The $D&P$ Shapley value: a weighted extension
First, we propose a weighted extension of the D&P Shapley value and then study several equivalences among the potentializability and some properties. On the basis of these equivalences and consistency, two axiomatizations are also proposed.
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| Дата: | 2016 |
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Institute of Mathematics, NAS of Ukraine
2016
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507696057286656 |
|---|---|
| author | Yu-Hsien, Liao Ю-Ґсіен, Ляо |
| author_facet | Yu-Hsien, Liao Ю-Ґсіен, Ляо |
| author_sort | Yu-Hsien, Liao |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T09:28:56Z |
| description | First, we propose a weighted extension of the D&P Shapley value and then study several equivalences among the potentializability and some properties. On the basis of these equivalences and consistency, two axiomatizations are also proposed. |
| first_indexed | 2026-03-24T02:13:25Z |
| format | Article |
| fulltext |
UDC 517.9
Yu-Hsien Liao (Department Appl. Mah., Nat. Pingtung Univ. Education, Taiwan)
THE D&P SHAPLEY VALUE: A WEIGHTED EXTENSION
D&P ЗНАЧЕННЯ ШЕПЛI: ЗВАЖЕНЕ РОЗШИРЕННЯ
First, we propose a weighted extension of the D&P Shapley value and then study several equivalences among the
potentializability and some properties. On the basis of these equivalences and consistency, two axiomatizations are also
proposed.
Спочатку запропоновано зважене розширення D&P значення Шеплi, а потiм вивчено кiлька властивостей еквiвалент-
ностi мiж потенцiалiзовнiстю та деякими iншими властивостями. На основi цих еквiвалентностей та узгодженостi
також отримано двi аксiоматизацiї.
1. Introduction. For traditional games, weights are assigned to the ”players” to modify the discrimi-
nations among players. Since players in multichoice transferable-utility (TU) games could be allowed
to have more than one activity levels, it is reasonable that weights could be assigned to the ”activity
levels” to modify the discriminations among activity levels. The weights have different significance
in different fields. For example, weights could be treated as parameters to modify the discriminations
among different activity levels of investment strategies. Here we propose a weighted extension of the
multichoice solution introduced by Derks and Peters [2], which we name the weighted D\&P value.
In the framework of traditional games, Hart and Mas-Colell [3] proposed the potential to show
that the Shapley value can be resulted as the marginal contributions vector of an unique potential.
Hart and Mas-Colell [3] also defined the self-reduced game and related consistency to characterize
the Shapley value. Subsequently, Ortmann [6, 7] and Calvo and Santos [1] propose some equivalent
relations to characterize the collection of all traditional solutions that admit a potential.
Here we build on the results of Hart and Mas-Colell [3], Ortmann [6, 7] and Calvo and Santos [1]
on 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] on multichoice
TU games, and show that the weighted D\&P value can be resulted as the marginal contributions vector
of a weighted potential.
2. Inspired by the results due to Ortmann [6, 7] and Calvo and Santos [1], we characterize
the collection of all multichoice solutions that admit a weighted potential. Here we provide some
equivalences among the potentializability of a solution, the properties of the weighted balanced
contributions and the equal loss. Further, we adopt the weighted potential to characterize the
weighted D\&P value of an auxiliary game.
3. By considering the players and the activity levels simultaneously, we propose an extended
self-reduction and related consistency. Different from the potential approach of Hart and Mas-
Colell [3], we show that the weighted D\&P value satisfies consistency based on ”dividend”. Finally,
we characterize the weighted D\&P value by means of the result (2) and consistency respectively.
2. Preliminaries. Let U be the universe of players. Suppose each player i \in U could be allowed
to havemi \in \BbbN actively levels. Also, we setMi = \{ 0, 1, . . . ,mi\} as the actively level space of player
i, where 0 means not participating, and M+
i =Mi \setminus \{ 0\} . For N \subseteq U,N \not = \varnothing , let MN =
\prod
i\in N
Mi
c\bigcirc YU-HSIEN LIAO, 2016
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 1 131
132 YU-HSIEN LIAO
be the product set of the actively level spaces for players in N, and MS
+ =
\prod
i\in S
M i
+ for all S \subseteq N.
Denote the zero vector in \BbbR N by 0N .
A multichoice TU game is a triple (N,m, v), where N is a finite and nonempty set of players,
m = (mi)i\in N is the vector that describes the amount of activity levels for each player, and v :
MN \rightarrow \BbbR is a characteristic function which assigns to each action vector x = (xi)i\in N \in MN the
worth when each player i participates at activity level xi \in Mi with v(0N ) = 0. Given a game
(N,m, v) and x \in MN , we write (N, x, v) for the multichoice TU subgame obtained by restricting
v to \{ y \in MN | yi \leq xi \forall i \in N\} . Denote the class of all multichoice TU games by MC.
Let w : \BbbN \cup \{ 0\} \rightarrow \BbbR + be a nonnegative function such that 0 = w(0) < w(l) \leq w(k) for all
l \leq k, then w could be called a weight function. Given (N,m, v) \in MC, let LN,m = \{ (i, j) | i \in
\in N, j \in M+
i \} .
Given a a weight function w for the actions. A solution on MC is a map\psi w assigning to each
(N,m, v) \in MC an element
\psi w(N,m, v) =
\bigl(
\psi wi,j(N,m, v)
\bigr)
(i,j)\in LN,m \in \BbbR L
N,m
.
