Abstract This paper focuses on compensation rules for multi-stage sequencing games.These rules distribute the implicit losses and gains when the players move from one stageof a multi-stage sequencing game to a subsequent stage. They are necessary because al-location rules that yield core elements for the games in each stage are not guaranteed toyield core elements for the multi-stage game. Three approaches to find compensation rulesthat yield core elements are studied. Two compensation rules that lead to stable divisionrules for multi-stage sequencing games are found as a result of these approaches. Axiomaticcharacterizations of these division rules are discussed.
Keywords Multi-stage sequencing game · Compensation rule · Core
1 Introduction
Sequencing games introduced in Curiel et al. (1989) model situations in which customerswaiting in line to be serviced can decrease their total costs by rearranging their positions inthe queue. The costs of each customer depends linearly on his completion time. The worthof a coalition in a sequencing game is defined as the maximal cost savings that the coalitioncan achieve by switching its members without jumping over non-members. The EGS-rule(Equal Gain Splitting rule) was introduced as a rule for allocating the cost savings generatedby moving from the starting order to an optimal order among the players. It was shownthat the allocation generated by the EGS-rule is an element of the core of the sequencinggame. Several modifications and generalizations of the original model have been studied inthe literature. These can be found in Borm et al. (2002), Calleja et al. (2002), Hamers et al.(1995, 1999), and Slikker (2005). An overview is given in Curiel et al. (2002).
Multi-stage sequencing games were introduced in Curiel (2010). These games model co-operation in sequencing situations that consist of several stages that follow each other. The
I. Curiel (B)School of Engineering, University of the Netherlands Antilles, Jan Noorduynweg 111, P.O. Box 3059,Curaçao, Netherlands Antillese-mail: [email protected]
starting position of a customer in the first stage is given, just as in a sequencing situation. Ineach subsequent stage the starting position of a customer is the position he obtained at theend of the preceding stage. For two-stage sequencing games it was shown in Curiel (2010)that applying the EGS-rule to each of the two stages need not yield a core element of thetwo-stage sequencing game. A modification of the EGS-rule called the MEGS-rule (Modi-fied Equal Gain Splitting rule) was defined and it was proven that the allocation given by theMEGS-rule will always be in the core of the two-stage sequencing game. The MEGS-rulecompensates a player who moves backwards in the first stage for the loss of his advanta-geous position on top of allocating the gains made in each stage according to the EGS-rule.In Curiel (2010) it was shown that, in general, allocation rules that yield core outcomes foreach stage separately need not yield stable outcomes for the overall situation. It turned out tobe necessary to consider rules that explicitly take into account compensations for the loss orgain of an advantageous position when switching occurs in a certain stage. This is an exam-ple of the following statement in a paper by Kranich et al. (2005) on dynamic transferableutility games: “Typically, a solution to a dynamic game will exploit intertemporal linkagesrather than simply apply a standard solution at each point in time.” In this paper we wantto focus on the part of an allocation rule that describes how these compensations shouldbe carried out. We will study m-stage sequencing games with m ≥ 2. When m > 2 thereare several multistage sequencing games that can be considered depending on the baselinechosen. That means, depending on how far back a coalition will look when evaluating itsvalue in the m-stage game. We will look at two extreme cases. In the first case the valueof a coalition will be based on the order at the very beginning of the game. In the secondcase the value of a coalition will be based on the order at the start of the stage that precedesthe current stage. For m = 2 the two cases coincide. We will take three approaches in orderto find suitable compensation rules. The first one is to consider likely agreements that twoplayers can make about compensations for the one who gives up an advantageous positionto the other. The second one is to define interstage games that describe the implicit lossesand gains of the players in the subsequent stage when moving from the starting order to anoptimal order in the present state. We will study compensation rules that arise by applyingwell known solution concepts to these interstage games. The third approach considers com-pensation rules that divide the implicit losses and gains generated by switching two playersbetween them. This can be done in several ways. The MEGS-rule from Curiel (2010) willappear as a result of the first approach. The second and the third approach will both lead toanother rule that is guaranteed to yield a core element of the multi-stage game. Furthermore,we will show which solution concepts and division methods are not suitable if stability ofthe resulting rule is a property that we do not want to dispense with.
Multi-stage sequencing games are examples of dynamic cooperative TU-games in a dis-crete time setting. Compared to the vast amount of work done in static cooperative gametheory, the work done in dynamic cooperative game theory is limited. Haurie (1975) studiesthe characteristic function and core of a game described by a multi-stage control system.Filar and Petrosjan (2000) investigate situations in which the characteristic function evolvesover time in accordance with a difference or differential equation that depends on the cur-rent characteristic function and the solution concept that is used to allocate the gains amongthe players. Lehrer (2002) describes allocation processes that converge to the least squarevalue of the game, the core, or the least core. Predtetchinski et al. (2002) look at the strongsequential core for two-period economies with uncertainty. Kranich et al. (2005) considerthree definitions for the core of a dynamic TU-game. Habis and Herings (2010) observe thatthere is a problem with the definition of the weak sequential core in Kranich et al. (2005)and propose a modification to solve this. Predtetchinski et al. (2006) define the weak se-quential core to deal with dynamic situations that involve uncertainty. Bauso and Timmer
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(2009) study dynamic cooperative games with uncertainty with the property that the valuesof the coalitions are bounded by a polyhedron.
In the multi-stage sequencing games that we consider the service times and costs in eachstage are exogenously given. As described above, the starting order in a stage depends onwhat happened in the preceding stage. This is different from the model of Kranich et al.(2005) who consider a finite sequence of exogenously specified TU-games and differentfrom the model of Filar and Petrosjan (2000) who consider situations where the game ineach stage depends on the payoff in the preceding stage. We are looking for division rulesthat generate allocations that are stable in each stage and that are also stable for the overallsituation.
The model that we study can be used to describe more general situations than the one ofcustomers waiting in line to be served. We can also think of a priority listing of n companies.In each stage the companies can choose among n goods or services. The company with thehighest priority has the first choice, the one with the second highest priority has the secondchoice, etcetera. A company can trade its priority with another. In this trade-off a companycan receive payment from another company.
