CC BY
2Efficient CS-Values Based on Consensus and Shapley Values
Alexandra B. Zinchenko, Polina P. Provotorova and George V. Mironenko
South Federal University,
Faculty of Mathematics, Mechanics and Computer Science, Milchakova 8”a”, Rostov-on-Don, 344-000, Russia E-mail: zin46@mail.ru prov-pa@inbox.ru georim89@mail.ru
Abstract. Two efficient values for transferable utility games with coalition structure are introduced and axiomatized by means of modified versions of null player property and four standard axioms (efficiency, additivity, external symmetry and internal symmetry). The first value uses the consensus value in game between coalitions and the Shapley value in games within coalitions.
The second one uses the consensus and Shapley values in inverse order.
Keywords: coalition structure, coalition value, consensus value, Shapley value, axiomatization.
1. Introduction
In cooperative transferable utility games with coalition structure (CS-game) it is supposed that players are already partitioned into groups. As the allocation rule for such game a coalitional value (CS-value) can be chosen. If it is component efficient, the worth of every structural component is distributed among its members. The player’s payoff does not depend on the coalitions formed by players outside his component. The efficient CS-values predict the grand coalition to form, i.e., the sum of individual payoffs equals to the grand coalition’s worth. The components of coalition structure are interpreted as a priori unions (blocks, pressure groups, superplayers) which make decisions as a single player.
In this paper, we focus on efficient CS-values. The first one, proposed by Owen (Owen, 1977), is a generalization of the Shapley value (Shapley, 1953) to the coalitional context. Owen defined the CS-value by decomposing a CS-game into an external game played by the structural components (quotient game) and an internal game that is induced on the players within a component. The Owen value splits the grand coalition’s worth among the components and then total reward of each component is shared among its members. Both payoffs are given by the Shapley value. In addition to efficiency, the Owen value is characterized by the additivity, external symmetry, internal symmetry and null player property (every null player gets nothing even though its coalition is in very strong position). But in real life the null player axiom does not seem good (Ju, et al., 2006; Kamijo, 2009; Hernandez-Lamoneda, et al., 2008). After the Owen value alternative CS-values have been presented in the literature (their description can be found, for example, in (Gomez-Rua and Vidal-Puga, 2008)), however they either satisfy the null player axiom or do not reflect the outside options of players within the same structural coalition.
Recently, Kamijo (Kamijo, 2007; Kamijo, 2009) introduced and axiomatized two efficient CS-values without null player property. They are called the two-step Shapley value and the collective value. An alternative characterization of the two-step Shapley value (Kamijo, 2009) is proposed in (Calvo and Gutierrez, 2010). The two-step Shapley value, as well as the Owen value, assumes that each structural component acts like a single player and receives its Shapley value in the quotient game for coalitions. Firstly, every player gets his Shapley value in the game restricted within his structural component. Secondly, the pure surplus received by the component in the quotient game is equally shared among the intra-coalition members. The collective value (Kamijo, 2007) differs from the two-step Shapley value that the Shapley value is replaced with weighed Shapley value. Both coalition values are insensitive to outside options. Let’s assume that two players belong to the same structural component and one of them dominates another. If these players are symmetric in the restricted game inside component, then according to the two-step Shapley value and the collective value they receive equal payoffs.
The consensus CS-value proposed in (Zinchenko, et al., 2010) is Owen-type extension of the consensus value (Ju, et al., 2006), i.e. this solution concept applies the consensus value to games inter- and intra-coalitions. The consensus CS-value satisfies the axioms that traditionally are used to characterize the Owen value except for the null player axiom. This axiom has been replaced the modified dummy player property. In present paper, we introduce two new solution concepts: the consensus-Shapley CS-value and the Shapley-consensus CS-value. These CS-values are obtained by means of a composition of the consensus value with the Shap-ley value. Together with the Owen and consensus CS-values new concepts cover the possible variations of the application of the consensus and Shapley values to games with coalition structure. For their characterization we introduce two axioms which can be seen as modifications of the classical null player property. For similar characterization all CS-values using the consensus value, we present also a third modification of null player axiom.