Here \psi wi,j(N,m, v) is the value of the player i when he takes actively level j to participate in game
(N,m, v). For convenience, we define \psi wi,0(N,m, v) = 0 for all (N,m, v) \in MC and for all i \in N.
Given S \subseteq N, let | S| be the number of elements in S and let eS(N) be the binary vector in \BbbR N
whose component eSi (N) satisfies
eSi (N) =
\left\{ 1 if i \in S,
0 otherwise.
Note that eS(N) will be denoted by eS if no confusion can arise.
Given (N,m, v) \in MC, x \in MN and i \in N, we define \| x\| w =
\sum n
k=1
w(xk), \| x\| =
=
\sum
k\in N
xk and S(x) = \{ k \in N | xk \not = 0\} .
For all x, y \in \BbbR N , we say y \leq x if yi \leq xi for all i \in N. The analogue of unanimity games for
multichoice TU games are minimal effort games (N,m, uxN ), where x \in MN \setminus \{ 0N\} , defined by
for all y \in MN ,
uxN (y) =
\left\{ 1 if y \geq x,
0 otherwise.
It is known that for (N,m, v) \in MC, it holds that v =
\sum
x\in MN\setminus \{ 0N\}
ax(v)uxN , where ax(v) =
=
\sum
S\subseteq S(x)
( - 1)| S| v(x - eS) is called to be the dividend among the necessary levels in x.
Definition 1. The weighted D&P value \Theta w is the solution on MC which associates with each
(N,m, v) \in MC, each weight function w, each player i \in N and each level j \in M+
i the value
\Theta w
i,j(N,m, v) =
\sum
x\in MN
xi\geq j
w(xi)
\| x\| w
\cdot ax(v) .
By the definition of \Theta w, all players allocate the dvivdend based on weights proportionably. The
weighted D&P value \Theta w
i,j is the ”weighted-marginal accumulation” of player i from level j to mi.
The weight w(j) could be treated as a prior reward of the activity level j.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 1
THE D&P SHAPLEY VALUE: A WEIGHTED EXTENSION 133
Remark 1. Derks and Peters [2] proposed the D\&P Shapley value \Theta . For each (N,m, v) \in MC,
each player i \in N and each level j \in M+
i ,
\Theta i,j(N,m, v) =
\sum
x\in MN
xi\geq j
ax(v)
\| x\|
.
Klijn et al. [5] and Hwang and Liao [4] provided several axiomatizations of the D\&P Shapley value
respectively.
3. Potentializability. Let N \subseteq U. For x \in \BbbR N and S \subseteq N, we write xS to be the restriction of
x at S. For (i, j), (k, l) \in LN,m, we introduce the substitution notation x - i to stand for xN\setminus \{ i\} and
let y = (x - i, j) \in \BbbR N be defined by y - i = x - i and yi = j. Moreover, x - ik to stand for xN\setminus \{ i,k\}
and let z = (x - ik, j, l) \in \BbbR N be defined by z - ik = x - ik, zi = j and zk = l.
Given a weight function w and (N,m, v) \in MC, we define a function Pw : MC - \rightarrow IR which
associates a real number Pw(N,m, v). Moreover,
Di,jPw(N,m, v) =
mi\sum
k=j
w(k)
\Bigl[
Pw
\bigl(
N, (m - i, k), v
\bigr)
- Pw
\bigl(
N, (m - i, k - 1), v
\bigr) \Bigr]
.
Definition 2. Let w be a weight function. A solution \psi w admits a w-potential if there exists a
function Pw : MC \rightarrow \BbbR satisfies for all (N,m, v) \in MC with N \not = \varnothing and for all (i, j) \in LN,m,
\psi wi,j(N,m, v) = Di,jPw(N,m, v).
Moreover, a function Pw : MC - \rightarrow \BbbR is 0-normalized if Pw(N, 0N , v) = 0 for each N \subseteq U.
Pw is efficient if for all (N,m, v) \in MC,
\sum
i\in N
mi\sum
j=1
Di,jPw(N,m, v) = v(m). (1)
Theorem 1. Let w be a weight function. A solution \psi w admits a uniquely 0-normalized and
efficient w-potential Pw if and only if \psi w is the solution \Theta w on MC. For all (N,m, v) \in MC and
for all (i, j) \in LN,m,
\Theta w
i,j(N,m, v) = Di,jPw(N,m, v).
Proof. Given a weight function w and (N,m, v) \in MC. Formula (1) can be rewritten as
Pw(N,m, v) =
=
1
\| m\| \| m\| w
\left[ v(m) +
\sum
i\in S(m)
mi - 1\sum
j=0
j
\Bigl(
w(j) - w(j + 1)
\Bigr)
Pw
\Bigl(
N, (m - i, j), 0), v
\Bigr) \right] . (2)
Starting with Pw(N, 0N , v), it determines Pw(N,m, v) recursively. This shows the existence of
the weighted potential Pw, and moreover that Pw(N,m, v) is uniquely determined by (1) (or (2))
applied to (N, x, v) for all x \in MN . Let
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 1
134 YU-HSIEN LIAO
Pw(N,m, v) =
\sum
x\in MN\setminus \{ 0N\}
1
\| x\| w
\cdot ax(v) . (3)
It is easy to check that (1) is satisfied by this Pw; hence (3) defines the uniquely 0-normalized
and efficient weighted potential. The result now follows since for all (i, j) \in LN,m,
\Theta w
i,j(N,m, v) = Di,jPw(N,m, v) =
\sum
x\in MN
xi\geq j
w(j)
\| x\| w
\cdot ax(v).
