (yi1 , · · · , yin ) ∈ Rn++ , qi ∈ (0, 1), ri ∈ R++ .

Here is the construction of g[a] for each a ∈ Rn++ . Rule 1: If (wi , xi , yi ) = (v, l, c) for all i, where (l, c) ∈ φ(a, R), vk = M RS(Rk , (lk , ck )), ∀k = 1, · · · , n for some R ∈ Rn , then g[a](d) = (l, c). Rule 2: If (wj , xj , yj ) = (v, l, c) for all j ̸= i, where (l, c) ∈ φ(a, R), vk = M RS(Rk , (lk , ck )), ∀k = 1, · · · , n for some R ∈ Rn , then  (xi , yi ), if y i − vi xi = ci − vi li i i g[a](d) = (l, c), otherwise. 31

Rule 3: If neither of Rule 1 or 2 applies, run a modulo game based on x. Let i denote the ∑ winner, then he receives (qi , ri ) if ri ≤ ai qi + j̸=i aj and (1, 0) otherwise, and everyone else receives (1, 0). First we show g[a] (N E((g[a], D), R) ⊃ φ(a, R) for all a ∈ Rn++ and R ∈ Rn . Suppose (l, c) ∈ φ(a, R). Let d give (wi , xi , yi ) = (v, l, c) for all i, where vk = M RS(Rk , (lk , ck )), ∀k = 1, · · · , n. From Rule 1, g[a](d) = (l, c). Then the only possible deviation for any i is through Rule 2, but it is unprofitable because of strict convexity of preference. Next we show g[a] (N E((g[a], D), R) ⊂ φ(a, R) for all a ∈ Rn++ and R ∈ Rn . Suppose d ∈ g[a] (N E((g[a], D), R). We show that Nash equilibrium strategy cannot be described by Rule 2 or 3. Suppose that the strategy profile falls in Rule 2. Then there is some i who is deviating from unanimity. Since n ≥ 3 there exists j ̸= i who can get whatever he likes by deviating so that Rule 3 applies and he gets whatever he likes by making the others work for nothing. This forms a profitable deviation. The equilibrium strategy profile cannot fall in Rule 3, since there is no equilibrium in such case. Hence the equilibrium situation has to fall in Rule 1. That is, d gives (wi , xi , yi ) = (v, l, c) for all i, where e (l, c) ∈ φ(a, R) ek , (lk , ck )) ∀k = 1, · · · , n vk = M RS(R e for some R. In order that d ∈ g[a] (N E((g[a], D), R) it holds vk = M RS(Rk , (lk , ck )) for all k = 1, · · · , n, since otherwise somebody has a profitable deviation by making Rule 2 apply. Then by LI we have (l, c) ∈ φ(a, R). It is straightforward to show that (g, D) = (g[a], D)a∈Rn++ is indeed an economic mechanism.

32

Given m, let i ⊂ Rm . Consider a message space for i given in the form n−i+1 Si (ai ) = Rn−i × Di ++ × (0, ai ] × R++ ) ∏ ∏ ( ∪ n−i+1 Let S(a) = ni=1 Si (a), A(a) = ni=1 Rn−i , A = a∈Rn A(a), and ++ × (0, ai ] × R++ ++ ∏ ∪ D = ni=1 Di . Let S = a∈Rn S(a), where its element is typically denoted by s = (α, d). ++

Consider an outcome function h : S → [0, 1]n × Rn+ . Say that (h, S) = (h, S(a))a∈Rn++ is an economic mechanism with skill-reporting if h(α, ·) : D → [0, 1]n × Rn+ forms an economic mechanism together with D for all a ∈ Rn++ , for all α ∈ A(a). Definition 5 φ is said to be Nash implementable via an economic mechanism with skillreporting if there exists a pair (h, S) such that h (N E((h, S(a)), R) = φ(a, R) for all a ∈ Rn++ and R ∈ Rn . Theorem 15 Suppose φ satisfies IS and n ≥ 3. Then it satisfies LI if and only if it is implementable via an economic mechanism with skill-reporting. Proof. Proof is similar to the one for production economy as in Hurwicz, Maskin and Postlewaite [10]. For each i, let n−i+1 Si (ai ) = Rn−i × [0, 1] × R+ × Di ++ × (0, ai ] × R++

where its element si = (αi , di ) consists of n−i+1 αi = (αi1 , · · · , αii , · · · , αin ) ∈ Rn−i ++ × [0, ai ] × R++

di = (wi , xi , yi , qi , ri ) ∈ Di Given s ∈ S, let M (s) = {i ∈ {1, · · · , n} : ∀j ̸= i, αji < αii } ti (a) = #{αji : j ∈ {1, · · · , n}} t(a) = max ti (a) i ∑∑ j βi (s) = |αj − αkj | j̸=i k̸=i

βi∗ (s)

= βi (s) 33

Note that

∑n i=1

βi (s) = 0 if and only if αi = αj for all i, j.