The paper is divided into four sections. The next section contains notations and definitions which are needed in the paper. In section 3, we introduce new CS-values. Section 4 is devoted the axiomatic characterization of CS-values based on consensus and Shapley values.
2. Preliminaries
A cooperative game with transferable utility (TU-game) is a pair (N, v) , where N = {1,...,n} is a player set and v G GN, where
Gn = {g :2,n ^ R | g(0) = °},
is a characteristic function. A subset S Ç N of player set is called a coalition and v(S) is interpreted as the worth that is available to coalition S. The cardinality of set S is written as s. A vector x G RN is called a payoff vector and xi denotes the payoff of player i. Given 0 = T Ç N a T-unanimity game is denoted by (N,uT) and determined as: uT(S) = 1 if T Ç S and uT(S) = 0 otherwise. Given a G R, let (N, auT) be the unanimity game multiplied by a scalar a, i.e. (auT)(S) = auT(S) for all S Ç N. The game (N,v), determined by v(S) = 0 for all S Ç N, is a zero-game.
Two players i,j € N are symmetric in (N,v) if v(S U i) = v(S U j) for every
S C N \ {i,j}. We say that players of coalition S with s > 2 are symmetric in
(N, v) if each pair of players of the coalition is symmetric in (N, v). A player i € N is dummy in (N, v) if he adds v(i) to any coalition non-containing him. Denote by
Du(N, v) = {i € N | v(S U i) - v(S) = v(i), S C N \ i}
a set of all dummy players in (N, v). A player i € N is a null in (N, v) if he adds
nothing to any coalition non-containing him. Denote by
Nu(N, v) = {i € N I v(S U i) = v(S), S C N \ i}
a set of all null players in (N, v). For any set G C GN a value on G is a function
$ : G ^ Rn which assigns to every TU-game v € G a vector $(N, v), where (N, v) represents the payoff to player i in (N, v). Denote by &(N, v) a set of all values of game (N, v). Consider the following properties of value $ € &(N, v).
Axiom 2.1 (efficiency). J2ieN $i(N,v) = v(N) for all v € G.
Axiom 2.2 (additivity). For any two v,w € G, p(N, v + w) = y>(N, v) + p(N, w),
where (v + w)(S) = v(S) + w(S) for all S C N.
Axiom 2.3 (symmetry). For all v € G and every symmetric players i,j € N in (^ v), ^i(N, v) = (^ v).
Axiom 2.4 (null player property). For all v € G and every i € Nu(N,v), $i (N,v) = 0.
Axiom 2.5 (neutral dummy property) (Ju, et al., 2006). For all v € G and every i € Du(N, v),
v) = v{i) H-----------^----------.
One of the most important values for GN is the Shapley value (Shapley, 1953) which is denoted as Sh. The Shapley value is given by
Shi(N,v)= Y, Pns(v(S U i) - v(S)), i € N, (1)
SCN\i
where
k!(i - k - 1)! pik =----------------, k < I and k,l € N.
The Shapley value is characterized by the axioms 2.1-2.4. For every v € GN the equal surplus division solution E € &(N, v) and the consensus value K € &(N, v) (Ju, et al., 2006) are given by the formulas
Ei{N, v) = is(i) + % e N (2)
n
(3)
Since the consensus value equals the average of the Shapley value and the equal surplus solution, it: "takes a neutral stand between the two polar opinions of utilitarianism and egalitarianism, and balances the tensions of the four fundamental principles of distributive justice" (Ju, et al., 2006). The consensus value is characterized by the axioms 2.1-2.3 and 2.5.
