DOI 10.1007/s11253-016-1214-5 Ukrainian Mathematical Journal, Vol. 68, No. 1, June, 2016 (Ukrainian Original Vol. 68, No. 1, January, 2016)

THE D&P SHAPLEY VALUE: A WEIGHTED EXTENSION Y.-H. Liao

UDC 517.9

First, we propose a weighted extension of the D&P Shapley value and then study several equivalences between the potentializability and other properties. On the basis of these equivalences and consistency, we also propose two axiomatizations.

1. Introduction In traditional games, the weights are assigned to the “players” with an aim to modify the discriminations among the players. Since the players in multichoice transferable-utility (TU) games could be allowed to have more than one activity level, it is reasonable that the weights should be assigned to the “activity levels” in order to modify the discriminations among the activity levels. The weights have different levels of significance in different fields. Thus, the weights could be regarded as parameters aimed at modifying the discriminations among different activity levels in the investment strategies. Here, we propose a weighted extension of the multichoice solution introduced by Derks and Peters [2]. We call this solution a weighted D&P value. Within the framework of traditional games, Hart and Mas-Colell [3] proposed a potential to show that the Shapley value can result as the vector of marginal contributions of a unique potential. Hart and Mas-Colell [3] also defined the self-reduced game and related consistency to characterize the Shapley value. Later, Ortmann [6, 7] and Calvo and Santos [1] propose some equivalent relations to characterize the collection of all traditional solutions that admit a potential. In the present paper, we generalize the results of Hart and Mas-Colell [3], Ortmann [6, 7], and Calvo and Santos [1] for the multichoice TU games. Three main results are as follows. 1. We propose a weighted extension of the potential due to Hart and Mas-Colell [3] for multichoice TU games and show that the weighted D&P value can result as the vector of marginal contributions of the weighted potential. 2. Inspired by the results of Ortmann [6, 7] and Calvo and Santos [1], we characterize the collection of all multichoice solutions that admit weighted potentials. Thus, we propose some equivalences among the potentializability of solutions, the properties of weighted balanced contributions, and the equal loss. Further, we use the weighted potential to characterize the weighted D&P value of an auxiliary game. 3. We consider the players and activity levels simultaneously and propose the extended self-reduction and related consistency. Unlike the potential approach of Hart and Mas-Colell [3], we show that the weighted D&P value satisfies the consistency based on the “dividend”. Finally, we characterize the weighted D&P value by means of result (2) and, hence, consistency. 2. Preliminaries Let U be the universe of players. Suppose that each player i 2 U could be allowed to have mi 2 N activity levels. In addition, we take Mi = {0, 1, . . . , mi } as the space of activity levels of player i, where 0 means that Department of Applied Mathematics, National Pingtung University Education, Taiwan. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 68, No. 1, pp. 131–141, January, 2016. Original article submitted July 30, 2012. 144

0041-5995/16/6801–0144

c 2016 

Springer Science+Business Media New York

T HE D&P S HAPLEY VALUE : A W EIGHTED E XTENSION

the player does not participate, and

+ i

145

= Mi \ {0}. For N ✓ U, N 6= ?, let MN =

Y

Mi

i2N

be the product set of the spaces of activity levels for the players in N, and Y M+S = M+i i2S

for all S ✓ N. We denote the zero vector in RN by 0N . A multichoice TU game is defined as a triple (N, m, v), where N is a finite but nonempty set of players, m = (mi )i2N is a vector that describes the number of activity levels for each player, and v : M N ! R is a characteristic function that assigns worth to each action vector x = (xi )i2N 2 M N and each player i participates on the activity level xi 2 Mi with v(0N ) = 0. Given a game (N, m, v) and x 2 M N , we write (N, x, v) for the multichoice TU subgame obtained by restricting v to {y 2 M N | yi  xi 8i 2 N }. Denote the class of all multichoice TU games by M C. Let w : N [ {0} ! R+ be a nonnegative function such that 0 = w(0) < w(l)  w(k) for all l  k. Then w is called a weight function. Given (N, m, v) 2 M C, let LN,m = {(i, j) | i 2 N, j 2 Mi+ }. Given a weight function w for the actions, a solution on M C is a map w assigning to each (N, m, v) 2 M C an element � w � N,m w (N, m, v) = i,j (N, m, v) (i,j)2LN,m 2 RL .

