A set of agents with different waiting costs have to receive a service of different length of time from a single provider which can serve only one agent at a time. One needs to form a queue and set up monetary transfers to compensate the agents who have to wait. We prove that no rule satisfies efficiency of queues and coalitional strategy-proofness.mathematical economics;

Kayi, Ãağatay

Ramaekers, Eve

English

Research Papers in Economics

Çağatay Kayı, Eve Ramaekers 
 
An Impossibility in Sequencing Problems 
 
RM/08/040 
 
JEL code: D63, C72 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Maastricht research school of Economics 
of TEchnology and ORganizations 
 
Universiteit Maastricht 
Faculty of Economics and Business Administration 
P.O. Box 616 
NL - 6200 MD Maastricht 
 
phone  : ++31 43 388 3830 
fax  : ++31 43 388 4873 
 
 
 
 An Impossibility in Sequencing Problems∗
C ¸a˘ gatay Kayı† and Eve Ramaekers‡
Abstract
A set of agents with diﬀerent waiting costs have to receive a service of diﬀerent length of
time from a single provider which can serve only one agent at a time. One needs to form a
queue and set up monetary transfers to compensate the agents who have to wait. We prove
that no rule satisﬁes eﬃciency of queues and coalitional strategy-proofness.
JEL Classiﬁcation: D63, C72.
Keywords: Sequencing problems, Pareto-eﬃciency, coalitional strategy-proofness.
1 Introduction
A set of agents simultaneously arrive at a service facility that can only serve one agent at
a time. Agents require service for diﬀerent lengths of time. The waiting cost may vary
from one agent to the other. Each agent is assigned a “consumption bundle” consisting
of a position in the queue and a positive or negative transfer. Each agent has quasi-linear
preferences over positions and transfers. For such a sequencing problem, a rule assigns each
agent a position in the queue and a positive or negative transfer such that no two agents are
assigned the same position, and the sum of the transfers is not positive.
Our objective is to identify rules that are well-behaved from the normative and strategic
viewpoints. The ﬁrst requirement is eﬃciency. It says that if an allocation is selected, there
∗We thank Paulo Barelli, Walter Bossert, Bettina Klaus, Fran¸ cois Maniquet, Manipushpak Mitra, Jay
Sethuraman, William Thomson, G´ abor Vir´ ag, and seminar participants at the University of Maastricht,
Social Choice and Welfare Conference 2006, University of Rochester for their helpful discussions and sugges-
tions.
†Maastricht University, Department of Economics, 6200 MD Maastricht, The Netherlands. E-mail:
C.Kayi@algec.unimaas.nl
‡Center for Operations Research and Econometrics (CORE)Universit catholique de Louvain (UCL) Voie
du Roman Pays, 34 B-1348 Louvain-la-Neuve Belgium E-mail: Eve.Ramaekers@uclouvain.be
1should be no other feasible allocation that each agent ﬁnds at least as desirable and at least
one agent prefers. Since preferences are linear, Pareto-eﬃciency can be decomposed into two
axioms: on the one hand, eﬃciency of queues, which says that a queue should minimize the
