A Mechanism Design for a Solution to theTragedy of Commons∗

Akira Yamada† Naoki Yoshihara‡

September 2001, This version April 2002.

AbstractWe consider the problem of the tragedy of commons in coopera-

tive production economies, and propose a mechanism to resolve thetragedy, taking seriously non-negligible aspects of that problem andthe real application of mechanisms. The mechanism permits agents tochoose their own labor time freely, and to announce their labor skillsfreely not only if they choose to understate them, but also if they over-state them. It doubly implements the proportional solution [Roemerand Silvestre (1989, 1993)] in Nash and strong equilibria when it isplayed as a normal game form, while it triply implements the solutionin Nash, subgame-perfect, and strong equilibria when it is played asa two-stage extensive game form.

Journal of Economic Literature Classification Numbers: C72, D51,D78, D82Keywords: triple implementation, proportional solution, unknownand possibly overstated labor skills, labor sovereigntyCorresponding Author: Naoki Yoshihara

∗An earlier version of this paper was presented at the annual meeting of the JapaneseEconomic Association held at Hitotsubashi University in October 2001. We are gratefulto Professor Takehiko Yamato for his useful comments in this conference. We are alsothankful to Professors Tatsuyoshi Saijo, Kotaro Suzumura, and Yoshikatsu Tatamitani fortheir kind comments.

†Department of Economics, Hitotsubashi University, Naka 2-1, Kunitachi, Tokyo 186-0004, Japan. (e-mail: ged9207 @srv.cc.hit-u.ac.jp)

‡Department of Political Science, Yale University, 70 Sachem, New Haven CT, 06511,USA, and Institute of Economic Research, Hitotsubashi University, Naka 2-1, Kunitachi,Tokyo, 186-0004, Japan. (e-mail: yosihara @ier.hit-u.ac.jp)

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1 Introduction

It is well-known as the “tragedy of commons,” that in cooperative productioneconomies, the resource allocations under free access to technology result in“overproduction” and inefficient Nash equilibria. In this paper, we providea mechanism which can solve the problem of the tragedy of commons. Asa normative solution for the tragedy in these economies, we adopt the pro-portional solution [Roemer and Silvestre (1989, 1993)] under joint ownershipof the technology, which assigns Pareto efficient allocations such that eachagent’s share in output consumption is proportional to his share in laborcontribution. Since the equilibrium allocations of the tragedy of commonsare Pareto inefficient proportional allocations, adopting this seems to be verynatural. So, our main concern will be restricted to constructing an incentivecompatible mechanism which implements the proportional solution.It is implementation theories which give us a framework to treat such

a mechanism design problem, and in fact, a few works provide mechanismswhich implement the proportional solution, such as Suh (1995), Yoshihara(1999, 2000), and Tian (2000). Also, there are several works on implementa-tion of other social choice correspondences in production economies.1 How-ever, in most of the literature on implementation in production economies, anon-negligible problem of asymmetric information involved in the productionprocess seems to be treated as a “black box.”Usually, the mechanisms in the implementation literature consist of pairs

of strategy spaces and outcome functions, where each agent is required toannounce some information, and the outcome function assigns an allocationto each profile of agents’ strategies. So, in the case of production economieswhere one of main productive factors is labor, once the mechanism assignsan allocation, then each agent seems to be implicitly required to supply theassigned labor time. In otherwords, it seems to be implicitly assumed thatthe mechanism coordinator is authorized to make agents work to realize theassigned allocation.2 This happens because the original concern of imple-mentation theory has been in adverse selection problems, and such a focusseems to be valid whenever we consider decentralized resource allocationsin exchange economies and/or production economies with no labor input.

1For example, Hurwicz et al. (1995), Hong (1995), Tian (1999) for private ownershipproduction economies with only private goods, Varian (1994) for production economieswith externality, and Kaplan and Wettstein (2000) for cooperative production economies.

2Roemer (1989) pointed out this implicit assumption explicitly.

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However, it seems to be more realistic that, in production economies withlabor input, the coordinator is not necessarily authorized to force agents toprovide the amount of labor time assigned by the outcome function. Thus, itseems we should be concerned about not only the adverse selection but alsothis kind of moral hazard in considering the problem of mechanism designunder production economies with labor input.Taking seriously the real application process of mechanisms in production

economies, we may suppose that the coordinator is not authorized to makeagents work as he wants, and also that although he can watch how much timeeach agent works in the factory by time-records, he cannot perfectly monitoreach agent’s labor contribution measured in efficiency units. In otherwords,the coordinator cannot know each agent’s level of labor skill or labor intensityexercised in the production process. Thus, there may be an incentive foreach agent to overstate his own labor skill or labor intensity, since such amisrepresentation may reward him under the proportional solution.3 Thispaper provides an incentive compatible mechanism which works well andimplements the solution even under such a more realistic model of the tragedyof commons.The mechanism we propose here is a type of sharing mechanism: each

agent can freely work in the factory as he wants,4 and he is asked to givesome information about his demand for the consumption good and his laborskill. After that, the outcome function only distributes the produced outputto agents, according to the information they gave and the record of theirsupply of labor time. Here, there is no restriction on strategy spaces whichprohibits agents from understating or overstating their labor skills.5 Thus, in

3Tian (2000) constructed a mechanism which is workable to implement the proportionalsolution even if the coordinator does not know agents’ endowment vectors of commoditiesunder the assumption that agents cannot overstate their endowments. As Tian (2000)mentioned himself, such an assumption may be justified when endowments consist only ofmaterial goods, for the coordinator can require agents to “place the claimed endowmentson the table” (Hurwicz et al. (1995)). In our setting where endowments are labor skills,however, such a requirement is no longer forceful, since the coordinator may not inspectthe amount of labor skills in advance of production.

4Thus, our mechanism has labor sovereignty (Kranich (1994)) which is not satisfied inthe previous mechanisms (Suh (1995), Yoshihara (1999, 2000), Tian (2000)).

5Even for theWalrasian or the Lindahl solution, there have not been found a mechanismwhich Nash-implements the solution when agents can overstate their endowments. Severalpapers discussed implementation of the solutions where underreported endowments areeither destroyed or withheld (Hurwicz et al. (1995), Hong (1995), Tian (1989, 1990, 1991,

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this mechanism, the coordinator should not make agents work as he wants,nor monitor each agent’s labor intensity exercised in production process. Wewill show that this mechanism doubly implements the proportional solutionin Nash and strong Nash equilibria.Taking seriously the real application of mechanisms in production economies

again, we may mention that there is a timing problem of agents’ strategy de-cisions: the time they announce information on their consumption demandand labor skill may be in advance of the production process or may be afterthe production process. Thus, each agent may be able to decide his strategyon what to announce after knowing all of the agents’ supply of labor times,or he may be able to choose his own supply of labor time after hearing all ofthe agents’ announcements on the two matters. This implies that the gamedefined by the sharing mechanism may be played with Nash-like behaviorsor with subgame-perfect behaviors. Since the coordinator cannot know inadvance which of the equilibrium notions the agents will adopt to play thegame, it is desirable that the mechanism can implement the solution indepen-dently of this problem. We will demonstrate that our mechanism can workwell even if it is played as a two-stage game form, by showing that the twotypes of two-stage game forms derived from the original sharing mechanismcan doubly implement the proportional solution in Nash and subgame-perfectequilibria.Our proposed mechanism has several other desirable properties. Saijo et

al. (1996) proposed properties that would be desired in a natural mechanism.Ours meets most of them: it satisfies feasibility (Hurwicz et al. (1995)),forthrightness (Saijo et al. (1996)), self-relevancy (Hurwicz (1960)), and bestresponse property (Jackson et al. (1994)). Also, our mechanism is a simplequantity mechanism in which each agent is only required to announce hislabor skill and his demand for the consumption good, and can provide anylabor time he wants. Finally, the mechanism does not depend on the numberof agents, and it works even in economies of two agents.There are a few other works (Varian (1994), Kaplan andWettstein (2000))

which also consider production economies with externality as this paper does,and propose mechanisms which implement efficient allocations. The technicalstructure of the mechanism here is quite different from these, however. Theirmechanisms do not satisfy labor sovereignty, so that they do not seem to befree from the “black box” problem entailed in production economies. More-

1992, 1993, 1999), Tian and Li (1995)).

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over, their mechanisms cannot implement the proportional solution, whileours can.In the following discussion, the economy is defined in Section 2. Section

3 discusses some of the main issues of mechanism design for the tragedyof commons, and defines sharing mechanisms. Section 4 provides a sharingmechanism which implements the proportional solution. Concluding remarksare in Section 5. All the proofs of the theorems here will be relegated to theAppendix.

2 Economic Environments

There are two goods, one of which is an input good (labor time) x ∈ R+

to be used to produce the other good y ∈ R+.6 The population in thesociety is given by the set N = 1, . . . , n, where 2 ≤ n < +∞. Eachagent i0s consumption vector is denoted by zi = (xi, yi), where xi denotes hislabor time, and yi denotes his assigned amount of output consumption. It isassumed that all agents face a common upper bound of labor time x , where0 < x < +∞, so that they have the same consumption set [0, x]×R+. Eachagent i0s preference is defined on [0, x] × R+ and represented by a utilityfunction ui : [0, x] × R+ → R which is continuous and quasi-concave on[0, x]× R+, and strictly monotonic (decreasing in labor time and increasingin the share of output) on [0, x) × R++.7 We denote by U the class of suchutility functions. Each agent i is also characterized by a labor skill whichis represented by a positive real number, sti ∈ R++.8 The universal set oflabor skills for all agents is denoted by S = R++.9 The labor skill sti ∈ Simplies i0s labor endowment per unit of labor time, which is measured inefficiency units. It can also be interpreted as i0s labor intensity which wouldbe exercised in the production process: that is, i0s labor input per unitof labor time measured in efficiency units.10 Thus, if his supply of labor

6The symbol R+ denotes the set of non-negative real numbers.7The symbol R++ denotes the set of positive real numbers.8The superscript t on st

i indicates “true,” so that sti denotes agent i0s true labor skill.