4. Equivalences and axiomatization. Here we provide some equivalences to characterize the
weighted D&P value. Let w be a weight function and \psi w be a solution on MC.
Efficiency (EFF): For all (N,m, v) \in MC,
\sum
i\in N
\sum mi
j=1
\psi wi,j(N,m, v) = v(m). \psi w is said
to be weak efficiency (WEFF) if for all (N,m, v) \in MC with | S(m)| = 1, \psi w satisfies EFF.
Weighted balanced contributions (WBC): For all (N,m, v) \in MC and all i, k \in N, i \not = k,
1
w(mi)
\Bigl[
\psi wi,mi
\bigl(
N,m, v
\bigr)
- \psi wi,mi
\bigl(
N,m - e\{ k\} , v
\bigr) \Bigr]
=
=
1
w(mk)
\Bigl[
\psi wk,mk
\bigl(
N,m, v
\bigr)
- \psi wk,mk
\bigl(
N,m - e\{ i\} , v
\bigr) \Bigr]
.
Equal loss (EL)1: For all (N,m, v) \in MC and all (i, j) \in LN,m, j \not = mi,
\psi i,j
\bigl(
N,m, v
\bigr)
- \psi i,j
\bigl(
N,m - e\{ i\} , v
\bigr)
= \psi i,mi
\bigl(
N,m, v
\bigr)
.
\psi w is said to be weak equal loss (WEL) if for all (N,m, v) \in MC with | S(m)| = 1, \psi w satisfies EL.
Definition 3. Given (N,m, v) \in MC and \psi w be a solution. The auxiliary multichoice TU game
(N,m, v\psi w) is defined by
v\psi w(x) =
\sum
i\in S(x)
xi\sum
j=1
\psi wi,j(N, x, v)
for all x \in MN . Note that v = v\psi w if \psi w satisfies efficiency.
Theorem 2. Let w be a weight function and \psi w be a solution. The following are equivalent:
(a) \psi w admits a w-potential;
(b) \psi w satisfies WBC and EL;
(c) \psi w(N,m, v) = \Theta w(N,m, v\psi w) for all (N,m, v) \in MC.
Proof. Let w be a weight function and \psi w be a solution. To verify (a)\Rightarrow (b), suppose \psi w admits
a w-potential Pw. For all (N,m, v) \in MC and for all i, k \in N, i \not = k,
1
w(mi)
\Bigl[
\psi wi,mi
\bigl(
N,m, v
\bigr)
- \psi wi,mi
\bigl(
N,m - e\{ k\} , v
\bigr) \Bigr]
=
=
1
w(mi)
w(mi)
\Bigl[
Pw
\bigl(
N,m, v
\bigr)
- Pw
\bigl(
N,m - e\{ i\} , v
\bigr) \Bigr]
-
1This axiom was introduced by Klijn, Slikker and Zazuelo [5].
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 1
THE D&P SHAPLEY VALUE: A WEIGHTED EXTENSION 135
- 1
w(mi)
w(mi)
\bigl[
Pw
\bigl(
N,m - e\{ k\} , v
\bigr)
- Pw
\bigl(
N,m - e\{ i\} - e\{ k\} , v
\bigr) \bigr]
=
=
\Bigl[
Pw
\bigl(
N,m, v
\bigr)
- Pw
\bigl(
N,m - e\{ k\} , v
\bigr) \Bigr]
-
-
\Bigl[
Pw
\bigl(
N,m - e\{ i\} , v
\bigr)
- Pw
\bigl(
N,m - e\{ i\} - e\{ k\} , v
\bigr) \Bigr]
=
=
1
w(mk)
w(mk)
\bigl[
Pw
\bigl(
N,m, v
\bigr)
- Pw
\bigl(
N,m - e\{ k\} , v
\bigr) \bigr]
-
- 1
w(mk)
w(mk)
\Bigl[
Pw
\bigl(
N,m - e\{ i\} , v
\bigr)
- Pw
\bigl(
N,m - e\{ i\} - e\{ k\} , v
\bigr) \Bigr]
=
=
1
w(mk)
\Bigl[
\psi wj,mk
\bigl(
N,m, v
\bigr)
- \psi wj,mk
\bigl(
N,m - e\{ i\} , v
\bigr) \Bigr]
.
Hence, \psi w satisfies WBC. Next, we show that \psi w satisfies EL. For all (N,m, v) \in MC and for all
(i, j) \in LN,m, j \not = mi,
\psi wi,j
\Bigl(
N,m, v
\bigr)
- \psi wi,j
\bigl(
N,m - e\{ i\} , v
\Bigr)
=
=
mi\sum
t=j
w(t)
\Bigl[
Pw
\bigl(
N, (m - i, t), v
\bigr)
- Pw
\bigl(
N, (m - i, t - 1), v
\bigr) \Bigr]
-
-
mi - 1\sum
t=j
w(t)
\Bigl[
Pw
\bigl(
N, (m - i, t), v
\bigr)
- Pw
\bigl(
N, (m - i, t - 1), v
\bigr) \Bigr]
=
= w(mi)
\bigl[
Pw
\Bigl(
N,m, v
\bigr)
- Pw
\bigl(
N,m - e\{ i\} , v
\bigr) \Bigr]
= \psi wi,j(N,m, v).