w (N, m, v) is the value of the player i who actively takes the level j in order to participate in the Here i,j w (N, m, v) = 0 for all (N, m, v) 2 M C and all i 2 N . game (N, m, v). For the sake of convenience, we define i,0 Given S ✓ N, let |S| be the number of elements in S and let eS (N ) be a binary vector in RN whose component eSi (N ) satisfies the relation 8 <1 if i 2 S, eSi (N ) = : 0, otherwise.

Note that eS (N ) is denoted by eS if this does not lead to any confusion. Given (N, m, v) 2 M C, x 2 M N and i 2 N, we define kxkw =

n X

w(xk ),

k=1

kxk =

X

xk ,

k2N

and S(x) = {k 2 N | xk 6= 0}. For any x, y 2 RN , we say y  x if yi  xi for all i 2 N. The analog of unanimity games for the multichoice TU games are the minimal effort games (N, m, uxN ), where x 2 M N \ {0N }, defined for all y 2 M N , uxN (y) =

8 <1, :

0

for y ≥ x, otherwise.

Y.-H. L IAO

146

It is known that, for (N, m, v) 2 M C, we have v=

X

ax (v)uxN ,

x2M N \{0N }

where ax (v) =

X

S✓S(x)

(−1)|S| v(x − eS )

is called the dividend among the necessary levels in x. Definition 1. The weighted D&P value ⇥w is the solution on M C that associates with every (N, m, v) 2 M C, every weight function w, each player i 2 N, and each level j 2 Mi+ with the value ⇥w i,j (N, m, v) =

X w(xi ) · ax (v) . kxk w N

x2M xi ≥j

By the definition of ⇥w , all players allocate the dividend based on weights proportionably. The weighted D&P value ⇥w i,j is the “weighted-marginal accumulation” of player i from level j to mi . The weight w(j) can be regarded as a prior reward for the activity level j. Remark 1. Derks and Peters [2] proposed the D&P Shapley value ⇥. For any (N, m, v) 2 M C, each player i 2 N, and each level j 2 Mi+ , ⇥i,j (N, m, v) =

X ax (v) . kxk N

x2M xi ≥j

Klijn, et al. [5] and Hwang and Liao [4] provided several axiomatizations of the D&P Shapley value respectively. 3. Potentializability Let N ✓ U. For x 2 RN and S ✓ N, let xS be the restriction of x at S. For (i, j), (k, l) 2 LN,m , we introduce a substitution notation x−i to stand for xN \{i} and let y = (x−i , j) 2 RN be defined by y−i = x−i and yi = j. Moreover, x−ik stands for xN \{i,k} and z = (x−ik , j, l) 2 RN is defined by z−ik = x−ik , zi = j and zk = l. Given a weight function w and (N, m, v) 2 M C, we define a function Pw : M C −! IR that associates a real number Pw (N, m, v). Moreover, i,j

D Pw (N, m, v) =

mi X k=j

h � � �i � w(k) Pw N, (m−i , k), v − Pw N, (m−i , k − 1), v .

Definition 2. Let w be a weight function. The solution w admits a w -potential if there exists a function Pw : M C ! R satisfying, for all (N, m, v) 2 M C with N 6= ? and all (i, j) 2 LN,m , the relation w i,j (N, m, v)

= Di,j Pw (N, m, v).

T HE D&P S HAPLEY VALUE : A W EIGHTED E XTENSION

147

Moreover, the function Pw : M C −! R is 0-normalized if Pw (N, 0N , v) = 0 for each

N ✓ U.