9For any two sets X and Y , X ⊆ Y whenever any x ∈ X also belongs to Y , and X ( Yif and only if X ⊆ Y and not (Y ⊆ X).10Often, it might be more natural to define labor endowment and labor intensity in a

discriminative way: for example, if sti ∈ S is i’s labor endowment per unit of labor time,

then i’s labor intensity is a variable sti, where 0 < st

i ≤ sti. In such a formulation, we

may view the amount of sti as being determined endogenously based upon the agent’s

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time is xi ∈ [0, x] and his labor intensity is sti ∈ S, then it is stixi ∈ R+

which implies his labor contribution to the production process measured inefficiency units. The production technology is described by a productionfunction f : R+ → R+ which is assumed to be continuous, strictly increasing,concave, and f (0) = 0. For simplicity, we fix a production function f for alleconomies. Thus, the economy is characterized by a pair of profiles e ≡ (u, st)with u = (u1, . . . , un) ∈ Un and st = (st1, . . . , stn) ∈ Sn. Denote the class ofsuch economies by E ≡ Un × Sn.Given st = (st1, . . . , s

tn) ∈ Sn, an allocation z = (xi, yi)i∈N ∈ ([0, x]×R+)

n

is feasible for st ifPyi ≤ f (

Pstixi). We denote by Z (s

t) the set of feasibleallocations for st ∈ Sn. An allocation z = (z1, . . . , zn) ∈ ([0, x]×R+)

n isPareto efficient for e = (u, st) ∈ E if z ∈ Z (st) and there does not existz0 = (z01, . . . , z

0n) ∈ Z (st) such that for all i ∈ N , ui (z0i) ≥ ui (zi), and for

some i ∈ N , ui (z0i) > ui (zi). Given st ∈ Sn, an allocation z =(xi, yi)i∈N ∈([0, x]×R+)

n is contribution-proportional for st if, for all i ∈ N , yi =st

ixiPst

jxjf¡P

stjxj¢. The proportional solution [Roemer and Silvestre (1989,

1993)] is a correspondence PR : E ³ ([0, x]×R+)n such that for each econ-

omy e = (u, st) ∈ E , PR (e) is the subset of Pareto efficient allocations fore, each of which is contribution-proportional for st.A normal-form game form is a pair Γ = (M,h), where M = M1 × · · · ×

Mn, Mi being the strategy space of agent i ∈ N , and h :M → ([0, x]×R+)n

being the outcome function which associates each m ∈ M with a uniqueelement h(m) ∈ ([0, x]×R+)

n. The i-th component of h(m) will be denotedby hi(m) ≡ (hi1(m), hi2(m)), where hi1(m) ∈ [0, x] and hi2(m) ∈ R+. Givenm ∈M , let m−i = (m1, · · · ,mi−1,mi+1, · · · ,mn). A normal-form game formΓ = (M,h) is labor sovereign if for every i ∈ N and every xi ∈ [0, x], thereexists a strategy mi ∈Mi such that for any m−i ∈M−i, hi1(mi,m−i) = xi.

characteristic of preference ordering (utility function). In spite of this more natural view,we will assume in the following discussion that the labor intensity is a constant value,st

i = sti, for the sake of simplicity. The main theorems in the following discussion would

remain valid with a few changes in the settings of the economic environments even if thelabor intensity were assumed to be variable.

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3 Problems of Mechanism Design for the Tragedyof Commons

Faced with the “tragedy of commons” problem in cooperative productioneconomies, we should first discuss what the main issues are in designingmechanisms for solving it. As a natural normative solution for the tragedy,we focus on the proportional solution PR, and design an incentive compat-ible mechanism which implements PR. To do this, we should mention thatthere are essentially two main considerations. First, in situations where themechanism coordinator is unaware of the agents’ characteristics, the designedmechanism should work to find a true PR-allocation for each economy. Thisis a well-known problem of implementation theories in economic environ-ments. However, even if the coordinator can find the true PR-allocation foreach economy through application of the mechanism, he might have to forceagents to work in the production process so as to be consistent with thetrue PR-allocation, since, in production economies, each vector of feasibleallocations has the component of each agent’s supply of labor time. Thus,if the coordinator cannot force agents to work in the production process theamount of labor assigned by the mechanism, the true PR-allocation couldn’tbe implemented. That is the second consideration about which we shouldbe concerned in designing incentive compatible mechanisms in productioneconomies. In otherwords, we should consider a mechanism which can im-plement PR even if the coordinator cannot force agents to work as he wants.In considering the above problem of enforceability, we should note the

meaning of an agent’s skill level sti as his labor intensity. Although the co-ordinator cannot force agents to supply their labor time as he wants, he cansee how much time each agent works in the production process by examininghis time-cards. But, this does not imply that he can perfectly monitor eachagent’s labor contribution evaluated in efficiency units, since it is likely thatthe coordinator cannot perfectly monitor each agent’s labor intensity. Thisimplies that there is an incentive for each agent to overstate his true labor in-tensity, since under the rule of PR his share of output is monotone increasingwith respect to the level of labor intensity. Thus, the enforceability problemof the coordinator should be taken more seriously, since, in the productionprocess, not only can he not force agents their supply of labor time as hisdesire, but also he couldn’t enforce agents to reveal their true labor intensity.In the following, we assume that in applying a designed mechanism, the

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coordinator can determine the true amount of each agent’s labor time inthe production process, but cannot force agents to work his desired amountof labor time, and also cannot monitor the amount of each agent’s laborcontribution measured in efficiency units. Then, we propose a mechanismwhich can implement PR even if such a lack of enforceability is entailed inthe production process.

3.1 Sharing mechanisms

Our proposed mechanism is a type of sharing mechanism: we consider thateach agent’s strategy consists of announcing two real numbers (s, w), (oneof which, s, can be counted as his labor skill or his labor intensity while theother, w, can be counted as his demand for the consumption good), and ofsupplying his own labor time, x, as he wants. Given each agent’s strategyprofile (s,x,w) ∈ Sn× [0, x]n×Rn+, the outcome function of the mechanismassigns a feasible distribution of total output f

¡Pstjxj

¢: the range is not

the set of allocations, but of distributions of output, since every agent’ssupply of labor time is determined by his own strategy choice, and the roleof the outcome function is only to distribute the total output produced byagents’ joint production. Thus, in such a mechanism, there is no problemof enforceability in the sense of the above discussion, since the mechanismcoordinator does not need to force all agents to work as he desires. Even ifwe restrict our consideration to mechanisms of this type, we can still proposea mechanism which implements the proportional solution.A sharing mechanism is a function g : Sn× [0, x]n×Rn+ → Rn+ satisfying

the following property: for any s ∈ Sn, any x ∈ [0, x]n, and any w ∈ Rn+,g (s,x,w) = y, where s = (s1, . . . , sn) denotes the agents’ reported skills andw = (w1, . . . , wn) their desired amounts of output consumption. A sharingmechanism g is feasible if for any st ∈ Sn, any s ∈ Sn, any x ∈ [0, x]n,and any w ∈ Rn+, (x, g (s,x,w)) ∈ Z (st). Note that the feasible sharingmechanism needs not refer to st in dividing f

¡Pstjxj

¢. Note also that any

feasible sharing mechanism is a labor sovereign normal-form game form.11

We denote by G the class of such feasible sharing mechanisms.11More correctly, any feasible sharing mechanism g is also expressed as a game form

Γg =¡Sn × [0, x]n ×Rn

+, h¢such that for any (s, x, w) ∈ Sn × [0, x]n × Rn

+, h (s, x, w) =(x, g (s, x, w)). Clearly, Γg is a labor sovereign normal-form game form, which is uniquelycorresponding to g.

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Given a feasible sharing mechanism g ∈ G, a feasible sharing game is de-fined for each economy e ∈ E as a non-cooperative game (N, (S × [0, x]×R+)

n , g, e).Fixing the set of players N and their strategy sets (S × [0, x]×R+)

n, we sim-ply denote a feasible sharing game (N, (S × [0, x]× R+)

n , g, e) by (g, e).Given a strategy profile (s,x,w) ∈ Sn× [0, x]n×Rn+, let

¡ss0i ,xx0i ,ww0i

¢ ∈Sn × [0, x]n × Rn+ be another strategy profile which is obtained by replacingthe i-th component (si, xi, wi) of (s,x,w) with (s0i, x

0i, w

0i). A strategy profile

(s∗,x∗,w∗) ∈ Sn × [0, x]n × Rn+ is a (pure-strategy) Nash equilibrium of thefeasible sharing game (g, e) if, for any i ∈ N and any (si, xi, wi) ∈ S× [0, x]×R+,

ui (x∗i , gi (s

∗,x∗,w∗)) ≥ ui¡xi, gi

¡s∗si,x∗xi

,w∗wi

¢¢.

