Different from the potential approach of Hart and Mas-Colell (1989), we provide the dividend approach to multi-choice games. Also, we define the level-reduced game by reducing the number of the activity levels and define related consistency on multi-choice games.

Yu-Hsien Liao

English

Research Papers in Economics

     
 
 
  
  
Volume 29, Issue 2 
  
Dividend approach and level consistency for the Derks and Peters value 
  
 
 
Yu-Hsien Liao  
Department of Mathematics, Chung Jen College of Nursing, Health Science and Management 
Abstract 
Different from the potential approach of Hart and Mas-Colell (1989), we provide the dividend approach to multi-
choice games. Also, we define the level-reduced game by reducing the number of the activity levels and define related 
consistency on multi-choice games.
Dear Sir, I am sorry to bother you. I would like to submit my paper "Dividend approach and level consistency for the Derks and Peters value" 
to the Economics Bulletin . I hope that it can be considered for publication in it. Please find attached files of the paper "Dividend approach and 
level consistency for the Derks and Peters value". Thanks for your patience and assistance! Please acknowledge receipt of this email. With best 
regards, Yu-Hsien Liao 
Citation: Yu-Hsien Liao, (2009) ''Dividend approach and level consistency for the Derks and Peters value'', Economics Bulletin, Vol. 29 
no.2 pp. 1054-1062. 
Submitted: Dec 29 2008.   Published: May 19, 2009. 
 
     1 Introduction
The Shapley value (Shapley, 1953) is a well-known solution concept in
cooperative game theory. A multi-choice TU game, introduced by Hsiao
and Raghavan (1992), is a generalization of a TU game in which each
player has several activity levels. There are several extensions of the
Shapley value in the framework of multi-choice games. Here we focus
one of these extensions proposed by Derks and Peters (1993), which we
name the D&P Shapley value.
Hart and Mas-Colell (1989) were the ﬁrst to introduce the potential
approach to TU games. In consequence, they proved that the Shapley
value can result as the vector of marginal contributions of a potential
function. The potential approach is also shown to yield a characterization
for the Shapley value, particularly in terms of an internal consistency
property. In Section 3, we introduce the dividend approach to multi-
choice games. The dividend approach is a dual view of the potential
approach. By the dividend approach, we show that the D&P Shapley
value can result as the vector of aggregate of a dividend.
There are two important factors, the players and their activity levels,
for multi-choice games. By reducing the number of the players, Hwang
and Liao (2008) proposed an extension of the reduced game due to Hart
and Mas-Colell (1989) on multi-choice games. By reducing the number of
the activity levels, we deﬁne the level-reduced game and the level consis-
tency in Section 4. Finally, we show that the D&P Shapley value satisﬁes
the level consistency based on the dividend approach.
2 Deﬁnitions and Notations
Let U be the universe of players. Let N = {1,··· ,n} ⊆ U be a set
of players. Suppose each player i has mi ∈ N levels at which he can
actively participate. Let m = (m1,··· ,mn) be the vector that describes
the number of activity levels for each player, at which he can actively
participate. For i ∈ U, we set Mi = {0,1,··· ,mi} as the action space of
player i, where action 0 means not participating, and M
+
i = Mi \ {0}.
For N ⊆ U, N 6= ∅, let MN =
Q
i∈N Mi be the product set of the action
spaces for players N. Denote the zero vector in RN by 0N.
A multi-choice TU game is a triple (N,m,v), where N is a non-
empty and ﬁnite set of players, m is the vector that describes the number
of activity levels for each player, and v : MN → R is a characteristic
function which assigns to each action vector x = (x1,··· ,xn) ∈ MN the
worth that the players can obtain when each player i plays at activity
1level xi ∈ Mi with v(0N) = 0. If no confusion can arise a game (N,m,v)
will sometimes be denoted by its characteristic function v. Given a multi-
choice game (N,m,v) and x ∈ MN, we write (N,x,v) for the multi-
choice TU subgame obtained by restricting v to {y ∈ MN | yi ≤
xi ∀i ∈ N} only. Denote the class of all multi-choice TU games by MC.
Given (N,m,v) ∈ MC, let LN,m = {(i,j) | i ∈ N,j ∈ M
+
i }. A
solution on MC is a map ψ assigning to each (N,m,v) ∈ MC an
element
ψ(N,m,v) =
 