In addition, Pw is efficient if, for all (N, m, v) 2 M C, mi XX

Di,j Pw (N, m, v) = v(m).

(1)

i2N j=1

Theorem 1. Let w be a weight function. The solution w admits a uniquely 0-normalized and efficient w -potential Pw if and only if w is the solution ⇥w on M C. For all (N, m, v) 2 M C and all (i, j) 2 LN,m , i,j ⇥w i,j (N, m, v) = D Pw (N, m, v).

Proof. Given a weight function w and (N, m, v) 2 M C, formula (1) can be rewritten as # " i −1 ⇣ ⌘ ⇣ ⌘ X mX 1 Pw (N, m, v) = j w(j) − w(j + 1) Pw N, (m−i , j), 0), v . v(m) + kmkkmkw

(2)

i2S(m) j=0

Starting from Pw (N, 0N , v), it enables us to determine Pw (N, m, v) recursively. This proves the existence of the weighted potential Pw , and, in addition, that Pw (N, m, v) is uniquely determined by the application of (1) [or (2)] to (N, x, v) for all x 2 M N . Let Pw (N, m, v) =

X

x2M N \{0

N}

1 · ax (v) . kxkw

(3)

It is easy to see that (1) is satisfied by this Pw ; hence, (3) specifies a uniquely 0-normalized and efficient weighted potential. The required result now follows because, for all (i, j) 2 LN,m , i,j ⇥w i,j (N, m, v) = D Pw (N, m, v) =

X w(j) · ax (v). kxk w N

x2M xi ≥j

4. Equivalences and Axiomatization In the present section, we propose some equivalences to characterize the weighted D&P value. Let w be a weight function and let w be a solution on M C. Efficiency (EFF): For all (N, m, v) 2 M C, mi XX

w i,j (N, m, v)

= v(m).

i2N j=1

The solution w is said to be weakly efficient (WEFF) if, for all (N, m, v) 2 M C with |S(m)| = 1, w satisfies EFF.

Y.-H. L IAO

148

Weighted Balanced Contributions (WBC): For all (N, m, v) 2 M C and all i, k 2 N, i 6= k, 1 h w(mi )

w i,mi

�

� N, m, v − =

w i,mi

1 h w(mk )

�

N, m − e{k} , v w k,mk

�

�i

� N, m, v −

w k,mk

�i N, m − e{i} , v .

�

Equal Loss (EL):1 For all (N, m, v) 2 M C and all (i, j) 2 LN,m , j 6= mi , i,j

The solution satisfies EL.

w

�

� N, m, v −

i,j

� N, m − e{i} , v =

�

i,mi

�

� N, m, v .

is said to be weak equal loss (WEL) if, for all (N, m, v) 2 M C with |S(m)| = 1,

Definition 3. Given (N, m, v) 2 M C. Let is defined by v

w

w

(x) =

be a solution. The auxiliary multichoice TU game (N, m, v xi X X

w

w

)

w i,j (N, x, v)

i2S(x) j=1

for all x 2 M N . Note that v = v

w

if

satisfies the efficiency.

w

Theorem 2. Let w be a weight function and (a)

w

admits a w -potential;

(b)

w

satisfies WBC and EL;

(c)

w (N, m, v)

= ⇥w (N, m, v

w

w

be a solution. The following assertions are equivalent:

) for all (N, m, v) 2 M C.

Proof. Let w be a weight function and let w be a solution. To verify (a) ) (b), we suppose that a w -potential Pw . For all (N, m, v) 2 M C and all i, k 2 N, i 6= k, 1 h w(mi )

w i,mi

�

� N, m, v − =

w i,mi

�

N, m − e{k} , v

�i

h � �i � � 1 w(mi ) Pw N, m, v − Pw N, m − e{i} , v w(mi ) −

⇥ � � �⇤ � 1 w(mi ) Pw N, m − e{k} , v − Pw N, m − e{i} − e{k} , v w(mi )

h � �i � � = Pw N, m, v − Pw N, m − e{k} , v

1

h � � �i � − Pw N, m − e{i} , v − Pw N, m − e{i} − e{k} , v

This axiom was introduced by Klijn, Slikker, and Zazuelo [5].