That is, \psi w satisfies EL.
To verify (b)\Rightarrow (c), suppose \psi w satisfies WBC and EL. Let (N,m, v) \in MC. The proof proceeds
by induction on the number \| m\| . If \| m\| = 1, let S(m) = \{ i\} and mi = 1, then by the definition of
v\psi w and efficiency of \Theta w, \psi wi,1(N,m, v) = v\psi w(m) = \Theta w
i,1(N,m, v\psi w). Suppose that \psi w(N,m, v) =
= \Theta w(N,m, v\psi w) for \| m\| \leq k, where k \geq 1.
Case \| m\| = k + 1 : Let i \in S(m). By induction hypotheses and WBC of \psi w and \Theta w, for all
k \in S(m) with k \not = i,
1
w(mi)
\psi wi,mi
(N,m, v) - 1
w(mk)
\psi wk,mk
(N,m, v) =
=
1
w(mi)
\psi wi,mi
(N,m - e\{ k\} , v) - 1
w(mk)
\psi wk,mk
(N,m - e\{ i\} , v) =
=
1
w(mi)
\Theta w
i,mi
(N,m - e\{ k\} , v\psi w) - 1
w(mk)
\Theta w
k,mk
(N,m - e\{ i\} , v\psi w) =
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 1
136 YU-HSIEN LIAO
=
1
w(mi)
\Theta w
i,mi
(N,m, v\psi w) - 1
w(mk)
\Theta w
k,mk
(N,m, v\psi w).
So we have that,
1
w(mi)
\bigl[
\psi wi,mi
(N,m, v) - \Theta w
i,mi
(N,m, v\psi w)
\bigr]
=
=
1
w(mk)
\bigl[
\psi wk,mk
(N,m, v) - \Theta w
k,mk
(N,m, v\psi w)
\bigr]
. (4)
By induction hypotheses and EL of \psi w and \Theta w, for every (h, l) \in LN,m, l \not = mh,
\psi wh,l
\bigl(
N,m, v
\bigr)
- \Theta w
h,l
\bigl(
N,m, v\psi w
\bigr)
=
=
\Bigl[
\psi wh,mh
\bigl(
N,m, v
\bigr)
+ \psi wh,l
\bigl(
N,m - e\{ h\} , v
\bigr) \Bigr]
-
-
\Bigl[
\Theta w
h,mh
\bigl(
N,m, v\psi w
\bigr)
+\Theta w
h,l
\bigl(
N,m - e\{ h\} , v\psi w
\bigr) \Bigr]
=
= \psi wh,mh
\bigl(
N,m, v
\bigr)
+\Theta w
h,l
\bigl(
N,m - e\{ h\} , v\psi
\bigr)
-
- \Theta w
h,mh
\bigl(
N,m, v\psi w
\bigr)
- \Theta w
h,l
\bigl(
N,m - e\{ h\} , v\psi w
\bigr)
=
= \psi wh,mh
\bigl(
N,m, v
\bigr)
- \Theta w
h,mh
\bigl(
N,m, v\psi w
\bigr)
. (5)
By definition of v\psi w , efficiency of \Theta w, equations (4), (5) and the induction hypotheses,
0 = v\psi w(m) - v\psi w(m) =
=
\sum
h\in S(m)
mh\sum
l=1
\psi wh,l(N,m, v) -
\sum
h\in S(m)
mh\sum
l=1
\Theta w
h,l(N,m, v) =
=
\sum
h\in S(m)
\bigl[
\psi wh,mh
(N,m, v) - \Theta w
h,mh
(N,m, v\psi w)
\bigr]
=
=
\sum
h\in S(m)
mhw(mh)
w(mi)
\bigl[
\psi wi,mi
(N,m, v) - \Theta w
i,mi
(N,m, v\psi w)
\bigr]
. (6)
Hence, \psi wi,mi
(N,m, v) - \Theta w
i,mi
(N,m, v\psi w) = 0. By equations (4), (5) and (6), \psi wk,l(N,m, v) =
= \Theta w
k,l(N,m, v\psi w) for all (k, l) \in LN,m.
To verify (c)\Rightarrow (a), suppose that \psi w(N,m, v) = \Theta w(N,m, v\psi w) for all (N,m, v) \in MC. Since
the weighted D&P value \Theta w admits a unique w-potential P\Theta w , we define a w-potential of \psi w as
P\psi w(N,m, v) = P\Theta w(N,m, v\psi w) for all (N,m, v) \in MC. Then for every (i, j) \in LN,m,
Di,jP\psi w(N,m, v) =
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 1
THE D&P SHAPLEY VALUE: A WEIGHTED EXTENSION 137
=
mi\sum
k=j
w(k)
\bigl[
P\psi w
\bigl(
N, (m - i, k), v
\bigr)
- P\psi w
\bigl(
N, (m - i, k - 1), v
\bigr) \bigr]
=
=
mi\sum
k=j
w(k)
\bigl[
P\Theta w
\bigl(
N, (m - i, k), v\psi w
\bigr)
- P\Theta w
\bigl(
N, (m - i, k - 1), v\psi w
\bigr) \bigr]
=
= \Theta w
i,j(N,m, v\psi w) = \psi wi,j(N,m, v).