Denote by NE (g, e) the set of Nash equilibria of (g, e). An allocation z =(xi, yi)i∈N ∈ ([0, x]×R+)

n is a Nash equilibrium allocation of the feasiblesharing game (g, e) if there exists (s,w) ∈ Sn × Rn+ such that (s,x,w) ∈NE (g, e) and y = g (s,x,w) where x = (xi)i∈N and y = (yi)i∈N . Denote byNA (g, e) the set of Nash equilibrium allocations of (g, e). A feasible sharingmechanism g ∈ G is said to implement the proportional solution on E inNash equilibria, if for all e ∈ E , NA (g, e) = PR (e).A strategy profile (s∗,x∗,w∗) ∈ Sn×[0, x]n×Rn+ is a (pure-strategy) strong

(Nash) equilibrium of the feasible sharing game (g, e), if for any T ⊆ N andany (si, xi, wi)i∈T ∈ S#T × [0, x]#T ×R#T

+ , there exists j ∈ T such that12

uj¡x∗j , gj (s

∗,x∗,w∗)¢ ≥ uj ¡xj, gj ¡(si, xi, wi)i∈T , (s∗i , x∗i , w∗i )i∈T c

¢¢.

Denote by SNE (g, e) the set of strong equilibria of (g, e). An allocationz = (xi, yi)i∈N ∈ ([0, x]×R+)

n is a strong equilibrium allocation of the feasi-ble sharing game (g, e) if there exists (s,w) ∈ Sn×Rn+ such that (s,x,w) ∈SNE (g, e) and y = g (s,x,w) where x = (xi)i∈N and y = (yi)i∈N . Denoteby SNA (g, e) the set of strong equilibrium allocations of (g, e). A feasiblesharing mechanism g ∈ G is said to implement the proportional solutionon E in strong equilibria, if for all e ∈ E , SNA (g, e) = PR (e). More-over, a feasible sharing mechanism g ∈ G is said to doubly implement theproportional solution on E in Nash and strong equilibria, if for all e ∈ E ,NA (g, e) = SNA (g, e) = PR (e).

12For any T ⊆ N , #T denotes the number of agents in T . For any T ⊆ N , T c denotesthe set of agents i ∈ N who does not belong to T .

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3.2 Timing Problem in Sharing Mechanisms

Before discussing our own sharing mechanism for implementing the propor-tional solution, we should mention the timing problem of strategy-decisionin real applications of sharing mechanisms, which is particularly relevent tothe case of production economies. Note that the strategic action for (s, w)is only to announce a pair of two real numbers, while the strategic actionfor x is to engage in production activity by supplying that amount of labortime. Thus, there may be a time difference between the point when (s,w)is announced and the period when x is exercised: the profile of two num-bers (s,w) may be announced before agents engage in production activity, or(s,w) may be announced after each agent supplies his labor time x. In theformer case, each agent i can decide his supply of labor time with the knowl-edge of the announcements (s,w), while in the latter case, he can decide hisannouncement (si, wi) with the knowledge of agents’ actions in productionprocess x.So, we must consider at least two types of two-stage game forms:

(1) The first type is that in the first stage, every agent i simultaneouslyannounces information on his labor skill or intensity, si, and his demand forthe consumption good, wi, and in the second stage, every agent i engagesin the production activity and provides his labor time, xi, as he wants, andafter the production process, the outcome function assigns a distribution ofthe output produced.(2) The second type has the converse sequence of strategic-decision. In thefirst stage, every agent i engages in the production activity and provideshis labor time, xi, as he wants, and after the production, every agent isimultaneously announces his labor skill or intensity, si, and his demand forshare of the output, wi, in the second stage. Finally, the outcome functionassigns a distribution of the output produced.The type (1) may be more relevant to the situation that each agent’s sti is histrue labor endowment, where the announcement of si implies the informationon one aspect of i’s characteristics, while the type (2) may be more relevantto the situation that each agent’s sti is his true labor intensity, where theannouncement of si implies the information on i’s labor intensity exercisedin production process.In spite of this argument, we provided the formal definition of the sharing

mechanism not as an extensive-form game form, but as only a normal-formgame form. This is because, once we define one specific sharing mechanism

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as a normal-form game form, then we can naturally derive from this, twoextensive game forms of the above types (1) and (2). Moreover, this is alsobecause we can show in the following discussion that the set of subgame-perfect equilibrium allocations are always invariant between the two types ofextensive game forms whenever they are derived from the same normal-formsharing mechanism.Given a feasible sharing mechanism g ∈ G, the (1) type g-implicit two-

stage mechanism Γ1g is a two-stage extensive game form in which the first

stage consists of selecting (s,w) from Sn × Rn+, the second stage consists ofselecting x from [0, x]n, and the final stage assigns an outcome which is thesame value as g (s,x,w) for every choice (s,w) ∈ Sn × Rn+ in the first stageand every choice x ∈ [0, x]n in the second stage. Given g ∈ G, the (2) type g-implicit two-stage mechanism Γ2

g is a two-stage extensive game form in whichthe first stage consists of selecting x from [0, x]n, the second stage consists ofselecting (s,w) from Sn ×Rn+, and the final stage assigns an outcome whichis the same value as g (s,x,w) for every choice x ∈ [0, x]n in the first stageand every choice (s,w) ∈ Sn ×Rn+ in the second stage.Given the feasible (1) type g-implicit two-stage game

¡Γ1g, e¢and each

strategy profile (s,w) ∈ Sn × Rn+ in the first stage of the game¡Γ1g, e¢, let

us denote its corresponding second stage subgame by¡Γ1g (s,w) , e

¢. Let

xe : Sn × Rn+ → [0, x]n be a Nash equilibrium mapping such that for each(s,w) ∈ Sn × Rn+, xe (s,w) is a (pure-strategy) Nash equilibrium of thesubgame

¡Γ1g (s,w) , e

¢. Denote the set of such Nash equilibrium mappings

in second stage subgames of the game¡Γ1g, e¢by Xe. A strategy profile

(s∗,w∗,xe∗) ∈ Sn × Rn+ × Xe is a (pure-strategy) subgame-perfect (Nash)equilibrium of the feasible (1) type g-implicit two-stage game

¡Γ1g, e¢, if for

any i ∈ N and any (si, wi) ∈ S × R+,

ui (xe∗i (s

∗,w∗) , gi (s∗,xe∗ (s∗,w∗) ,w∗))≥ ui

¡xe∗i (s

∗si,w∗wi

), gi¡s∗si,xe∗(s∗si

,w∗wi),w∗wi

¢¢,

where xe∗i (s

∗,w∗) is the i-th component of the Nash equilibrium strategyprofile xe∗ (s∗,w∗) in the second stage subgame induced from the strategychoice (s∗,w∗) in the first stage.Given the feasible (2) type g-implicit two-stage game

¡Γ2g, e¢and each

strategy profile x ∈ [0, x]n in the first stage of the game ¡Γ2g, e¢, let us denote

its corresponding second stage subgame by¡Γ2g(x), e

¢. Let ωe : [0, x]n →

Sn×Rn+ be a Nash equilibrium mapping such that for each x ∈ [0, x]n, ωe (x)

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is a (pure-strategy) Nash equilibrium of the subgame¡Γ2g (x) , e

¢. Denote the

set of such Nash equilibrium mappings in second stage subgames of the game¡Γ2g, e¢by Ωe. A strategy profile (x∗,ωe∗) ∈ [0, x]n ×Ωe is a (pure-strategy)

subgame-perfect (Nash) equilibrium of the feasible (2) type g-implicit two-stage game

¡Γ2g, e¢, if for any i ∈ N and any xi ∈ [0, x],

ui (x∗i , gi (x

∗,ωe∗ (x∗))) ≥ ui¡xi, gi

¡x∗xi,ωe∗ ¡x∗xi

¢¢¢.

Denote by SPE¡Γ1g, e¢(resp. SPE

¡Γ2g, e¢) the set of subgame-perfect

equilibria of¡Γ1g, e¢(resp.

¡Γ2g, e¢). An allocation z = (xi, yi)i∈N ∈ ([0, x]×R+)

n

is a subgame-perfect equilibrium allocation of the game¡Γ1g, e¢(resp.

¡Γ2g, e¢)

if there exists (s,w,xe) ∈ SPE ¡Γ1g, e¢(resp. (x,ωe) ∈ SPE ¡Γ2

g, e¢) such

that xe (s,w) = x and y = g (s,xe (s,w) ,w) (resp. y = g (x,ωe (x))) wherex = (xi)i∈N and y = (yi)i∈N . Denote by SPA

¡Γ1g, e¢(resp. SPA

¡Γ2g, e¢)

the set of subgame-perfect equilibrium allocations of¡Γ1g, e¢(resp.

¡Γ2g, e¢).

A feasible (1)-type (resp. (2)-type) g-implicit two-stage mechanism Γ1g (resp.

Γ2g) is said to implement the proportional solution on E in subgame-perfectequilibria, if for all e ∈ E , SPA ¡Γ1

g, e¢= PR (e) (resp. SPA

¡Γ2g, e¢=

PR (e)). Moreover, a feasible (1)-type (resp. (2)-type) g-implicit two-stagemechanism Γ1

g (resp. Γ2g) is said to doubly implement the proportional solu-

tion on E in Nash and subgame-perfect equilibria, if for all e ∈ E, NA (g, e) =SPA

¡Γ1g, e¢= PR (e) (resp. NA (g, e) = SPA

¡Γ2g, e¢= PR (e)). Fi-

nally, a feasible (1)-type (resp. (2)-type) g-implicit two-stage mechanismΓ1g (resp. Γ2

g) is said to triply implement the proportional solution on E inNash, subgame-perfect, and strong equilibria, if for all e ∈ E , NA (g, e) =SPA

¡Γ1g, e¢= SNA (g, e) = PR (e) (resp. NA (g, e) = SPA

¡Γ2g, e¢=

SNA (g, e) = PR (e)).