Rule 1: If there exists a ∈ Rn++ such that αi = a for all i, then h(s) = g[a](d). Suppose Rule 1 does not apply (which implies

∑n j=1

βj (s) > 0). Then, either Rule 2 or

Rule 3 applies. Rule 2: If M (s) = ∅, then (i) If t(s) = 2 then

(

βi (s) 1 − ∑n ,0 j=1 βj (s)

hi (s) =

)

for all i ∈ {1, · · · , n}. (ii) If t(s) ≥ 3 then

) βi∗ (s) ,0 1 − ∑n ∗ j=1 βj (s)

( hi (s) =

for all i ∈ {1, · · · , n}. Rule 3: (i) If M (s) ̸= ∅ and

∑

ri ≤

∑

αii qi +

αjj

j ∈M / (s)

i∈M (s)

i∈N (s)

∑

then hi (s) = (qi , ri ), ∀i ∈ M (s) hj (s) = (1, 0), ∀j ∈ / M (s)

(ii) If M (s) ̸= ∅ and

∑ i∈N (s)

ri >

∑

αii qi +

i∈M (s)

∑

αjj

j ∈M / (s)

then hj (s) = (1, 0), ∀j ∈ {1, · · · , n} Claim 1: Suppose s = (s1 , · · · , sn ) has the form si = (a, di ) for every i such that d = (d1 , · · · , dn ) ∈ N E((g[a], D), R). Then s ∈ N E((h, S(a)), R). 34

Proof. Suppose s ∈ / N E((h, S(a)), R). Since d ∈ N E((g[a], D), R), any deviation for i in the form sei = (a, dei ) is not profitable. Hence we consider a deviation for i in the form sei = (e αi , dei ) with α ei ̸= a. Since α eii ≤ ai we have M (e si , s−i ) = ∅ and Rule 2 applies. Then it holds βi (e si , s−i ) = 0 because all the others are unanimous with regard to skill reporting. Therefore we have hi (e si , s−i ) = (1, 0). Since h(s) = g[a](d) ∈ φ(a, R), and because of IS we have hi (s)Pi (1, 0), which implies that the deviation is making i strictly worse off. Claim 2: Suppose s = (s1 , · · · , sn ) has the form si = (a′ , di ) for every i such that a′ ̸= a. Then s ∈ / N E((h, S(a)), R). Proof. Suppose s ∈ N E((h, S(a)), R). Since no exaggeration is possible we have a′ ≤ a. Since a′ ̸= a there is i with a′i < ai . Then, for all j ̸= i we have αji = a′i < ai . Let sei = (e αi , dei ) be α eii = ai α eij = αij = aj , ∀j ̸= i w ei = wi x ei = xi yei = yi qei = li (s) rei = ai li (s) +

∑ j̸=i

Then M (e si , s−i ) = {i}

35

a′j

By Rule 3 (i), we have ( hi (e si , s−i ) =

li (s), ai li (s) +

∑

) a′j

j̸=i

hj (e si , s−i ) = (1, 0), ∀j ̸= i Then we have ci (e si , s−i ) = ai li (s) +

∑

a′j

j̸=i

> a′i li (s) +

∑

a′j lj (s)

j̸=i

= ci (s) +

∑

cj (s)

j̸=i

> ci (s), where the strict inequalities follow from IS. Therefore hi (e si , s−i )Pi hi (s), which contradicts to the assumption of NE. Claim 3: If there is no unanimity with regard to skill-reporting, then there is no NE. Proof. Case 1: Suppose M (s) = {1, · · · , n}. Pick any i ∈ {1, · · · , n}. Let sei = (e αi , dei ) be α eij = αjj , ∀j ∈ {1, · · · , n} w ei = wi x ei = xi yei = yi qei = li (s) rei = ai li (s) +

∑

a′j

j̸=i

Then j ∈ / M (e si , s−i ) for all j ̸= i. Now we show i ∈ M (e si , s−i ). Since M (s) = {1, · · · , n}, we have αii > αji for all j ̸= i. si , s−i ). Hence it holds α eii = αii > αji for all j ̸= i, which implies i ∈ M (e 36

Thus we obtain M (e si , s−i ) = {i} By Rule 3 (i), we have ( hi (e si , s−i ) =

li (s), ai li (s) +

∑

) a′j

j̸=i

hj (e si , s−i ) = (1, 0), ∀j ̸= i By the similar argument as before this forms a profitable deviation for i. Case 2: Suppose M (s) ̸= {1, · · · , n} and M (s) ̸= ∅. Then there is i ∈ / M (s), who receives hi (s) = (1, 0). Let sei = (e αi , dei ) be α eij = αjj , ∀j ∈ {1, · · · , n} dei = di Then, by construction we have j ∈ / M (e si , s−i ) for all j ̸= i. Also, since any j ̸= i is not deviating we have i ∈ / M (e si , s−i ) again. Thus we have M (e si , s−i ) = ∅. Subcase 2-(i): Suppose t(e si , s−i ) = 2 By Rule 2,