total waiting cost, and on the other hand, balancedness, which says that transfers should sum
up to zero. Second is a minimal fairness requirement, no-envy, which requires that no agent
should prefer another agent’s assignment to her own. No-envy implies eﬃciency of queues.
Third is immunity to strategic behavior. As unit waiting costs may not be known, the
rule should provide agents the incentive to reveal these costs truthfully. Strategy-proofness
requires that each agent should ﬁnd her assignment when she truthfully reveals her unit
waiting cost at least as desirable as her assignment when she misrepresents it.1 We are
also concerned about possible manipulations by groups, and consider coalitional strategy-
proofness: no group of agents should make each of its members at least as well oﬀ, and at
least one of them better oﬀ, by jointly misrepresenting their waiting costs. Finally is non-
bossiness: if an agent’s change in her announcement does not aﬀect her assignment, then it
should not aﬀect any other agent’s assignment.
For sequencing problems, Pareto-eﬃciency and strategy-proofness are compatible (Suijs,
1996). For the subdomain of sequencing problems in which agents require service for the
same length of time2, Pareto-eﬃciency, equal treatment of equals in welfare, and strategy-
proofness are compatible. (Kayı and Ramaekers, 2007). We show that if we impose the
stronger incentive property of coalitional strategy-proofness with eﬃciency of queues,w eh a v e
an impossibility result. Independently, Mutuswami and Mitra (2006) show that coalitional
strategy-proofness, eﬃciency of queues and feasibility–which says that transfers should sum
up less or equal to zero– are not compatible. In our proof, we make use of non-bossiness
and get some corollaries: We show that no rule satisﬁes Pareto-eﬃciency and coalitional
strategy-proofness, no rule satisﬁes Pareto-eﬃciency, non-bossiness, and strategy-proofness,
and no rule satisﬁes no-envy, non-bossiness, and strategy-proofness.
In Section 2, we formally introduce the model. In Section 3, we deﬁne the properties on
rules. In Section 4, we give the impossibility result.
1For an extensive survey on strategy-proofness, see Thomson, 2006.
2For queueing problems, see Mitra and Sen, 1998; Maniquet, 2003; Katta and Sethuraman, 2006; Chun
2006.
22 Model
There is a ﬁnite set of agents N. Let |N| = n ≥ 2. For each agent i ∈ N, let ri ∈ R++ be the
servicing time and ci ∈ R+ be the unit waiting cost of i ∈ N. Let r =( ri)i∈N ∈ RN
++ and
c =( ci)i∈N ∈ RN
+ be the lists of the servicing times and unit waiting costs respectively. A
queue is a bijection σ : N →{ 1,2,...,n}. Let Q be the set of all queues. For each agent i ∈ N,
we will denote σ(i)a sσi. Each agent i ∈ N has to be assigned a position σi ∈{ 1,2,...,n}
in a queue σ and may receive a positive or negative monetary transfer ti ∈ R. Preferences
are linear over X ≡{ 1,2,...,n}×R. For each queue σ ∈Qand each i ∈ N, let Pi(σ)
be the set of agents preceding agent i in queue σ, i.e., Pi(σ) ≡{ j ∈ N|σj <σ i}.I f i is
served σi-th, her total waiting cost is