4 Implementation of the proportional solu-tion

The existence of the proportional solution was already shown by Roemer andSilvestre (1993) in general convex economies. Moreover, we can provide analternative proof in our setting by employing Theorem 1 of Yoshihara (2000a)under a moderate assumption.In the following, we obey two additional assumptions.

12

Assumption 1 (boundary condition):13

∀i ∈ N,∀zi ∈ [0, x)×R++,∀z0i ∈ [0, x]× 0 , ui (zi) > ui (z0i) .Assumption 2: The production function f is continuously differentiable.

We denote by f 0 (x) the derivative of f at x.

4.1 Nash and strong implementability

In this subsection, we will set aside the timing problem of sharing mechanismsand propose a sharing mechanism as a normal form game form, which dou-bly implements the proportional solution in Nash and strong equilibria. Oursharing mechanism will work well even when the coordinator does not knowthe agents’ labor skills, and therefore cannot observe their individual laborcontributions in efficiency units, but only each agent’s supply of labor timeand the total amount of output produced by all the agents. The correspond-ing two types of implicit extensive game forms of this sharing mechanism willbe argued in the next subsection.To propose our mechanism, let us introduce four feasible sharing mecha-

nisms defined as follows:

• gPR is such that for each st ∈ Sn and each strategy profile (s,x,w) ∈Sn × [0, x]n ×Rn+,gPRi (s,x,w) = sixiP

sjxjf¡P

stjxj¢for all i ∈ N .

• g(z,bs) with bs ∈ Sn and z =(x, y) ∈ [0, x]n × Rn+ is such that for eachst ∈ Sn and each strategy profile (s,x,w) ∈ Sn × [0, x]n ×Rn+,

g(z,bs)i (s,x,w) =

min©max 0, yi + bsi · f 0 (Pbsjxj) · (xi − xi) , f ¡P stjxj

¢ªif z ∈Z (bs) and xj = xj, sj = bsj (∀j 6= i),0 otherwise,

for all i ∈ N .• gm is such that for each st ∈ Sn and each strategy profile (s,x,w) ∈Sn × [0, x]n ×Rn+, and for all i ∈ N ,

13The boundary condition is usually assumed in the literature of implementation ineconomic environments. Under this assumption, the proportional solution assigns only in-terior allocations, and satisfiesMaskin Monotonicity [Maskin; (1977)], which is a necessarycondition for Nash implementability.

13

gmi (s,x,w) =

½f¡P

stjxj¢if xi = µ (x−i) and wi > max wjj 6=i

0 otherwise.

where µ (x−i) ≡maxxj<x

xjj 6=i +x−max

xj <xxjj 6=i

2.

• gd is such that for each st ∈ Sn and each strategy profile (s,x,w) ∈Sn × [0, x]n ×Rn+, and for all i ∈ N ,gdi (s,x,w) =

½f¡P

stjxj¢if Nm (s) = i ⊆ N0 (x) ,

0 otherwise,

whereNm (s) ≡ni ∈ N : si = max sjj∈N

oandN0 (x) ≡ i ∈ N : xi = 0.

Note that, first, gPR distributes the produced output among the agents inproportion to their reported shares of labor contribution. Second, g(z,bs) isdesigned to implement z in Nash equilibrium under some economy with bs.If z is Pareto efficient for some economy with bs, say (bu,bs), then z becomesa Nash equilibrium allocation of the game

¡g(z,bs), (bu,bs)¢, since each agent’s

attainable allocations by his unilateral deviation from z are the points inthe “budget set” defined by the supporting price f 0 (

Pbsjxj) at z. Third,gm assigns all of the produced output to only one agent who provides themaximal interior amount of labor time and reports a maximal amount ofdemand for the output, where the scheme µ (x−i) is introduced to have agentsfind their best response strategies. Fourth, gd also assigns all of the producedoutput to only one agent who reports the highest labor skill and does notwork at all.Given (s,x,w) = (si, xi, wi)i∈N ∈ Sn × [0, x]n ×Rn+, let PR (s,x,w)−1 ≡

u ∈ Un : (x,w) ∈ PR (u, s). If PR (s,x,w)−1 6= ∅, then (x,w) shouldbe a PR-optimal allocation for some economy with s. Let us call such a(s,x,w) a PR-consistent strategy profile. Note that if gPR (s,x,w) = wholds and (x,w) is an interior allocation, then PR (s,x,w)−1 6= ∅ holds.Given (s,x,w) ∈ Sn × [0, x]n ×Rn+, let N (s,x,w) ≡ i ∈ N : ∃ (s0i, x0i, w0i) ∈S×[0, x]×R+ s.t. PR

¡ss0i ,xx0i ,ww0i

¢−1 6= ∅. ThisN (s,x,w) is the set of po-tential deviators under strategy profile (s,x,w), since any i ∈ N (s,x,w) canconstitute a PR-consistent strategy profile with the others’ fixed strategiesby changing his strategy from (si, xi, wi). Given (s,x,w) ∈ Sn× [0, x]n×Rn+and i ∈ N (s,x,w), letSi (s,x,w) ≡ ©s0i ∈ S : ∃ (x0i, w0i) ∈ [0, x]× R+ s.t. PR(ss0i ,xx0i ,ww0i)

−1 6= ∅ª .14

Note that Si (s,x,w) is closed and bounded from below, or otherwise, Si (s,x,w) =S. The latter case occurs if and only if f is linear on [0, b] such thatPk 6=iskxk < b.

We introduce a feasible sharing mechanism g∗ ∈ G which works in eachgiven st ∈ Sn as follows:

For any (s,x,w) = (si, xi, wi)i∈N ∈ Sn × [0, x]n ×Rn+,Rule 1: If f (

Psjxj) = f

¡Pstjxj

¢, then

1-1: if PR (s,x,w)−1 6= ∅, then g∗ (s,x,w) = gPR (s,x,w),1-2: if PR (s,x,w)−1 = ∅, and N (s,x,w) 6= ∅, then1-2-1: if #N (s,x,w) > 1, then g∗ (s,x,w) = 0,1-2-2: if N (s,x,w) = j for some j ∈ N , then g∗j (s,x,w) = g(bz,bs)

j (s,x,w)and g∗i (s,x,w) = 0 for any i 6= j, where bz = (xbxj

,w bwj) and bs = sbsj

suchthat bsj = arg min

s0j∈Sj(s,x,w)| s0j − sj | & PR(sbsj

,xbxj,w bwj

)−1 6= ∅,

1-3: in any other case, g∗ (s,x,w) = gm (s,x,w).

Rule 2: If f (Psjxj) 6= f

¡Pstjxj

¢, then g∗ (s,x,w) = gd (s,x,w).

It is easy to check that g∗ is feasible and does not refer to st in dividingf¡P

stjxj¢. This mechanism g∗ is a combination of the four sharing mech-

anisms defined above: First, g∗ computes the expected amount of producedoutput f (

Psjxj) from the data (s,x,w) and compares this with the real

amount of produced output f¡P

stjxj¢. In the case that these two values

coincide, if a strategy profile (s,x,w) is a PR-consistent, then g∗ applies gPR

inRule 1-1; if (s,x,w) is not PR-consistent, and there is a unique potentialdeviator, then g∗ applies g(bz,bs) in Rule 1-2-2 so as to punish him; for anyother case, g∗ either applies gm in Rule 1-3 or assigns nothing to everyonein Rule 1-2-1. Finally, if f (

Psjxj) and f

¡Pstjxj

¢are different, then g∗

applies gd in Rule 2.Note that the data (bz,bs) of g(bz,bs) in Rule 1-2-2 is obtained by replacing

the unique potential deviator’s strategy with some appropriate one, wheresuch an operation is possible by the definition of N (s,x,w). Note also thatsuch (bz,bs) is essentially uniquely determined: first, bsj is uniquely determined,since if sj ∈ Sj (s,x,w), then bsj = sj, while otherwise, then Sj (s,x,w) isbounded from below and bsj = minSj (s,x,w). Second, once bsj is uniquely

15

determined, then the other agents’ strategies together with bsj give us theunique information about j’s potential consumption vector (bxj, bwj), wheneverthe production function f is strictly concave, because of the proportionalityof the PR-optimal allocation. Even if f is linear, the ratios between inputand output of j’s potential consumption vectors should be the same value,by which the corresponding supporting price is uniquely determined.Before we formally show the performance of g∗, let us briefly explain how

the mechanism induces true information of labor skills in the following (A),and how it attains desirable allocations in the following (B):

(A) Since g∗ is a feasible sharing mechanism, it only distributes total amountsof output f

¡Pstjxj

¢among agents according to the agents’ supplies of labor

time x, reported labor skills s, and demands for the outputw. As discussed insection 3, since the agents’ true labor contributions in efficiency units are notobservable, each agent may misrepresent his labor skill so as to increase hisshare of output. To prove this problem, a scheme of reward-and-punishmentis set up in the mechanism as follows. First, if f (