The paper is organized as follows. In Sect. 2 we will introduce the model formally andgive the necessary definitions. In Sect. 3 we will concentrate on compensation rules. Threeapproaches to define compensation rules are discussed. A compensation rule that arises nat-urally from the given situation will be defined and we will show that it need not lead to acore element when combined with a stable division rule for sequencing situations. A modi-fication of this compensation rule that will always yield a core element when combined withthe EGS-rule will be given. We will show that when we combine this modification with theShapley-value it need not yield a core-element. We will define an interstage game to modelthe situation while moving from one stage to another. We will study compensation rules thatarise when we apply well-known solution concepts to the interstage game. We will intro-duce a class of compensation rules and we will show that only one member of this class isguaranteed to yield a core element of the multistage game. An axiomatic characterization ofthis rule will be given.
2 Multi-stage sequencing games
Let N = {1,2, . . . , n} be the set of customers. For every S ⊂ N the set of permutations ofS is denoted by ΠS . A sequencing situation is an ordered triplet (σ ;α; s) with σ ∈ ΠN ,α ∈ Rn and s ∈ Rn+. The initial order of the customers in the queue is given by σ . Thecosts of customer i ∈ N if his job is completed at time t is given by αit . The time it takesto process the job of customer i is si . By changing their order in the queue the customerscan decrease the total costs of N . The urgency index ui of customer i is given by ui := αi
si.
Arranging the customers in decreasing urgency index order will minimize the total costsof N . Switching two neighbours i and j with i in front of j will result in a cost change ofαj si − αisj . We will denote this amount by glij . The gain gij that two neighbours i and j
with σ(i) = σ(j) − 1 can achieve is given by
gij = (glij )+ = max{glij ,0}.By successive switches of neighbours i and j with i in front of j and ui < uj an optimalorder for N can be reached.
The sequencing game associated with this sequencing situation is defined by taking thevalue of coalition S to be equal to the maximal cost savings that S can achieve by rearranging
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its members without jumping over non-members. In this context it is useful to talk aboutconnected coalitions. A coalition T is called connected with respect to a permutation σ ifi, j ∈ T and k ∈ N with σ(i) < σ(k) < σ(j) imply k ∈ T . By P (σ, i) we denote the setof predecessors of i in the order given by σ . Let S be a coalition that is not connected.A maximal (with respect to set inclusion) connected subset of S is called a component of S.The components of S form a partition of S which we denote by S/σ . The sequencing gamev corresponding to the sequencing situation (σ ;α; s) is defined as follows.
v(T ) =∑
i∈T
∑
k∈P(σ,i)∩T
gki for a connected coalition T .
v(S) =∑
T ∈S/σ
v(T ) for a coalition S that is not connected.
An m-stage sequencing situation is an ordered (2m+1)-tuplet (σ ;α1; s1;α2; s2; . . . ;αm; sm)
with σ ∈ ΠN , αq ∈ Rn and sq ∈ Rn+ for 1 ≤ q ≤ m. The order in which the customers startin stage 1 is given by σ . In each subsequent stage q the customers start in the order thatthey reached at the end of the preceding stage q − 1. The costs and processing times instage q are given by αq and sq , respectively. At the beginning of each stage all the jobsthat should be processed during that stage are ready. One can think of several definitionsof a multistage sequencing game depending on how long a history a coalition considers inevaluating its value. In this paper we will consider two definitions. In the first definition, thevalue of a coalition in an m-stage sequencing game equals the maximal cost savings thatthe coalition can achieve in each stage by rearranging its members (without jumping overnon-members) with respect to the original order, i.e., σ . We call this game the long historym-stage sequencing game. We denote the characteristic function of this game by v. Let(σ ;α1; s1; . . . ;αm; sm) be an m-stage sequencing situation. The value v(T ) of a connectedcoalition in the corresponding long history m-stage sequencing game is given by
v(T ) =∑
i∈T
∑
k∈P(σ,i)∩T
(g1
ki + g2ki + · · · + gm
ki
).
The value of a non-connected coalition S is given by
v(S) =∑
T ∈S/σ
v(T ).
In this definition a coalition goes back all the way to the beginning to evaluate its value ineach stage.
Let σqopt be an optimal order for the sequencing situation occurring in stage q for q > 0
and σ 0opt = σ . In the second definition, the value of a coalition in an m-stage sequencing
game equals the maximal cost savings that the coalition can achieve in each stage q ≥ 2 byrearranging its members (without jumping over non-members) with respect to the startingorder in stage q − 1, i.e., the order σ
q−2opt , (for example, in stage 2 consider σ 0
opt ) togetherwith the maximal cost savings it can achieve in stage 1. We call this game the short historym-stage sequencing game. We denote the characteristic function of this game by w. Let(σ ;α1; s1; . . . ;αm; sm) be an m-stage sequencing situation. The value w(T ) of a coalitionin the corresponding short history m-stage sequencing game is given by
w(T ) =∑
T ′∈T/σ
∑
i∈T ′
∑
k∈P(σ,i)∩T ′g1
ki +m∑
q=2
∑
T ′∈T/σq−2opt
∑
i∈T ′
∑
k∈P(σq−2opt ,i)∩T ′
gq
ki .
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Table 1 The cost changings forthe 3-stage sequencing situationof Example 1
12 13 14 23 24 34
gl1ij
1 2 3 1 2 1
gl2ij
−5 −7 2 −2 7 9
gl3ij
−5 −3 −2 2 3 1
Table 2 The sequencing and3-stage sequencing games ofExample 1
{1,2} {1,4} {2,3} {3,4} {1,2,3}v1(S) 1 0 1 1 4
v2(S) 5 0 2 0 14
v3(S) 0 2 2 0 2
v(S) 1 0 3 11 6
w(S) 6 0 1 10 12
{1,2,4} {1,3,4} {2,3,4} {1,2,3,4}v1(S) 1 1 4 10
v2(S) 5 0 2 14
v3(S) 2 2 2 4
v(S) 1 11 26 34
w(S) 6 10 20 38
In this definition a coalition evaluates its value by considering subsequent two-stage peri-ods. Note that the connected coalitions change since the starting orders under considerationchange.
When there are only 2 stages v and w coincide.In this situation one can also consider the m sequencing games that occur in each stage.
In stage q this will be the sequencing game corresponding to the sequencing situation(σ
q−1opt ;αq; sq). In general neither of the m-stage sequencing games defined above equals
the sum of the m separate sequencing games. The following example illustrates this.