w

admits

T HE D&P S HAPLEY VALUE : A W EIGHTED E XTENSION

=

⇥ � �⇤ � � 1 w(mk ) Pw N, m, v − Pw N, m − e{k} , v w(mk ) −

=

h � � �i � 1 w(mk ) Pw N, m − e{i} , v − Pw N, m − e{i} − e{k} , v w(mk )

1 h w(mk )

w j,mk

�

� N, m, v −

Hence, w satisfies WBC. Further, we show that j 6= mi , we get w i,j

⇣

� N, m, v − =

w i,j

w

�

mi X t=j

N, m − e{i} , v

w j,mk

�i N, m − e{i} , v .

�

satisfies EL. For all (N, m, v) 2 M C and all (i, j) 2 LN,m , ⌘

h � � � �i w(t) Pw N, (m−i , t), v − Pw N, (m−i , t − 1), v

−

i.e.,

149

m i −1 X t=j

h � � � �i w(t) Pw N, (m−i , t), v − Pw N, (m−i , t − 1), v

⇥ ⇣ �i � � = w(mi ) Pw N, m, v − Pw N, m − e{i} , v =

w i,j (N, m, v),

satisfies EL. To check (b) ) (c), we suppose that w satisfies WBC and EL. Let (N, m, v) 2 M C. The proof proceeds by induction on the number kmk. Assume that kmk = 1, S(m) = {i}, and mi = 1. Then, by the definition of v w and the efficiency of ⇥w , we get w

w i,1 (N, m, v)

=v

w

(m) = ⇥w i,1 (N, m, v

w

).

Suppose that w (N, m, v) = ⇥w (N, m, v w ) for kmk  k, where k ≥ 1. Case kmk = k + 1: Let i 2 S(m). By the induction hypotheses and the WBC of k 2 S(m) with k 6= i, we have 1 w(mi )

w i,mi (N, m, v)

−

1 w(mk )

w

and ⇥w , for all

w k,mk (N, m, v)

=

1 w(mi )

=

1 ⇥w (N, m − e{k} , v w(mi ) i,mi

=

1 ⇥w (N, m, v w(mi ) i,mi

w i,mi (N, m

− e{k} , v) −

w

)−

w

1 w(mk )

)−

w k,mk (N, m

− e{i} , v)

1 ⇥w (N, m − e{i} , v w(mk ) k,mk

1 ⇥w (N, m, v w(mk ) k,mk

w

).

w

)

Y.-H. L IAO

150

Thus, we conclude that 1 ⇥ w(mi )

w i,mi (N, m, v)

− ⇥w i,mi (N, m, v

By the induction hypotheses and the EL of w h,l

�

w

w

� � N, m, v − ⇥w h,l N, m, v =

=

= By the definition of v

w

h

1 ⇥ w(mk )

w

�

w h,l

h � − ⇥w h,mh N, m, v

w h,mh

w

�

− ⇥w k,mk (N, m, v

�

N, m − e{h} , v

�

�i

� {h} + ⇥w ,v h,l N, m − e

w

� � � {h} ,v N, m, v + ⇥w h,l N, m − e

� − ⇥w h,mh N, m, v

w h,mh

w k,mk (N, m, v)

w

⇤ ) .

(4)

and ⇥w , for any (h, l) 2 LN,m , l 6= mh , we can write

� N, m, v +

w h,mh

�

⇤ ) =

w

�

�

� {h} − ⇥w ,v h,l N, m − e

� � N, m, v − ⇥w h,mh N, m, v

w

�

.

w

�i

�

(5)

, the efficiency of ⇥w , equations (4), (5), and the induction hypotheses, we find

0=v

=

w

(m) − v

mh X X

w

(m)

w h,l (N, m, v)

h2S(m) l=1

=

X ⇥

w h,mh (N, m, v)

h2S(m)

=

X mh w(mh ) ⇥ w(mi )

−

mh X X

⇥w h,l (N, m, v)

h2S(m) l=1

− ⇥w h,mh (N, m, v

w i,mi (N, m, v)

h2S(m)

w

)

⇤

− ⇥w i,mi (N, m, v

w

⇤ ) .