ψi,j(N,m,v)

(i,j)∈LN,m ∈ R
LN,m
.
Here ψi,j(N,m,v) is the power index or the value of the player i when he
takes action j to play game v. For convenience, given (N,m,v) ∈ MC
and a solution ψ on MC, we deﬁne ψi,0(N,m,v) = 0 for all i ∈ N.
To state the D&P Shapley value, some more notations will be needed.
Given S ⊆ N, let |S| be the number of elements in S and let eS(N) be
the binary vector in RN whose component eS
i (N) satisﬁes
e
S
i (N) =

1 if i ∈ S ,
0 otherwise .
Note that if no confusion can arise eS
i (N) will be denoted by eS
i .
Given (N,m,v) ∈ MC and x ∈ MN, we deﬁne kxk =
P
k∈N xk and
S(x) = {k ∈ N | xk 6= 0}.
Let x,y ∈ RN, we say y ≤ x if yi ≤ xi for all i ∈ N. The analogue
of unanimity games for multi-choice games are minimal eﬀort games
(N,m,ux
N), where x ∈ MN, x 6= 0N, deﬁned by for all y ∈ MN,
u
x
N(y) =

1 if y ≥ x ;
0 otherwise.
Hsiao and Raghavan (1992) showed that for (N,m,v) ∈ MC it holds
that v =
P
x∈MN\{0N}
ax(v) ux
N, where ax(v) =
P
S⊆S(x)
(−1)|S| v(x − eS).
Deﬁnition 1 (Derks and Peters, 1993) The D&P Shapley value
Θ is the solution on MC which associates with each (N,m,v) ∈ MC and
each player i ∈ N and each j ∈ M
+
i the value
Θi,j(N,m,v) =
X
x∈MN,xi≥j
ax(v)
kxk
.
Note that the so called dividend ax(v) is divided equally among the
necessary levels.
23 Potential and Dividend
For x ∈ RN, we write xS to be the restriction of x at S for each S ⊆ N.
Let N ⊆ U, i ∈ N and x ∈ RN, for convenience we introduce the
substitution notation x−i to stand for xN\{i} and let y = (x−i,j) ∈ RN
be deﬁned by y−i = x−i and yi = j.
Given a function P : MC −→ R which associates a real number
P(N,m,v) to each game (N,m,v). Then for each (i,j) ∈ LN,m we deﬁne
D
i,jP(N,m,v) = P(N,m,v) − P
 