Example 1 Let N = {1,2,3,4}. Consider the 3-stage sequencing situation (σ ;α1; s1;α2; s2;α3; s3) with σ(i) = i for i ∈ N , α1 = (1,2,3,4), α2 = (10,5,3,12), α3 = (7,2,4,5), ands1 = s2 = s3 = (1,1,1,1). The gl
q
ij ’s for this situation are given in Table 1.The sequencing games v1, v2, v3, the long history three-stage sequencing games v, and
the short history three-stage sequencing game w are given in Table 2.The coalitions that have value 0 in all the 5 games in the table do not appear in the table.
At the end of stage 1 the order will have changed from 1234 to 4321 which is the startingorder of stage 2. At the end of stage 2 the order will have changed to 4123. This is thestarting order of stage 3. The order at the end of the three stages is 1432. Straightforwardverification shows that v �= v1 + v2 + v3 and w �= v1 + v2 + v3.
A sequencing situation need not have a unique optimal order. For the one-stage case thisdoes not pose a problem. The sequencing game stays the same regardless of which optimalorder is reached. For the multi-stage case the actual optimal order which is reached at theend of stage q can influence the situation in stage q + 1. In the following we will assumethat in the case of the existence of multiple optimal orders in a stage, the optimal order thatis reached at the end of that stage is the one which does not require two neighbours to switchif they have the same urgency index in that stage. So among all optimal orders in a stage itis the one that requires the least number of neighbour switches.
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Sequencing games are convex. The core is not empty and the Shapley-value is in thecore. In Curiel et al. (1989) it is shown that the Shapley-value φ(v) of the sequencing gamev associated with the sequencing situation (σ ;α; s) is given by
φi(v) =∑
k,j :σ(k)≤σ(i)≤σ(j)
gkj
σ (j) − σ(k) + 1for all i ∈ N.
The Shapley-value distributes the gain generated by two players who switch, equally amongthemselves and all the players standing between them. By the Shapley-value of the sequenc-ing situation (σ ;α; s) we mean the Shapley-value of the corresponding sequencing game.Another core element is given by the Equal Gain Splitting or EGS-rule which was definedspecifically for sequencing situations. The EGS-rule divides the gain gij equally between i
and j . Let F(σ, i) be the set of followers of i in the order given by σ . The EGS-rule is givenby
EGSi (σ ;α; s) = 1
2
∑
k∈P(σ,i)
gki + 1
2
∑
j∈F(σ,i)
gij .
In Curiel (2010) it was shown that applying the EGS-rule to the separate sequencing situa-tions need not yield a stable allocation for a 2-stage sequencing game. The same holds forthe Shapley-value. Example 2 gives an illustration of this for a 3-stage sequencing situation.It should be noted here that when player i evaluates the allocation that he receives in thelong history m-stage sequencing game v when a rule f for sequencing situations is used toallocate the cost savings in each stage he will consider
m∑
q=1
fi
(σ
q−1opt ;αq; sq
) −m∑
q=2
αq
i
∑
j∈F(σ,i)∩P(σq−1opt ,i)
sq
j +m∑
q=2
αq
i
∑
j∈P(σ,i)∩F(σq−1opt ,i)
sq
j .
So, in his evaluation, player i takes into account the implicit loss or gain he starts with at thebeginning of stage q compared to his starting position at the very beginning of the multistagesituation.
When player i evaluates the allocation that he receives in the short history m-stage se-quencing game w when a rule f for sequencing situations is used to allocate cost savings ineach stage he will consider
m∑
q=1
fi
(σ
q−1opt ;αq; sq
) −m∑
q=2
αq
i
∑
j∈F(σq−2opt ,i)∩P(σ
q−1opt ,i)
sq
j +m∑
q=2
αq
i
∑
j∈P(σq−2opt ,i)∩F(σ
q−1opt ,i)
sq
j .
So, in his evaluation, player i takes into account the implicit loss or gain he starts with at thebeginning of stage q ≥ 2 compared to his starting position in the stage directly preceding q .
Example 2 Consider the 3-stage sequencing situation and game of Example 1. For the se-quencing situation (σ ;α1; s1) of the first stage we have EGS(σ ;α1; s1) = (3,2,2,3). Forthe sequencing situation (σ 1
opt ;α2; s2) of the second stage we have EGS(σ 1opt ;α2; s2) =
(6,3 12 ,4 1
2 ,0). For the sequencing situation (σ 2opt ;α3; s3) of the third stage we have
EGS(σ 2opt ;α3; s3) = (1,1,1,1). We will consider the adjustments of the players in the mul-
tistage game v first and then in the multistage game w. Player 1 had position 1 at thebeginning while he starts the second stage in position 4 and the third stage in position 2.
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Therefore, he makes the following adjustment: −3α21 − α3
1 = −37. Player 2 had position 2at the beginning while he starts the second stage and third stages in position 3. He makesthe adjustment: −α2
2 − α32 = −7. Player 3 had position 3 at the beginning while he starts
the second stage in position 2 and the third stage in position 4. He makes the adjustment:α2
3 − α33 = −1. Player 4 had position 4 at the beginning while he starts the second stage
and the third stage in position 1. He makes the adjustment: 3α24 + 3α3
4 = 51. So the al-location corresponding to giving each player the value according to the EGS-rule in eachstage becomes (10,6 1
2 ,7 12 ,4)+ (−37,−7,−1,51) = (−27,− 1
2 ,6 12 ,55). It is clear that this
allocation is not in the core of the long history multistage game v.The Shapley-value yields the following results. For the sequencing game of the first stage
we have φ(v1) = (1 1112 ,3 1
12 ,3 112 ,1 11
12 ). For the game of the second stage we have φ(v2) =(4 5
6 ,5 56 ,3 1
3 ,0). For the game of the third stage we have φ(v3) = (1,1,1,1). After adjustingusing the Shapley-value in each state yields the allocation (−29 1
4 ,2 1112 ,6 5
12 ,53 1112 ). One can
easily see that this allocation is not in the core of the long history multistage game v.In the multistage game w the adjustments become as follows. Player 1 makes the adjust-
ment: −3α21 + 2α3
1 = −16. Player 2 makes the adjustment: −α22 = −5. Player 3 makes the
adjustment: α23 − 2α3
3 = −5. Player 4 makes the adjustment: 3α24 = 36. So the allocation ac-
cording to the EGS-rule becomes (10,6 12 ,7 1
2 ,4) + (−16,−5,−5,36) = (−6,1 12 ,2 1
2 ,40).It is clear that this allocation is not in the core of the short history multistage game w.