Hence, w i,mi (N, m, v)

− ⇥w i,mi (N, m, v

w

) = 0.

By using equations (4), (5) and (6), we obtain w k,l (N, m, v)

= ⇥w k,l (N, m, v

w

)

for all (k, l) 2 LN,m .

To check (c) ) (a), we suppose that w

(N, m, v) = ⇥w (N, m, v

w

)

for all (N, m, v) 2 M C.

(6)

T HE D&P S HAPLEY VALUE : A W EIGHTED E XTENSION

151

Since the weighted D&P value ⇥w admits a unique w -potential P⇥w , we define the w -potential of P

w

(N, m, v) = P⇥w (N, m, v

w

w

as

)

for all (N, m, v) 2 M C. Thus, for any (i, j) 2 LN,m , Di,j P

w

(N, m, v) =

mi X

⇥ w(k) P

mi X

� ⇥ w(k) P⇥w N, (m−i , k), v

k=j

=

k=j

= ⇥w i,j (N, m, v Hence, P

w

is a w -potential of

Theorem 3. A solution

w

w

�

w

� N, (m−i , k), v − P w

�

w

�

N, (m−i , k − 1), v

�⇤

� − P⇥w N, (m−i , k − 1), v

w

�⇤

w i,j (N, m, v).

)=

w.

satisfies the EFF, EL and WBC if and only if

w

= ⇥w .

Proof. Given a weight function w. By equation (1) and Theorem 1, it is teasy to see that ⇥w satisfies the EFF. Since ⇥w admits a w -potential, ⇥w satisfies the WBC and EL by Theorem 2. By Definition 3 and the EFF of ⇥w , we get v⇥w = v. By Theorem 2, the proof is completed. 5. Player-Action Reduction and Axiomatization In the present section, we propose the player-action reduction and related consistency to characterize the weighted D&P value. N \S Given (N, m, v) 2 M C, a solution w on M C, S ✓ N, S 6= ?, and γ 2 M+ . The player-action w reduced game (S, mS , vS,γ ) with respect to S, m, γ, and w is defined as follows: For all ↵ 2 M S , w

vS,m,γ (↵) = v(↵, γ) −

γk X X

w k,t (N, (↵, γ), v).

k2N \S t=1

The player-action reduction is based on the idea that, when reapportioning the payoff allotment within S, all members in N \S take nonzero levels based on the action vector γ to cooperate. Then, in the case of the playeraction reduction, the coalition S takes an activity level ↵ to cooperate with the coalition N \ S with the activity level γ. The solution w satisfies the player-action consistency (PACON) if, for all S ✓ N, all (i, j) 2 LS,mS N \S and all γ 2 M+ , we get w i,j

�

� w S, mS , vS,m,γ =

w i,j w

� N, (mS , γ), v .

�

⇥ Lemma 1. Let (N, m, v) 2 M C and let (S, mS , vS,m,γ ) be a player-action reduced game. If

v=

X

↵2M N \{0N }

a↵ (v) · u↵N ,

Y.-H. L IAO

152 w

⇥ can be expressed in the form then vS,γ w

⇥ = vS,m,γ

X

w

⇥ a↵ (vS,m,γ )u↵S ,

↵2M S \{0S }

where w

⇥ a↵ (vS,m,γ )=

X

βγ

k↵kw · a(↵,t) (v), k↵kw + ktkw

for all ↵ 2 M S .

N \S

Proof. Let (N, m, v) 2 M C, S ✓ N and γ 2 M+ ⇥w vS,m,γ (↵)

= v(↵, γ) −

. For all ↵ 2 M S , we obtain

γk X X

k2N \S t=1

� � ⇥w k,t N, (↵, γ), v .