Psjxj) 6= f

¡Pstjxj

¢, then

clearly s 6= st holds, and there must be at least one agent, say j ∈ N , who hasmisreported his labor skill, sj 6= stj, and supplied a positive amount of labortime xj > 0. Then, this agent is definitely punished under the application ofRule 2. Moreover, we can also show in the proof of Theorem 1 later thatsuch a case does not correspond to an equilibrium situation.Second, the remaining problem to see, is the case where someone lies

about his labor skill, s 6= st, but still f (P sjxj) = f¡P

stjxj¢holds. In this

case, there must be at least two agents who have misreported their laborskills respectively, and either supplied together positive amounts of labortime or chosen together “non-working.” If they are in the latter, they will bepunished under the application of Rule 1-3. Then, by the definition of Rule1-3, such a situation does not constitute an equilibrium. Consider the former,and suppose that one of such agents, say j ∈ N , changes from xj > 0 tox0j = 0, together with reporting a sufficiently high level of labor skill. Then, it

follows that f (Psixi) = f (

Pstixi) changes to f

ÃPi6=jsixi

!6= f

ÃPi6=jstixi

!,

thereby j may be better off under the application of Rule 2, while the othermisreporting agents remain punished. Thus, this second case may also notcorrespond to an equilibrium situation, and the following lemma actuallyconfirms such an insight:

16

Lemma 1: Let the feasible sharing mechanism g∗ ∈ G be given as above.Given an economy (u, st) ∈ E , let a strategy profile (s,x,w) ∈ Sn× [0, x]n×Rn+ be a Nash equilibrium of the game (g∗,u, st) such that f (

Psjxj) =

f¡P

stjxj¢. Then, it follows that si = sti for all i ∈ N with xi > 0.

(B) What is left to be explained is how the mechanism implements the pro-portional solution when all agents report their true labor skills, s = st. Todo this, we adopt a scheme developed by Yoshihara (2000a).Since s = st, the strategy profile (s,x,w) ∈ Sn× [0, x]n×Rn+ corresponds

only toRule 1, and among the three subrules of Rule 1, only Rule 1-1 canrealize a desirable allocation in equilibrium, and the other two are to punishagents who have deviated from the situation of Rule 1-1. Let us supposethat (s,x,w) is PR-consistent. Then, (x,w) becomes the outcome by Rule1-1, which is a PR-optimal allocation for some economy with s. However,this does not necessarily imply that (x,w) is PR-optimal for the currenttrue economy: (x,w) may not be Pareto efficient for the current economy.In such a case, (x,w) should not be an equilibrium allocation of the gameunder the current economy, and Rule 1-2-2 is necessary for solving thisproblem: by deviating from (s,x,w) to induce Rule 1-2-2, any agent canenjoy at most the budget line determined by the supporting price f 0 (

Pstixi)

at (x,w). Thus, if (x,w) is an equilibrium allocation, then it implies that(x,w) is Pareto efficient.Now, we analyze the performance of g∗.

Theorem 1: Let Assumptions 1 and 2 hold. Let the feasible sharing mecha-nism g∗ ∈ G be given as above. Then, g∗ implements the proportional solutionon E in Nash equilibria.Next, we may also consider the possibility of the agents’ behaving coali-

tionally, but where the coordinator does not know in advance, which of sub-groups in the population will behave coalitionally. The following theoremgives a solution to this problem:

Theorem 2: Let Assumptions 1 and 2 hold. Let the feasible sharing mecha-nism g∗ ∈ G be given as above. Then, g∗ doubly implements the proportionalsolution on E in Nash and strong equilibria.Remark 1: The mechanism allows agents to choose their labor time freely,which implies the mechanism is labor sovereign (Kranich (1994)).

17

Remark 2: In the mechanism each agent is required to announce only hislabor skill and his desired amount of output consumption. Thus the mech-anism is a quantity-announcing mechanism, and is self-relevant (Hurwicz1960).

Remark 3: In the mechanism each agent receives his announced consump-tion bundle in equilibria by inducing Rule 1-1. Thus, the mechanism meetsforthrightness (Saijo et al. 1996).

Remark 4: Each agent has a best response to every strategy combina-tion of the other agents. Suppose an agent j is given a strategy profile¡s−j,x−j,w−j

¢of the other agents. If he can induce Rule 2 with xj = 0,

then his best respose in Rule 2 is a combination of xj = 0 and sj >max sii6=j. If he can induce Rule 2 with xj > 0 but not with xj =0, then he may as well induce the other rules with xj = 0. If he caninduce Rule 1-3, then his best response in Rule 1-3 is a combinationof xj = µ (x−j), wj > max

nf (Pstixi) , max wii6=j

o, and sj such that

f

ÃPi6=jsixi + sjxj

!= f (

Pstixi). If he can induce Rule 1-2-1 and j /∈

N (s,x,w), then by wj > maxnf (Pstixi) , max wii 6=j

o, he can also in-

duce Rule 1-3. If he can induce Rule 1-2-1 and j ∈ N (s,x,w), then hecan be better off by moving to Rule 1-2-2. If he can induce Rule 1-2-2but j /∈ N (s,x,w), then by wj > max

nf (Pstixi) , max wii6=j

o, he can

also induce 1-3. If he can induce Rule 1-2-2 and j ∈ N (s,x,w), then oneof his best resposes in Rule 1-2-2 is a combination of wj = 0 and (sj, xj),the latter of which is a solution of the following optimization problem:

max(sj ,xj)

uj

Ãxj,min

(max

(0, yj + bsj · f 0ÃX

i6=jsixi + bsjxj! · (xj − xj)) , f ³X stixi

´)!

subject to f (Psixi) = f (

Pstixi). If he can induce Rule 1-1, then he can

also induce Rule 1-2-2, and one of his best resposes among Rules 1-1 and1-2-2 lies in Rule 1-2-2. Thus, the mechanism satisfies the best responseproperty (Jackson et al. 1994).

Remark 5: Each rule in the mechanism does not refer to the number ofagents in the given economy. That is, the mechanism does not depend onthe number of agents, and it works even in economies of two agents.

18

4.2 Implementation of the proportional solution withthe timing problem

Given the feasible sharing mechanism g∗, we can also induce the (1) typeand the (2) type two-stage g∗-implicit extensive game forms Γ1

g∗ and Γ2g∗

respectively from g∗. As discussed in the previous section, in real applicationsof the mechanism in production economies, g∗ works as either Γ1

g∗ or Γ2g∗,

because of the timing problem in strategic actions. So, given either Γ1g∗ or

Γ2g∗ , agents may play the game by adopting not the Nash equilibrium notionbut the subgame-perfect notion. Then, the full implementation by g∗ maynot be warranted. However:

Theorem 3: Let Assumptions 1 and 2 hold. Let the feasible sharing mech-anism g∗ ∈ G be given as above. Then, the (1)-type g∗-implicit extensivegame form doubly implements the proportional solution on E in Nash andsubgame-perfect equilibria.

Theorem 4: Let Assumptions 1 and 2 hold. Let the feasible sharing mech-anism g∗ ∈ G be given as above. Then, the (2)-type g∗-implicit extensivegame form doubly implements the proportional solution on E in Nash andsubgame-perfect equilibria.

These two theorem are appealing when we see the real application processof the mechanisms carefully. Although g∗ may work as either Γ1

g∗ or Γ2g∗

because of the timing problem in strategic actions, the coordinator cannotmake sure in advance which of these two equilibrium notions the agentsshould adopt in playing the game. Double implementability, as in the abovetheorems, implies that the mechanism works well even in such a situation.By the above four theorems in this section, we can summarize as follows:

Corollary: Let Assumptions 1 and 2 hold. Let the feasible sharing mech-anism g∗ ∈ G be given as above. Then, both the (1)-type and the (2)-typeg∗-implicit extensive game forms respectively triply implement the propor-tional solution on E in Nash, subgame-perfect, and strong equilibria.

This result implies that the mechanism g∗ permits each agent variouskinds of freedom: he can choose his own supply of labor time freely; he ispermitted to overstate his labor skill; he can behave unilaterally or coalition-ally; and he can behave strong-rationally, as in the subgame-perfect response,

19

or weak-rationally, as in the Nash-like response. Even if all of these kindsof freedom are given to agents, the mechanism g∗ still works well and im-plements the solution in a decentralized way. If we carefully examine thesituation of the tragedy of commons in cooperative production economies,this result might be very appealing.

5 Concluding remarks

We have proposed a feasible sharing mechanism which successfully triplyimplements the proportional solution in Nash, subgame-perfect, and strongequilibria, even when the coordinator does not know each agent’s labor skillor his preference, and consequently, agents can not only understate, but alsooverstate their labor skills. The performance of our proposed mechanismis summarized in Table 1, which provides a comparison with other relevantmechanisms.

Insert Table 1 around here.

As shown in Table 1, our proposed mechanism has two undesirable fea-tures. First, it lacks continuity. Second, the mechanism fails to meet bal-ancedness or non-wastefulness. Out of equilibria, it sometimes realizes anallocation for which the sum of agents’ consumption of output is strictly lessthan the total amount of the product. One reason is that the mechanismpermits agents to both overstate and/or understate their labor skills: tobe informed of their true labor skills, the coordinator has to punish agentswho misrepresent their skills. It is difficult, however, to find out who has liedabout his labor skill when only aggregate information

¡f (Psjxj) and f

¡Pstjxj

¢¢is available. Therefore, the mechanism basically punishes all agents whenthere must be a deviator

¡f (Psjxj) 6= f

¡Pstjxj

¢¢. The other reason is that

this mechanism is labor sovereign. The labor sovereign mechanism shouldaccept a profile of the agents’ choice of labor time as an outcome, even whenit may constitute a non-desirable allocation. Thus, when the true deviatorcannot be identified, the mechanism can punish all potential deviators onlyby reducing their shares of output, which leads it to violate balancedness.We conjecture that there may be a trade-off between labor sovereignty andbalancedness. However, it is still an open question whether or not thereexists a mechanism which satisfies labor sovereignty and balancedness, andimplements the proportional solution in the economic environment that wehave focused upon.