l∈Pi(σ) rlci. Her preferences can be represented by
the function ui deﬁned as follows: for each (σi,t i) ∈ X, ui(σi,t i)=−

l∈Pi(σ) rlci + ti.
We use the following notational shortcut. If her waiting cost is c 
i, then her preferences are
represented by the function u 
i, deﬁned by u 
i(σi,t i)=−

l∈Pi(σ) rlc 
i + ti;i fi ti s˜ ci, then
we use ˜ ui(σi,t i)=−

l∈Pi(σ) rl˜ ci + ti, and so on. A sequencing problem is deﬁned as a list
s ≡ (ri,c i)i∈N ∈ RN
++ × RN
+. Let S≡RN
++ × RN
+ be the class of all problems.
An allocation for s ∈Sis a pair (σ,t) ≡ (σi,t i)i∈N ∈ XN. An allocation (σ,t) ∈ XN
is feasible for s ∈Sif no two agents are assigned the same position in σ, (i.e., for each
{i,j}⊆N with i  = j,w eh a v eσi  = σj), and the sum of the coordinates of t is non-positive,
(i.e.,

i∈N ti ≤ 0). Let Z(s) be the set of all feasible allocations for s ∈S . An (allocation)
rule ϕ is a function that associates with each problem s ∈Sa feasible allocation ϕ(s) ∈ Z(s).
Given s ∈Sand S ⊆ N, rS ≡ (cl)l∈S and cS ≡ (cl)l∈S are restrictions of servicing times r
and unit waiting costs c to S respectively. Given i ∈ N, r−i ≡ (rl)l∈N\{i} and c−i ≡ (cl)l∈N\{i}
are the restrictions of r and c to N\{i} respectively.
3 Properties of rules
In this section, we deﬁne properties of rules. Let ϕ be a rule. First, if an allocation is selected,
there should be no other feasible allocation that each agent ﬁnds at least as desirable and
at least one agent prefers.
Pareto-eﬃciency: For each s ∈Sand each (σ,t) ∈ ϕ(s), there is no (σ ,t  ) ∈ Z(s) such
that for each i ∈ N, ui(σ 
i,t  
i) ≥ ui(σi,t i) and for at least one j ∈ N, uj(σ 
j,t  
j) >u j(σj,t j).
3If an allocation is Pareto-eﬃcient for s, any other allocation at which the queue is the
same is also Pareto-eﬃcient. Therefore, it is meaningful to speak of the eﬃciency of queues.
It requires to minimize the total waiting cost. Thus, an allocation (σ,t)i sPareto-eﬃcient
for s if and only if for each σ  ∈Q , we have

i∈N

l∈Pi(σ) rlci ≥

i∈N

l∈Pi(σ) rlci, i.e., σ
is eﬃcient for s and

i∈N ti = 0, i.e., t is balanced for s. Let Q∗(s) be the set of all eﬃcient
queues for s. For each s ∈Sand each (σ,t) ∈ Z(s), we have σ ∈Q ∗(s) if and only if for
each {i,j}⊂N with i  = j,i fσi <σ j, then ci/ri ≥ cj/rj. (Smith, 1956) For simplicity,
throughout the paper, we will assume that agents have unequal ratios of unit waiting costs
and servicing time. Therefore, there is only one eﬃcient queue.
Summarizing the discussion above, Pareto-eﬃciency can be decomposed into two axioms:
Eﬃciency of queues: For each s ∈Sand each (σ,t) ∈ ϕ(s), we have σ ∈ Q∗(s).
Balancedness: For each s ∈Sand each (σ,t) ∈ ϕ(s), we have