The Shapley-value yields the following allocation after adjusting: (−8 14 ,4 11
12 ,2 512 ,3 11
12 ).One can easily see that this allocation is not in the core of the short history multistagegame w.
In fact, it was shown in Curiel (2010) that there does not exist an allocation rule forsequencing situations that will yield a core element for each of the separate sequencinggames which summed together and adjusted for the implicit losses and gains will alwaysresult in a core allocation of the multistage sequencing game.
To obtain a rule that will yield a core element of the multistage sequencing game it isnecessary to consider not only the division of the gij ’s but also of the implicit loss and gainthat player i incurs when the order of the players is changed from the starting order of astage to an optimal order for that stage. In the next section we will focus on this last aspect.
3 Compensation rules
Let (σ ;α1; s1; . . . ;αm; sm) be an m-stage sequencing situation. A compensation rule CR forsuch a situation is a function which assigns to every two subsequent stages q − 1 and q adivision of the implicit losses and gains of the players when they change from the startingorder τ of the first stage to the optimal order σ
q−1opt of the first stage. Formally,
CR(τ ;αq−1; sq−1;αq; sq
) = (CR1
(τ ;αq−1; sq−1;αq; sq
), . . . , . . . ,
CRn
(τ ;αq−1; sq−1;αq; sq
))
satisfying the efficiency property
∑
i∈N
CRi
(τ ;αq−1; sq−1;αq; sq
) =∑
i∈N
(α
q
i
∑
j∈P(τ,i)∩F(σq−1opt ,i)
sq
j
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− αq
i
∑
j∈F(τ,i)∩P(σq−1opt ,i)
sq
j
).
Depending on whether we are considering the long history multistage game v or the shorthistory multistage game w the order τ in the definition above will be given by the order σ
which is the order at the very beginning of the multistage situation, or the order σq−2opt which
is the order at the beginning of stage q − 1.For our first approach to find a suitable compensation rule we consider the situation
from the following perspective. Let τ(i) < τ(j), and σq−1opt (i) > σ
q−1opt (j). Here τ is either σ
or σq−2opt depending on whether we are considering the long history case or the short history
case. So, originally i is standing in front of j and they switch in stage q −1. The implicit lossof i equals α
q
i sq
j . He can ask j to recompense him for this. If σqopt (i) > σ
qopt (j) then j will
agree to this and i’s implicit loss will be 0, while j ’s implicit gain will be αq
j sq
i −αq
i sq
j = gq
ij .If σ
qopt (i) < σ
qopt (j) then it is not likely that j will agree to compensate i fully for his implicit
loss since this would leave j with a loss of αq
i sq
j − αq
j sq
i = gq
ji > 0. Player j would bewilling to give his total implicit gain of α
q
j sq
i to i hereby making his gain/loss equal to 0and making i’s loss equal to α
q
i sq
j − αq
j sq
i = gq
ji . We call the compensation rule describedby this procedure the CRμ-rule. Notational efficacy in giving the formal definition of theCRμ-rule is obtained by using the following notation introduced in Curiel (2010). The orderpreserving index OPij (q − 1, q) of players i and j with respect to two subsequent stagesq − 1 and q is defined as follows.
OPij (q − 1, q) ={
1 if (σq−1opt (i) − σ
q−1opt (j))(σ
qopt (i) − σ
qopt (j)) > 0
0 otherwise.
As the name suggests the order preserving index indicates if the order of i and j is reversedwhen moving from the optimal order in stage q − 1 to the optimal order in stage q . If it isnot reversed the order preserving index equals 1, if it is reversed the order preserving indexequals 0.
Formally, the CRμ-rule is given by
CRμi
(τ ;αq; sq;αq+1; sq+1
) =∑
j∈F(τ,i)∩P(σqopt ,i)
(OPij (q, q + 1) − 1
)g
q+1j i
+∑
k∈P(τ,i)∩F(σqopt ,i)
OPki(q, q + 1)gq+1ki (1)
for all i ∈ N .We can combine the CRμ compensation rule with a division rule f for sequencing situ-
ations to obtain a division rule for multistage sequencing situations. We make a distinctionbetween a rule generated in this way for the long history case and one for the short historycase. The first we denote by μf l and the second by μf s . So,
μf li
(σ ;α1; s1; . . . ;αm; sm
)
:=m∑
q=1
fi
(σ
q−1opt ;αq; sq
) +m∑
q=2
CRμi
(σ ;αq−1; sq−1;αq; sq
)(2)
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for all i ∈ N in the long history case and
μf si
(σ ;α1; s1; . . . ;αm; sm
)
:=m∑
q=1
fi
(σ
q−1opt ;αq; sq
) +m∑
q=2
CRμi
(σ
q−2opt ;αq−1; sq−1;αq; sq
)(3)
for all i ∈ N in the short history case.Although the justification given above for the CRμ compensation rule seems reasonable,
combining it with the EGS-rule or the Shapley-value need not yield a core-element. In fact,the following proposition shows that if we take an individual rational division rule f forsequencing situations for which fi(σ ;α; s) = 0 if and only if i is a dummy in (σ ;α; s) andcombine it with the CRμ compensation rule this combination need not yield a core elementfor the multi-stage sequencing game. A dummy in a sequencing situation is a player whodoes not change position with any player during the move to an optimal order.
Proposition 1 Let (σ ;α1; s1; . . . ;αm; sm) be an m-stage sequencing situation. Let v be thecorresponding long history m-stage sequencing game. Let w be the corresponding shorthistory m-stage sequencing game. Let f be an individual rational division rule for sequenc-ing situations which assigns 0 only to dummies. Let μf l and μf s be the division rules formulti-stage sequencing situations that we obtain by combining f with CRμ. Then
(a) μf l(σ ;α1; s1; . . . ;αm; sm) need not be an element of C(v), and(b) μf s(σ ;α; s1; . . . ;αm; sm) need not be an element of C(w).
