An axiomatization of the Shapley Value using a fairness property

TLDR: In this article, an axiomatization of the Shapley value for TU-games using a fairness property is provided. But this property states that if to a game we add another game in which two players are symmetric then their payoffs change by the same amount.

Abstract: In this paper we provide an axiomatization of the Shapley value for TU-games using a fairness property. This property states that if to a game we add another game in which two players are symmetric then their payoffs change by the same amount. We show that the Shapley value is characterized by this fairness property, efficiency and the null player property. These three axioms also characterize the Shapley value on the class of simple games.

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Tilburg University
An Axiomatization of the Shapley Value Using a Fairness Property
van den Brink, J.R.
Publication date:
1999
Link to publication in Tilburg University Research Portal
Citation for published version (APA):
van den Brink, J. R. (1999).
An Axiomatization of the Shapley Value Using a Fairness Property
. (CentER
Discussion Paper; Vol. 1999-120). Econometrics.
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Download date: 09. aug.. 2022

An Axiomatization of the Shapley Value
using a Fairness Prop erty

Renevan den Brink
Department of Econometrics
and CentER
Tilburg University
P.O. Box 90153
5000 LE Tilburg
The Netherlands
Decemb er 1999

This paper is a revised version of TI-discussion pap er 8-95-249, \ An axiomatization of the Shapley
value using component eciency and fairness". I would like to thank Gerard van der Laan and Eric van
Damme for useful remarks on a previous draft of this pap er. Financial support from the Netherlands
organization for scientic research (NWO) ESR-grant 510-01-0504 is gratefully acknowledged.

Abstract
In this paper we provide an axiomatization of the Shapley value for TU-
games using a
fairness
property. This prop erty states that if to a game we
add another game in whichtwo players are
symmetric
then their payos change
by the same amount. We show that the Shapley value is characterized by this
fairness property, eciency and the null player property. These three axioms
also characterize the Shapley value on important sub classes of games, suchas
the class of simple games or the class of apex games.
Keywords
: TU-game, Shapley value, fairness, simple games.
JEL classication number: C71
1 Intro duction
A situation in which a nite set of players can obtain certain payos by cooperation
can b e described bya
cooperative game with transferable utility
{or simply a TU-game{
being a pair (
N; v
), where
N
=
f
1
;:::;n
g
is the set of players and
v
:2
N
!
IRisa
characteristic function
such that
v
(
;
) = 0. Since we take the set of players
N
to be
xed, we represent a TU-game by its characteristic function
v
. The collection of all
characteristic functions on
N
is denoted by
G
N
.
A (single valued) solution for TU-games is a function
f
:
G
N
!
IR
N
which as-
signs an
j
N
j
-dimensional real vector to every TU-game. This vector can b e seen as
a distribution of the payos that can b e obtained by co operation over the individual
players in the game. A famous solution is the
Shapley value
(Shapley (1953a)). Var-
ious axiomatizations of the Shapley value have been given. In this paper we provide
an axiomatization of the Shapley value using eciency, the null player property and a
fairness
property. This last property states that if to a game
v
2G
N
we add a game
w
2G
N
in which players
i
and
j
are
symmetric
then the payos of players
i
and
j
change by the same amount, i.e., if
w
(
S
[f
i
g
)=
w
(
S
[f
j
g
) for all
S

N
nf
i; j
g
then
f
i
(
v
+
w
)
,
f
i
(
v
)=
f
j
(
v
+
w
)
,
f
j
(
v
).
This concept of fairness is related to fairness as introduced byMyerson (1977)
for games in which the possibilities of coalition formation in a TU-game are limited
because of the fact that players are part of a limited
communication structure
. In that
1

model fairness means that deleting a communication relation b etween two players has
the same eect on b oth their payos. A similar fairness axiom is used in van den
Brink (1997) for games in which the cooperation possibilities in a TU-game are limited
because the players are part of a hierarchical
permission structure
in which there are
players who need p ermission from certain other players before they are allowed to
coop erate. In that model fairness means that deleting a p ermission relation between
two players has the same eect on b oth their payos. In van den Brink (1995a) a
fairness axiom for
relational power measures
for
directedgraphs
1
is introduced. In that
context fairness means that deleting a relation between two nodes in a digraph changes
their relational p ower by the same amount.
As already noted by Dubey (1975), axiomatizations of the Shapley value on
G
N
not necessarily characterize the Shapley value on important subclasses of games such
as the class of
simple games
. A TU-game
v
is simple if
v
(
S
)
2f
0
;
1
g
for all
S

