fair groves mechanisms
TRANSCRIPT
Fair Groves Mechanisms∗
Murat Atlamaz† Duygu Yengin‡
December 2005
Abstract
We study allocation problems in which costly tasks are to be assigned, and moneytransfers are used to achieve fairness among agents. We consider a series of fairnessnotions of decreasing restrictiveness that are based on the Rawl’s maximin equitycriterion. These fairness notions impose welfare lower bounds. We show that eachlevel of fairness and a corresponding upper bound on the budget deficit together withassignment-efficiency and strategy-proofness characterize a unique mechanism up toPareto-indifference. As fairness gets stronger, the upper bound on the budget deficitincreases.We generalize our results to the multiple-task and the multi-attribute public good
settings.Keywords: Strategy-proofness, fairness, Groves mechanisms, allocation of indivisi-
ble tasks, imposition problems, Rawlsian maximin criterion, public good with multipleattributes, welfare bounds.
1 Introduction
We study the problem of allocating tasks among agents when money transfers are allowed.We characterize a class of parameterized Groves mechanisms that satisfy a maximin typeof fairness criterion and respect upper bounds on total transfer. Each agent incurs a costfor performing each combination of tasks, and these costs are her private information. A“center” assigns the tasks and determines monetary transfers among agents based on thereported costs.1 Agents have preferences over the sets of tasks and a perfectly divisiblegood to which we refer as “money”. Preferences are represented by utility functions thatare linear with respect to the perfectly divisible good. Agents do not have the option ofrefusing the tasks assigned to them. These problems are referred to as imposition problems.As an example, in times of natural disasters and wars, governments impose certain servicessuch as supplying food, medicine, or transportation on private firms.
∗An earlier version of this paper was circulated under the title “Characterizations of Strategy-Proofand Fair Mechanisms in Imposition Problems”. We are grateful to William Thomson for his guidance andadvice.
†Department of Economics, University of Rochester, Rochester, NY 14627, USA; e-mail:[email protected].
‡Department of Economics, University of Rochester, Rochester, NY 14627, USA; e-mail:[email protected].
1 If the transfer of an agent is positive, this is a payment from the center to the agent, and if the transferis negative, this is a payment from the agent to the center.
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We consider mechanisms that minimize the total cost of performing the tasks(assignment-efficiency) and induce agents to report their true costs (strategy-proofness).The so-called Groves mechanisms are the only mechanisms that satisfy these two proper-ties.2 It is well-known that the transfers induced by a Groves mechanism do not alwayssum up to zero (i.e., no Groves mechanism is budget-balanced). We want to determine theGroves mechanisms that are fair.
A commonly used notion of fairness is no-envy (Foley, 1967), which requires that noagent prefers another agent’s bundle to her own. However, if the number of tasks is atleast two, no Groves mechanism is envy-free on the domain of unrestricted (in particular,monotone) preferences.3 Therefore, no-envy is too strong as a notion of fairness for theGroves mechanisms. Another problem with imposing no-envy is that, at an envy-freeallocation, the utility of an agent may be arbitrarily small. In other words, agents cannotbe guaranteed a minimal utility level. Hence, there may be great discrepancies in theutilities of the agents.
In this paper, we consider an alternative fairness notion which is introduced by Porter,Shoham, and Tennenholtz (JET, 2004), hereafter PST. For simplicity, consider the allo-cation of a single task. Agents are identical in all respects other than the costs they incurwhen performing the task. However, in an imposition problem, since agents do not havethe freedom to refuse the tasks assigned to them, they should not be held responsible forthe costs. Therefore, one may argue that at a fair allocation the agents should experienceequal utilities. One may also consider assignment-efficiency and budget-balancedness asdesirable properties. We refer to an assignment-efficient and budget-balanced allocationat which utilities are equilized through transfers as a first-best allocation.4
Since no Groves mechanism is budget-balanced, no strategy-proof mechanism existsthat always chooses a first-best allocation. Yet, Groves mechanisms exist that guaranteeeach agent a utility at least as high as her utility at a first-best allocation. PST introducea fairness notion, 1-fairness, which requires that common utility achieved at a first-bestallocation be a lower bound on the actual utilities. In particular, a mechanism is 1-fairif for each economy, it respects the corresponding lower bound. Since transfers can be ofany size, it is always possible to ensure that each agent’s utility exceeds this lower bound.However, such transfers may lead to large (budget) deficits.5 Hence, our concern is to limitthe deficit while ensuring 1-fairness. However, if we insist on the non-positivity of totaltransfer (no-deficit), no Groves mechanism satisfies this requirement and 1-fairness. Infact, strategy-proofness, no-deficit, and 1-fairness are incompatible (Corollary 1 of PST).
One way to obtain compatibility of strategy-proofness, no-deficit, and fairness is toweaken 1-fairness by reducing the lower bound on the utilities. To do that, for eachinteger k between 1 and the number of agents, we define a k-th best allocation as onewhich assigns the task to an agent with the k-th lowest cost and equalizes the utilitiesthrough transfers for which budget balances. The resulting utility of each agent is thenegative of the k-th lowest cost divided by the number of agents. One may use thatcommon utility achieved in a k-th best allocation as a lower bound on the actual utilities.
2This result does not hold for any domain of preferences. The convexity of the domain is sufficient forthe result to hold.
3 If there is a single task to be assigned or the costs are superadditive, envy-free Groves mechanismsexist (Pápai, 2004).
4Since there may be several assignment-efficient allocations, several first-best allocations may exist.However, each agent’s utility is the same at each first-best allocation.
5Note that the deficit is the same as total transfer.
2
A mechanism is k-fair if for each economy, it respects the corresponding lower bound. Fork ∈ {3, . . . , n}, Groves mechanisms that satisfy k-fairness and no-deficit exist.
Our main objective is to characterize, for each k ∈ {1, . . . , n}, the Groves mechanismsthat are k-fair and respect certain upper bounds on total transfer. For each k ∈ {1, . . . , n},there exists a minimal upper bound on total transfer that is compatible with k-fairness6.This corresponding upper bound on the deficit depends on the k-th smallest cost. Obvi-ously, there is a trade-off between the lower bound on the utilities and the correspondingminimal upper bound on the deficit. We show that k-fairness and the correspondingminimal upper bound on the deficit together characterize a unique Groves mechanism,up to Pareto indifference.7 Hence, we obtain a class of characterizations parametrizedby k ∈ {1, . . . , n}. We also show that the unique mechanism (up to Pareto indifference)characterized for k = 2 is the only one that minimizes the deficit among all 1-fair Grovesmechanisms, and, for each k ∈ {1, . . . , n}, among all k-fair and “order-preserving” Grovesmechanisms.
Our model can be easily adapted to the problem of allocating a desirable good when allagents have equal claims over the good. As an example, consider distribution of a certainnumber of computers to the public schools by the school district. The schools may differin their needs for computers. The district wants to assign computers to the schools ingreater needs. The district may use transfers to have the schools reveal their true needs.If the number of computers is not large enough, some schools may receive no computers.Fairness can be restored by compensating those schools with money so that their utilitiesare above a certain utility level.
We prove our results in the single-task setting in Section 2. In Section 3, we show thatif the cost functions are additive, Theorems 1 and 2 generalize to the multiple-task setting(Theorems 4 and 5). In Section 4, Theorems 1, 2, and 3 generalize to the multi-attributepublic good setting (Theorems 6, 7, and 8). We omit the proofs of the results in Sections 3and 4 since these proofs are straightforward extensions of those of the results in Section 2.