3. New coalition values
We shall recall some facts which are useful later. A coalition structure C = {C\,..., Cm} on a player set is an exogenous partition of players into a set of groups, i.e. U'm=1Ci = N and Ci n Cj = 0 for i = j. The sets making up the partition are called components. We also assume Ce = 0 for all Ce € C. Denote by 3N a set of all coalition structures on a fixed player set N. A TU-game with coalition structure (CS-game) (N, v, C) consists of TU-game (N, v), where v € GN, and the coalition structure C € 3N. A family of all TU-games with coalition structure and player set N is denoted by UN. For any set G C GN and any set 3 € 3N a coalition value (CS-value) on G is a function f : G x 3 ^ Rn that associates with each game (N,v,C) a vector f (N,v,C) € Rn, where fi(N,v,C) represents the player i’s payoff in game (N, v) with coalition structure C. Denote by F(N, v, C) a set of all CS-values of game (N, v, C).
Let
M = {e I Ce € C}
is a set of coalitional indices in C and let p € M be fixed. For every S € Cp € C let
CS = {C1,...,Cp-1,S,Cp+1,...,Cm}
is a partition of set N \ (Cp \ S). Similarly (Owen, 1977) we consider the following types of games.
1. Given a game (N, v, C) € UN the quotient game (or the external game) is the TU-game (M, vc) determined by
vc(Q) = v(U Ce), Q C M. (4)
e<EQ
This is interpreted as the game between the components of C in which each coalition Cp € C acts as a player.
2. Given a game (N, v, C) € UN and a coalition 0 = S € Cp € C the modified quotient game is the TU-game (M,vcs ) determined by
vcS (Q) = v(S ^ y Ce) forQ 3 p, vc S (Q) = vc (Q) for Q C M \ p. (5)
eeQ\p
A game (M,vcs) is played between the subcoalition S of component Cp and the remaining components of structure C. Assume that as the solution concept of this game the value
■0 € &(M, vcs)
is chosen.
3. Given a game (N, v, C) € UN and a component Cp the reduced game (or the internal game) is the TU-game (Cp, ) within coalition Cp. Worth (S) of every nonempty coalition S C Cp is equal to its payoff in the game (M,vcs) between components of structure CS, i.e. (S) = 0p(M,vcs) , whenever 0 = S C Cp. Assume that as the solution concept of this game the value
V G $(Cp ,vPp )
is chosen. The corresponding CS-value ^0 € F(N, v, C) is determined by
^0i(N, v, C) = <f(Cp, v$), i € Cp € C. (6)
Let f € F(N,v,C). We will use the following translations of axioms 2.1-2.4 to the coalitional framework.
Axiom 3.1 (efficiency). For all (N,v,C) € UN, ^ieN fj,(N,v,C) = v(N).
Axiom 3.2 (additivity). For any two (N,v,C), (N,w,C) € UN, f (N,v+w,C) = f (N, v, C)+ f (N,w,C).
Axiom 3.3 (external symmetry). For all (N,v,C) € UN and any two symmetric in (M,vc) players r, e € M the total values for coalitions Cr,Ce, are equal, i.e. 12ic fi(N, v,C) = J2ic fi(N,v,C).
Axiom 3.4 (internal symmetry). For all (N,v,C) € UN, any two players i,j who are symmetric in (N, v) and belong to the same component in C, get the same payoffs, i.e. fi(N, v, C) = fj(N, v, C).
Axiom 3.5 (null player property). For all (N,v,C) € UN and every i € Nu(N, v), fi(N, v, C) = 0.
The consensus CS-value (Zinchenko, et al., 2010) is obtained by replacing the functions 0 and ^ in (6) by the consensus value.
Definition 1. The consensus CS-value KK for (N, v, C) € UN is determined by KKi(N, v, C) = K(Cp, vK), i € Cp € C,
where
vK(S) = Kp(M, vcs), S C Cp € C. (7)
So the consensus CS-value relates to the consensus value as the Owen value relates to the Shapley value. We can say that consensus CS-value is a composition of the consensus value with itself. It is easily verified that the consensus CS-value coincides with the consensus value in case that all unions are singletons and satisfies the quotient game property (the total payoff assigned to the players of an a component equals the payoff of this component in the quotient game). The consensus CS-value is the unique value on the class of CS-games satisfying the axioms 3.1-3.4 and modified dummy axiom (Zinchenko, et al., 2010). Similarly to consensus CS-value we will introduce two new value for games with coalition structure.