Proof Since v = w and μf l = μf s in the 2-stage case both (a) and (b) are proved by the fol-lowing example. Because of efficiency and the fact that only dummies obtain 0, if (σ ;α; s) isa sequencing situation with maximal cost savings equal to gij obtained by switching neigh-bours i and j there is a 0 < λ < 1 (λ may depend on (σ ;α; s)) such that fi(σ ;α; s) = λgij .Let (σ ;α1; s1;α2; s2) be a 2-stage sequencing situation with σ(i) = σ(j) − 1, σ 1
opt (k) =σ(k) for all k ∈ N \ {i, j}, σ 1
opt (i) = σ 1opt (j) + 1, σ 2
opt = σ , and g1ij < (1 − λ2)g2
j i . Here λ2 issuch that fi(σ
1opt ;α2; s2) = λ2g2
j i . Then
μf li
(σ ;α1; s1; . . . ;αm; sm
) = fi
(σ ;α1; s1
) + fi
(σ 1
opt ;α2; s2) − g2
j i
< g1ij + λ2g2
j i − g2j i
= g1ij − (
1 − λ2)g2
j i < 0.
It follows that μf l(σ ;α1; s1; . . . ;αm; sm) /∈ C(v). �
The fact that using the compensation rule CRμ does not necessarily yield an element ofthe core stems from the fact that it does not compensate player i enough when τ(i) < τ(j),σ
q−1opt (i) > σ
q−1opt (j), and σ
qopt (i) < σ
qopt (j). In the following we will consider a rule that gives
additional compensation to i in this case. We will call this compensation rule the CRM-rule. In the case that τ(i) < τ(j), and OP(q − 1, q) = 1, the rule CRM equals CRμ. Whenτ(i) < τ(j) and OP(q − 1, q) = 0 the rule CRM prescribes that j compensates i with theamount α
q
j sq
i plus j ’s share of the gains arising in stage q when they switch, i.e., 12g
q
ji .
Ann Oper Res
Formally,
CRMi
(τ ;αq; sq;αq+1; sq+1
) :=∑
j∈F(τ,i)∩P(σqopt ,i)
(OPij (q, q + 1) − 1
)1
2g
q+1j i
+∑
k∈P(τ,i)∩F(σqopt ,i)
(OPki(q, q + 1)g
q+1ki
+ (OPki(q, q + 1) − 1
)1
2g
q+1ik
)(4)
for all i ∈ N .The allocation rule for multistage sequencing situations that we obtain when we combine
the CRM compensation rule with the EGS-rule will be called modified equal gain splittingrule and will be denoted by MEGSl in the long history case and by MEGSs in the shorthistory case. So,
MEGSli
(σ ;α1; s1; . . . ;αm; sm
)
:=m∑
q=1
EGSi
(σ
q−1opt ;αq; sq
) +m∑
q=2
CRMi
(σ ;αq−1; sq−1;αq; sq
)(5)
for all i ∈ N in the long history case and
MEGSsi
(σ ;α1; s1; . . . ;αm; sm
)
:=m∑
q=1
EGSi
(σ
q−1opt ;αq; sq
) +m∑
q=2
CRMi
(σ
q−2opt ;αq−1; sq−1;αq; sq
)(6)
for all i ∈ N in the short history case.In Curiel (2010) the MEGS-rule was defined as a modification of the EGS-rule for 2-
stage sequencing situations and it was shown that the MEGS-rule yields an element of thecore of the 2-stage sequencing game. A similar proof can be given for m > 2.
The following example shows that combining the CRM compensation rule with theShapley-value is not guaranteed to give a core element.
Example 3 Let N = {1,2,3}, σ(i) = i for all i ∈ N , α1 = (2,3,5), α2 = (18,17,3),s1 = s2 = (1,1,1). For the 2-stage sequencing situation (σ ;α1; s1;α2; s2) and the sequenc-ing games v1 and v2 of stage 1 and stage 2 respectively we have φ(v1) = (1 1
2 ,2 12 ,2),
φ(v2) = (5 12 ,12 1
2 ,12), CRM(σ ;α1; s1;α2; s2) = (−8,−7 12 ,−14 1
2 ). Combining the allo-cations given by the Shapley-value and the CRM compensation rule yields (−1,7 1
2 ,− 12 )
which is clearly not an element of the core of the 2-stage sequencing game correspondingto (σ ;α1; s1;α2; s2).
In the following we will study compensation rules that arise by defining an interstagegame and applying well known solution concepts to this game. If we consider the situation(τ ;αq−1; sq−1;αq; sq) then the implicit loss or gain when players i, j with τ(j) = τ(i) + 1switch position equals gl
q
ij . Let (σ ;α1; s1; . . . ;αm; sm) be an m-stage sequencing situation
and let v be the corresponding long history m stage sequencing game. For 1 ≤ q ≤ m − 1we define the q-th long history interstage game ψq by
ψq(T ) =∑
i∈T
∑
j∈T ∩F(σ,i)∩P(σqopt ,i)
glq+1ij
for all T ⊂ N that are connected with respect to σ . For a coalition S that is not connectedwith respect to σ we define
ψq(S) =∑
T ∈S/σ
ψq(T ).
The q-th short history interstage game γ q is defined by
γ q(T ) =∑
i∈T
∑
j∈T ∩F(σq−1opt ,i)∩P(σ
qopt ,i)
glq+1ij
for all T ⊂ N that are connected with respect to σq−1opt . For a coalition S that is not connected
with respect to σq−1opt we define
γ q(S) =∑
T ∈S/σq−1opt
γ q(T ).
It will be clear that these interstage games need be neither superadditive nor subadditive normonotonic.
Example 4 describes the interstage games that occur in the 3-stage sequencing situationof Example 1.
Example 4 Let (σ ;α1; s1;α2; s2;α3; s3) be the 4-person 3-stage sequencing situation ofExample 1. The games ψ1, ψ2, γ 1, γ 2 are given in Table 3.
The coalitions that have value 0 in all four games do not appear in the table.
Although an interstage game is not a sequencing game and does not have an underlyingsequencing situation we can still apply the basic idea of the EGS-rule for sequencing situa-tions to an interstage game by changing the gij ’s in the definition of the EGS-rule into glij ’s.We will call the compensation rule that we obtain in this way the CRE compensation rule.