2 Single task
A single indivisible task α is to be assigned to one of n agents by a “center”. Let N ={1, ..., n} be the set of the agents, where n ≥ 2. There is a perfectly divisible good thatcan be transfered between the agents and the center. Each agent i incurs a private costci ∈ Ci = [c, c] for performing the task, where 0 < c < c ≤ ∞. For each i ∈ N , letai = α if agent i is assigned the task, and ai = ∅ if she is not. Her utility from consuming(ai, ti) ∈ {∅, α} × R is
u(ai, ti; ci) =
½−ci + ti if ai = α,ti if ai = ∅.
Transfer ti can be of any size. If ti < 0, this transfer is a payment from agent i to thecenter, and if ti > 0, this transfer is a subsidy to agent i.
6 In particular, for k = 3, the minimal upper bound on deficit is zero. Hence, 3-fairness is compatiblewith no-deficit. For k > 3, the corresponding upper bound is negative for some economies. Hence, fork ≥ 3, the center may obtain a surplus.
7The Groves mechanisms for k = 1 and 3 are introduced by PST.
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An economy is specified by c = (c1, ..., cn). Let C = C1×· · ·×Cn be the set of economies.As usual, (c1, . . . , ci−1, ci+1, . . . , cn) is written as c−i.8
An allocation is a list (a, t) = (ai, ti)i∈N ∈ {∅, α}N × RN such that ai = α for somei ∈ N , and aj = ∅ for each j ∈ N\{i}. The allocations (a, t) and (a0, t0) are Pareto-indifferent if
Pi ti =
Pi t0i, and for each c ∈ C and each i ∈ N , u(ai, ti; ci) = u(a0i, t
0i; ci).
A mechanism is a function ϕ = (a, t) that associates with each economy c ∈ C anallocation, that is, for each i ∈ N , ϕi(c) = (ai(c), ti(c)) ∈ {∅, α} × R. A mechanism ϕconsists of an assignment function a : C → {∅, α}N and a transfer function t : C → RN .The mechanisms ϕ and ϕ0 are Pareto-indifferent if for each c ∈ C, ϕ(c) and ϕ0(c) arePareto-indifferent.
For each k ∈ N , let c[k] be the k-th cost in the ascending order of the costs in{c1, . . . , cn}. All ties are taken into account in this order. For instance, if there aretwo agents whose costs are the lowest, then c[1] = c[2]. For each i ∈ N , let (c−i)[k] be thek-th cost in the ascending order of the costs in {c1, . . . , ci−1, ci+1, . . . , cn}.
The most natural objective is Pareto-efficiency. Since there is no restriction on the sizeof the transfers, any allocation is Pareto dominated by some other allocation. Hence, thereis no Pareto-efficient allocation. However, we can still define a notion of efficiency amongthe allocations for which the total transfer is the same. An allocation that assigns the taskto an agent whose cost is the lowest is not Pareto dominated by another allocation withthe same or smaller total transfer. We require mechanisms to only choose such allocations:
Assignment-efficiency: For each c ∈ C and each i ∈ N , ai(c) = α implies ci = c[1].
Since costs are private information, an assignment-efficient mechanism assigns thetask to an agent with the lowest actual cost if the agents report their true costs. Then, adesirable property for a mechanism is that no agent can ever benefit by misreporting hercost:
Strategy-proofness:9 For each c ∈ C, each i ∈ N , and each c0i ∈ Ci, u(ϕi(c); ci) ≥u(ϕi(c
0i, c−i); ci).
A well-known class of mechanisms are the so-called Groves mechanisms. Let H ={(hi)i∈N | for each i ∈ N , hi : C−i → R}. In any economy, there may be more thanone agent whose cost is the lowest. So, we assume that each Groves mechanism uses atie-breaking rule τ to decide on which of these agents is to be assigned the task. The ruleτ specifies, for each economy c ∈ C, an agent τ(c) among the agents whose costs are thelowest. Let T = {τ | τ : C → N , and cτ(c) = c[1] for each c ∈ C} be the set of tie-breakingrules. Tie-breaking rules allow us to work with single-valued Groves mechanisms.10
The Groves mechanism associated with h = (hi)i∈N ∈ H and τ ∈ T , Gh,τ = (ah,τ , th,τ ),is defined as follows: for each c ∈ C and each i ∈ N ,
ah,τi (c) = α if and only if i = τ(c), and
th,τi (c) =
(hi(c−i) if ah,τi (c) = α,
−c[1] + hi(c−i) if ah,τi (c) = ∅.Note that for each h ∈ H, the mechanisms in {Gh,τ}τ∈T are Pareto-indifferent. We
have the following lemma.
8For each i ∈ N , let C−i be the cross product of C1, . . . , Ci−1, Ci+1, . . . , Cn.9See Thomson (2005) for an extensive survey on strategy-proofness.10 In the mechanism design literature, the Groves mechanisms are not necessarily restricted to be single-
valued (see, for instance, Green and Laffont, 1977; Walker, 1980).
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Lemma 1. A mechanism ϕ is assignment-efficient and strategy-proof on C if and only ifit is a Groves mechanism.
Proof: Since C is convex, the proof follows from Holmström (1979). 2
The following property requires that the center neither incurs a deficit nor obtains asurplus. (Note that the deficit is the same as total transfer).
Budget-balancedness: For each c ∈ C,P
i ti(c) = 0.11
It is well-known that Groves mechanisms are not budget-balanced (Green and Laffont,1979; Schummer, 2000). Therefore, no mechanism is assignment-efficient, strategy-proof,and budget-balanced. We may still want to ensure that the center does not incur a deficit.The following property formalizes this idea.
No-deficit: For each c ∈ C,P
i ti(c) ≤ 0.We can generalize no-deficit by imposing an upper bound on the total transfer. For each
economy, we consider an upper bound which depends on this economy. Let m : C → R.
m-Bounded deficit: For each c ∈ C,P
i ti(c) ≤ m(c).
Another objective is fairness. Agents differ only in their costs. One of the basic notionsof fairness is equal treatment of equals: agents whose costs are the same experience equalutilities. When agents incur different costs, in principle, the center can treat them differ-ently, e.g. the agents with lower costs can be guaranteed higher utilities. Even if costsdiffer, there is still another reason why fairness may require that agents be treated equally:the fact that the tasks are imposed on them. Since they do not have the freedom to refusethe tasks assigned to them, they should not be held responsible for their costs. Responsi-bility goes hand in hand with freedom. Therefore, in imposition problems, fairness requiresthat agents experience equal utilities unless it is possible to have Pareto improvement onthe equal utility distribution. We borrow Rawls’ difference principle known also as themaximin criterion:
“The difference principle is a strongly egalitarian conception in the sensethat unless there is a distribution that makes both persons better off (limitingourselves to the two-person case for simplicity), an equal distribution is to bepreferred.”(Rawls, 1971)
The maximin criterion implies that resources are allocated to maximize the minimalutility among all the agents.
Let us consider the implications of the properties, no-deficit and the maximin criterion.Clearly, no-deficit and the maximin criterion together imply budget-balancedness. Anallocation satisfies these properties if it assigns the task to an agent with the lowest costand equalizes utilities through transfers. We refer to such an allocation as a first-bestallocation. The resulting utility of each agent is − c[1]
n . Even if we do not impose no-deficit or the maximin criterion, we may still require − c[1]
n to be a lower bound on utilities.A mechanism is 1-fair if for each economy, it respects the corresponding lower bound
11Various papers consider the allocation of objects when there is a given amount of money to be exactlydistributed (i.e. the budget is balanced). See Alkan, Demange, and Gale (1991); and Tadenuma andThomson (1991).