i∈N ti =0 .
The next requirement is necessary for no agent to prefer another agent’s assignment to her
own.
No-envy: For each s ∈S , each (σ,t) ∈ ϕ(s), and each i ∈ N, there is no j ∈ N\{i} such
that ui(σj,t j) >u i(σi,t i).
The last requirements are motivated by strategic considerations. The planner may not
know the agents’ cost parameters. If agents behave strategically when announcing them,
neither eﬃciency nor equity may be attained. Thus, we require that each agent should ﬁnd
her assignment when she truthfully reveals her unit waiting cost at least as desirable as her
assignment when she misrepresents it.
Strategy-proofness: For each s ∈S , each i ∈ N, and each c 
i ∈ R+,i f(σ,t)=ϕ(s) and
(σ ,t  )=ϕ(r,c 
i,c −i), then ui(σi,t i) ≥ ui(σ 
i,t  
i).
We also consider the requirement that no group of agents should be able to make each of its
members at least as well oﬀ, and at least one of them better oﬀ, by jointly misrepresenting
its members’ waiting costs.
Coalitional strategy-proofness: For each s ∈Sand each S ⊆ N, there is no c 
S ∈ RS
+
such that if (σ,t)=ϕ(s) and (σ ,t  )=ϕ(r,c 
S,c N\S), then for each i ∈ S, we have ui(σ 
i,t  
i) ≥
ui(σi,t i) and for some j ∈ S, we have uj(σ 
j,t  
j) >u j(σj,t j).
The next requirement is that if an agent’s change in her announcement does not aﬀect her
assignment, then it should not aﬀect any other agent’s assignment.
4Non-bossiness: For each s ∈S , each i ∈ N, and each c 
i ∈ R+,i fϕi(s)=ϕi(r,c 
i,c −i),
then ϕ(s)=ϕ(r,c 
i,c −i).
4 Result
First, we establish a relationship between eﬃciency of queues and coalitional strategy-
proofness, and non-bossiness.
Proposition 1. Let ϕ be a rule satisfying eﬃciency of queues and coalitional strategy-
proofness, then it satisﬁes non-bossiness.
Proof. Let ϕ be a rule satisfying the axioms of Proposition 1. We need to show that for each
s ∈S , each i ∈ N, each c 
i ∈R +,i fz = ϕ(s) and z  = ϕ(r,(c 
i,c −i)) are such that zi = z 
i,
then z = z . Indeed, let s =( r,c) ∈S , i ∈ N, c 
i ∈R +, z = ϕ(s), and z  = ϕ(r,(c 
i,c −i))
be such that zi = z 
i. By contradiction, suppose that there is j ∈ N such that zj  = z 
j.
Then, since zi = z 
i,w eh a v eui(σ,ti)=ui(σ ,t  
i). By eﬃciency of queues, σj = σ 
j. Since
zj  = z 
j,w eh a v etj  = t 
j. First, suppose tj >t  
j. Then, uj(σ,tj) >u j(σ ,t  
j) and there
is (c 
i,c j) ∈R
{i,j}
+ such that ui(σ,ti)=ui(σ ,t  
i) and uj(σ,tj) >u j(σ ,t  
j), contradicting
coalitional strategy-proofness. Second, suppose tj <t  
j. Then, uj(σ,tj) <u j(σ ,t  
j) and
there is (ci,c j) ∈R
{i,j}
+ such that u 
i(σ,ti)=u 
i(σ ,t  
i) and uj(σ ,t  
j) >u j(σ,tj), contradicting
coalitional strategy-proofness.
Then, we prove our main result.
Theorem 1. No rule satisﬁes eﬃciency of queues and coalitional strategy-proofness.
Proof. By contradiction, let ϕ be a rule satisfying the axioms of Theorem 1. Then, by
Proposition 1, ϕ satisﬁes non-bossiness. Now, assume that for each i ∈ N,w eh a v eri =1 .
We establish two claims:
Claim 1: For each s ∈S , each i ∈ N, and each c 
i ∈ R+,
if (σ,t)=ϕ(r,c) and (σ ,t  )=ϕ(r,(c 
i,c −i)) are such that σi = σ 
i, then (σ,t)=( σ ,t  ).
Let s ∈S , i ∈ N, c 
i ∈ R+,( σ,t)=ϕ(r,c) and (σ ,t  )=ϕ(r,(c 
i,c −i)) are such that σi = σ 
i.
By strategy-proofness, −

