Borm, P.E.M.

Fernández, C.

Hendrickx, R.L.P.

Tijs, S.H.

English

Research Papers in Economics

Drop Out Monotonic Rules for Sequencing
Situations
Cristina Fern´ andez1 Peter Borm2 Ruud Hendrickx2;3
Stef Tijs2
Abstract
This note introduces a new monotonicity property for sequencing situa-
tions. A sequencing rule is called drop out monotonic if no player will be
worse oﬀ whenever one of the players decides to drop out of the queue before
processing starts. This intuitively appealing property turns out to be very
strong: we show that there is at most one rule satisfying both stability and
drop out monotonicity. For the standard model of linear cost functions, the
existence of this rule is established.
1 Introduction
For various classes of economic situations in which agents can cooperate, allocation
rules have been developed, which handle the problem of allocating the rewards
or cost savings from cooperation among the agents involved. Often, for such an
economic situation, one can deﬁne a corresponding cooperative game. In this setting,
stable rules are interesting, which assign allocations that are core elements of the
corresponding games.
In this note, we consider stable rules, which behave well in case agents drop out,
leaving a reduced economic situation. We say that a rule is drop out monotonic if
applying the rule to the reduced situation yields an allocation, which, depending
on the context, either makes all remaining players better oﬀ or all players worse oﬀ
than in the original situation.
1Facultad de Ciencias Econ´ omicas y Empresariales, Departamento de Economia Aplicada I,
Universidad de Sevilla, Av. Ram´ on y Cajal, n±1. This author acknowledges ﬁnancial support
through DGICYT grant BFM 2001-2378.
2CentER and Department of Econometrics and Operations Research, Tilburg University, PO
Box 90153, 5000 LE Tilburg, The Netherlands.
3Corresponding author. E-mail: ruud@uvt.nl.
1If the cooperative games corresponding to reduced situations are subgames of the
the original game, then a stable and drop out monotonic rule generates a popula-
tion monotonic allocation scheme (pmas) for the original game (Sprumont (1990)).
In the cases of linear production situations (Owen (1975)), airport situations (Lit-
tlechild and Owen (1973)) and holding situations (Tijs et al. (2000)), the game
corresponding to a reduced situation after one player drops out is a subgame of the
original game. So here, the existence of stable and monotonic rules boils down to the
existence of a pmas. Such pmas-es do not always exist for linear production games.
However, for airport situations, the Shapley value induces one of many stable and
drop out monotonic rules. For holding games, the rule which gives all gains to the
so-called holding house keeper is a pmas.
The property of drop out monotonicity introduced in this note is inspired by the
so-called fairness condition introduced in Ambec and Sprumont (2002). They study
the problem of water management from a game theoretical point of view: given a
river of certain capacity ﬂowing through a number of countries with certain demand
for water, how should the water of the river be allocated?
The fairness condition states that whenever one of the countries ceases to demand
water (drops out), all other countries should be better oﬀ. Contrary to the examples
mentioned before, the reduced situation after a player drops out does not give rise to
a subgame of the original game. Ambec and Sprumont show that there is a unique
allocation rule which satisﬁes both stability (ie, generates a core element) and the
fairness condition. This rule (the ¹ rule) is the marginal vector corresponding to
the ordering of the countries along the river (from upstream to downstream).
This note studies the drop out monotonicity property in the context of sequencing
situations, as introduced in Curiel et al. (1989), in which there is also a natural
ordering of the players forming the initial queue. Indeed, in the most basic class
of sequencing situations (with linear cost functions), a result similar to Ambec and
Sprumont is established. Within a more general class of sequencing situations (with
regular cost functions), it turns out that there is at most one stable and drop out
monotonic rule, which must be the analogue of the ¹ rule.
22 Sequencing situations
In this section, we brieﬂy review the model of sequencing situations as introduced in
Curiel et al. (1989). A sequencing situation is a triple (N;p;k), where N = f1;:::;ng
is a ﬁnite set of players waiting in front of a machine in order to have a job processed,
p 2 RN
++ is a vector of processing times, where pi > 0 represents the time that the
job of player i 2 N requires to be processed, and k = (ki)i2N is a vector of cost
functions. For each player i 2 N, ki(t) represents the costs of player i if his job
is completed in t time units. Costs are assumed to be additive: the total costs
of a coalition S ½ N equal the sum of the individual costs of the members of S.
Furthermore, the cost functions are regular, ie, for all i 2 N, ki(t) is increasing in t
and ki(0) = 0.
We pay special attention to the class of sequencing situations with linear cost
functions: ki(t) = ®it for all t 2 R+ with ®i ¸ 0. A sequencing situation with linear
cost functions is denoted by (N;p;®) with ® = (®i)i2N.
In a sequencing situation, there is an initial ordering on the players in the queue,
which without loss of generality we assume to put player 1 at the front and player n
at the back. By swapping adjacent places in the queue, players are allowed to save
costs (cf. Curiel et al. (1989)). As only neighbours can switch, the cost savings of a