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(PST). By making large transfers, we can ensure that each agent’s utility exceeds thislower bound. Hence, many 1-fair Groves mechanisms exist. However, 1-fairness, togetherwith no-deficit characterizes a unique mechanism (up to Pareto-indifference) which alwaysselects a first-best allocation. Unfortunately, this mechanism is not strategy-proof (seeCorollary 1 of PST).
One way to obtain the compatibility of strategy-proofness, no-deficit, and fairness isto weaken the fairness property by reducing the lower bound on the utilities. To do that,for each k ∈ {1, . . . , n}, we define a k-th best allocation as the one which assigns the taskto an agent with the k-th lowest cost and equalizes the utilities through transfers thatbalance the budget. The resulting utility of each agent is − c[k]
n . We require the utilityachieved in a k-th best allocation to be a lower bound on the utilities:
k-Fairness: For each c ∈ C and each i ∈ N , u(ϕi(c); ci) ≥ −c[k]n .
As k increases, we obtain a series of fairness notions of decreasing restrictiveness: thehigher k is, the smaller the utility lower bound required by k-fairness.
Unfortunately, strategy-proofness, no-deficit, and 2-fairness are incompatible (see The-orem 1 of PST). Since 1-fairness is stronger than 2-fairness, the incompatibility persistsif we replace 2-fairness with 1-fairness. For each k ∈ {3, . . . , n}, mechanisms that satisfyassignment-efficiency, strategy-proofness, no-deficit, and k-fairness exist. Let us present aclass of such mechanisms, parametrized by k ∈ {2, . . . , n}.Mechanism Fk,τ : Let k ∈ {2, . . . , n} and τ ∈ T . Let F k,τ = (ak,τ , tk,τ ) be such that foreach c ∈ C and each i ∈ N ,
ak,τi (c) = α if and only if i = τ(c), and
tk,τi (c) =
(− (c−i)[k−1]n + c[2] if ak,τi (c) = α,
− (c−i)[k−1]n if ak,τi (c) = ∅.
For each k ∈ {2, . . . , n} and τ ∈ T , the transfer of mechanism F k,τ differs from that of
a pivotal mechanism by the term − (c−i)[k−1]n .12 Let Fk = {F k,τ}τ∈T .13 Clearly, for eachk ∈ {2, . . . , n}, the mechanisms in Fk are Pareto-indifferent. Also, for each k ∈ {2, . . . , n}and each τ ∈ T , F k,τ is a Groves mechanism: if for each i ∈ N , and each c−i ∈ C−i,hi(c−i) = −
(c−i)[k−1]n + (c−i)[1], then F k,τ = Gh,τ .
Our main characterization is the following:
Theorem 1. Let k ∈ {2, . . . , n}. For each c ∈ C, let mk(c) = c[2] − c[k−1]. A mechanismsatisfies assignment-efficiency, strategy-proofness, mk-bounded deficit, and k-fairness ifand only if it belongs to Fk.
Proof: Let k ∈ {2, . . . , n}. It is easy to show that the mechanisms in Fk satisfy theproperties listed in Theorem 1. Now, let ϕ satisfy these properties. By Lemma 1, ϕis a Groves mechanism associated with some h = (hi)i∈N ∈ H and some τ ∈ T . Letϕ = (ah,τ , th,τ ). Note that for each c ∈ C and each i ∈ N , ah,τi (c) = α if and only ifi = τ(c), and
th,τi (c) =
(hi(c−i) if ah,τi (c) = α,
−c[1] + hi(c−i) if ah,τi (c) = ∅.(1)
12Pivotal mechanisms are introduced by Clarke (1971).13PST introduce F1 and F3. However, they do not mention Fk for any k ∈ N\{1, 3}.
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Claim 1: For each i ∈ N and each c−i ∈ C−i,
hi(c−i) ≥ −(c−i)[k−1]
n+ (c−i)[1]. (2)
Proof: Assume, by contradiction, that for some i ∈ N and some c−i ∈ C−i,
hi(c−i) < −(c−i)[k−1]
n+ (c−i)[1]. (3)
By k-fairness, for each ci ∈ Ci,
−c[1] + hi(c−i) ≥ −c[k]
n.
This inequality and (3) together imply
(c−i)[1] − c[1] +c[k] − (c−i)[k−1]
n> 0. (4)
Note that (4) holds no matter what ci is. Let ci = (c−i)[1]. Then,
(c−i)[1] − c[1] +c[k] − (c−i)[k−1]
n= 0,
which contradicts (4).Claim 2: For each i ∈ N and each c−i ∈ C−i,
hi(c−i) = −(c−i)[k−1]
n+ (c−i)[1].
Proof: Assume, by contradiction, that for some i ∈ N and some c−i ∈ C−i,
hi(c−i) 6= −(c−i)[k−1]
n+ (c−i)[1].
Then, by Claim 1, Xi∈N
hi(c−i) >Xi∈N
∙−(c−i)[k−1]
n+ (c−i)[1]
¸. (5)
Note thatXi∈N
∙−(c−i)[k−1]
n+ (c−i)[1]
¸= −n− k + 1
nc[k−1] −
k − 1n
c[k] + (n− 1)c[1] + c[2]. (6)
Also, by mk-bounded deficit,
−(n− 1)c[1] +Xi∈N
hi(c−i) ≤ c[2] − c[k−1]. (7)
Then, (5), (6), and (7) together imply
−n− k + 1
nc[k−1] −
k − 1n
c[k] + c[2] < c[2] − c[k−1]. (8)
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Now, let ci = (c−i)[k−1]. Then,
n− k + 1
nc[k−1] +
k − 1n
c[k] = c[k−1].
This equality contradicts (8).By Claim 2, for each i ∈ N and each c−i ∈ C−i,
hi(c−i) = −(c−i)[k−1]
n+ (c−i)[1].
Since ah,τ = ak,τ , substituting hi(c−i) into (1) we obtain F k,τ . 2
For k = 3, mk-bounded deficit is equivalent to no-deficit. For each k ∈ {4, . . . , n} andeach c ∈ C, mk(c) ≤ 0 (i.e., the surplus is non-negative). Furthermore, for some c ∈ C, thesurplus is positive. The idea that the center wants to make at least a certain surplus makessense especially in our second motivation, in which a desirable object is to be assigned. Inthat case, the center may want to guarantee a certain surplus in return for assigning theobject.
Consider a mechanism that satisfies assignment-efficiency, strategy-proofness, and k-fairness. Then, this is a Groves mechanism associated with h = (hi)i∈N where for eachi ∈ N , hi satisfies (2). If for each i ∈ N , (2) holds as equality, the deficit is minimized.So, we have the following corollary of Theorem 1.
Corollary 1. Let k ∈ {2, . . . , n}. A mechanism has the smallest deficit among all mecha-nisms satisfying assignment-efficiency, strategy-proofness, and k-fairness if and only if itbelongs to Fk.
For each k ∈ {2, . . . , n}, each τ ∈ T , and each c ∈ C, the total transfer of mechanismF k,τ is
Pi∈N
tk,τi (c) = −n−k+1n c[k−1]− k−1
n c[k]+c[2]. This is also the minimal deficit mentioned
in Corollary 1. Let k = 2. Consider an economy c0 ∈ C such that c0[2] > c0[1]. Then,Pi∈N
t2,τi (c) > 0. Hence, the mechanisms in F2 violate no-deficit. On the other hand, for
k ∈ {3, . . . , n}, the mechanisms in Fk satisfy no-deficit sincePi∈N
tk,τi (c) ≤ c[2]− c[k−1] ≤ 0.