(

) βi (e si , s−i ) 1 − ∑n ,0 si , s−i ) j=1 βj (e

hi (e si , s−i ) = Note that

βi (e si , s−i ) ≥ max |αjj − αkj | > 0 j,k̸=i

follows because there is some j ∈ M (s). Therefore hi (e si , s−i )Pi hi (s). Subcase 2-(ii): Suppose t(e si , s−i ) ≥ 3. Then we have βk∗ (e si , s−i ) > 0 for all k ∈ {1, · · · , n}, ) si , s−i ) βi∗ (e ,0 1 − ∑n ∗ si , s−i ) j=1 βj (e

(

and hi (e si , s−i ) =

37

dominates (1, 0). Hence hi (e si , s−i )Pi hi (s). Case 3: Suppose M (s) = ∅. Subcase 3-(i) : t(s) = 2 and βk (s) > 0 for all k ∈ {1, · · · , n}. Because reports are not unanimous there exist i, j ∈ {1, · · · , n} such that αij ̸= αjj . Let sei = (e αi , dei ) be α eii = αii α eij = αjj α eik = αik ∀k ̸= i, j dei = di By construction, we have βi (e si , s−i ) = βi (s) > 0 and βj (e si , s−i ) = βj (s) > 0 and βk (e si , s−i ) < βk (s) Also, we have M (e si , s−i ) = M (s) = ∅ and t(e si , s−i ) = t(s) = 2. Hence by Rule 2-(i) we have (

) βi (e si , s−i ) 1 − ∑n ,0 si , s−i ) j=1 βj (e

hi (e si , s−i ) = (

which dominates hi (s) =

) βi (s) ,0 , 1 − ∑n j=1 βj (s)

Therefore hi (e si , s−i )Pi hi (s). Subcase 3-(ii) : t(s) = 2 and βi (s) = 0 for some i. By Rule 2-(i) we have hi (s) = (1, 0) . 38

Subsubcase 3-(ii)-(a): Suppose there exists sei = (e αi , dei ) such that t(e si , s−i ) ≥ 3, (e αi , α−i ) is not unanimous, and M (e si , s−i ) = ∅. Then By Rule 2-(ii) we have ( hi (e si , s−i ) =

) βi (e si , s−i ) 1 − ∑n ,0 si , s−i ) j=1 βj (e

which dominates hi (s) = (1, 0). Therefore hi (e si , s−i )Pi hi (s). Subsubcase 3-(ii)-(b): Suppose (a) is false, then it has to be that αk = αj ̸= αi for all k, j ̸= i. Then the case is either (1) αii = αji for all j ̸= i, or (2) αii ̸= αji for all j ̸= i. Subsubsubcase 3-(ii)-(b)-(1): Suppose αi = αi for all j ̸= i. Let sei = (e αi , dei ) be i

j

α eii = αii α eij = αjj ∀j ̸= i dei = di Then (e αi , α−i ) is unanimous (all being equal to some a′ ), hence h(e si , s−i ) = g[αi , α−i ](d) ∈ φ(a′ , R). By NC, hi (e si , s−i ) strictly dominates (1, 0). On the other hand, since hi (s) = (1, 0) we have hi (e si , s−i )Pi hi (s). Subsubsubcase 3-(ii)-(b)-(2): Suppose αii ̸= αji for all j ̸= i. Then βj (s) > 0 for all j ̸= i. Pick any j ̸= i. Let sej = (e αj , de−j ) be α eji = αii α ejk = αjk ∀k ̸= i dei = di Then we have t(e sj , s−j ) = 2, (e αj , α−j ) is not unanimous and M (e sj , s−j ) = ∅. Also we have βi (e sj , s−j ) = βi (s) = 0 39

and βj (e sj , s−j ) = βj (s) > 0 and βk (e sj , s−j ) < βk (s) for some k ̸= i, j. This implies β (e s ,s ) βj (s) ∑n j j −j > ∑n sj , s−j ) k=1 βk (e k=1 βk (s) Therefore hj (e sj , s−j )Pj hj (s). Subcase 3-(iii) : Suppose t(s) ≥ 3. Then there exist i, j, k such that αii , αji and αki are different. Let sej = (e αj , de−j ) be α eji ∈ (αii − |αji − αii |, αii + |αji − αii |) α ejk = αjk ∀k ̸= i dei = di Then we have t(e sj , s−j ) = t(s), (e αj , α−j ) is not unanimous and M (e sj , s−j ) = ∅. Also we have βj∗ (e sj , s−j ) = βj∗ (s) and βi∗ (e sj , s−j ) = βi∗ (s) and βk∗ (e sj , s−j ) < βk∗ (s) for some k ̸= i, j. This implies

By Rule 2-(ii) it holds

sj , s−j ) β ∗ (e βj∗ (s) ∑n j ∗ > ∑n ∗ sj , s−j ) k=1 βk (e k=1 βk (s) ( ) βj∗ (s) hj (s) = 1 − ∑n ,0 ∗ k=1 βk (s)

40

Therefore we obtain hj (e sj , s−j )Pj hj (s).

41