coalition of players equals the sum of the savings made by its connected components.
In this note, we consider the resulting sequencing cost game (N;c), where for a
coalition S ½ N, c(S) is deﬁned as the minimal total costs of the members of S over
all their admissible rearrangements in the queue.1
The core of a cost game (N;c) is deﬁned by
C(c) = fx 2 R
N j
X
i2N
xi = c(N);8S½N :
X
i2S
xi · c(S)g:
Core elements are stable in the sense that if such a vector is proposed as cost
allocation for the grand coalition, no coalition will have an incentive to split oﬀ and
cooperate on their own.
A sequencing rule is a function f assigning to every sequencing situation (N;p;k)
a vector f(N;p;k) 2 RN
+ such that
P
i2N fi(N;p;k) = c(N). A rule f is called
stable if ¹(N;p;k) 2 C(c) for every sequencing situation (N;p;k). In this note, we
investigate the ¹ rule, deﬁned by
1Curiel et al. (1989) consider the related cost savings game. The deﬁnition of the cost game is
analogous and omitted for the sake of brevity.
3¹j(N;p;k) = c(Pj) ¡ c(Pj¡1)
for all j 2 N, where Pj = f1;:::;jg. In case the cost functions are linear, we can
rewrite this as
¹j(N;p;®) = c(fjg) ¡
X
i2N:i<j
gij:;
where gij = maxf0;®jpi ¡ ®ipjg equals the cost savings attainable by player i and
j when i is directly in front of j, regardless of the exact position in the order.
According to the ¹ rule, the gain gij goes fully to player j, who is behind i in the
queue.
Since every sequencing game is ¾-component additive (cf. Curiel et al. (1995)),
the ¹ rule is stable. So letting the players at the front of the queue pay the highest
costs and attributing the gains to the players at the back of the queue results in a
stable outcome.
3 Drop out monotonicity
Suppose that one player in the queue decides to wait no longer and drops out. One
natural question in this situation is how the costs of the other players will be aﬀected
by this. It seems natural that none of the players should be worse oﬀ if one of them
drops out of the queue. Formally, a rule f is called drop out monotonic if for all
sequencing situations (N;p;k) and all q 2 N we have
fj(N;p;k) ¸ fj((N;p;k)
¡q)
for all j 2 Nnfqg, where (N;p;k)¡q = (Nnfqg, (pi)i2Nnfqg, (ki)i2Nnfqg) is the reduced
situation without player q, in which the initial ordering on the remaining players is
the same as in the original situation.
Proposition 3.1 ¹ is drop out monotonic on the class of sequencing situations with
linear cost functions.
Proof: Let (N;p;®) be a sequencing situation with linear cost functions, let
q 2 N and let j 2 Nnfqg. If j < q, then ¹j(N;p;®) = c(fjg) ¡
P
i2N:i<j gij =
¹j((N;p;®)¡q). If j > q, then ¹j((N;p;®)¡q) = (
Pj
i=1 pi ¡ pq)®j ¡
P
i2N:i<j gij +
gqj = ((
Pj
i=1 pi)®j ¡
P
i2N:i<j gij) + (gqj ¡ pq®j) = ¹j(N;p;®) ¡ minfpj®q;pq®jg ·
¹j(N;p;®). ¤
4Proposition 3.1 shows that the ¹ rule is drop out monotonic in case the cost functions
are linear. The question now arises whether this is the only rule satisfying this
property. In the following theorem, we show that within the class of sequencing
situations with regular cost functions (not necessarily linear), the ¹ rule is the only
possible stable and drop out monotonic rule.
Theorem 3.2 Let f be a sequencing rule. If f is stable and drop out monotonic,
then f equals the ¹ rule.
Proof: Let (N;p;k) be a sequencing situation with regular cost functions and let
f be a stable and drop out monotonic rule. Denote the corresponding game by
(N;c) and denote fS
i = fi(S;(pj)j2S;(kj)j2S) and ¹i = ¹i(N;p;k) for all i 2 N and
S ½ N;S 6= ;. We show that f = ¹ by an inductive argument.
First, from drop out monotonicity it follows that fN
1 ¸ f
f1g
1 . From stability we have
fN
1 · c(f1g) = f
f1g
1 . Hence, fN
1 = f
f1g
1 = c(f1g) = ¹1.
Next, let j 2 f2;:::ng. Assume fN
i = f
Pj¡1
i = ¹i for all i 2 Pj¡1. From drop out
monotonicity we have fN
i ¸ f
Pj
i for all i 2 Pj, so
P
i2Pj fN
i ¸
P
i2Pj f
Pj
i = c(Pj).
By stability,
P
i2Pj fN
i · c(Pj). So
P
i2Pj fN
i = c(Pj) and, using the induction
hypothesis, fN
j = c(Pj) ¡
P
i2Pj¡1 fN
i = c(Pj) ¡ c(Pj¡1) = ¹j.
Hence, we may conclude that f = fN = ¹. ¤
It follows from Proposition 3.1 and Theorem 3.2 that drop out monotonicity and
stability together characterise the ¹ rule on the class of sequencing situations with
linear cost functions.
References
Ambec, S. and Y. Sprumont (2002). Sharing a river. Journal of Economic The-
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Curiel, I., G. Pederzoli, and S. Tijs (1989). Sequencing games. European Journal of
Operational Research, 40, 344–351.
Curiel, I., J. Potters, V. Rajendra Prasad, S. Tijs, and B. Veltman (1995). Sequenc-
ing and cooperation. Operations Research, 42, 566–568.
Littlechild, S. and G. Owen (1973). A simple expression for the Shapley value in a
special case. Management Science, 20, 370–372.
5Owen, G. (1975). On the core of linear production games. Mathematical Program-
ming, 9, 358–370.
Sprumont, Y. (1990). Population monotonic allocation schemes for cooperative
games with transferable utility. Games and Economic Behavior, 2, 378–394.
Tijs, S., A. Meca, and M. L´ opez (2000). Beneﬁt sharing in holding situations. CIO
Discussion Paper I-2000-01, Universidad Miguel Hern´ andez, Elche, Spain. (To
appear in European Journal of Operational Research).
6

A simple expression for the Shapley value in a special case.

Beneﬁt sharing in holding situations.

On the core of linear production games.

Population monotonic allocation schemes for cooperative games with transferable utility.

Sequencing and cooperation.

Sequencing games.

Sharing a river.

Drop out monotonic rules for sequencing situations.

Drop out monotonic rules for sequencing situations.

Abstract

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