The following is an immediate consequence of Theorem 1.
Corollary 2. A mechanism satisfies assignment-efficiency, strategy-proofness, no-deficit,and 3-fairness if and only if it belongs to F3.
Corollary 2 states that the example given by PST to prove the compatibility ofassignment-efficiency, strategy-proofness, no-deficit, and 3-fairness is in fact the uniquesuch mechanism, up to Pareto-indifference.
So far, to obtain compatibility of strategy-proofness, no-deficit, and fairness, we haveweakened the fairness property by imposing k-fairness for k ∈ {2, . . . , n}. Another way toobtain compatibility is to weaken no-deficit while keeping 1-fairness. In Theorem 1, wecharacterize the class F2 by assignment-efficiency, strategy-proofness, m-bounded deficit,and 2-fairness, where for each c ∈ C, m(c) = c[2] − c[1]. The same class is also characteri-zated by strengthening 2-fairness to 1-fairness, and keeping the other properties.14
14Even if we strengthen 2-fairness to 1-fairness, we do not have to increase the upper bound on thebudget deficit to characterize F2.
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Theorem 2. For each c ∈ C, let m(c) = c[2] − c[1]. A mechanism satisfies assignment-efficiency, strategy-proofness, m-bounded deficit, and 1-fairness if and only if it belongsto F2.
Proof: Consider the proof of Theorem 1 for k = 2. If we impose 1-fairness instead of2-fairness, inequality (4) becomes
(c−i)[1] − c[1] +c[1] − (c−i)[1]
n> 0. (9)
Similarly as in the proof of Theorem 1, let ci = (c−i)[1]. Then, (c−i)[1] − c[1] = 0, whichcontradicts (9). The rest of the proof is identical to that of the proof of Theorem 1. 2
Theorem 2 states that the example given by PST to prove the compatibility ofassignment-efficiency, strategy-proofness, m2-bounded deficit, and 1-fairness, where foreach c ∈ C, m2(c) = c[2] − c[1], is in fact the unique such mechanism, up to Pareto-indifference. The following corollary of Theorem 2 is in the same spirit as Corollary 1.
Corollary 3. A mechanism has the smallest deficit among all mechanisms satisfyingassignment-efficiency, strategy-proofness, and 1-fairness if and only if it belongs to F2.
For each k ∈ {3, . . . , n}, the mechanisms in Fk have the following drawback: an agentwith a cost lower than c[k] experiences a lower utility than that experienced by an agentwith a cost higher than c[k]. In a sense, she is penalized for having a lower cost. We definethe following property to avoid this kind of penalties.
Order-preservation: For each c ∈ C and each i, j ∈ N , ci ≤ cj implies u(ϕi(c); ci) ≥u(ϕj(c); cj).
PST impose a similar requirement that the utilities of the losers be ordered. Theyrefer to this requirement as “no-competence penalty”. Order-preservation is stronger thanno-competence penalty since the former also compares the winner’s utility with the losers’utilities. The mechanisms in F2 satisfy order-preservation.
Let k ∈ {2, . . . , n}. We know that k-fairness places a common lower bound on theutilities of all agents. The following lemma states that k-fairness together with assignment-efficiency, strategy-proofness, and order-preservation places a cost-dependent lower boundon the utility of those agents whose costs are at most c[k].
Lemma 2. Let k ∈ {2, . . . , n}. Let ϕ = (a, t) satisfy assignment-efficiency, strategy-proofness, k-fairness, and order-preservation. For each c ∈ C, each j ∈ {2, . . . , k}, andeach i ∈ N such that ci = c[j], u(ϕi(c); ci) ≥ −
c[j−1]n .
Proof: Let ϕ satisfy the first four properties listed in Lemma 2. By Lemma 1, ϕ is a Grovesmechanism associated with some h = (hi)i∈N ∈ H and some τ ∈ T . Let ϕ = (ah,τ , th,τ ).We proceed by induction on j ∈ {2, . . . , k}.Base step: For each c ∈ C and each i ∈ N such that ci = c[k],
u(ϕi(c); ci) ≥ −c[k−1]n
.
To prove this, assume, by contradiction, that for some c ∈ C and some i ∈ N such thatci = c[k],
u(ϕi(c); ci) < −c[k−1]n
. (10)
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Let bc ∈ C be such that bci = c[k−1], and for each l ∈ N\{i}, bcl = cl. Note that u(ϕi(bc);bci) =−bc[1] + hi(bc−i). Since bc[1] = c[1] and hi(bc−i) = hi(c−i), then
u(ϕi(bc);bci) = u(ϕi(c); ci). (11)
Note that bc[k] = c[k−1]. This equality, (10), and (11) together imply u(ϕi(bc);bci) < −bc[k]n .This inequality contradicts k-fairness.Induction step: Suppose that for each c ∈ C, each j ∈ {r + 1, . . . , k}, and each i ∈ Nsuch that ci = c[j], u(ϕi(c); ci) ≥ −
c[j−1]n . We want to show that for each c ∈ C and each
i ∈ N such that ci = c[r], u(ϕi(c); ci) ≥ −c[r−1]n . To prove this, assume, by contradiction,
that for some c ∈ C and some i ∈ N such that ci = c[r],
u(ϕi(c); ci) < −c[r−1]n
. (12)
Let bc ∈ C be such that bci = c[r−1], and for each l ∈ N\{i}, bcl = cl. As in Base step, wehave
u(ϕi(bc);bci) = u(ϕi(c); ci). (13)
Note that bc[r] = c[r−1]. This equality, (12), and (13) together imply u(ϕi(bc);bci) < −bc[r]n .Let s ∈ N\{i} be such that bcs = bc[r+1]. By order-preservation, u(ϕs(bc);bcs) ≤ u(ϕi(bc);bci).Thus, u(ϕs(bc);bcs) < −bc[r]n . This inequality contradicts the induction hypothesis. 2
By Corollary 3, we already know that the mechanisms in F2 have the smallest deficitamong the mechanisms which satisfy assignment-efficiency, strategy-proofness, and 1-fairness. In the following theorem, we show that in Corollary 3, if we weaken 1-fairness tok-fairness for any k ∈ {2, . . . , n}, and impose order-preservation, we still characterize theclass F2.
Theorem 3. Let k ∈ {2, . . . , n}. A mechanism has the smallest deficit among allmechanisms satisfying assignment-efficiency, strategy-proofness, k-fairness, and order-preservation if and only if it belongs to F2.
Proof: We already know that the mechanisms in F2 satisfy the four properties listedin Theorem 3. Let ϕ be a mechanism that satisfies these properties. By Lemma 1, ϕis a Groves mechanism associated with some h = (hi)i∈N ∈ H and some τ ∈ T . Letϕ = (ah,τ , th,τ ). Let c ∈ C and i ∈ N .Claim 1: If ah,τi (c) = ∅, then th,τi (c) ≥ − c[1]
n .Proof: Assume, by contradiction, that ah,τi (c) = ∅, and th,τi (c) < − c[1]
n . This inequality isequivalent to
u(ϕi(c); ci) < −c[1]
n. (14)
Let bc ∈ C be such that bci = c[2], and for each l ∈ N\{i}, bcl = cl. By the same argument as
in the proof of Lemma 2, we obtain (11). Let j ∈ N be such that bcj = bc[2] and ah,τj (bc) = ∅.By order-preservation, u(ϕj(bc);bcj) = u(ϕi(bc);bci). This equality and (11) together implyu(ϕj(bc);bcj) = u(ϕi(c); ci). By Lemma 2, u(ϕj(bc);bcj) ≥ −bc[1]n . Note that bc[1] = c[1]. Hence,u(ϕi(c); ci) ≥ −
c[1]n , which contradicts (14).