Hence, P\psi w is a w-potential of \psi w.
Theorem 3. A solution \psi w satisfies EFF, EL and WBC if and only if \psi w = \Theta w.
Proof. Given a weight function w. By equation (1) and Theorem 1, it easy to check that \Theta w
satisfies EFF. Since \Theta w admits a w-potential, \Theta w satisfies WBC and EL by Theorem 2.
By Definition 3 and EFF of \Theta w, v\Theta w = v. By Theorem 2, the proof is completed.
5. Player-action reduction and axiomatization. Here we propose the player-action reduction
and related consistency to characterize the weighted D&P value.
Given (N,m, v) \in MC, a solution \psi w on MC, S \subseteq N,S \not = \varnothing and \gamma \in M
N\setminus S
+ . The player-
action reduced game (S,mS , v
\psi w
S,\gamma ) with respect to S, m, \gamma and \psi w is defined as follows. For all
\alpha \in MS ,
v\psi
w
S,m,\gamma (\alpha ) = v(\alpha , \gamma ) -
\sum
k\in N\setminus S
\gamma k\sum
t=1
\psi wk,t(N, (\alpha , \gamma ), v).
The player-action reduction asserts that, when reapportioning the payoff allotment within S, all
members in N \setminus S take nonzero levels based on the action vector \gamma to cooperate. Then in the
player-action reduction, the coalition S takes activity level \alpha to cooperate with the coalition N \setminus S
with activity level \gamma . A solution \psi w satisfies player-action consistency (PACON) if for all S \subseteq N,
for all (i, j) \in LS,mS and for all \gamma \in M
N\setminus S
+ , \psi wi,j
\bigl(
S,mS , v
\psi w
S,m,\gamma
\bigr)
= \psi wi,j
\bigl(
N, (mS , \gamma ), v
\bigr)
.
Lemma 1. Let (N,m, v) \in MC and (S,mS , v
\Theta w
S,m,\gamma ) be a player-action reduced game. If v =
=
\sum
\alpha \in MN\setminus \{ 0N\}
a\alpha (v)\cdot u\alpha N , then v\Theta
w
S,\gamma can be expressed to be v\Theta
w
S,m,\gamma =
\sum
\alpha \in MS\setminus \{ 0S\}
a\alpha (v\Theta
w
S,m,\gamma )u
\alpha
S ,
where for all \alpha \in MS ,
a\alpha (v\Theta
w
S,m,\gamma ) =
\sum
\beta \leq \gamma
\| \alpha \| w
\| \alpha \| w + \| t\| w
\cdot a(\alpha ,t)(v).
Proof. Let (N,m, v) \in MC, S \subseteq N and \gamma \in M
N\setminus S
+ . For all \alpha \in MS ,
v\Theta
w
S,m,\gamma (\alpha ) = v(\alpha , \gamma ) -
\sum
k\in N\setminus S
\gamma k\sum
t=1
\Theta w
k,t
\bigl(
N, (\alpha , \gamma ), v
\bigr)
. (7)
By EFF of \Theta w, v\Theta
w
S,\gamma (0S) = 0 and for all \alpha \in MS \setminus \{ 0S\} ,
(7) =
\sum
k\in S(\alpha )
\alpha k\sum
t=1
\Theta w
k,t(N, (\alpha , \gamma ), v) =
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 1
138 YU-HSIEN LIAO
=
\sum
k\in S(\alpha )
\alpha k\sum
t=1
\sum
\mu \leq (\alpha ,\gamma )
\mu k\geq t
w(\mu k)a
\mu (v)
\| \mu \| w
=
=
\sum
k\in S(\alpha )
\left[ \sum
\mu \leq (\alpha ,\gamma )
\mu k\geq 1
w(\mu k)a
\mu (v)
\| \mu \| w
+ . . .+
\sum
\mu \leq (\alpha ,\gamma )
\mu k\geq \alpha k
w(\mu k)a
\mu (v)
\| \mu \| w
\right] =
=
\sum
k\in S(\alpha )
\left[ \sum
p\leq \alpha
pk\geq 1
\sum
\beta \leq \gamma
w(pk)a
(p,\beta )(v)
\| p\| w + \| \beta \| w
+ . . .+
\sum
p\leq \alpha
pk\geq \alpha k
\sum
\beta \leq \gamma
w(pk)a
(p,\beta )(v)
\| p\| w + \| \beta \| w
\right] =
=
\sum
p\leq \alpha
\sum
\beta \leq \gamma
\| p\| w
\| p\| w + \| \beta \| w
a(p,\beta )(v). (8)
Set
ap(v\Theta
w
S,m,\gamma ) =
\sum
\beta \leq \gamma
\| p\| w
\| p\| w + \| \beta \| w
\cdot a(p,\beta )(v).
By equation (8), for all \alpha \in MS ,
v\Theta
w
S,m,\gamma (\alpha ) =
\sum
p\leq \alpha
\sum
\beta \leq \gamma
\| p\| w
\| p\| w + \| \beta \| w
\cdot a(p,\beta )(v) =
\sum
p\leq \alpha
ap(v\Theta
w
S,m,\gamma ).