Definition 2. The consensus-Shapley CS-value KSh for (N,v,C) € UN is determined by
KShi(N, v, C) = Shi(Cp, vK), i C Cp € C, where function v^ is determined by (7).
The consensus-Shapley CS-value reflects the result of a bargaining procedure by which, in the quotient game, each a component receives a payoff determined by the consensus value and, within each component, the members share this payoff in
accordance with the Shapley value. In the grand coalition, there are no outside options, hence for trivial structure C = {N} the consensus-Shapley CS-value coincides with the two-step Shapley value.
In order to axiomatize the consensus and the consensus-Shapley CS-values we need to express v^ through characteristic functions of original game. For every coalition 0 = S C Cp we have
(7),(3) Ep(M,vc S)+ Shp(M,vc S)
v,
W.P) 1 r ^ , V°p (M) - SeGM VCS (e) 1 ^ f f
— — b 2^ Pmq(vcs[P U Q) - vCs{Q))\
QÇM\p
(A) 1 r / o\ U (W \ Cp)) - ¡/(S') -- 2HS)+ m
+ ^ Pmq(v (S U y Ce) - v ( y Ce))]-
QÇM\p eeQ eeQ
Because pmq = ^ for q = 0
:(S ) =
mv(S ) + v (S U (N \ Cp)) - £ e€M\p v (Ce )
2m
+ E U Ce)-v(\J Ce)), 9^SCCpeC. (8)
QGM\p e£Q e£Q
Q=9
Definition 3. The Shapley-consensus CS-value ShK for (N, v, C) € UN, is determined by
ShKi(N, v, C) = Ki(Cp, vSph), i C Cp € C,
where
vSh(S) = Shp(M, vcs), S C Cp € C. (9)
Hence, the Shapley-consensus CS-value uses Shapley value in the quotient game between components and the consensus value in reduced game within component.
4. Axiomatization
First, we introduce three modification of axiom 3.5.
Axiom 4.1 (first modified null player property). Let (N, v, C) € UN, Cp € C and i € Cp n Nu(N, v) = 0. Then
fi(N,v,C) = Apj>M’l'c), (10)
2cpm
where
Ap(M,vc ) = v (N \ Cp) - £ v (Ce). (11)
e£M\p
2
Axiom 4.2 (second modified null player property). Let (N,v,C) G UN, Cp G C and i G Cp O Nu(N, v) = 0. Then
Shp(M, vc) ~J2jec Shp(M,vCj) fi(N,v, C) = —---------- eCv------------------------------ -
ZCp
Axiom 4.3 (third modified null player property). Let (N, v, C) G UN and Cp G C. Then for every i G Cp O Nu(N, v) = 0 it holds that:
(i) if Cp C Nu(N,v), then fi(N,v,C) is determined by (10),
(ii) if Ap(M,vC) =0 then
Kp(M,vc) -£~c Kp(M,vcj) fi(N, v, C) = ——-------- ^ V---------(12)
ZCp
It is easily to prove that if both conditions Cp C Nu(N, v) and Ap(M, vc) = 0 hold for some p G M, then formulas (10) and (12) are reduced to fi(N, v, C) = 0, i G Cp. The axioms 4.1-4.3 show that even a null player can receive some portion of bargaining surplus if a coalition that he belongs to generates it. Notice, that axioms 4,1 and 4.2 determine the payoffs of all null players in (N, v), but the axiom 4.3 determines payoffs of players which belong to null component Cp C Nu(N, v) or are null (see Lemma 2 below) in reduced game within Cp. The following lemma shows that the null player i G Cp O Nu(N,v) always is null in reduced game (Cp,vSh) inside his components Cp.
Lemma 1. Let (N,v,C) G UN, p G M and i G Cp O Nu(N,v) = 0. Then
?p,vSh
i G Nu(Cp, vSh).