Claim 2: If ah,τi (c) = α, then th,τi (c) ≥ − c[2]n + c[2].
10
Proof: Assume, by contradiction, that ah,τi (c) = α, and
th,τi (c) < −c[2]n+ c[2]. (15)
Let bc ∈ C be such that bci = c[2], and for each l ∈ N\{i}, bcl = cl. Note that u(ϕi(bc);bci) =−bc[1] + hi(bc−i). Since, bc[1] = c[2] and hi(bc−i) = hi(c−i), then
u(ϕi(bc);bci) = −c[2] + hi(c−i). (16)
Note that th,τi (c) = hi(c−i). This equality, (15), (16), and the fact that bc[2] = c[2]
together imply u(ϕi(bc);bci) < −bc[2]n . Using Claim 1, for each j ∈ N\{i} such thatah,τj (bc) = ∅, u(ϕj(bc);bcj) ≥ −bc[2]n . Hence, u(ϕi(bc);bci) < u(ϕj(bc);bcj), which contradictsorder-preservation.If the inequalities in Claim 1 and Claim 2 hold as equalities, the deficit is minimized. Thetransfers specified in these equalities correspond to those of a mechanism in F2. 2
Theorem 3 states that for each k ∈ {2, . . . , n}, if we also impose order-preservationin Corollary 1, we characterize the class F2 instead of Fk. Hence, the trade-off betweenfairness and bounded-deficit disappears.
3 Multiple tasks
In this section, we extend the model to the multiple-task setting where each agent canperform more than one task. Let Λ be the finite set of tasks. Let A be the set of subsetsof Λ. Each agent i has a cost function ci : A → [0, w] where 0 < w ≤ ∞. Let ci(∅) = 0.We refer to the domain of such cost functions as the unrestricted domain.
Let, for each i ∈ N and each α ∈ Λ, cαi = ci({α}). A cost function ci is additive iffor each Ai ∈ A, ci(Ai) =
Pα∈Ai
cαi . For each i ∈ N , let Ci be the set of such cost functions
for agent i. From now on, we consider the domain of additive cost functions, if not statedotherwise.
Agent i’s utility from consuming (Ai, ti) ∈ A×R is
u(Ai, ti; ci) = −Xα∈Ai
cαi + ti.
An economy is specified by c = (c1, ..., cn). Let C = C1×· · ·×Cn be the set of economies.An allocation is a list (A, t) = (Ai, ti)i∈N ∈ AN × RN such that ∪
i∈NAi = Λ, and for each
i, j ∈ N , Ai ∩ Aj = ∅. A mechanism is a function ϕ = (A, t) that associates with eacheconomy c ∈ C an allocation, that is, for each i ∈ N , ϕi(c) = (Ai(c), ti(c)) ∈ A×R.
For each α ∈ Λ and each k ∈ N , let cα[k] be the k-th cost in the ascending order ofthe costs in {cα1 , . . . , cαn}. Let cα−i[k] be the k-th cost in the ascending order of the costs in{cα1 , . . . , cαi−1, cαi+1, . . . , cαn}.
By additivity, it is meaningful to adapt assignment-efficiency as follows:
Assignment-efficiency: For each c ∈ C and each i ∈ N , α ∈ Ai(c) implies cαi = cα[1].
11
The definitions of strategy-proofness and m-bounded deficit remain the same. Lemma1 still holds because the domain C of cost profiles remains convex.
Now, we adapt the Groves mechanisms. Let H = {(hi)i∈N | for each i ∈ N, hi : C−i →R}. Let T = {τ | τ : C ×Λ→ N , and cατ(c,α) = cα[1] for each c ∈ C and α ∈ Λ} be the set oftie-breaking rules. The Groves mechanism associated with h = (hi)i∈N ∈ H and τ ∈ T ,Gh,τ = (ah,τ , th,τ ), is defined as follows: for each c ∈ C and each i ∈ N ,
α ∈ Ah,τi (c) if and only if i = τ(c, α), and
th,τi (c) = −Pj 6=i
Pα∈Ah,τ
j (c)
cαj + hi(c−i).
We adapt k-fairness as follows:
k-Fairness: For each c ∈ C and each i ∈ N , u(ϕi(c); ci) ≥ − 1nPα∈Λ
cα[k].
Finally, we adapt F k,τ to the multiple-task setting.
Mechanism Fk,τ : Let k ∈ {2, . . . , n} and τ ∈ T . Let F k,τ = (Ak,τ , tk,τ ) be such that foreach c ∈ C and each i ∈ N ,
α ∈ Ak,τi (c) if and only if i = τ(c, α), and
tk,τi (c) = −Pj 6=i
Pα∈Ak,τj (c)
cαj − 1n
Pα∈Λ
cα−i[k−1] +Pj 6=i
Pα∈Ak,τ
j (c−i)
cαj .
Let Fk = {F k,τ}τ∈T . Clearly, for each k ∈ {2, . . . , n}, the mechanisms in Fk arePareto-indifferent. Also, for each k ∈ {2, . . . , n} and each τ ∈ T , F k,τ is a Groves mech-anism: if for each i ∈ N , and each c−i ∈ C−i, hi(c−i) = − 1n
Pα∈Λ
cα−i[k−1] +Pj 6=i
Pα∈Ak,τj (c−i)
cαj ,
then F k,τ = Gh,τ . This transfer differs from the transfer of a pivotal mechanism by theterm − 1n
Pα∈Λ
cα−i[k−1].
Theorem 1 extends to the multiple-task setting as follows:
Theorem 4. Let k ∈ {2, . . . , n}. For each c ∈ C, let mk(c) ≡Pα∈Λ
(cα[2] − cα[k−1]). A
mechanism satisfies assignment-efficiency, strategy-proofness, mk-bounded deficit, and k-fairness if and only if it belongs to Fk.
Theorem 2 extends to the multiple-task setting as follows:
Theorem 5. For each c ∈ C, let m(c) ≡Pα∈Λ
(cα[2]−cα[1]). A mechanism satisfies assignment-
efficiency, strategy-proofness, m-bounded deficit, and 1-fairness if and only if it belongsto F2.
Corollaries 1, 2, and 3 extend to the multiple-task setting in a straightforward way.
One can define assignment-efficiency for the unrestricted domain in a straightforwardway. PST adapt the mechanisms in F2 and F3 to the unrestricted domain, and show thatthese mechanisms satisfy the properties listed in Theorems 4 and 5, respectively. At thispoint, it is an open question whether the characterizations in Theorems 4 and 5 still holdon the unrestricted domain.
12
4 Multi-attribute public good
So far, we have not considered the possibility that agents derive benefits from the per-formance of the tasks. However, there are real-world examples in which costly tasks areto be assigned to some agents and all agents benefit from the performance of the tasksindependently of the identities of agents who are assigned the tasks. For instance, considera group of stores in a shopping mall. Suppose the stores together want to advertise themall. Each store may have its own advertisement department or work with an agency.The stores assign the task to one among themselves that can accomplish this task at thelowest cost. Another example is the preparation of an end-of-year show at a school. Someparents design costumes and decorations while others contribute money. A third exampleis carpooling by parents to drive their children to school. In general, job definitions arenot clear-cut in informal groups such as small non-profit organizations, social clubs, familyfirms, etc. When a task is to be performed, one of the group members is assigned the task.