N
.
It turns out that eciency, the null player property, and fairness also characterize the
Shapley value on the class of simple games. Van den Brink (1995a) shows that these
three axioms characterize the Shapley value on the even smaller class of
apex games
.
Besides the literature on fairness started in Myerson (1977), this paper also is
related to the axiomatization of the Shapley value by eciency, symmetry and strong
monotonicity given in Young (1985). A solution satises
strong monotonicity
if for
every pair of games
v; w
2G
N
and
i
2
N
, the payo of
i
in
v
is at least equal
to its payo in
w
if the
marginal contribution
of player
i
to any coalition in
v
is
at least equal to its corresponding marginal contribution in
w
, i.e.,
f
i
(
v
)

f
i
(
w
)if
v
(
S
[f
i
g
)
,
v
(
S
)

w
(
S
[f
i
g
)
,
w
(
S
) for all
S

N
nf
i
g
. As argued byChun (1991),
it is sucient to require that
f
i
(
v
)=
f
i
(
w
)if
v
(
S
[f
i
g
)
,
v
(
S
)=
w
(
S
[f
i
g
)
,
w
(
S
) for
all
S

N
nf
i
g
. So, strong monotonicity essentially compares the payo of a player if
we add a game in which this player is a
nul l player
, while fairness compares the change
in payo of two players if we add a game in which these players are
symmetric
.
1
A directed graph is a pair (
N; D
) where
N
is a nite set of nodes and
D

N

N
is a binary
relation on
N
. A relational p ower measure for directed graphs is a function that assigns real values to
all nodes in a directed graph. For a general discussion about relational power measures for directed
graphs we refer to van den Brink (1994).
2

The paper is organized as follows. In Section 2 we dene fairness and show that the
Shapley value is the unique solution on
G
N
that satises eciency, the null player
property and fairness. We also show that these three axioms characterize the Shapley
value on the class of simple games. We end Section 2 by comparing fairness with
strong monotonicity and
balancedcontributions
as considered in, e.g., Myerson (1980)
and Hart and Mass-Colell (1989). In Section 3 we generalize the characterization of
the Shapley value to
weighted Shapley values
as considered in, e.g., Shapley (1953b)
and Kalai and Samet (1987). Finally, there is an appendix that discusses
components
in TU-games which are used in the proof of the main theorem.
2 An axiomatization of the Shapley value
In this section we provide an axiomatization of the Shapley value using eciency, the
null player prop erty and fairness. The
Shapley value
(Shapley (1953a)) is the function
Sh
:
G
N
!
IR
N
given by
Sh
i
(
v
)=
X
S
3
i

v
(
S
)
j
S
j
for all
i
2
N;
with
dividends

v
(
S
)=
P
T

S
(
,
1)
j
S
j,j
T
j
v
(
T
) for all
S

N
(see Harsanyi (1959)). We
rst state the well-known eciency and null player axioms for solutions
f
:
G
N
!
IR
N
.
Player
i
2
N
is a
nul l player
in
v
2G
N
if
v
(
S
)=
v
(
S
nf
i
g
) for all
S

N
.
Axiom 2.1 (Eciency)
For every
v
2G
N
it holds that
P
i
2
N
f
i
(
v
)=
v
(
N
)
.
Axiom 2.2 (Null player property)
If
i
2
N
is a nul l player in game
v
2G
N
then
f
i
(
v
)=0
.
Players
i; j
2
N
are
symmetric
in
v
2G
N
if
v
(
S
[f
i
g
)=
v
(
S
[f
j
g
) for all
S

N
nf
i; j
g
.
Fairness states that if to a game
v
2G
N
we add a game
w
2G
N
in which players
i
and
j
are symmetric, then the payos of players
i
and
j
change by the same amount.
Axiom 2.3 (Fairness)
If
i; j
2
N
are symmetric players in
w
2G
N
, then
f
i
(
v
+
w
)
,
f
i
(
v
)=
f
j
(
v
+
w
)
,
f
j
(
v
)
for all
v
2G
N
:
3

Frequently asked questions

Q1. What are the contributions in this paper?

In this paper the authors provide an axiomatization of the Shapley value for TUgames using a fairness property. The authors show that the Shapley value is characterized by this fairness property, e ciency and the null player property.