Hence v\Theta
w
S,m,\gamma can be expressed to be v\Theta
w
S,m,\gamma =
\sum
\alpha \in MS\setminus \{ 0S\}
a\alpha (v\Theta
w
S,m,\gamma ) \cdot u\alpha S .
Different from the potential approach of Hart and Mas-Colell [3], we investigate the player-action
consistency of the weighted D&P value by applying dividend.
Lemma 2. The solution \Theta w satisfies PACON.
Proof. Let (N,m, v) \in MC, S \subseteq N and \gamma \in M
N\setminus S
+ . By Definition 1 and Lemma 1, for all
(i, j) \in LS,mS ,
\Theta w
i,j(S,mS , v
\Theta w
S,m,\gamma ) =
\sum
\alpha \in MS
\alpha i\geq j
w(\alpha i)a
\alpha (v\Theta
w
S,m,\gamma )
\| \alpha \| w
=
=
\sum
\alpha \in MS
\alpha i\geq j
w(\alpha i)
\| \alpha \| w
\sum
t\leq \gamma
\| \alpha \| w
\| \alpha \| w + \| t\| w
\cdot a(\alpha ,t)(v) =
=
\sum
\beta \leq (mS,\gamma )
\beta i\geq j
w(\alpha i)a
\beta (v)
\| \beta \| w
= \Theta w
i,j(N, (mS , \gamma ), v).
Hence, the solution \Theta w satisfies PACON.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 1
THE D&P SHAPLEY VALUE: A WEIGHTED EXTENSION 139
Lemma 3. If a solution \psi w satisfies PACON and WEFF, then \psi w satisfies EFF.
Proof. Let w be a weight function and \psi w be a solution. Assume that \psi w satisfies WEFF and
PACON. Let (N,m, v) \in MC. It is trivial for | S(m)| = 1 by WEFF. Assume that | S(m)| \geq 2. Let
k \in S(m). By the definition of the reduction,
v\psi
w
\{ k\} ,m,mN\setminus \{ k\}
(mk) = v(m) -
\sum
i\in N\setminus \{ k\}
mi\sum
j=1
\psi wi,j(N,m, v).
Since \psi w satisfies PACON, for all j \in M+
k ,
\psi wk,j(N,m, v) = \psi wk,j
\Bigl(
N,mk, v
\psi w
\{ k\} ,m,mN\setminus \{ k\}
\Bigr)
.
By WEFF of \psi w,
v\psi
w
\{ k\} ,m,mN\setminus \{ k\}
(mk) =
mk\sum
j=1
\psi wk,j
\Bigl(
N,mk, v
\psi w
\{ k\} ,m,mN\setminus \{ k\}
\Bigr)
=
mk\sum
j=1
\psi wk,j(N,m, v).
Hence
\sum
i\in N
\sum mi
j=1
\psi wi,j(N,m, v) = v(m), i.e., \psi w satisfies EFF.
Lemma 4. Given a weight function w, a solution \psi w, (N,m, v) \in MC, S \subseteq N, and y \in
\in MS \setminus \{ 0S\} . Then \Bigl(
S, y, v\psi
w
S,m,mN\setminus S
\Bigr)
=
\Bigl(
S, y, v\psi
w
S,(y,mN\setminus S),mN\setminus S)
\Bigr)
.
Proof. It is easy to derive this result by the definitions of a subgame and a reduced game, we
omit it.
Lemma 5. If a solution \psi w satisfies WEL and PACON , then it also satisfies EL.
Proof. Let w be a weight function and \psi w be a solution on MC. Suppose that a solution \psi w on
MC satisfies WEL and PACON. Let (N,m, v) \in MC , i \in N and j \in M+
i \setminus \{ mi\} . Let y = m - e\{ i\} ,
consider the reduction
\Bigl(
\{ i\} , (mi - 1), v\psi \{ i\} ,y,mN\setminus S
\Bigr)
of the subgame (N, y, v) of (N,m, v) with respect
to \{ i\} , y, mN\setminus S and \psi w, and the reduction
\Bigl(
\{ i\} ,mi, v
\psi w
\{ i\} ,m,mN\setminus S
\Bigr)
of (N,m, v) with respect to \{ i\} ,
m, mN\setminus S and \psi w, respectively. By Lemma 4, it is easy to see that
\Bigl(
\{ i\} , (mi - 1), v\psi
w
\{ i\} ,y,mN\setminus S
\Bigr)
is
the subgame of
\Bigl(
\{ i\} ,mi, v
\psi w
\{ i\} ,m,mN\setminus S
\Bigr)
, i.e.,\Bigl(
\{ i\} , (mi - 1), v\psi
w
\{ i\} ,y,mN\setminus S
\Bigr)
=
\Bigl(
\{ i\} , (mi - 1), v\psi
w
\{ i\} ,m,mN\setminus S
\Bigr)
.