Ann Oper Res
Formally,
CREi
(τ ;αq−1; sq−1;αq; sq
) =∑
j∈F(τ,i)∩P(σq−1opt ,i)
1
2gl
q
ij +∑
k∈P(τ,i)∩F(σq−1opt ,i)
1
2gl
q
ki .
In the long history case τ equals σ and in the short history case τ equals σq−1opt . The rule we
obtain when we combine the CRE compensation rule with the EGS-rule in we will call theEEGSl-rule in the long history case and the EEGSs -rule in the short history case. The “E” inCRE and the extra “E” in EEGS stand for the equal division of the implicit gains and losseswhen moving from one stage to another. The EEGSl-rule and the EEGSs -rule will alwaysyield allocations that are in the core of the corresponding multi-stage sequencing games.Before giving the proof of this statement we will describe a different way to arrive at theEEGSl- and EEGSs -rule.
In the following we will consider a class of compensation rules that divide the implicitlosses and gains that occur when two players switch, between them. If we consider thesituation (τ ;αq−1; sq−1;αq; sq) then the implicit loss or gain when players i, j with τ(j) =τ(i) + 1 switch position equals gl
q
ij . Let 0 ≤ λ ≤ 1. We will divide glq
ij between i and j bygiving λgl
q
ij to i and (1 − λ)glq
ij to j . Formally, we define CRλ for each 0 ≤ λ ≤ 1 by
CRλi
(τ ;αq−1; sq−1;αq; sq
) =∑
j∈F(τ,i)∩P(σq−1opt ,i)
λglq
ij +∑
k∈P(τ,i)∩F(σq−1opt ,i)
(1 − λ)glq
ki .
The question that arises is whether the use of a particular compensation rule to distributethe implicit losses and gains that occur between two stages together with a rule to distributethe cost savings in the sequencing game in each stage will lead to a core element of themultistage sequencing game under consideration. In the following we will consider a CRλ-compensation rule together with the EGS-rule to allocate the cost savings in a multistagesequencing game. We will denote such a rule by λEGSl in the long history case and λEGSl
in the short history case. Let (σ ;α1; s1; . . . ;αm; sm) be an m-stage sequencing situation. Wedefine
λEGSli
(σ ;α1; s1; . . . ;αm; sm
)
:=m∑
q=1
EGSi
(σ
q−1opt ;αq; sq
) +m∑
q=2
CRλi
(σ ;αq−1; sq−1;αq; sq
)
for all i ∈ N in the long history case and
λEGSsi
(σ ;α1; s1; . . . ;αm; sm
)
:=m∑
q=1
EGSi
(σ
q−1opt ;αq; sq
) +m∑
q=2
CRλi
(σ
q−2opt ;αq−1; sq−1;αq; sq
)
for all i ∈ N in the short history case.Theorem 2 states that λEGSl and λEGSs are guaranteed to give core elements in the
corresponding games if and only if λ = 12 .
Theorem 2 Let (σ ;α1; s1; . . . ;αm; sm) be an m-stage sequencing situation and let v,w bethe corresponding long history, respectively, short history m-stage sequencing games. Then
Ann Oper Res
(a) λEGSl (σ ;α1; s1; . . . ;αm; sm) ∈ C(v) if and only if λ = 12 and
(b) λEGSs(σ ;α1; s1; . . . ;αm; sm) ∈ C(w) if and only if λ = 12 .
Proof We prove (a). The proof of (b) is similar. First we prove the “if” part. Let v bea long history m-stage sequencing game arising from the m-stage sequencing situation(σ ;α1; s1; . . . ;αm; sm) and let λ = 1
2 . Let T ⊂ N , then
∑
i∈T
λEGSli
(σ ;α1; s1; . . . ;αm; sm
)
=∑
i∈T
(m∑
q=1
EGSi
(σ
q−1opt ;αq; sq
)
+m∑
q=2
CRλi
(σ ;αq−1; sq−1;αq; sq
))
=∑
i∈T
(m∑
q=1
(1
2
∑
k∈P(σq−1opt ,i)
gq
ki + 1
2
∑
j∈F(σq−1opt ,i)
gq
ij
)
+m∑
q=2
(1
2
∑
k∈P(σ,i)∩F(σq−1opt ,i)
glq
ki + 1
2
∑
j∈F(σ,i)∩P(σq−1opt ,i)
glq
ij
))(7)
and
v(T ) ≤∑
i∈T
∑
k∈P(σ,i)∩T
(g1
ki + g2ki + · · · + gm
ki
). (8)
Let i ∈ T . We consider the contribution of a player k ∈ N \ {i} to the sums in (8) and (7). Wedistinguish two cases: (I) k /∈ T and (II) k ∈ T . We start with case (I), k /∈ T . Then k doesnot contribute anything to the sum in (8). To evaluate the contribution of k to the sum in (7)we distinguish four subcases, namely,
(Ia) k ∈ P (σ, i) ∩ P (σq−1opt , i).
(Ib) k ∈ P (σ, i) ∩ F(σq−1opt , i).
(Ic) k ∈ F(σ, i) ∩ P (σq−1opt , i).
(Id) k ∈ F(σ, i) ∩ F(σq−1opt , i).
In case (Ia) the contribution of k to the sum in (7) equals 12g
q
ki ≥ 0 for all 1 ≤ q ≤ m. In case(Ib) the contribution of k to the sum in (7) equals 1
2gq
ik + 12 gl
q
ki ≥ 0 for q ≥ 2 and 12 g1
ik ≥ 0for q = 1. In case (Ic) the contribution of k to the sum in (7) equals 1
2gq
ki + 12 gl
q
ik ≥ 0 forq ≥ 2 and 1
2 g1ki ≥ 0 for q = 1. In case (Id) the contribution of k to the sum in (7) equals
12g
q
ik ≥ 0 for all 1 ≤ q ≤ m. So, whenever k /∈ T his contribution to the sum in (7) is greaterthan or equal to his contribution to the sum in (8).