20

Although we focused only on the proportional solution, our method ofmechanism design for the proportional solution is also applicable to other so-lutions proposed in cooperative production economies. For example, it is easyto construct a sharing mechanism which has the same desirable propertiesas those our mechanism in this paper has, and triply implements the equalbenefit solution [Roemer and Silvestre (1989)] in Nash, subgame-perfect, andstrong equilibria. However, it still remains to be an interesting task to char-acterize the class of social choice correspondences implementable by feasiblesharing mechanisms as proposed here, each of which works well even whenthe coordinator cannot prohibit agents from overstating their labor skills.

6 Appendix

Proof of Lemma 1. Suppose there exists j ∈ N with sj 6= stj and xj > 0.Let NL (s) be the set of such j. Since f (

Psixi) = f (

Pstixi), NL (s) is not

a singleton. Moreover, any j ∈ NL (s) can obtain y0j = fÃPi6=jstixi

!> 0 with

s0j > max sii6=j and x0j = 0 under Rule 2. Note that

Xj∈NL(s)

y0j =X

j∈NL(s)

f

ÃXi6=j

stixi

!=X

j∈NL(s)

f

Xi∈NL(s)\j

stixi+X

i/∈NL(s)

stixi

≥

Xj∈NL(s)

f

stjxj+ Xi/∈NL(s)

stixi

(since NL (s) is not a singleton)

≥ f

Xj∈NL(s)

stjxj+ Xi/∈NL(s)

stixi

(since f is concave and f (0) ≥ 0)

≥ f

Xj∈NL(s)

stjxj+X

i/∈NL(s)

stixi

= f

³Xstixi

´≥X

j∈NL(s)

yj ≡X

j∈NL(s)

g∗j (s,x,w) .

21

IfP

j∈NL(s)

y0j >P

j∈NL(s)

yj, then there must be j ∈ NL (s) who has an incentiveto induce Rule 2 by s0j > max sii6=j and x0j = 0. If

Pj∈NL(s)

y0j =P

j∈NL(s)

yj

and there exists one individual j ∈ NL (s) who has y0j > yj, then he has anincentive to change xj to x0j = 0 and report s0j > max sii6=j. Finally, ifPj∈NL(s)

y0j =P

j∈NL(s)

yj and y0j = yj for all j ∈ NL (s), then every j ∈ NL (s) hasan incentive to change xj to x0j = 0, since uj(x

0j, y

0j) = uj(0, yj) > uj(xj, yj) by

the strict monotonicity of utility functions. Thus, in any case, it contradictsthe fact that (s,x,w) is a Nash equilibrium.

Proof of Theorem 1. Let (u, st) ∈ E be any given.

(1) First, we show that PR (u, st) ⊆ NA (g∗,u, st). Let z = (x,y) ∈PR (u, st). Then, if the strategy profile of agents is (st,x,y) = (sti, xi, yi)i∈N ∈(S × [0, x]×R+)

n, then g∗ (st,x,y) = y by Rule 1-1. Since st À 0 and zis an efficient proportional allocation, Assumption 1 implies x À 0 andg∗i (s

t,x,y) > 0 for all i ∈ N . Suppose that an individual j ∈ N deviatesfrom

¡stj, xj, yj

¢to¡s0j, x

0j, w

0j

¢ ∈ S × [0, x]×R+.

Note first that the deviation cannot induce Rule 1-3. If the deviationresults in Rule 1-2-1, then g∗j

³sts0j,xx0j ,yw0j

´= 0. If the deviation induces

Rule 2, then x0j > 0. In fact, if x0j = 0, thenPi 6=jstixi + s

0jx0j =Pi6=jstixi + s

tjx0j,

so that f

ÃPi6=jstixi + s

0jx0j

!= f

ÃPi6=jstixi + s

tjx0j

!, which contradicts the fact

thatRule 2 is induced. Hence, it must be the case that g∗j³sts0j,xx0j ,yw0j

´= 0

under Rule 2.

If the deviation induces Rule 1-2-2 with s0j 6= stj, then x0j = 0. So,

g∗j³sts0j ,xx

0j,yw0j

´≤ bwj + bsj · f 0ÃX

i6=jstixi + bsjbxj

!· ¡x0j − bxj¢

= bwj − bsjbxj · f 0ÃXi6=j

stixi + bsjbxj!

22

with some (bsj, bxj, bwj) ∈ S × [0, x]×R+ such that PR³stbsj,xbxj

,y bwj

´−1

6= ∅.

Suppose f

ÃPi6=jstixi + bsjbxj

!6= f

ÃPi6=jstixi + s

tjxj

!. Since PR (st,x,y)−1 6=

∅, it implies that both of the points

ÃPi6=jstixi + bsjbxj, f

ÃPi6=jstixi + bsjbxj

!!

and

ÃPi6=jstixi + s

tjxj, f

ÃPi6=jstixi + s

tjxj

!!must be on the same line that

passes through (0, 0). Since f is concave, f must be linear on a closed

interval

"0,max

(Pi6=jstixi + bsjbxj,P

i6=jstixi + s

tjxj

)#. Hence,

bwj − bsjbxj · f 0ÃXi6=j

stixi + bsjbxj!= 0.

Thus, g∗j³sts0j,xx0j ,yw0j

´≤ 0. Next, suppose that f

ÃPi6=jstixi + bsjbxj

!=

f

ÃPi6=jstixi + s

tjxj

!. This implies bsjbxj = stjxj and bwj = yj, since PR (st,x,y)−1 6=

∅ and PR³stbsj,xbxj

,y bwj

´−1

6= ∅. Thus,

g∗j³sts0j ,xx

0j,yw0j

´= bwj − bsjbxj · f 0ÃX

i6=jstixi + bsjbxj

!= yj − stjxj · f 0

³Xstixi

´= yj + s

tj · f 0

³Xstixi

´· ¡x0j − xj¢ .

Since z is Pareto efficient, the deviation gives no additional benefit to j.Consider the case that the deviation induces Rule 1-2-2 with s0j = stj. If

g∗j³st,xx0j ,yw0j

´= g

(bz,bs)j

³st,xx0j ,yw0j

´underRule 1-2-2 where bz = ³(bxj,bwj) , (xi, yi)i6=j´, then bs = st and the pointÃPi6=jstixi + s

tjbxj, f

ÃPi6=jstixi + s

tjbxj!!

must be on the line that starts from

23

(0, 0) and passes through (Pstixi, f (

Pstixi)), because we have (bwj,y−j) =

gPR (st, (bxj,x−j) , (bwj,y−j)) as well as y = gPR (st,x,y). Since f is concave,f must be linear on a closed interval

"0,max

(Pstixi,

Pi6=jstixi + s

tjbxj)#,

which impliesg∗j³st,xx0j ,yw0j

´= g

(z,st)j

³st,xx0j ,yw0j

´.

Sinceg

(z,st)j

³st,xx0j ,yw0j

´≤ yj + stj · f 0

³Xstixi

´· ¡x0j − xj¢

and z is Pareto efficient, the deviation gives no additional benefit to j.

Next, if the deviation induces Rule 1-1 with s0j 6= stj, then x0j = 0. So,g∗j³sts0j,xx0j ,yw0j

´= 0. Therefore, s0j = s

tj follows when Rule 1-1 is induced.

However, if the deviation induces Rule 1-1, then f must be linear on a

closed interval

"0,max

(Psixi,

Pi6=jsixi + s

0jx0j

)#, and we obtain

g∗j³st,xx0j ,yw0j

´≤ yj + stj · f 0

³Xsixi

´· ¡x0j − xj¢ .

The Pareto efficiency of z implies no additional benefit for j.

(2) Second, we will show that NA (g∗,u, st) ⊆ PR (u, st). Let (s,x,w) =(si, xi, wi)i∈N ∈ (S × [0, x]× R+)

n be a pure-strategy Nash equilibrium ofthe feasible sharing game (g∗,u, st).

Suppose that (s,x,w) inducesRule 2. If N0 (x) = ∅, then g∗i (s,x,w) =0 for all i ∈ N . Note that when Rule 2 is induced, there exists at least anindividual j ∈ N such that

Pi6=jsixi 6=

Pi6=jstixi. In fact, if not, then (n− 1) ·

(Psixi) = (n− 1) · (P stixi), which contradicts the fact that Rule 2 is

induced. Thus, there exists an individual j ∈ N in this case who can enjoyg∗j³ss0j ,xx0j ,ww0j

´> 0 with s0j > max sii∈N and x0j = 0 under Rule 2.

If N0 (x) 6= ∅, then everyone j ∈ N0 (x) can monopolize all f (Pstixi)

by supplying xj = 0 while reporting his labor skill as s0j so that s0j > si for

all i 6= j under Rule 2. Thus, if #N0 (x) ≥ 2, then no profile of agents’strategies can constitute a Nash equilibrium in Rule 2.

