PART IV. Games in Coalitional Form
1. Many-Person TU Games
We now consider many-person
cooperative games
. In such games there are no
restrictions on the agreements that may be reached among the players. In addition, we
assume that all payofs are measured in the same units and that there is a
transferrable
utility
which allows
side payments
to be made among the players. Side payments may
be used as inducements For some players to use certain mutually bene±cial strategies.
Thus, there will be a tendency For players, whose objectives in the game are close, to
Form alliances or coalitions. The structure given to the game by coalition Formation is
conveniently studied by reducing the game to a Form in which coalitions play a central
role. AFter de±ning the coalitional Form oF a many-person TU game, we shall learn how to
transForm games From strategic Form to coalitional Form and vice versa.
1.1 Coalitional Form. Characteristic Functions.
Let
n
≥
2 denote the number
oF players in the game, numbered From 1 to
n
,andlet
N
denote the set oF players,
N
=
{
1
,
2
,...,n
}
.A
coalition
,
S
, is de±ned to be a subset oF
N
,
S
⊂
N
,andth
es
e
to
F
all coalitions is denoted by 2
N
. By convention, we also speak oF the empty set,
∅
,a
sa
coalition, the
empty coalition
.These
t
N
is also a coalition, called the
grand coalition
.
IF there are just two players,
n
= 2, then there are Four coalition,
{∅
,
{
1
}
,
{
2
}
,N
}
.I
F
there are 3 players, there are 8 coalitions,
{∅
,
{
1
}
,
{
2
}
,
{
3
}
,
{
1
,
2
}
,
{
1
,
3
}
,
{
2
,
3
}
}
.²
o
r
n
players, the set oF coalitions, 2
N
,has2
n
elements.
De±nition.
The
coalitional form
of an
n
-person game is given by the pair
(
N,v
)
,
where
N
=
{
1
,
2
}
is the set of players and
v
is a real-valued function, called the
characteristic function
of the game, deFned on the set,
2
N
, of all coalitions (subsets of
N
), and satisfying
(i)
v
(
∅
)=0
,and
(ii) (superadditivity) if
S
and
T
are disjoint coalitions (
S
∩
T
=
∅
), then
v
(
S
)+
v
(
T
)
≤
v
(
S
∪
T
)
.
Compared to the strategic or extensive Forms oF
n
-person games, this is a very simple
de±nition. Naturally, much detail is lost. The quantity
v
(
S
) is a real number For each
coalition
S
⊂
N
, which may be considered as the value, or worth, or power, oF coalition
S
when its members act together as a unit. Condition (i) says that the empty set has value
zero, and (ii) says that the value oF two disjoint coalitions is at least as great when they
work together as when they work apart. The assumption oF superadditivity is not needed