In case (II) we consider the same 4 cases but now with k ∈ T . We call them (IIa), (IIb),(IIc), and (IId). In case (IIa) the contribution of k to the sum in (8) equals g
q
ki for all 1 ≤ q ≤m. The contribution of k to the sum in (7) also equals g
q
ki , where 12g
q
ki comes from λEGSi (v),and 1
2gq
ki comes from λEGSk(v). In case (IIb) the contribution of k to the sum in (8) equalsg
q
ki for all 1 ≤ q ≤ m. The contribution of k to the sum in (7) equals gq
ik + glq
ki = gq
ki where
Ann Oper Res
the equality follows from the fact that if gq
ki > 0 then gq
ik = 0 and glq
ki = gq
ki , and if gq
ki = 0then gl
q
ki = −gq
ik . In case (IIc) the contribution of k to the sum in (8) equals gq
ik for all1 ≤ q ≤ m. The contribution of k to the sum in (7) equals g
q
ki +glq
ik = gq
ik for similar reasonsas used in case (IIb). In case (IId) the contribution of k to the sum in (8) equals g
q
ik . Thecontribution of k to the sum in (7) also equals g
q
ik . So whenever k ∈ T his contribution to thesum in (7) equals his contribution to the sum in (8). Together with the result in case (I) thisleads to the conclusion that for λ = 1
2
∑
i∈T
λEGSli
(σ ;α1; s1; . . . ;αm; sm
) ≥ v(T )
for all T ⊂ N and therefore, λEGSl (σ ;α1; s1; . . . ;αm; sm) ∈ C(v).For the “only if” part we distinguish the two cases λ > 1
2 en λ < 12 . Let λ > 1
2 . Weconsider a 2 player 2-stage sequencing situation (σ ;α1; s1;α2; s2) with σ(i) = i for i ∈{1,2}, s1 = s2 = (1,1), α1 = (α1
1, α12) with α1
2 > α11 , and α2 = 1
2λ−1 (α12 + 1, α1
1). Then
λEGSl1
(σ ;α1; s1;α2; s2
) = 1
2
(α1
2 − α11
) +12
2λ − 1
(α1
2 + 1 − α11
)
+ λ
2λ − 1
(α1
1 − (α1
2 + 1))
=(
1
2+
12
2λ − 1− λ
2λ − 1
)(α1
2 − α11
) +12 − λ
2λ − 1< 0. (9)
It follows that λEGSl (σ : α1; s1;α2; s2) /∈ C(v).Let λ < 1
2 . We consider almost the same 2 player 2-stage sequencing situation as above.Only α2 is different and equals 1
1−2λ(α1
2 + 1, α11). Then
λEGSl2
(σ ;α1; s1;α2; s2
) = 1
2
(α1
2 − α11
) +12
1 − 2λ
(α1
2 + 1 − α11
)
+ 1 − λ
1 − 2λ
(α1
1 − (α1
2 + 1))
=(
1
2+
12
1 − 2λ+ λ − 1
1 − 2λ
)(α1
2 − α11
) + λ − 12
1 − 2λ< 0. (10)
It follows that λEGSl (σ ;α1; s1;α2; s2) /∈ C(v) which completes the proof. �
Let f be a division rule for sequencing situation given by f (σ ;α; s) = F(v) where F isan individual rational solution concept for games which possesses the anonymity propertyand v is the sequencing game arising from the sequencing situation (σ ;α; s). Then f willyield the same outcome as the EGS-rule for any 2-player sequencing situation and it followsfrom the proof of the “only if” part that combining a CRλ compensation rule for λ �= 1
2 withany such rule will not guarantee a core element. For λ = 1
2 it is straightforward to see thatCRλ = CRE and that λEGSl = EEGSl and λEGSs = EEGSs .
Comparing the MEGSl- and MEGSs -rule with the EEGSl- and the EEGSs -rule we seethat the compensation rule CRM associated with the MEGSl- and MEGSs -rule makes adistinction between player i and j that depends on who was first in the original orderwhile the CRE compensation rule does not make this distinction. For the 2-stage case the
Ann Oper Res
MEGSl-rule equals the MEGSs -rule and the EEGSl-rule equals the EEGSs -rule and wewill leave the superscript out when considering 2-stage sequencing situations. In Curiel(2010) the MEGS-rule for 2-stage sequencing situations was characterized by the stage-1-dummy property, the equivalence property, and the modified switch property. In the fol-lowing we will show that the EEGS-rule is characterized by the stage-1-dummy prop-erty, the equivalence property, and the switch property. Let us recall the definition of theproperties mentioned above. A division rule f for 2-stage sequencing situations is saidto possess the stage-1-dummy-property if fi(σ
1opt ;α2; s2) = EGSi (σ
1opt ;α2; s2) whenever
i is a dummy in stage 1 of the 2-stage sequencing situation (σ ;α1; s1;α2; s2). That is,whenever i does not switch places with any player in stage 1. Let (σ ;α1; s1;α2; s2) and(τ ;α1; s1;α2; s2) be two 2-stage sequencing situations with P (σ, i) = P (τ, i). We saythat these two 2-stage sequencing situations are i-equivalent. A division rule f is saidto possess the equivalence property if fi(σ ;α1; s1;α2; s2) = fi(τ ;α1; s1;α2; s2) for anytwo i-equivalent 2-stage sequencing situations (σ ;α1; s1;α2; s2) and (τ ;α1; s2;α2; s2).Let (σ ;α1; s1;α2; s2) be a 2-stage sequencing situation with |σ(i) − σ(j)| = 1. We callthe 2-stage sequencing situation (τ ;α1; s1;α2; s2) the ij -inverse of (σ ;α1; s1;α2; s2) if τ
arises from σ by switching i and j . A division rule is said to possess the switch propertyif fi(τ ;α1; s1;α2; s2) − fi(σ ;α1; s1;α2; s2) = fj (τ ;α1; s1;α2; s2) − fj (σ ;α1; s1;α2; s2)
whenever (σ ;α1; s1;α2; s2) and (τ ;α1; s1;α2; s2) are ij -inverses of each other.
Theorem 3 The EEGS-rule is the unique rule for 2-stage sequencing situations that pos-sesses the stage-1-dummy property, the equivalence property, and the switch property.
Proof From the definition of the EEGS-rule it follows that its possesses the stage-1-dummyproperty, the equivalence property, and the switch property. Let f be a division rule for 2-stage sequencing situations that also possesses these three properties. We define the set ofmisplaced pairs of neighbours in the 2-stage sequencing situation (σ ;α1; s1;α2; s2) by
Mσ = {(i, j)|σ(i) = σ(j) − 1, u1
i < u1j
}.