Proof. We have to prove that under maded assumptions vSh(S U i) = vSh(S) for all S C Cp \ i. Take S C Cp G C, then
vSph(s) (9=(D £ Pmq[vcs(Q Up) - vcs (Q)]
QCM\p
Pmq [v(S UU Ce) - v(U Ce)],
QCM\p eeQ eeQ
vSh(S U i) - vSh(S) = Pmq [v(S U i U U Ce) - v(S ^ Ce)] 0
QCM\p eeQ eeQ
for all S C Cp \ i. □
Next lemma gives the necessary and sufficient condition at which the null player in (N, v) remains null in reduced game within his component.
Lemma 2. Let (N,v,C) G UN, p G M and i G Cp O Nu(N,v) = 0. Then
i G Nu(Cp,vK) iff Ap(M,vc) = 0, where Ap(M,vc) is determined by (11).
Proof. It is sufficient to show that under maded assumptions vK (S U i) = vK (S)
for all S G Cp \ i iff Ap(M, vc ) = 0. We consider two cases.
Case (a). S = 0, S C Cp \ i. Then
Kic,, (8) m[v (S U i) + v (S)]+ v(S U i U (N \ Cp)) - v(S U (N \ Cp))
vKK (S U i) - vK (S)
2m
+ £ ^fMSu.uUc.)-KSu|Jcy] o.
QGM\p e£Q e£Q
Q=9
This gives
vK(S U i)= vK(S), p G M, i G Cp O Nu(N, v), 0 = S G Cp \ i. (13)
Case (b). S = 0. Then
K K (8) mv(i)+ v(i U (N \ Cp)) - EeeM\p v(Ce)
vp (S U i) = vp (i)
2m
, Pmq, /-,,1 /| I^ym (ieNu(N,u)) v(N\Cp) J2eeM\p v(Ce)
+ \J ce) -KU ^)] = -^-•
QGM\p e£Q e£Q
Q=0
Hence,
v?(i) = VC\ ieCpn Nu(N, i/), p e M, (14)
■> vv - 2m , • -vp (V = 0 iff Ap
i.e. vK(i) = 0 iff Ap(M, vc) =0. □
Lemma 3. For any (N,v,C) G UN, the consensus CS-value KK G F(N,v,C) satisfies the axioms 3.1 - 3.4 and 4.3.
Proof. Because the consensus value satisfies the axioms 2.1 - 2.3, the consensus CS-value satisfies the axioms 3.1 - 3.4 (Gomez-Rua and Vidal-Puga, 2008). Let us see that KK satisfies the axiom 4.3. Assume p G M and Cp O Nu(N,v) = 0. We consider two cases.
Case (a). Cp C Nu(N, v). By Definition 1 and axiom 2.1, we have
]T KKi(N, v, C) =J2 Ki(Cp, vK) = vK (Cp) = Kp(M, vc).
i£Cp i£Cp
Since Cp contains the null players only, the player p G M is null in (M, vc). By axiom 2.5
TS ^ , ^c(M)-£eeMi/c(e) (4) v(N)-J2eGMv{Ce)
Kp{M,vc) = vc{p) +--------------^----------- = HCP) +-------------^-------------
_ V(N \ CP) ~ Y,eeM\p "(Ce) (11) AP(M, vc)
2m 2m
All players of coalition Cp are symmetric in (N, v), therefore
KKi(N,v, C)
(axiom 3.4) Kp(M,vc) Ap(M,vc)
2cpm
for all i G Cp, i.e. KK satisfies (10).
Case (b). Ap(M,vC) = 0. Take i G Cp O Nu(N,v). By Lemma 2 we have
i G Nu(Cp, vpK). As the consensus value satisfies the neutral dummy property and Nu(Cp, vK) C Du(Cp, vK), then
TSTSfAT (def-^ ts (n K\ (axiom 2.5) K vp (Cp) ~ J2jeCp Vp C?)
KKi(N, v, C) = Ki(Cp, vp ) = vp(i)-\-------------—-------------
c
p
for each i G Cp O Nu(N,v). It is follows from vK(i) = 0, vK(Cp) (==) Kp(M,vC) and vK(j) == Kp(M,vCj), j G Cp, that KK satisfies (12). □
We are going to characterize the consensus CS-value by replacing the modified dummy axiom (Zinchenko, et al., 2010) by the third modified null player property (axiom 4.3).