(7)

By the EFF of ⇥w and the equality w

⇥ vS,γ (0S ) = 0,

for all ↵ 2 M S \ {0S }, we find (7) =

↵k X X

⇥w k,t (N, (↵, γ), v)

k2S(↵) t=1

=

↵k X X X w(µk )aµ (v) kµkw

k2S(↵) t=1

=

3

X 6 X w(µk )aµ (v) X w(µk )aµ (v) 7 6 7 + . . . + 4 5 kµkw kµkw µ(↵,γ)

µ(↵,γ)

µk ≥1

µk ≥↵k

2

3

X X w(pk )a(p,β) (v) 7 X 6 X X w(pk )a(p,β) (v) 6 7 + ... + 4 kpkw + kβkw kpkw + kβkw 5

k2S(↵)

=

µk ≥t

2

k2S(↵)

=

µ(↵,γ)

XX

p↵ βγ

p↵

pk ≥1

βγ

p↵

pk ≥↵k

βγ

kpkw a(p,β) (v). kpkw + kβkw

We set w

⇥ ap (vS,m,γ )=

X

βγ

kpkw · a(p,β) (v). kpkw + kβkw

(8)

T HE D&P S HAPLEY VALUE : A W EIGHTED E XTENSION

153

By equation (8), for all ↵ 2 M S , we get w

⇥ vS,m,γ (↵) =

XX

p↵ βγ

X kpkw ⇥w · a(p,β) (v) = ap (vS,m,γ ). kpkw + kβkw p↵

w

⇥ Hence, vS,m,γ can be expressed in the form

X

w

⇥ vS,m,γ =

w

⇥ a↵ (vS,m,γ ) · u↵S .

↵2M S \{0S }

Unlike the potential approach of Hart and Mas-Colell [3], we investigate the player-action consistency of the weighted D&P value by applying the dividend. Lemma 2. The solution ⇥w satisfies PACON. N \S

Proof. Let (N, m, v) 2 M C, S ✓ N and γ 2 M+ we have ⇥w ⇥w i,j (S, mS , vS,m,γ )

=

⇥w ) X w(↵i )a↵ (vS,m,γ

k↵kw

↵2M S ↵i ≥j

=

. By Definition 1 and Lemma 1, for all (i, j) 2 LS,mS ,

X w(↵i ) X k↵kw · a(↵,t) (v) k↵k k↵k + ktk w w w S tγ

↵2M ↵i ≥j

X

=

β(mS ,γ)

βi ≥j

w(↵i )aβ (v) = ⇥w i,j (N, (mS , γ), v). kβkw

Hence, the solution ⇥w satisfies PACON. Lemma 3. If a solution

w

satisfies PACON and WEFF, then

w

satisfies EFF.

Proof. Let w be a weight function and let w be a solution. Assume that w satisfies WEFF and PACON. Let (N, m, v) 2 M C. The assertion is trivial for |S(m)| = 1 by WEFF. Assume that |S(m)| ≥ 2. Let k 2 S(m). By the definition of reduction, w

v{k},m,m Since

w

N \{k}

(mk ) = v(m) −

mi X X

w i,j (N, m, v).

i2N \{k} j=1

satisfies PACON, for all j 2 Mk+ , we get w k,j (N, m, v)

By the WEFF of

w k,j

=

w, w

v{k},m,m

N \{k}

(mk ) =

mk X j=1

w k,j

⇣

⇣

w

N, mk , v{k},m,m

w

N, mk , v{k},m,m

N \{k}

N \{k}

⌘

=

⌘

.

mk X j=1

w k,j (N, m, v).

Y.-H. L IAO

154

Hence, mi XX

w i,j (N, m, v)

= v(m),

i2N j=1

i.e.,

w

satisfies EFF.