N,(m−i,j − 1),v

.
Deﬁnition 2 A solution ψ on MC admits a potential if there exists
a function P : MC → R satisﬁes for all (N,m,v) ∈ MC and for all
(i,j) ∈ LN,m,
ψi,j(N,m,v) = D
i,jP(N,m,v).
Solutions that admit a potential assign a scalar evaluation to each
game in such a way that a player’s payoﬀ is his marginal contribution
to this evaluation. Moreover, a function P : MC −→ R is said to be 0-
normalized if P(N,0N,v) = 0 for each N ⊆ U. And we say it is eﬃcient
if it satisﬁes the following condition: For all (N,m,v) ∈ MC,
X
i∈N
mi X
j=1
D
i,jP(N,m,v) = v(m).
Theorem 1 (Hwang and Liao, 2008) A solution ψ on MC admits
a uniquely 0-normalized and eﬃcient potential P if and only if ψ is the
D&P Shapley value Θ on MC. For all (N,m,v) ∈ MC and (i,j) ∈ LN,m,
Θi,j(N,m,v) = D
i,jP(N,m,v).
Given a function d : MC −→ R which associates a real number
d(N,m,v) to each (N,m,v) ∈ MC. Then for each (i,j) ∈ LN,m we
deﬁne Z
i,j
d(N,m,v) =
X
x∈MN,xi≥j
d(N,x,v).
Deﬁnition 3 A solution ψ on MC admits a dividend if there exists a
function d : MC → R satisﬁes for all (N,m,v) ∈ MC, N 6= ∅ and for
all (i,j) ∈ LN,m,
ψi,j(N,m,v) =
Z
i,j
d(N,m,v).
3Solutions that admit a dividend assign a scalar evaluation to each
game in such a way that a player’s payoﬀ is his marginal accumulation
to this evaluation. Moreover, a function d : MC −→ R is said to be 0-
normalized if d(N,0N,v) = 0 for each N ⊆ U. And we say it is eﬃcient
if it satisﬁes the following condition: For all (N,m,v) ∈ MC,
X
i∈N
mi X
j=1
Z
i,j
d(N,m,v) = v(m).
Theorem 2 Let ψ be a solution on MC. ψ admits a potential if and only
if ψ admits a dividend.
Proof. Assume that ψ be a solution on MC and ψ admits a dividend d.
Deﬁne P : MC −→ R to be that P(N,m,v) =
P
x∈MN d(N,x,v) for all
(N,m,v) ∈ MC. Since ψ admits the dividend d, for all (N,m,v) ∈ MC
and (i,j) ∈ LN,m,
ψi,j(N,m,v) =
P
x∈MN,xi≥j
d(N,x,v)
=
P
x∈MN
d(N,x,v) −
P
x∈MN,xi≤j−1
d(N,x,v)
= P(N,m,v) − P
 
N,(m−i,j − 1)

.
Hence, ψ admits the potential P.
Assume that ψ be a solution on MC and ψ admits a potential P.
Deﬁne d : MC −→ R to be that for all (N,m,v) ∈ MC, d(N,m,v) = P
T⊆S(m)(−1)|T|P(N,m − eT,v). It is easy to check that P(N,m,v) = P
x∈MN d(N,x,v). Since ψ admits the potential P, for all (N,m,v) ∈
MC and (i,j) ∈ LN,m,
ψi,j(N,m,v) = P(N,m,v) − P
 
N,(m−i,j − 1)

=
P
x∈MN
d(N,x,v) −
P
x∈MN,xi≤j−1
d(N,x,v)
=
P
x∈MN,xi≥j
d(N,x,v).
Hence, ψ admits the dividend d. The proof is completed.
Theorem 3 A solution ψ on MC admits a uniquely 0-normalized and
eﬃcient dividend d if and only if ψ is the D&P Shapley value Θ on MC.
For each multi-choice game (N,m,v) ∈ MC and (i,j) ∈ LN,m
Θi,j(N,m,v) =
X
x∈MN,xi≥j
d(N,x,v).
4Proof. It is easy to derive this result by Theorems 1 and 2.
Remark 1 Given (N,m,v) ∈ MC, by Deﬁnition 1 and Theorem 3,
d(N,x,v) =
ax(v)
kxk
for all x ∈ MN.
4 Level-reduced Game and Level Consis-
tency
In this section we deﬁne the level-reduced game and the level consis-
tency which are putting the accent on ” level ”. Based on the dividend
approach, we also show that the D&P value satisﬁes the level consistency.
By reducing the number of the players, Hwang and Liao (2008) pro-
posed an extension of the reduced game due to Hart and Mas-Colell
(1989) on multi-choice games as follows. For S ⊆ N, we denote Sc =
N \S. Given a solution ψ, a game (N,m,v) ∈ MC, and S ⊆ N, the re-
duced game
 
N,(mS,0Sc),v
ψ
S,m

with respect to ψ, S and m is deﬁned
by for all x ∈ MS,
v
ψ
S,m(x,0Sc) = v(x,mSc) −
X
i∈Sc
mi X
j=1
ψi,j
 