We will show by induction on the cardinality of Mσ that f = EEGS. Let (σ ;α1; s1;α2; s2)
be a two-stage sequencing situation with Mσ = ∅. Then every player is a dummy instage 1 and hence f (σ ;α1; s1;α2; s2) = EGS(σ 1
opt ;α2; s2) = EEGS(σ ;α1; s1;α2; s2). Sup-pose f (σ ;α1; s1;α2; s2) = EEGS(σ ;α1; s1;α2; s2) for all 2-stage sequencing situations(σ ;α1; s1;α2; s2) with |Mσ | ≤ m. Let (τ ;α1; s1;α2; s2) have Mτ = m+1. Then there existsa 2-stage sequencing situation (σ ;α1; s1;α2; s2) and a pair (k, l) ∈ Mτ such that σ(i) = τ(i)
for all i ∈ N \ {k, l}, and σ(k) = τ(l), σ(l) = τ(k). So |Mσ | = m. From the equivalenceproperty and the induction assumption it follows that
fi
(τ ;α1; s1;α2; s2
) = fi
(σ ;α1; s1;α2; s2
)
= EEGSi
(σ ;α1; s1;α2; s2
)
= EEGSi
(τ ;α1; s1;α2; s2
)for all i ∈ N \ {k, l}.
Let Cτ and Cσ be the total costs associated with the 2-stage sequencing situations(τ ;α1; s1;α2; s2) and (σ ;α1; s1;α2; s2) respectively. Then Cτ − Cσ = g1
kl + gl2kl . This is
also the difference in the total cost savings between the two situations. From efficiency, theswitch property, and the induction assumption it follows that
fk
(τ ;α1; s1;α2; s2
) = fk
(σ ;α1; s1;α2; s2
) + 1
2
(g1
kl + gl2kl
)
Ann Oper Res
= EEGSk
(σ ;α1; s1;α2; s2
) + 1
2
(g1
kl + gl2kl
)
= EEGSk
(τ ;α1; s1;α2; s2
).
Similarly, it follows that fl(τ ;α1; s1;α2; s2) = EEGSl (τ ;α1; s1;α2; s2). So f = EEGS. �
Combining the CRE compensation rule with the Shapley-value need not yield a coreelement. Example 3 shows this because for this example CRE(σ ;α1; s1;α2; s2) equalsCRM(σ ;α1; s1;α2; s2).
In the following we will consider what happens when we apply the rule given by theShapley-value for sequencing games to the interstage games with the same adaptations thatwe used in the definition of the CRE compensation rule. We will call the compensation rulethat we obtain when doing this the CRS-rule. The CRS-rule does not belong to the classof CRλ compensation rules since it distributes the implicit losses or gains generated by theswitch of two players among all the players standing between them. The next example showsthat combining the CRS compensation rule with the EGS-rule or the Shapley-value need notyield a core element.
Example 5 Consider the 3-person 2-stage sequencing situation (σ ;α1; s1;α2; s2) given byσ(i) = i for all i ∈ N , α1 = (4,3,5), α2 = (10,3,4), and s1 = s2 = (1,1,1). Then
EGS(σ ;α1; s1
) + EGS(σ 1
opt ;α2; s2) + CRS
(σ ;α1; s1;α2; s2
)
=(
1
2,1,1
1
2
)+ (3,0,3) +
(−2,−1
1
2,−1
1
2
)
=(
11
2,−1
2,3
),
which is clearly not in the core of the 2-stage sequencing game. For the combination withthe Shapley-value we have
φ(v1
) + φ(v2
) + CRS(σ ;α1; s1;α2; s2
)
=(
1
3,1
1
3,1
1
3
)+ (3,0,3) +
(−2,−1
1
2,−1
1
2
)
=(
11
3,−1
6,2
5
6
),
which is also not in the core of the 2-stage sequencing game. Since for 2-stage situations thelong history game and the short history game coincide this example is valid for both games.
Summarizing, we see that of the compensation rules that we studied two yield a coreelement when combined with the EGS-rule, namely the CRE compensation rule and theCRM compensation rule. The CRE-rule combined with the EGS-rule gives the EEGS-rule,and the CRM-rule combined with the EGS-rule gives the MEGS-rule.
4 Concluding remarks
In this paper we concentrated on the use of compensation rules to distribute the implicitgains and losses when moving from one stage to another in a multistage sequencing game.
Ann Oper Res
Such rules are necessary because allocation rules that yield stable outcomes for the separatesequencing games in each stage need not yield stable outcomes for the multistage game.We considered three approaches to construct compensation rules. The first approach giverise to the CRμ compensation rule and the CRM compensation rule, the second to the CREcompensation rule and the CRS-compensation rule, and the third to the class of CRλ com-pensation rules. We showed that the CRμ compensation rule is not guaranteed to yield acore element when combined with a reasonable division rule for sequencing situations. TheCRM compensation rule will always yield a core element when combined with the EGS-rulebut is not guaranteed to yield a core element when combined with the Shapley-value. TheCRE compensation rule will always yield a core element when combined with the EGS-rulebut is not guaranteed to yield a core element when combined with the Shapley-value. TheCRS compensation rule is not guaranteed to yield a core element when combined with theEGS-rule or the Shapley-value. For λ �= 1
2 it was shown that the CRλ-rule is not guaranteedto yield a core element when combined with the EGS-rule or any other reasonable divisionrule for sequencing situations. For λ = 1
2 the CRλ rule equals the CRE-rule. We called thecombination of the CRE compensation rule and the EGS-rule the EEGS-rule and we gavean axiomatic characterization of the EEGS-rule in the 2-stage case.
Typical for multistage sequencing games is the fact that compensation is needed for theloss of an advantageous position. A future line of research is to consider multistage permu-tation games where this aspect is also present. Another future line of research is to considerm-stage sequencing situations with uncertainty. Several choices with respect to the model-ing of the uncertainty in each stage and the ability to change the choice of a compensationrule along the way will need to be made.
Acknowledgements The author expresses her gratitude to two anonymous reviewers for their valuablecomments on a previous version of this paper.
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