Lemma 4. Given a weight function w, a solution w , (N, m, v) 2 M C, S ✓ N, and y 2 M S \ {0S }. Then ⌘ ⇣ ⌘ ⇣ w w S, y, vS,m,mN \S = S, y, vS,(y,m ),m ) . N \S

N \S

Proof. It is easy to prove this result by using the definitions of a subgame and a reduced game. Thus, we omit the proof. Lemma 5. If a solution

w

satisfies WEL and PACON, then it also satisfies EL.

Proof. Let w be a weight function and let w be a solution on M C. Suppose that a solution w on M C satisfies WEL and PACON. Let (N, m, v) 2 M⌘C, i 2 N, and j 2 Mi+ \ {mi }. Also let y = m − e{i} . ⇣ of the subgame (N, y, v) of (N, m, v) with respect to {i}, y, Consider the reduction {i}, (mi −1), v{i},y,m N \S ⇣ ⌘ w mN \S , and w , and the reduction {i}, mi , v{i},m,m of (N, m, v) with respect to {i}, m, mN \S , and w , N \S respectively. By Lemma 4, it is easy to see that

i.e.,

⇣

w

{i}, (mi − 1), v{i},y,m ⇣

Hence, w i,j (N, m, v)

−

N \S

⌘

is a subgame of

w

{i}, (mi − 1), v{i},y,m

w i,j (N, m

⇣ w = {i}, (mi − 1), v{i},m,m

N \S

N \S

⌘

,

⌘

.

− e{i} , v)

=

w i,j (N, m, v)

=

i,j

=

i,j

=

i,mi

−

w i,j (N, y, v)

⇣

{i}, mi , v{i},m,m

⇣

N \S

⌘

⇣ w {i}, mi , v{i},m,m

{i}, mi , v{i},m,m ⇣

N \S

N \S

{i}, mi , v{i},m,m

⌘ ⌘

N \S

(by y = m − e{i} )

−

i,j

−

i,j

⌘

⇣

{i}, (mi − 1), v{i},y,m

⇣

N \S

{i}, (mi − 1), v{i},m,m

(by WEL) =

⌘

N \S

i,mi (N, m, v)

⌘

(by PACON) (by Lemma 4)

(by PACON).

Therefore, satisfies EL. We now characterize the weighted D&P value by means of the player-action consistency. A solution satisfies the standard for two-person games (ST) if, for all (N, m, v) 2 M C with |S(m)| = 2, we can write w

(N, m, v) = ⇥w (N, m, v).

w

T HE D&P S HAPLEY VALUE : A W EIGHTED E XTENSION

155

Remark 2. By adding a “dummy” player to one-person games, one can easily show that if a solution w satisfies PACON and ST, then w (N, m, v) = ⇥w (N, m, v) for all (N, m, v) 2 M C with |S(m)| = 1. Hence, if w satisfies PACON and ST, then it satisfies WEFF and WEL. Theorem 4. 1. A solution

w

satisfies WEFF, WEL, WBC, and PACON if and only if

2. A solution

w

satisfies ST and PACON if and only if

w

w

= ⇥w .

= ⇥w .

Proof. Given a weight function w. The proof of Assertion 1 of this theorem follows from Remark 2, Lemmas 2, 3, 5, and Theorem 3. We now prove Assertion 2 of this theorem. By Definition 1, it is easy to see that ⇥w satisfies ST. By Lemma 2, ⇥w satisfies PACON. To prove the uniqueness of Assertion 2 of this theorem, we suppose that the solution w on M C satisfies ST and PACON. By Remark 2 and Lemmas 3 and 5, w satisfies EFF and EL. By Assertion 1 of this theorem, it remains to show that w satisfies WBC. Given (N, m, v) 2 M C. The proof proceeds by induction on the number kmk. Assume that kmk = 1 and S(m) = {i}. By the EFF of w and ⇥w , we get w i,1 (N, m, v)

Moreover, assume that

w (N, m, v)

= v(m) = ⇥w i,1 (N, m, v).

= ⇥w (N, m, v) for kmk  l − 1, where l ≥ 2.