N,(x,mSc),v

.
Deﬁnition 4 (Hwang and Liao, 2008) A solution ψ on MC satisﬁes
consistency (CON) if for all (N,m,v) ∈ MC, for all S ⊆ N and for
all (i,j) ∈ LS,mS, ψi,j
 
N,(mS,0Sc),v
ψ
S,m

= ψi,j(N,m,v).
Theorem 4 (Hwang and Liao, 2008) The solution Θ satisﬁes CON.
Inspired by Hsiao, Yeh and Mo (1994), we deﬁne an alternative level-
reduced game and related consistency as follows. Given (N,m,v) ∈ MC
and a solution ψ. For each z ∈ MN \ {0N}, we deﬁne an action vector
z∗ = (z∗
i)i∈N where 
z∗
i = mi if zi < mi
z∗
i = 0 if zi = mi.
Furthermore, we deﬁne a new game vψ
z with z ∈ MN \ {0N} such that
for all y ≤ z,
v
ψ
z (y) = v(y ∨ z
∗) −
X
k∈S(z∗)
mk X
t=1
ψk,t(N,(y ∨ z
∗),v).
where (y ∨ z∗)i = max{yi,z∗
i} for all i ∈ N. We call (N,z,vψ
z ) a level-
reduced game of v with respect to z and the solution ψ.
5Deﬁnition 5 A solution ψ on MC satisﬁes level consistency (LCON)
if for all i ∈ N \ S(z∗) and for all j ≤ zi, ψi,j(N,m,v) = ψi,j(N,z,vψ
z ).
Remark 2 Given (N,m,v) ∈ MC, S ⊆ N and a solution ψ. Let z =
(mS,0Sc), by deﬁnitions of vψ
z and v
ψ
S,m, vψ
z (y) = v
ψ
S,m(y) for all y ≤ z.
Hence, if a solution ψ satisﬁes LCON, then ψ satisﬁes CON.
It is known that each (N,m,v) ∈ MC can be expressed as a lin-
ear combination of minimal eﬀort games and this decomposition ex-
ists uniquely. The following lemma relates the relation of coeﬃcients
of expressions between a (N,m,v) ∈ MC and its level-reduced game
(N,z,vΘ
z ) with respect to Θ and z.
Lemma 1 Let (N,m,v) ∈ MC and (N,z,vΘ
z ) be the level-reduced game
of (N,m,v) with respect to Θ and z ∈ MN \ {0N}. Obviously, z can be
written by z = (mS,zSc) for some S ⊆ N. If v =
P
y∈MN\{0N}
ay(v) · u
y
N,
then vΘ
z can be expressed to be vΘ
z =
P
0N6=y≤z
ay(vΘ
z ) · u
y
N, where for all
y ≤ z,
a
y(v
Θ
z ) =
( P
t≤mSc
k(yS,0Sc)k
k(yS,0Sc)k+k(t,0S)ka(yS,t)(v) if y = (yS,0Sc) ,
0 if y = (yS,ySc) with |S(ySc,0S)| 6= 0.
Proof. Let (N,m,v) ∈ MC and z ∈ MN \ {0N}. Obviously, z =
(mS,zSc) for some S ⊆ N where zi 6= mi for all i ∈ Sc. For any y ≤ z,
v
Θ
z (y) = v(y ∨ z
∗) −
X
k∈S(z∗)
mk X
t=1
Θk,t
 
N,(y ∨ z
∗),v

. (1)
Clearly, vΘ
z (y) = 0 if y = 0N. Since z∗ = (0S,mSc),
(1) = v(yS,mSc) −
P
k∈Sc
mk P
t=1
Θk,mk
 