24

If #N0 (x) = 1 and #N\N0 (x) ≥ 2, then there exists at least an in-dividual j ∈ N\N0 (x) such that

Pi6=j,i∈N\N0(x)

sixi 6=P

i6=j,i∈N\N0(x)

stixi. In

fact, if not, then (n− 2) ·Ã Pi∈N\N0(x)

sixi

!= (n− 2) ·

à Pi∈N\N0(x)

stixi

!,

which contradicts the fact that Rule 2 is induced. Thus, there exists anindividual j ∈ N\N0 (x) in this case who can enjoy g∗j

³ss0j ,xx0j ,ww0j

´> 0

with s0j > max sii∈N and x0j = 0 under Rule 2. If #N0 (x) = 1 withN0 (x) = i and #N\N0 (x) = 1 with N\N0 (x) = j, then j can enjoyg∗j³ss0j ,xx0j ,ww0j

´> 0 with s0j = stj, x

0j =

x2, and w0j > max

©wi, f

¡stjx

0j

¢ªunder Rule 1-3. Thus, if #N0 (x) = 1, no profile of agents’ strategies canconstitute a Nash equilibrium in Rule 2.

Suppose that (s,x,w) induces Rule 1-3. Then, g∗j (s,x,w) = 0 for some

j ∈ N . If xj = 0, then he can enjoy g∗j³ss0j ,xx0j ,ww0j

´> 0 with s0j = stj,

x0j = µ (x−j), and w0j > maxnf¡stjx

0j

¢, max wii6=j

o, under Rule 1-3.

Thus xj > 0. Then, if sj 6= stj, then he can enjoy g∗j³ss0j ,xx0j ,ww0j

´> 0 with

s0j > max sii∈N and x0j = 0 under Rule 2. At the same time, if sj = stj,

then he can enjoy g∗j³ss0j ,xx0j ,ww0j

´> 0 with s0j = stj, x

0j = µ (x−j), and

w0j > maxnf¡stjx

0j

¢, max wii6=j

o, under Rule 1-3.

Suppose that (s,x,w) induces Rule 1-2-1. Then, g∗i (s,x,w) = 0 forall i ∈ N . If xj = 0 for some j ∈ N , then N (s,x,w) = j or ∅. Hencex À 0. Any j ∈ N with sj = stj can enjoy g

∗j

³ss0j ,xx0j ,ww0j

´> 0 with

s0j = stj, x

0j = µ (x−j), and w

0j > max

nf¡stjx

0j

¢, max wii6=j

o, under Rule

1-3 or Rule 1-2-2. At the same time, any j ∈ N with sj 6= stj can enjoy

g∗j³ss0j ,xx0j ,ww0j

´> 0 with s0j > max sii∈N and x0j = 0 under Rule 2.

Suppose that (s,x,w) induces Rule 1-2-2. Then, N\N (s,x,w) 6= ∅,and xi > 0 and g∗i (s,x,w) = 0 for all i ∈ N\N (s,x,w). Any j ∈N\N (s,x,w) with sj = stj can enjoy g

∗j

³ss0j ,xx0j ,ww0j

´> 0 with s0j = stj,

x0j = µ (x−j), and w0j > maxnf¡stjx

0j

¢, max wii6=j

o, under Rule 1-

25

3. At the same time, any j ∈ N\N (s,x,w) with sj 6= stj can enjoy

g∗j³ss0j ,xx0j ,ww0j

´> 0 with s0j > max sii∈N and x0j = 0 under Rule 2.

Thus, (s,x,w) corresponds toRule 1-1. Then, for z =¡x, gPR (s,x,w)

¢,

we have g∗ (s,x,w) = gPR (s,x,w), whichmeans g∗i (s,x,w) =sixiPsjxjf¡P

stjxj¢

for each i ∈ N . Note next that if xj = 0 for some j ∈ N , then PR (s,x,w)−1 =∅ by Assumption 1, which contradicts the fact that Rule 1-1 is induced.Hence x À 0. Moreover, f (

Psjxj) = f

¡Pstjxj

¢when Rule 1-1 is in-

duced. Therefore, Lemma 1 implies it must be the case that s = st.

Since (s,x,w) is a Nash equilibrium, it holds that for all i ∈ N and all(s0i, x

0i, w

0i) ∈ S × [0, x] × R+, ui (xi, g∗i (s,x,w)) ≥ ui

¡x0i, g

∗i

¡ss0i ,xx0i ,ww0i

¢¢.

At the same time, any i ∈ N can enjoy

g∗i¡ss0i ,xx0i ,ww0i

¢= min

(max

n0, yi + s

ti · f 0

³Xstjxj

´· (x0i − xi)

o, f

ÃXj 6=istjxj + s

tix0i

!)

with s0i = sti and w0i = 0 under Rule 1-2-2. Since f is concave, the facts

together imply the Pareto efficiency of z, or z ∈ PR (u, st).

Proof of Theorem 2. Since NA (g∗, e) = PR (e) for all e ∈ E , we haveonly to show SNA (g∗, e) = NA (g∗, e) for all e ∈ E . Let e = (u, st) ∈ E beany given. By definition, SNA (g∗, e) ⊆ NA (g∗, e). Suppose SNA (g∗, e) (NA (g∗, e). Then, there exists (s,x,w) ∈ NE (g∗, e) such that for someT ( N and some (s0i, x

0i, w

0i)i∈T ∈ S#T × [0, x]#T × R#T

+ ,

uj¡xj, g

∗j (s,x,w)

¢< uj

³x0j, g

∗j

³(s0i, x

0i, w

0i)i∈T , (si, xi, wi)i∈N\T

´´for all j ∈ T . Note that T = N is eliminated by Pareto efficiency ofNA (g∗, e). By construction of g∗, there is at most one agent who can enjoya positive amount of output under Rules 1-2-1, 1-2-2, 1-3, and 2. Sinceg∗i (s,x,w) > 0 for all i ∈ N ,

³(s0i, x

0i, w

0i)i∈T , (si, xi, wi)i∈N\T

´should induce

Rule 1-1 by Assumption 1. Then, f must be linear on a closed interval"0,max

(Psixi,

Pi∈Ts0ix

0i +

Pi∈N\T

sixi

)#, and we obtain

g∗j³(s0i, x

0i, w

0i)i∈T , (si, xi, wi)i∈N\T

´≤ wj + stj · f 0

³Xsixi

´· ¡x0j − xj¢

26

for some j ∈ T . The Pareto efficiency of (xi, g∗i (s,x,w))i∈N implies noadditional benefit for this j, which implies a desired contradiction. Thus,NA (g∗, e) = SNA (g∗, e).

Proof of Theorem 3. Since NA (g∗, e) = PR (e) and SPA¡Γ1g∗ , e

¢ ⊆NA (g∗, e) for all e ∈ E , we have only to show PR (e) ⊆ SPA ¡Γ1

g∗, e¢for

all e ∈ E . First, we will show that in every second stage subgame, there isat least one Nash equilibrium strategy. Let us take a strategy mapping xe :Sn × Rn+ → [0, x]n such that for each second stage subgame

¡Γ1g∗ (s,w) , e

¢:

for all i ∈ N , xei (s,w) =

x2

if wi > max wkk 6=i , si = sti, andfor any j 6= i and any x0j,PR

¡s,¡x2, x0j,0−i,j

¢,w¢−1

= ∅0 otherwise

.

Then, we can see that xe (s,w) ∈ NE ¡Γ1g∗ (s,w) , e

¢. Note that g∗ (s,xe (s,w) ,w)

corresponds to Rule 1-3. To simplify the notation, let us denote x =xe (s,w) in the following discussion.Note first that no individual can induce Rule 1-1 by changing his strat-

egy, and that no individual can enjoy output consumption underRule 1-2-1.If i ∈ N can induce Rule 1-2-2 by changing his strategy, then the devia-tion gives him no additional benefit, or g∗i

¡s,xx0i ,w

¢= 0, because it must

be the case that i /∈ N ¡s,xx0i ,w¢. Moreover, if i ∈ N can induce Rule2 by changing his strategy, then it must be the case that xi > 0, whichimplies the deviation gives him no additional benefit, or g∗i

¡s,xx0i ,w

¢= 0.

Finally, if there exists i ∈ N who can enjoy a positive amount of outputunder Rule 1-3, then wi > max wkk 6=i, si = sti, and for any j 6= i

and any x0j, PR¡s,¡x2, x0j,0−i,j

¢,w¢−1

= ∅. His strategy is already x2,

which is necessary for him to get a positive output under Rule 1-3. Thus,x ∈ NE ¡Γ1

g∗ (s,w) , e¢.

Now, we will show that for each e = (u, st) ∈ E , if bz = (bx,by) ∈ PR (u, st),then there exists a subgame perfect equilibrium whose corresponding out-come is bz. Consider the following strategy profile of the extensive game¡Γ1g∗, e

¢:

(1) In the first stage, every individual i reports (si, wi) = (sti, byi).(2) In the second stage:(2-1): if (s,w) = (st, by) is the action profile of all individuals in the firststage, then any i ∈ N supplies xi = bxi;

27

(2-2): if (s,w) =¡¡s0j, s

t−j¢,¡w0j, by−j¢¢, where s0j = stj, w0j 6= byj, and for all

i 6= j, wi ≤ w0j, is the action profile of all individuals in the first stage, thenfor this j ∈ N ,

xj = arg maxx∗j∈χj

µst,bx−j ,byw0

j

¶ uj³x∗j , g

∗j

³st, bxx∗j , byw0j´´

and for all i 6= j, xi = bxi, whereχj³st, bx−j, byw0j´ ≡ ½x∗j | N ³st, bxx∗j , byw0j´ = j or PR³st, bxx∗j , byw0j´−1

6= ∅¾;

(2-3): in any other case,

for all i ∈ N , xi =

x2

if wi > max wkk 6=i , si = sti, andfor any j 6= i and any x0j,PR

¡s,¡x2, x0j,0−i,j

¢,w¢−1

= ∅0 otherwise

.