In the examples above, the tasks can be seen as public goods. Once a task is performedby an agent, all agents benefit from it without any rivalry.
A task may have different attributes. Consider the first example in the first paragraphin which a group of stores want to advertise the mall. The method and the content of theadvertisement, and the frequency of the advertisement’s display are some of the attributesof this task. The advertisement may appear on a local radio station, a newspaper, ornational TV. The frequency of the advertisement’s display may be high, moderate or low.We refer to each possible combination of the realizations of the attributes as a quality. Inprinciple, the benefits and the costs of agents depend on the qualities of the tasks. In theprevious sections, we have implicitly assumed that each task had a fixed quality.15 In thissection, we relax this assumption.
To simplify the analysis, we consider the single task case.16 A task is to be assignedto one of n ≥ 2 agents by a center. Let R be the set of finitely many attributes of thetask. Let Qr be the set of realizations of attribute r ∈ R. Let Q =
Qr∈R
Qr be the set of all
qualities. Let q denote a typical element in Q.Each agent i has a cost function ci : Q → R+. Let C be the set of all cost functions.
Let b : Q→ R+ be the benefit function, common to all agents. This function is commonknowledge. Throughout this section, we fix the benefit function. For each i ∈ N , let ai = qif agent i is assigned the task at quality q ∈ Q, and ai = ∅ if she is not assigned the task.Her utility from consuming (ai, ti) ∈ (Q ∪ {∅})×R is
u(ai, ti; ci) =
½b(q)− ci(q) + ti if ai = q,b(q) + ti if ai = ∅, and aj(c) = q for some j ∈ N\{i}.
An economy is specified by c = (c1, ..., cn) ∈ CN . An allocation is a list (a, t) =(ai, ti)i∈N ∈ (Q∪{∅})N ×RN such that ai = q for some i ∈ N and some q ∈ Q, and aj = ∅for each j ∈ N\{i}.15Even though we have not incorporated the benefits to the model in the previous sections, the analysis
would not change if each task had a fixed quality, and for each task the agents derived identical benefitsindependently of identity of the agent who was assigned the task.16We can extend the model and the results to the multiple public good setting if the benefit functions
and the cost functions are additive across tasks and qualities.
13
A mechanism is a function ϕ = (a, t) that associates with each economy c ∈ CN anallocation, that is, for each i ∈ N ,
ϕi(c) = (ai(c), ti(c)) ∈ (Q ∪ {∅})×R.
We impose the following assumption:
Assumption M: The functions b, c1, . . . , cn are such that for each i ∈ N , nb(q) − ci(q)has a maximizer on Q.
For each i ∈ N , let Q∗i = argmaxq∈Q
{ nb(q)− ci(q)}. Let si = nb(q)− ci(q) be the social
surplus if agent i is assigned the task at a quality q ∈ Q∗i . Clearly, Q∗i and si depend on
ci. However, to simplify notation, we do not display these dependencies in the rest of thepaper. For each k ∈ {2, . . . , n}, let s[k] be the k-th social surplus in the descending orderof the social surpluses in {s1, . . . , sn}.17
We adapt assignment-efficiency as follows:
Assignment-efficiency: For each c ∈ CN and each i ∈ N , ai(c) = q implies q ∈ Q∗i andsi = s[1].
The definitions of strategy-proofness and m-bounded deficit remain the same. Lemma1 still holds because the domain CN of cost profiles remains convex.
Now, we adapt the Groves mechanisms. Let H = {(hi)i∈N | for each i ∈ N , hi :CN\{i} → R}. Let T = {τ | τ : CN → N × Q such that for each c ∈ CN , τ(c) = (i, q)implies si = s[1] and q ∈ Q∗i } be the set of tie-breaking rules. The Groves mechanismassociated with h = (hi)i∈N ∈ H and τ ∈ T , Gh,τ = (ah,τ , th,τ ), is defined as follows: foreach c ∈ CN and each i ∈ N ,
ah,τi (c) = q if and only if (i, q) = τ(c), and
th,τi (c) =
⎧⎪⎨⎪⎩(n− 1)b(q) + hi(c−i) if ah,τi (c) = q,(n− 1)b(q)− cj(q) + hi(c−i) if ah,τi (c) = ∅, and
ah,τj (c) = q for some j ∈ N\{i}.We adapt k-fairness as follows:
k-Fairness: For each c ∈ CN and each i ∈ N , u(ϕi(c); ci) ≥ 1ns[k].
Finally, we adapt F k,τ to the multi-attribute public good setting.
Mechanism Fk,τ : Let k ∈ {2, . . . , n} and τ ∈ T . Let F k,τ = (ak,τ , tk,τ ) be such that foreach c ∈ CN and each i ∈ N ,
ak,τi (c) = q if and only if (i, q) = τ(c), and
tk,τi (c) =
⎧⎪⎨⎪⎩(n− 1)b(q) + (s−i)[k−1]
n − s[2] if ak,τi (c) = q,
−b(q) + (s−i)[k−1]n if ak,τi (c) = ∅, and
ak,τj (c) = q for some j ∈ N\{i}.
Let Fk = {F k,τ}τ∈T . Clearly, for each k ∈ {2, . . . , n}, the mechanisms in Fk arePareto-indifferent. Also, for each k ∈ {2, . . . , n} and each τ ∈ T , F k,τ is a Groves mecha-
nism: if for each i ∈ N , and each c−i ∈ C−i, hi(c−i) =(s−i)[k−1]
n −(s−i)[1], then F k,τ = Gh,τ .Theorem 1 extends to the multi-attribute public good setting as follows:
17Notice that in the previous sections we order costs in an ascending way.
14
Theorem 6. Let k ∈ {2, . . . , n}. For each c ∈ C, let mk(c) ≡ s[k−1] − s[2]. A mechanismsatisfies assignment-efficiency, strategy-proofness, mk-bounded deficit, and k-fairness ifand only if it belongs to Fk.
Theorem 2 extends to the multi-attribute public good setting as follows:
Theorem 7. For each c ∈ C, let m2(s) ≡ s[1] − s[2]. A mechanism satisfies assignment-efficiency, strategy-proofness, m2-bounded deficit, and 1-fairness if and only if it belongsto F2.
Corollaries 1, 2, and 3 extend to the multi-attribute public good setting in a straight-forward way.
We adapt order-preservation as follows:
Order-preservation: For each c ∈ CN and each i, j ∈ N , si ≥ sj implies u(ϕi(c); ci) ≥u(ϕj(c); cj).
The mechanisms in F2 satisfy order-preservation. On the other hand, for each k ∈{3, . . . , n}, the mechanisms in Fk do not satisfy order-preservation.
Lemma 2 and Theorem 3 extend to the multi-attribute public good setting as follows:
Lemma 3. Let k ∈ {2, . . . , n}. Let ϕ = (a, t) satisfy assignment-efficiency, strategy-proofness, k-fairness, and order-preservation. For each c ∈ CN , each j ∈ {2, . . . , k}, andeach i ∈ N such that si = s[j], u(ϕi(c); ci) ≥
s[j−1]n .
Theorem 8. Let k ∈ {2, . . . , n}. Among the mechanisms satisfying assignment-efficiency,strategy-proofness, k-fairness, and order-preservation, a mechanism minimizes the deficitif and only if it belongs to F2.
15
References
Alkan A, Demange G, Gale D (1991): Fair allocation of indivisible goods and criteria ofjustice. Econometrica 59:1023-1039.
Clarke EH (1971): Multi-part pricing of public goods. Public Choice 11:17-33.