N,(yS,mSc),v

=
P
k∈S(yS,0Sc)
yk P
t=1
Θk,t(N,(y ∨ z∗),v)
=
P
k∈S(yS,0Sc)
yk P
t=1
P
x≤(y∨z∗)
xk≥t
ax(v)
kxk
=
P
k∈S(yS,0Sc)
h P
x≤(y,mSc)
xk≥1
ax(v)
kxk + ··· +
P
x≤(y,mSc)
xk=yk
ax(v)
kxk
i
=
P
k∈S(yS,0Sc)
hP
p≤yS
pk≥1
P
t≤mS
a(p,t)(v)
k(p,0Sc)k+k(t,0S)k + ··· +
P
p≤yS pk=yk
P
t≤mS
a(p,t)(v)
k(p,0Sc)k+k(t,0S)k
i
=
P
x≤y
P
t≤mS
k(xS,0Sc)k
k(xS,0Sc)k+k(t,0S)k · a(xS,t)(v).
(2)
6By deﬁnition of vΘ
z , for any y ≤ z,
v
Θ
z (y) = v
Θ
z (yS,0Sc). (3)
Set
a
y(v
Θ
z ) =
( P
t≤mSc
k(yS,0Sc)k
k(yS,0Sc)k+k(t,0S)ka(yS,t)(v) if y = (yS,0Sc) ,
0 if y = (yS,ySc) with |S(ySc,0S)| 6= 0.
By (2) and (3), for all y ≤ z,
v
Θ
z (y) =
X
x≤y
X
t≤mS
k(xS,0Sc)k
k(xS,0Sc)k + k(t,0S)k
· a
(xS,t)(v) =
X
x≤y
a
x(v
Θ
z ).
Hence vΘ
z can be expressed to be vΘ
z =
P
0N6=y≤z
ay(vΘ
z ) · u
y
N.
Theorem 5 The solution Θ satisﬁes LCON.
Proof. Let (N,m,v) ∈ MC and z ∈ MN \ {0N}. Obviously, z =
(mS,zSc) for some S ⊆ N where zi 6= mi for all i ∈ Sc. For all i ∈
N \ S(z∗) and for all 0 < j ≤ zi,
Θi,j(N,z,vΘ
z ) =
P
y≤z,yi≥j
ay(vΘ
z )
kyk
=
P
y≤z,yi≥j
|S(ySc,0S)|=0
ay(vΘ
z )
kyk (By deﬁnition of vΘ
z )
=
P
y≤z,yi≥j
|S(ySc,0S)|=0
1
k(yS,0Sc)k ·
P
t≤mSc
k(yS,0Sc)k
k(yS,0Sc)k+k(t,0S)ka(yS,t)(v)
(By Lemma 1)
=
P
x∈MN,xi≥j
ax(v)
kxk
= Θi,j(N,m,v).
Hence the solution Θ satisﬁes LCON.
Remark 3 Hwang and Liao (2008) characterized the D&P value by
means of consistency. By Remark 2 and Theorem 5, it’s easy to check
that consistency could be replaced by level-consistency in those axiomati-
zations.
In fact, as some axioms are alterd to ﬁt solutions form focusing
on ”dividend”, the executions for characterizations among solutions on
multi-choice games are similar. The dividend forms not only oﬀer inter-
pretations for solutions but also provide motivations for axioms of solu-
tions.
7References
Derks J, Peters H (1993) A Shapley value for games with restricted coali-
tions. International Journal of Game Theory 21:351-360
Hart S, Mas-Colell A (1989) Potential, value and consistency. Economet-
rica 57:589-614
Hsiao CR, Raghavan TES (1992) Monotonicity and dummy free prop-
erty for multi-choice cooperative games. International Journal of Game
Theory 21:301-302
Hsiao CR, Yeh YN and Mo JP (1994) The potential of multi-choice coop-
erative game. Conference paper. International Mathematics Conference
’94, National Sun Yat-sen University, Kaohsiung, Taiwan, Dec. 2-5, 1994.
Hwang YA, Liao YH (2008) Potential approach and characterizations
of a Shapley value in multi-choice games. Mathematical Social Sciences
56,321-335
Shapley LS (1953) A value for n-person game, In: Kuhn, H.W., Tucker,
A.W.(eds.). Contributions to the Theory of Games II, Princeton:307-317
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Dividend approach and level consistency for the Derks and Peters value

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