Note that for the subgame of (2-1), x = bx ∈ NE ¡Γ1g∗ (s,w) , e

¢, since

any deviator can enjoy a positive amount of output only under Rule 1-2-2. Also, x ∈ NE ¡Γ1

g∗ (s,w) , e¢for the subgame of (2-3), as we have

already shown. Moreover, we can see that for the subgame of (2-2), x ∈NE

¡Γ1g∗ (s,w) , e

¢. Note that g∗ (s,x,w) corresponds to Rule 1-1 or Rule

1-2-2 if¡Γ1g∗ (s,w) , e

¢corresponds to (2-2). Note also that χj

³st, bx−j, byw0j´

is non-empty, since x∗j = 0 guarantees N³st, bxx∗j , byw0j´ = j.

Suppose that g∗ (s,x,w) corresponds to Rule 1-1 when¡Γ1g∗ (s,w) , e

¢corresponds to (2-2). Then, f must be linear on

"0,max

(Pstibxi,P

i6=jstibxi + stjxj

)#,

which implies the Pareto efficiency of¡(xj, bx−i) , ¡w0j, by−j¢¢. Thus, x ∈

NE¡Γ1g∗ (s,w) , e

¢.

Suppose that g∗ (s,x,w) corresponds toRule 1-2-2 when¡Γ1g∗ (s,w) , e

¢corresponds to (2-2). Note first that any individual cannot induce Rule 2.Moreover, any i 6= j cannot induce Rule 1-1, and he can induce Rule 1-3,but only to enjoy no output consumption. Finally, j cannot induce Rule1-3. Thus, x ∈ NE ¡Γ1

g∗ (s,w) , e¢.

28

Now, let us see that the above strategy profile (1)-(2) constitutes a sub-game perfect equilibrium of the extensive game

¡Γ1g∗, e

¢. By the strategy

profile (1)-(2) of the extensive game¡Γ1g∗ , e

¢, g∗ (s,x,w) = g∗ (st, bx,by) = by.

Suppose that individual j deviates from (sj, wj) to¡s0j, w

0j

¢in the first stage.

Then by (2-2) and (2-3), he only gets

g∗j³ss0j ,x

³ss0j ,ww0j

´,ww0j

´≤ byj + stj · f 0 ³X stkbxk´ · (xj − bxj) .

These arguments imply that no individual has any incentive to deviate from(sj, wj) in the first stage. Thus, bz = (bx,by) ∈ SPA ¡Γ1

g∗ ,u, st¢.

Proof of Theorem 4. Since NA (g∗, e) = PR (e) and SPA¡Γ2g∗ , e

¢ ⊆NA (g∗, e) for all e ∈ E , we have only to show PR (e) ⊆ SPA ¡Γ2

g∗, e¢for all

e ∈ E . First, we will show that in every second stage subgame, there is atleast one Nash equilibrium strategy.Let us take a strategy profile (s,w) of the given second stage subgame¡

Γ2g∗(x), e

¢as follows:

for all i ∈ N , (si, wi) =½(sti, 0) if xi 6= µ (x−i)(sti, f (s

tixi) + 1) otherwise

.

Then, we can see that (s,w) ∈ NE ¡Γ2g∗(x), e

¢. Note that g∗ (s,x,w) cor-

responds to Rule 1-3. Note first that no individual can induce Rule 1-1by changing his strategy, and that no individual can enjoy output consump-tion under Rule 1-2-1. If any i ∈ N can induce Rule 1-2-2 by chang-ing his strategy, then such a deviation gives him no additional benefit, org∗i¡ss0i ,x,ww0i

¢= 0, because it must be the case that i /∈ N ¡ss0i ,x,ww0i¢.

Moreover, if any i ∈ N can induce Rule 2 by changing his strategy, thenit must be the case that xi > 0, which implies this deviation gives him noadditional benefit, or g∗i

¡ss0i ,x,ww0i

¢= 0. Finally, if any i ∈ N can induce

Rule 1-3 by changing his strategy, then such a deviation gives him no ad-ditional benefit, since the individual i who has xi = µ (x−i) is already fixedin the first stage game. Thus, (s,w) ∈ NE ¡Γ2

g∗(x), e¢.

Now, we will show that for each e = (u, st) ∈ E , if bz = (bx,by) ∈ PR (u, st),then there exists a subgame perfect equilibrium whose corresponding out-come is bz. Consider the following strategy profile of the extensive game¡Γ2g∗, e

¢:

(1) In the first stage, every individual i supplies bxi > 0.29

(2) In the second stage:(2-1): if bx is the action profile of all individuals in the first stage, then forany i ∈ N , (si, wi) = (sti, byi);(2-2): if x =

¡x0j, bx−j¢ À 0, where x0j 6= bxj, is the action profile of all indi-

viduals in the first stage, then

for this j ∈ N , (sj, wj) =¡stj, f

¡stjx

0j

¢+ 1¢,

for all i 6= j, (si, wi) =½(sti, byi) if xi 6= µ (x−i)(sti, f (

Pstkxk)) otherwise

;

(2-3): in any other case,

for all i ∈ N , (si, wi) =½(sti, 0) if xi 6= µ (x−i)(sti, f (s

tixi) + 1) otherwise

.

Note that for the subgame of (2-1), (s,w) = (st, by) ∈ NE ¡Γ2g∗ (x) , e

¢,

since any deviator can enjoy a positive amount of output only under Rule1-2-2. Also, (s,w) ∈ NE ¡Γ2

g∗ (x) , e¢for any subgame

¡Γ2g∗ (x) , e

¢of the

case (2-3), as we have already shown. Moreover, we can see that for thesubgame of (2-2), (s,w) ∈ NE ¡Γ2

g∗ (x) , e¢.

(i) Consider the case that for all i 6= j, xi 6= µ (x−i). Then, g∗ (s,x,w)corresponds to Rule 1-2-2. Note that for any individual, changing his an-nouncement of his labor skill cannot make him better off. Note also any i 6= jcan induce Rule 1-3 by changing from wi = byi to w0i = f (P stkxk), whichcannot make him better off. Finally, j cannot induce Rule 1-3 by changingfrom wj = f

¡stjxj

¢+ 1 to any w0j ≥ 0. Thus, (s,w) ∈ NE

¡Γ2g∗ (x) , e

¢.

(ii) Consider the case that there exists i 6= j with xi = µ (x−i). Theng∗ (s,x,w) corresponds toRule 1-3. Note first that no individual can induceRule 1-1 by changing his strategy, and that no individual can enjoy outputconsumption under Rule 1-2-1. If any i ∈ N can induce Rule 1-2-2 bychanging his strategy, then such a deviation gives him no additional benefit,or g∗i

¡ss0i ,x,ww0i

¢= 0, because it must be the case that i /∈ N ¡ss0i,x,ww0i¢.

Moreover, if any i ∈ N can induce Rule 2 by changing his strategy, thenthe deviation gives him no additional benefit, or g∗i

¡ss0i ,x,ww0i

¢= 0, since

xÀ 0. Finally, if any i ∈ N can induce Rule 1-3 by changing his strategy,then such a deviation gives him no additional benefit, since the individual iwho has xi = µ (x−i) is already fixed in the first stage game. Thus, (s,w) ∈NE

¡Γ2g∗ (x) , e

¢.

30

Now, let us see that the above strategy profile (1)-(2) constitutes a sub-game perfect equilibrium of the extensive game

¡Γ2g∗, e

¢. By the strategy

profile (1)-(2) of the extensive game¡Γ2g∗ , e

¢, g∗ (s,x,w) = g∗ (st, bx, by) = by.

Suppose that individual j deviates from bxj to x0j 6= bxj in the first stage. Ifx0j = 0, then by (2-3), he only gets g

∗j

¡s¡x0j, bx−j¢ , ¡x0j, bx−j¢ ,w ¡x0j, bx−j¢¢ =

0. If x0j > 0, then by (2-2), he only gets

g∗j¡s¡x0j, bx−j¢ , ¡x0j, bx−j¢ ,w ¡x0j, bx−j¢¢ ≤ byj + stj · f 0 ³X stibxi´ · ¡x0j − bxj¢ .

These arguments imply that no individual has any incentive to deviate fromxj in the first stage. Thus, bz = (bx,by) ∈ SPA ¡Γ2

g∗,u, st¢.

7 References

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33

Suh(1995)

Yoshihara(1999)

Tian(2000)

Yamada-Yoshihara(2001)

Equilibriumnotions

·NA·SNA·UNA

·NA·SNA·UNA

·NA·SNA

·NA·SNA·SPA

# of goods 2 2 m ≥ 2 2# of agents n ≥ 2 n ≥ 3 n ≥ 2 n ≥ 2

endowmentinformation

known knownunknown;overstatementsare prohibited

unknown;overstatementsandunderstatementsare possible

laborsovereignty

no no no yes

feasibility yes yes yes yesself-relevancy no yes no yesbest responseproperty

no yes no yes

forthrightness no yes yes yesbalancedness no yes no nocontinuity no no yes no

Table 1: Performance of mechanisms implementing PR

where NA means “Nash implementability,” SNA means “strong Nashimplementability,” UNA means “undominated Nash implementability,”

and SPA means “subgame-perfect implementability.”

34