Theorem 1. KK is the unique value on the class of CS-games UN satisfying the axioms 3.1 - 3.4 and 4.3.
Proof. By Lemma 3 the consensus CS-value satisfies the axioms 3.1- 3.4 and 4.3. Using the standard scheme of characterization of CS-values through the additivity axiom, we will prove that these properties uniquely determine the payoffs for (multiplied) unanimity CS-game (N,auT,C), where 0 = T C N, a G R. The theorem statement will follows from that the set {uT}$=tcN form a basis for GN.
Let f be a CS-value on GN satisfying the mentioned axioms. First, we calculate Ap(M, vc), Kp(M, vc) and Kp(M, vcj), j G Cp, for (N, auT, C). Denote
D = {e G M| Ce n T = 9.} (15)
It is follows from
(our)(Ce) = i a IhèiWie for all Ce G C,
a, p G M \ D, 0, p G D,
that
(aur)(N \ Cp) =
Ap(M, (aur)c) (= (aur)(N\Cp)- ^ (aur)(Ce) = j
1\Æ\ ^ V
a, p G M \ D and d > 1, 0, otherwise.
EM\p
Hemce, in game (N, aur, C) for everyone component Cp G C at least one of conditions of axiom 4.3 holds true: Cp Ç Nu(N,aur) or Ap(M, (aur)C) = 0. From (3) and following formulas
Shi(N, aur)
0, i G N \ T, f. ieT,
0, i G N \ T, t = 1, Ei(N, aur) = ^ a, i G T, t = 1,
i G N, t > 1,
we have that
Ki(N, aur)
0, i G N \ T, t =1,
a, i G T, t =1,
tGN\T, t> 1, ^±i), i G T, t> I.
Clearly, the quotient game (M, (auT)c) is (multiplied) D-unanimity game, i.e. the games (M, (auT)c) and (M, auD) are equivalent. Therefore
Kp(M, (auT)c) = Kp(M, auD) for all p G M.
Since K(N, av) = aK(N, v) for all v G GN and a G R, we have
0, p G M \ D, d =1,
is ini i \ \ _ ) a p G D, d = 1 KP(M,(auT)c) - { PeM\D,d> 1,
n d > 1
2md ’ P ^ a ,> 1.
If j G Cp G C and structure Cp = {Ci,..., Cp-1, j, Cp+1,..., Cm} is received from C by removal of the players belonging to coalition T (null players in (N,auT)), then (M, (auT)cj) coincides with (M, (auT)c) and also with (M, auD). Otherwise (M, (auT)cj) is zero-game. Thus
Kp(M, Wut)c. ) = { KptM' (““T)c0; = 0' for alj G Cp G a
Let’s show now that f (N, auT, C) is uniquely determined by the axioms enumerated in the theorem. By the axioms 3.3 and 3.1,
EE fi(N, auT, C) + E E fi(N, auT, C)
pED iE Cp pEM\D iE Cp
= d E fi(N, auT,C) + (m - d) E fi(N,auT,C) = a.
i^Cp i^Cp
pED pEM\D
If p G M \ D = 0, then Cp C Nu(N, auT) and by the axiom 4.3
fi(N,auT,C) = (at(/r)c) = | ^ d for all i £ Cp, p £ M \D.
2Cpin { 2^-, a > i,
Summing up fi,(N, auT, C) over i G Cp gives
0, d = 1
, h{l\,auT,U) = < _
i£Cp ^ 2
If p G D, then Ap(M, (auT)c) = 0 and by the axiom 4.3
Kp(M> (<*uT)c) ~ Zj€Cp Kp(M>
/¿(AT, QiUj*, C) =------------------------------------------, I eCp\T, p £ D
2 cp
E fi(N,auT,C)={ j J’ for all p £ M \ D.
■,cn I 2m’ ® >
Denote tp = \Cp O T|. From formulas for Kp(M, (auT)c) and Kp(M, (auT)cj) we have that
0, tp = 1,
fi(N, auT, C) = ^ 2ip > 1, d = I, for an j g Cp\T, p £ D.