Foley D (1967): Resource allocation and public sector. Yale Economic Essays 7:45-98.
Green J, Laffont JJ (1979): Incentives in public decision making. North-Holland, Ams-terdam.
Groves T (1973): Incentives in teams. Econometrica 41:617-631.
Holmström B (1979): Groves’ scheme on restricted domains. Econometrica 47:1137-1144.
Pápai S (2004): Groves sealed bid auctions of heterogeneous objects with fair prices.Social Choice and Welfare 20:371-385.
Porter R, Shoham Y, Tennenholtz M (2004): Fair imposition. Journal of EconomicTheory 118:209—228.
Rawls J (1971): A Theory of Justice. Belknap Press. Cambridge, MA.
Schummer (2000): Eliciting preferences to assign position and compensation. Games andEconomic Behavior 30:293-318.
Tadenuma K, Thomson W (1991): No-envy and consistency in economies with indivisiblegoods. Econometrica 59:1755-1767.
Thomson W (2005): Strategy-proof allocation rules on economic domains. Mimeo, Uni-versity of Rochester.
Walker M (1980): On the nonexistence of a dominant strategy mechanism for makingoptimal public decisions. Econometrica 48:1521-1540.
16
This appendix is not intended for publication and is meant for the referees only. Itcontains the proofs of the results in Sections 3 and 4.
Appendix
Proof of Theorem 4: Let k ∈ {2, . . . , n}. It is easy to show that the mechanisms inFk satisfy the properties listed in Theorem 4. Now, let ϕ satisfy these properties. ByLemma 1, ϕ is a Groves mechanism associated with some h = (hi)i∈N ∈ H and someτ ∈ T . Let ϕ = (Ah,τ , th,τ ). Note that for each c ∈ C and each i ∈ N , α ∈ Ah,τ
i (c) if andonly if i = τ(c, α), and
th,τi (c) = −Xj 6=i
Xα∈Ah,τ
j (c)
cαj + hi(c−i). (17)
Claim 1: For each i ∈ N and each c−i ∈ C−i,
hi(c−i) ≥ −1
n
Xα∈Λ
cα−i[k−1] +Xj 6=i
Xα∈Ah,τ
j (c−i)
cαj .
Proof: Assume, by contradiction, that for some i ∈ N and some c−i ∈ C−i,
hi(c−i) < −1
n
Xα∈Λ
cα−i[k−1] +Xj 6=i
Xα∈Ah,τ
j (c−i)
cαj (18)
By k-fairness, for each ci ∈ Ci,
−Xj
Xα∈Ah,τj (c)
cαj + hi(c−i) ≥ −1
n
Xα∈Λ
cα[k]. (19)
Inequalities (18) and (19) together imply
Xj 6=i
Xα∈Ah,τj (c−i)
cαj −Xj
Xα∈Ah,τ
j (c)
cαj +1
n
"Xα∈Λ
cα[k] −Xα∈Λ
cα−i[k−1]
#> 0. (20)
Let, for each α ∈ Λ, cαi = cα−i[1]. Then,
Xj 6=i
Xα∈Ah,τj (c−i)
cαj −Xj
Xα∈Ah,τ
j (c)
cαj +1
n
"Xα∈Λ
cα[k] −Xα∈Λ
cα−i[k−1]
#= 0.
This equality contradicts (20).Claim 2: For each i ∈ N and each c−i ∈ C−i,
hi(c−i) = −1
n
Xα∈Λ
cα−i[k−1] +Xj 6=i
Xα∈Ah,τ
j (c−i)
cαj .
17
Proof: Assume, by contradiction, that for some i ∈ N and some c−i ∈ C−i,
hi(c−i) 6= −1
n
Xα∈Λ
cα−i[k−1] +Xj 6=i
Xα∈Ah,τ
j (c−i)
cαj .
Then, by Claim 1,
Xi∈N
hi(c−i) >Xi∈N
⎡⎢⎣− 1n
Xα∈Λ
cα−i[k−1] +Xj 6=i
Xα∈Ah,τ
j (c−i)
cαj
⎤⎥⎦ . (21)
Note that
Xi∈N
⎡⎢⎣− 1n
Xα∈Λ
cα−i[k−1] +Xj 6=i
Xα∈Ah,τ
j (c−i)
cαj
⎤⎥⎦ =Xα∈Λ
∙−n− k + 1
ncα[k−1] (22)
−k − 1n
cα[k] + (n− 1)cα[1] + cα[2]
¸.
Also, by mk-bounded deficit,
−(n− 1)Xα∈Λ
cα[1] +Xi∈N
hi(c−i) ≤Xα∈Λ
(cα[2] − cα[k−1]) (23)
Then, (21), (22), and (23) together imply
−n− k + 1
n
Xα∈Λ
cα[k−1] −k − 1n
Xα∈Λ
cα[k] +Xα∈Λ
cα[2] <Xα∈Λ
(cα[2] − cα[k−1]). (24)
Now, let, for each α ∈ Λ, cαi = cα−i[k−1]. Then,
n− k + 1
n
Xα∈Λ
cα[k−1] +k − 1n
Xα∈Λ
cα[k] =Xα∈Λ
cα[k−1].
This equality contradicts (24).By Claim 2, for each i ∈ N and each c−i ∈ C−i,
hi(c−i) = −1
n
Xα∈Λ
cα−i[k−1] +Xj 6=i
Xα∈Ah,τ
j (c−i)
cαj .
Since Ah,τ = Ak,τ , substituting hi(c−i) into (17) we obtain F k,τ . 2
Proof of Theorem 5: Consider the proof of Theorem 4 for k = 2. If we impose 1-fairnessinstead of 2-fairness, inequality (20) becomes
Xj 6=i
Xα∈Ah,τ
j (c−i)
cαj −Xj
Xα∈Ah,τj (c)
cαj +1
n
"Xα∈Λ
cα[1] −Xα∈Λ
cα−i[1]
#> 0. (25)
18
Similarly as in the proof of Theorem 4, let, for each α ∈ Λ, cαi = cα−i[1]. Then,
Xj 6=i
Xα∈Ah,τ
j (c−i)
cαj −Xj
Xα∈Ah,τj (c)
cαj +1
n
"Xα∈Λ
cα[1] −Xα∈Λ
cα−i[1]
#= 0,
which contradicts (25). The rest of the proof is identical to that of the proof of Theorem4. 2
Proof of Theorem 6: Let k ∈ {2, . . . , n}. It is easy to show that the mechanisms in Fk
satisfy the properties listed in Theorem 6. Now, let ϕ satisfy these properties. By Lemma1, ϕ is a Groves mechanism associated with some h = (hi)i∈N ∈ H and some τ ∈ T . Letϕ = (ah,τ , th,τ ). Note that for each c ∈ CN and each i ∈ N , ah,τi (c) = q if and only if(i, q) = τ(c), and
th,τi (c) =
⎧⎪⎨⎪⎩(n− 1)b(q) + hi(c−i) if ah,τi (c) = q,
(n− 1)b(q)− cj + hi(c−i) if ah,τi (c) = ∅, andah,τj (c) = q for some j ∈ N\{i}.
(26)
Claim 1: For each i ∈ N and each c−i ∈ CN\{i}, hi(c−i) ≥(s−i)[k−1]
n − (s−i)[1].Proof: Assume, by contradiction, that for some i ∈ N and some c−i ∈ CN\{i},
hi(c−i) <(s−i)[k−1]
n− (s−i)[1]. (27)
By k-fairness, for each ci ∈ C,s[1] + hi(c−i) ≥
s[k]
n.