Hence
\psi wi,j(N,m, v) - \psi wi,j(N,m - e\{ i\} , v) =
= \psi wi,j(N,m, v) - \psi wi,j(N, y, v) (by y = m - e\{ i\} ) =
= \psi i,j
\Bigl(
\{ i\} ,mi, v
\psi
\{ i\} ,m,mN\setminus S
\Bigr)
- \psi i,j
\Bigl(
\{ i\} , (mi - 1), v\psi \{ i\} ,y,mN\setminus S
\Bigr)
(by PACON) =
= \psi i,j
\Bigl(
\{ i\} ,mi, v
\psi
\{ i\} ,m,mN\setminus S
\Bigr)
- \psi i,j
\Bigl(
\{ i\} , (mi - 1), v\psi \{ i\} ,m,mN\setminus S
\Bigr)
(by Lemma 4) =
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 1
140 YU-HSIEN LIAO
= \psi i,mi
\Bigl(
\{ i\} ,mi, v
\psi
\{ i\} ,m,mN\setminus S
\Bigr)
(by WEL) = \psi i,mi(N,m, v) (by PACON).
So, \psi satisfies EL.
Subsequently, we characterize the weighted D&P value by means of player-action consistency.
A solution \psi w satisfies standard for two-person games (ST) if for all (N,m, v) \in MC with
| S(m)| = 2, \psi w(N,m, v) = \Theta w(N,m, v).
Remark 2. By adding a ”dummy” player to one-person games, it is easy to show that if a
solution \psi w satisfies PACON and ST, then \psi w(N,m, v) = \Theta w(N,m, v) for all (N,m, v) \in MC
with | S(m)| = 1. Hence, If \psi w satisfies PACON and ST, it satisfies WEFF and WEL.
Theorem 4. 1. A solution \psi w satisfies WEFF, WEL, WBC and PACON if and only if \psi w = \Theta w.
2. A solution \psi w satisfies ST and PACON if and only if \psi w = \Theta w.
Proof. Given a weight function w. The proof of Proposition 1 of this theorem follows from
Remark 2, Lemmas 2, 3, 5 and Theorem 3.
To prove Proposition 2 of this theorem. By Definition 1, it is easy to see that \Theta w satisfies ST.
By Lemma 2, \Theta w satisfies PACON. To prove uniqueness of Proposition 2 of this theorem, suppose
that the solution \psi w on MC satisfies ST and PACON. By Remark 2, Lemmas 3 and 5, \psi w satisfies
EFF and EL. By Proposition 1 of this theorem, it remains to show that \psi w satisfies WBC. Given
(N,m, v) \in MC. The proof proceeds by induction on the number \| m\| . Assume that \| m\| = 1
and S(m) = \{ i\} . By EFF of \psi w and \Theta w, \psi wi,1(N,m, v) = v(m) = \Theta w
i,1(N,m, v). Assume that
\psi w(N,m, v) = \Theta w(N,m, v) if \| m\| \leq l - 1, where l \geq 2.
Case \| m\| = l : Two cases may be distinguish:
Case 1: Assume that | S(m)| \leq 2. Since \psi w satisfies ST, \psi w(N,m, v) = \Theta w(N,m, v).
Case 2: Assume that | S(m)| \geq 3. Let i, k \in S(m) and S = \{ i, k\} . By PACON of \psi w and
Lemma 4,
1
w(mi)
\Bigl[
\psi wi,mi
(N,m, v) - \psi wi,mi
(N,m - e\{ k\} , v)
\Bigr]
=
=
1
w(mi)
\Bigl[
\psi wi,mi
\Bigl(
S,mS , v
\psi w
S,m,mN\setminus S
\Bigr)
- \psi wi,mi
\Bigl(
S,mS - e\{ k\} , v\psi
w
S,m - e\{ k\} ,mN\setminus S
\Bigr) \Bigr]
=
=
1
w(mi)
\Bigl[
\psi wi,mi
\Bigl(
S,mS , v
\psi w
S,m,mN\setminus S
\Bigr)
- \psi wi,mi
\Bigl(
S,mS - e\{ k\} , v\psi
w
S,m,mN\setminus S
\Bigr) \Bigr]
. (9)
By ST of \psi w, PACON and WBC of \Theta w and Lemma 4,
(9) =
1
w(mi)
\Bigl[
\Theta w
i,mi
\Bigl(
S,mS , v
\psi w
S,m,mN\setminus S
\Bigr)
- \Theta w
i,mi
\Bigl(
S,mS - e\{ k\} , v\psi
w
S,m,mN\setminus S
\Bigr) \Bigr]
=
=
1
w(mk)
\Bigl[
\Theta w
k,mk
\Bigl(
S,mS , v
\psi w
S,m,mN\setminus S
\Bigr)
- \Theta w
k,mk
\Bigl(
S,mS - e\{ i\} , v\psi
w
S,m,mN\setminus S
\Bigr) \Bigr]
=
=
1
w(mk)
\Bigl[
\psi wk,mk
\Bigl(
S,mS , v
\psi w
S,m,mN\setminus S
\Bigr)
- \psi wk,mk
\Bigl(
S,mS - e\{ i\} , v\psi
w
S,m,mN\setminus S
\Bigr) \Bigr]
=
=
1
w(mk)
\Bigl[
\psi wk,mk
\Bigl(
S,mS , v
\psi w
S,m,mN\setminus S
\Bigr)
- \psi wk,mk
\Bigl(
S,mS - e\{ i\} , v\psi
w
S,m - e\{ i\} ,mN\setminus S
\Bigr) \Bigr]
=
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 1
THE D&P SHAPLEY VALUE: A WEIGHTED EXTENSION 141
=
1
w(mk)
\Bigl[
\psi wk,mk
(N,m, v) - \psi wk,mk
(N,m - e\{ i\} , v)
\Bigr]
.