All players of coalition Cp \ T are symmetric in (N, auT). By the axiom 3.4
E fi(N,auT, C)
i'E Cp n T
pED
512 Alexandra B. Zinchenko, Polina P. Provotorova, George V. Mironenko
a - (m - d)Yl iECp: pEM\D fi(N, auT, C)
fi(N,auT ,C).
d
iECp\T :pED
All players of coalition Cp O T are also symmetric in (N, auT). From axioms 3.4 and simple calculations we have that
a, d = tp = 1,
fi(N,auT,C)={ d=1>tP>1> for all ieCpnT, pe D.
a(m+d)(cp + tp) I
Acpmdtp ’ ’
The received formulas uniquely determine fi(N, auT, C) for all i G N. □
Theorem 2. KSh is the unique value on the class of CS-games UN satisfying the axioms 3.1 - 3.4 and 4.1.
Proof. Because the consensus and Shapley values satisfy the axioms 2.1 - 2.3, KSh satisfies the axioms 3.1 - 3.4. Let p G M and i G CpONu(N, v) = 0. From Definition
2, formulas (1), (13), (14) and the equality pc s = — for s = 0, it follows that
KSh,(N,v,C)= Yl ft>?(SU.)-<-(S)l (13M“' ^w =
2Cp 2Cpm
SCCp\i p p
i.e. KSh satisfies the axiom 4.1.
Let f be a CS-value on GN satisfying the listed axioms. All members of coalition N \ T are null players in (N, auT). By axiom 4.1
fi(N, aur, C) = Ap(M’ for all ie N\T.
2cpm
Summing up last equality yields
Ev—^ ^ A (axioms 3.3) m — d . . ^ r . , ,
fi(N, auT, C) = —^—Ap(M,(auT)c),
pEM\D iECp
V' fi(N,auT,C) = ^——Ap(M,(auT)c)) for all p e M\D, z' 2cpm
iECp\T p
where tP = \Cp O T\ and D is determined by (15). All players of coalition D are symmetric in the quotient game (M, (auT)c)), hence
d[ fi(N,auT,C)~I—p--------Ap(M, (auT)c)}-\—7-----------------AP(M, (a.uT)c) ^ a
2 Cp 2
i'E Cp nT
pE D
E„ . _ _ a tnd — Cp^m \ \
fi(N, auT, C) = — -1--2c md paUT)c)•
itCpnT p
pE D
For p G D all players of coalition Cp O T are symmetric in (N, auT). By axiom 3.4 a c ___________________________t d
fi(N, cmT, C) = — - ^ ^ ^ AP(M, (auT)c) for all i e Cp n T, p G D.
Since Cp C N \ T for p G M \ D, and Ap(M, (auT)c) is determined uniquely for all p G M (see the proof of Theorem 1), CS-value fi(N, auT, C) is also determine uniquely for all i G N. □
Theorem 3. ShK is the unique value on the class of CS-games UN satisfying the axioms 3.1 - 3.4 and 4.2.
Proof. The axiom 4.2, as well as the axiom 4.1, determines uniquely the payoffs of all null players in (N, v), therefore the proof of this theorem is similar to that of Theorem 2. It only remains to show that ShK satisfies the axiom 4.2. Let p G M and i G Cp O Nu(N, v) = 0. By Lemma 1 we have i G Nu(Cp, vSh). Hence
ShKi(N, v, C) ( =/0 K^, vsph) {axi°= 2'5) vsph (i) + ^ {Cp)+^ecp^v 0?)
2cp
It is follows from vSh(i) = 0, vSh(Cp) = Shp(M,vC) and vSh(j) == Shp(M,vCj), j G Cp, that ShK satisfies the axiom 4.2. □
Notice that KK and KSh do not satisfy the coalitional null player property introduced in (Kamijo, 2009) to characterize the two-step Shapley value. Therefore, even the member of null component Cp C Nu(N, v) can get a positive KK and KSh CS-values.
References
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