The case kmk = l : Two subcases can be distinguished: Case 1. Assume that |S(m)|  2. Since w

w

satisfies ST, we have

(N, m, v) = ⇥w (N, m, v).

Case 2. Assume that |S(m)| ≥ 3. Let i, k 2 S(m) and let S = {i, k}. By the PACON of we find 1 h w(mi ) =

= By the ST of

w,

w i,mi (N, m, v)

1 h w(mi ) 1 h w(mi )

w i,mi

w i,mi

−

w i,mi (N, m

− e{k} , v)

⇣

⌘ w S, mS , vS,m,mN \S −

⇣

⌘ w S, mS , vS,m,mN \S −

i

w i,mi

w i,mi

⇣

w

w

S, mS − e{k} , vS,m−e{k} ,m

⇣

w

S, mS − e{k} , vS,m,mN \S

N \S

⌘i

.

PACON and WBC of ⇥w and Lemma 4, we obtain

(9) =

=

=

⌘ ⇣ ⌘i w w 1 h w ⇣ {k} ⇥i,mi S, mS , vS,m,mN \S − ⇥w , vS,m,mN \S i,mi S, mS − e w(mi )

⌘ ⇣ ⌘i w w 1 h w ⇣ {i} S, m ⇥k,mk S, mS , vS,m,mN \S − ⇥w − e , v S k,mk S,m,mN \S w(mk ) 1 h w(mk )

w k,mk

⇣

⌘ w S, mS , vS,m,mN \S −

w k,mk

⇣

w

S, mS − e{i} , vS,m,mN \S

⌘i

and Lemma 4,

⌘i (9)

Y.-H. L IAO

156

=

=

1 h w(mk ) 1 h w(mk )

w k,mk

⇣

⌘ w S, mS , vS,m,mN \S −

w k,mk (N, m, v)

−

w k,mk

w k,mk (N, m

⇣

w

S, mS − e{i} , vS,m−e{i} ,m

i − e{i} , v) .

N \S

⌘i

Hence, w satisfies WBC. The following examples show that each axiom used in Theorems 3 and 4 is logically independent of the other axioms. Example 1. We define a solution

w

on M C for all (N, m, v) 2 M C and all (i, j) 2 LN,m by w i,j (N, m, v)

Clearly,

w

= 0.

satisfies EL(WEL), WBC and PACON but violates EFF(WEFF) and ST.

Example 2. We define a solution

w i,j (N, m, v)

w

=

on M C for all (N, m, v) 2 M C and all (i, j) 2 LN,m by 8 > <⇥w i,j (N, m, v) + " > : ⇥w i,j (N, m, v) −

for j = mi ,

" mi − 1

for j 6= mi ,

where " 2 R \ {0}. Clearly, w satisfies EFF(WEFF), WBC, and PACON but violates EL(WEL). Example 3. We define a solution

w

on M C for all (N, m, v) 2 M C and all (i, j) 2 LN,m by w i,j (N, m, v)

=

X ax (v) . kxk N

x2M xi ≥j

Clearly,

w

satisfies EFF(WEFF), EL(WEL), and PACON but violates WBC.

Example 4. We define a solution

w

on M C for all (N, m, v) 2 M C and all (i, j) 2 LN,m by w i,j (N, m, v)

= ⇥w i,j (N, m, v)

if |S(m)| = 1 or mi = 1; otherwise, w i,j (N, m, v)

where " > 0. Clearly,

w

= ⇥w i,j (N, m, v) + ",

satisfies WEFF, WBC, and EL(WEL) but violates PACON.

Example 5. We define a solution

w

w i,j (N, m, v)

where " 2 R \ {0}. Clearly,

w

on M C for all (N, m, v) 2 M C and all (i, j) 2 LN,m by

=

8 <⇥w i,j (N, m, v) :

⇥w i,j (N, m, v) − ",

satisfies ST but violates PACON.

for |S(m)|  2, otherwise,

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157

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8.

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