Hence, \psi w satisfies WBC.
The following examples show that each of the axioms used in Theorems 3 and 4 is logically
independent of the remaining axioms.
Example 1. Define a solution \psi w on MC by for all (N,m, v) \in MC and for all (i, j) \in LN,m,
\psi wi,j(N,m, v) = 0.
Clearly, \psi w satisfies EL(WEL), WBC and PACON, but it violates EFF(WEFF) and ST.
Example 2. Define a solution \psi w on MC by for all (N,m, v) \in MC and for all (i, j) \in LN,m,
\psi wi,j(N,m, v) =
\left\{
\Theta w
i,j(N,m, v) + \varepsilon if j = mi,
\Theta w
i,j(N,m, v) -
\varepsilon
mi - 1
if j \not = mi,
where \varepsilon \in \BbbR \setminus \{ 0\} . Clearly, \psi wsatisfies EFF(WEFF), WBC and PACON, but it violates EL(WEL).
Example 3. Define a solution \psi w on MC by for all (N,m, v) \in MC and for all (i, j) \in LN,m,
\psi wi,j(N,m, v) =
\sum
x\in MN
xi\geq j
ax(v)
\| x\|
.
Clearly, \psi w satisfies EFF(WEFF), EL(WEL) and PACON, but it violates WBC.
Example 4. Define a solution \psi w on MC by for all (N,m, v) \in MC and for all (i, j) \in LN,m,
\psi wi,j(N,m, v) = \Theta w
i,j(N,m, v)
if | S(m)| = 1 or mi = 1; otherwise
\psi wi,j(N,m, v) = \Theta w
i,j(N,m, v) + \varepsilon ,
where \varepsilon > 0. Clearly, \psi w satisfies WEFF, WBC and EL(WEL), but it violates PACON.
Example 5. Define a solution \psi w on MC by for all (N,m, v) \in MC and for all (i, j) \in LN,m,
\psi wi,j(N,m, v) =
\left\{ \Theta w
i,j(N,m, v) if | S(m)| \leq 2,
\Theta w
i,j(N,m, v) - \varepsilon otherwise,
where \varepsilon \in \BbbR \setminus \{ 0\} . Clearly, \psi w satisfies ST, but it violates PACON.
References
1. Calvo E., Santos J. C. Potential in cooperative TU-games // Math. Soc. Sci. – 1997. – 34. – P. 175 – 190.
2. Derks J., Peters H. A Shapley value for games with restricted coalitions // Int. J. Game Theory. – 1993. – 21. –
P. 351 – 360.
3. Hart S., Mas-Colell A. Potential, value and consistency // Econometrica. – 1989. – 57. – P. 589 – 614.
4. Hwang Y. A., Liao Y. H. Potential approach and characterizations of a Shapley value in multichoice games // Math.
Soc. Sci. – 2008. – 56. – P. 321 – 335.
5. Klijn F., Slikker M., Zazuelo J. Characterizations of a multichoice value // Int. J. Game Theory. – 1999. – 28. –
P. 521 – 532.
6. Ortmann K. M. Preservation of differences, potential, conservity // Working paper. – Univ. Bielefeld, 1995. – № 236.
7. Ortmann K. M. Conservation of energy in nonatomic games // Working paper. – Univ. Bielefeld, 1995. – № 237.
8. Shapley L. S. A value for n-person game // Ann. Math. Stud. – Princeton: Princeton Univ. Press, 1953. – 28. –
P. 307 – 317.
Received 30.07.12
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 1
|
| id | umjimathkievua-article-1827 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:13:25Z |
| publishDate | 2016 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/ec/3fba3dba254ede7e8d0619d62334b4ec.pdf |
| spelling | umjimathkievua-article-18272019-12-05T09:28:56Z The $D&P$ Shapley value: a weighted extension D&P значення Шеплi: зважене розширення Yu-Hsien, Liao Ю-Ґсіен, Ляо First, we propose a weighted extension of the D&P Shapley value and then study several equivalences among the potentializability and some properties. On the basis of these equivalences and consistency, two axiomatizations are also proposed. Спочатку запропоновано зважене розширення D&P значення Шеплi, а потiм вивчено кiлька властивостей еквiвалент- ностi мiж потенцiалiзовнiстю та деякими iншими властивостями. На основi цих еквiвалентностей та узгодженостi також отримано двi аксiоматизацiї. Institute of Mathematics, NAS of Ukraine 2016-01-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1827 Ukrains’kyi Matematychnyi Zhurnal; Vol. 68 No. 1 (2016); 131-141 Український математичний журнал; Том 68 № 1 (2016); 131-141 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1827/809 Copyright (c) 2016 Yu-Hsien Liao |
| spellingShingle | Yu-Hsien, Liao Ю-Ґсіен, Ляо The $D&P$ Shapley value: a weighted extension |
| title | The $D&P$ Shapley value: a weighted extension |
| title_alt | D&P значення Шеплi: зважене розширення |
| title_full | The $D&P$ Shapley value: a weighted extension |
| title_fullStr | The $D&P$ Shapley value: a weighted extension |
| title_full_unstemmed | The $D&P$ Shapley value: a weighted extension |
| title_short | The $D&P$ Shapley value: a weighted extension |
| title_sort | $d&p$ shapley value: a weighted extension |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1827 |
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