This inequality and (27) together imply
s[1] − (s−i)[1] +(s−i)[k−1] − s[k]
n> 0. (28)
Let ci ∈ C be such that si = (s−i)[1]. Then,
s[1] − (s−i)[1] +(s−i)[k−1] − s[k]
n= 0.
This equality contradicts (28).
Claim 2: For each i ∈ N and each c−i ∈ CN\{i}, hi(c−i) =(s−i)[k−1]
n − (s−i)[1].Proof: Assume, by contradiction, that for some i ∈ N and some c−i ∈ CN\{i}, hi(c−i) 6=(s−i)[k−1]
n − (s−i)[1]. Then, by Claim 1,
Xi∈N
hi(c−i) >Xi∈N
∙(s−i)[k−1]
n− (s−i)[1]
¸. (29)
Note thatXi∈N
∙(s−i)[k−1]
n− (s−i)[1]
¸=
n− k + 1
ns[k−1] +
k − 1n
s[k] − (n− 1)s[1] − s[2]. (30)
19
Also, by mk-bounded deficit,
(n− 1)s[1] +Xi∈N
hi(c−i) ≤ s[k−1] − s[2]. (31)
Then, (29), (30), and (31) together imply
n− k + 1
ns[k−1] +
k − 1n
s[k] − s[2] < s[k−1] − s[2]. (32)
Now, let ci ∈ C be such that si = (s−i)[k−1]. Then,
n− k + 1
ns[k−1] +
k − 1n
s[k] = s[k−1].
This equality contradicts (32).
By Claim 2, for each i ∈ N and each c−i ∈ CN\{i}, hi(c−i) =(s−i)[k−1]
n − (s−i)[1]. Sinceah,τ = ak,τ , substituting hi(c−i) into (26) we obtain F k,τ . 2
Proof of Theorem 7: Consider the proof of Theorem 6 for k = 2. If we impose 1-fairnessinstead of 2-fairness, inequality (28) becomes
s[1] − (s−i)[1] +(s−i)[1] − s[1]
n> 0. (33)
Similarly as in the proof of Theorem 6, let ci ∈ C be such that si = (s−i)[1]. Then,
s[1] − (s−i)[1] +(s−i)[1]−s[1]
n = 0, which contradicts (33). The rest of the proof is identicalto that of the proof of Theorem 6. 2
Proof of Lemma 3: Let ϕ satisfy the first four properties listed in Lemma 3. By Lemma1, ϕ is a Groves mechanism associated with some h = (hi)i∈N ∈ H and some τ ∈ T . Letϕ = (ah,τ , th,τ ). We proceed by induction on j ∈ {2, . . . , k}.Base step: For each c ∈ CN and each i ∈ N such that si = s[k],
u(ϕi(c); ci) ≥s[k−1]n
.
To prove this assume, by contradiction, that for some c ∈ CN and some i ∈ N such thatsi = s[k],
u(ϕi(c); ci) <s[k−1]n
. (34)
Let bc ∈ CN be such that for each l ∈ N\{i}, bcl = cl, and bsi = s[k−1] where bsi = maxq∈Q
nb(q)−bci(q). Note that u(ϕi(bc);bci) = bs[1] + hi(bc−i). Since bs[1] = s[1] and hi(bc−i) = hi(c−i), then
u(ϕi(bc);bci) = u(ϕi(c); ci). (35)
Note that bs[k] = s[k−1]. This equality, (34), and (35) together imply u(ϕi(bc);bci) < bs[k]n .
This inequality contradicts k-fairness.Induction step: Suppose that for each c ∈ CN , each j ∈ {r + 1, . . . , k}, and each i ∈ Nsuch that si = s[j], u(ϕi(c); ci) ≥
s[j−1]n . We want to show that for each c ∈ CN and each
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i ∈ N such that si = s[r], u(ϕi(c); ci) ≥s[r−1]n . To prove this assume, by contradiction,
that for some c ∈ CN and some i ∈ N such that si = s[r],
u(ϕi(c); ci) <s[r−1]n
. (36)
Let bc ∈ CN be such that for each l ∈ N\{i}, bcl = cl, and bsi = s[r−1] where bsi =maxq∈Q
{nb(q)− bci(q)}. As in Base step, we haveu(ϕi(bc);bci) = u(ϕi(c); ci). (37)
Note that bs[r] = s[r−1]. This equality, (36), and (37) together imply u(ϕi(bc);bci) < bs[r]n .
Let z ∈ N\{i} be such that bsz = bs[r+1]. By order-preservation, u(ϕz(bc);bcz) ≤ u(ϕi(bc);bci).Thus, u(ϕz(bc);bcz) < bs[r]
n . This inequality contradicts the induction hypothesis. 2
Proof of Theorem 8: We already know that the mechanisms in Fk satisfy the fourproperties listed in Theorem 8. Let ϕ be a mechanism that satisfies these properties. ByLemma 1, ϕ is a Groves mechanism associated with some h = (hi)i∈N ∈ H and someτ ∈ T . Let ϕ = (ah,τ , th,τ ). Let c ∈ CN and i ∈ N .Claim 1: If ah,τi (c) = ∅, then th,τi (c) ≥ s[1]
n − b(q) where ah,τj (c) = q for some j ∈ N\{i}.Proof: Assume, by contradiction, that ah,τi (c) = ∅, and th,τi (c) <
s[1]n −b(q) where a
h,τj (c) =
q for some j ∈ N\{i}. This inequality is equivalent to
u(ϕi(c); ci) <s[1]
n. (38)
Let bc ∈ CN be such that for each l ∈ N\{i}, bcl = cl, and bsi = s[2] where bsi = maxq∈Q
nb(q)−bci(q). By the same argument as in the proof of Lemma 2, we obtain (35). Let j ∈ N besuch that bsj = bs[2] and ah,τj (bc) = ∅. By order-preservation, u(ϕj(bc);bcj) = u(ϕi(bc);bci). Thisequality and (35) together imply u(ϕj(bc);bcj) = u(ϕi(c); ci). By Lemma 3, u(ϕj(bc);bcj) ≥bs[1]n . Note that bs[1] = s[1]. Hence, u(ϕi(c); ci) ≥
s[1]n , which contradicts (38).
Claim 2: If ah,τi (c) = q, then th,τi (c) ≥ s[2]n − s[2] + (n− 1)b(q).
Proof: Assume, by contradiction, that ah,τi (c) = q, and
th,τi (c) <s[2]
n− s[2] + (n− 1)b(q). (39)
Let bc ∈ CN be such that for each l ∈ N\{i}, bcl = cl, and bsi = s[2] where bsi = maxq∈Q
nb(q)−bci(q). Note that u(ϕi(bc);bci) = bs[1] + hi(bc−i). Since, bs[1] = s[2] and hi(bc−i) = hi(c−i), then
u(ϕi(bc);bci) = s[2] + hi(c−i). (40)
Note that th,τi (c) = hi(c−i) + (n − 1)b(q). This equality, (39), (40), and the fact thatbs[2] = s[2] together imply u(ϕi(bc);bci) < bs[2]n . Using Claim 1, for each j ∈ N\{i} such
that ah,τj (bc) = ∅, u(ϕj(bc);bcj) ≥ bs[2]n . Hence, u(ϕi(bc);bci) < u(ϕj(bc);bcj), which contradicts
order-preservation.If the inequalities in Claim 1 and Claim 2 hold as equalities, the deficit is minimized. Thetransfers specified in these equalities correspond to those of a mechanism in F2. 2
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