l∈Pi(σ) rlci + ti ≥−

l∈Pi(σ) rlci + t 
i and −

l∈Pi(σ) rlc 
i + ti ≤
−

l∈Pi(σ) rlc 
i+t 
i. By assumption, for each i ∈ N, we have ri = 1. Thus, −(σi−1)ci+ti ≥
5−(σ 
i −1)ci +t 
i and −(σi −1)c 
i +ti ≤− (σ 
i −1)c 
i +t 
i. Thus, as σi = σ 
i,w eh a v eti = t 
i.B y
non-bossiness,( σ,t)=( σ ,t  ).
Claim 2: For each s ∈Ssuch that for each j,k ∈ N, we have cj  = ck if and only if j  = k,
for each i ∈ N, and each c 
i ∈ R+ such that for each j ∈ N\{i},w eh a v ec 
i >c j if and only
if ci >c j,i f( σ,t)=ϕ(s), then (σ,t)=ϕ(r,(c 
i,c −i)).
Let s ∈S , i ∈ N, c 
i ∈ R+ be such that for each j ∈ N\{i},w eh a v ec 
i  = cj and c 
i >c j if
and only if ci >c j, and (σ,t)=ϕ(s), (σ ,t  )=ϕ(r,c 
i,c −i)). By eﬃciency of queues, σ 
i = σi.
Thus, by Claim 1, (σ,t)=ϕ(r,(c 
i,c −i)).
Claims 1 to 2 being proved, we now come to a contradiction. Without loss of generality,
suppose N = {1,2,...,n}. Let {c,c }⊆RN
+ be such that
(i) c1 >c 2 >c 3... > cn,
(ii) c 
2 >c  
1 >c  
3 > ... > c 
n, and
(iii) for each i ∈ N\{1}, c 
i = ci.
Let (σ,t)=ϕ(s) and (σ ,t  )=ϕ(r,c ). By eﬃciency of queues, for each i ∈ N,w eh a v e
σi = i, whereas σ 
1 =2 ,σ 
2 = 1, and for each i ∈ N\{1,2},w eh a v eσi = σ 
i = i. Thus,
(σ,t)  =( σ ,t  ). By strategy-proofness, u1(σ1,t 1)=t1 ≥− c1+t 
1 = u1(σ 
1,t  
1) and u 
1(σ 
1,t  
1)=
−c 
1 + t 
1 ≥ t1 = u 
1(σ1,t 1). That is, t 
1 ∈ [t1 + c 
1,t 1 + c1]. Thus, agent 1’s transfer depends
either on a constant, i.e., t1 = t1 + c with c ∈ [c1,c  
1], or on its own announcement, i.e.,
t1 = t1 + f(c 
1,c 1) with f(c 
1,c 1) ∈ [c1,c  
1]. Clearly, this contradicts strategy-proofness.
The following paragraphs establish examples of rules that satisfy only one of the axioms
in Theorem 1.
(i) Equally Distributed Pairwise Pivotal rule (Kayı and Ramaekers, 2007 and Suijs, 1996)
satisﬁes eﬃciency of queues but not coalitional strategy-proofness.
(ii) Any rule that selects the same arbitrary queue and sets the transfer to each agent equal
to zero satisﬁes coalitional strategy-proofness, but not eﬃciency of queues.
Finally, by using the implication of properties, we have the following corollaries.
Corollary 1.
1. No rule satisﬁes Pareto-eﬃciency and coalitional strategy-proofness.
2. No rule satisﬁes Pareto-eﬃciency, non-bossiness, and strategy-proofness.
63. No rule satisﬁes no-envy, non-bossiness, and strategy-proofness.3
References
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[2] Katta, A. and J. Sethuraman (2006) “Cooperation in Queues,” mimeo, Columbia Uni-
versity, New York, NY, USA.
[3] Kayı, C ¸. and E. Ramaekers (2007) “Characterizations of Pareto-eﬃcient, fair, and
strategy-proof allocation rules in queueing problems,” mimeo, Center for Operations
Research and Econometrics (CORE)Universit catholique de Louvain (UCL).
[4] Maniquet, F. (2003) “A characterization of the Shapley value in queueing problems,”
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31. Since Pareto-eﬃciency implies eﬃciency of queues and by Theorem 1, we have the result. 2. Since the
contradiction at the end of Theorem 1 is to strategy-proofness, the result follow from Proposition 1. 3. Assume
that ϕ satisﬁes no-envy. Let c ∈C ,( σ,t)=ϕ(c), {i,j}⊂N, with i  = j be such that ci >c j but σi >σ j.B y
no-envy,w eh a v eui(σi,t i) ≥ ui(σj,t j) and uj(σj,t j) ≥ uj(σi,t i). Then, (σi−σj)ci+tj ≤ ti ≤ (σi−σj)ci that
contradicts ci >c j. It follows that no-envy implies eﬃciency of queues. The result follows from Theorem 1.
7

A characterization of the Shapley value in queueing problems,”

Characterizations of Pareto-eﬃcient, fair, and strategy-proof allocation rules in queueing problems,” mimeo, Center for Operations Research and Econometrics (CORE)Universit catholique de Louvain (UCL).

Cooperation in Queues,” mimeo,

Dominant strategy implementation of ﬁrst best public decision,” mimeo, Indian Statistical Institute,

Group Strategyproofness in Queueing Models,” Economics Discussion Papers 610,

Mechanism design in queueing problems,”

No-envy in queueing problems,”

On incentive compatibility and budget balancedness in public decision making,”

Strategy-proof resource allocation rules,” mimeo,

An Impossibility in Sequencing Problems

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