Values for Partially Defined
Cooperative Games
David Housman
Mathematics and Computer Science
Goshen College
Undergraduate Student Collaborators:
David Letscher, Notre Dame University (1990)
Thomas Ventrudo, Loyola College of Maryland (1991)
Jodi Wallman, Bryn Mawr College (1991)
Roger Lee, Harvard University (1992)
Jennifer Rich, Drew University (1993)
LeeAnne Brutt, Allegheny College (1994)
Anna Engelsone, Goshen College (1999)
Rachelle Ramer, Goshen College (2003)
1
Cooperative Game Review
Definition. A cooperative game is a set of players
N {1,2, , n} and a worth function w from coalitions
(nonempty subsets of players) to real numbers. A
cooperative game ( N , w) is zero-monotonic if
w(S )  w(S {i})  w({i}) for all iS  N .
Example. N {1,2,3} and w is as defined in the table.
{1,2,3} {1,2}
S
w(S )
20
14
{1,3}
8
{2,3}
6
{1}
2
{2}
0
{3}
0
Definition. An allocation method is a function  from
cooperative games to allocations ( n -vectors of reals).
Definition. The Shapley value is an allocation method
defined by
(s 1)! n  s !

i ( N , w)  
 w(S )  w(S {i})


n!
S N
where s is the size of the coalition S . We use the
convention w()  0 .
Example.
1( N , w)  2!0!20  6 1!1!14  0 1!1!8  0  0!2!2  0  9




3! 
3! 
3! 
3! 
2
Definitions. The allocation method  is
 efficient if iN i ( N , w)  w( N ) for all games ( N , w) . “All
of the potential savings are allocated.”
 symmetric if  (i) ( N , w) i ( N , w) for all games ( N , w) ,
permutations  of N , and players i N , where the worth
function  w is defined by ( w)( (S ))  w(S ) for all
coalitions S . “A player’s name is irrelevant.”
 dummy subsidy-free if i ( N , w)  0 for all games ( N , w)
and players i N satisfying w(S )  w(S {i}) for all
coalitions S  N . “Players who never contribute to the
worth of any coalition receive nothing.”
 additive if i ( N ,v  w) i ( N ,v) i ( N , w) for all games
( N , v) and ( N , w) . “Accounting procedures are irrelevant.”
Theorem (Shapley, 1953). The Shapley value is the unique
allocation method that is efficient, symmetric, dummy
subsidy-free, and additive.
S
v(S )
u12(S )
u1(S )
N
12
6
2
w(S )
20
{1,2} {1,3} {2,3}
6
6
6
6
0
0
2
2
0
14
8
6
{1}
0
0
2
2
1
2
3
4
3
2
4
3
0
4
0
0
9
7
4
3
Partially Defined Cooperative Games
Definition. A partially defined cooperative game (PDG) is
a set of players N {1,2, , n}, a collection of coalitions C
containing N , and a worth function w from C to reals.
Example. N {1,2,3,4,5}, C {S  N :| S |{1,4,5}}, and w is
as defined in the table.
12345 1234 1235 1245 1345 2345
i
S
w(S ) 600
480
480
360
180
60
0
Definition. An extension of the PDG ( N ,C, w) is a game
( N , wˆ ) satisfying ŵ(S )  w(S ) for all SC .
Example. N {1,2,3,4,5} and ŵ is as defined in the table.
12345 1234 1235 1245 1345 2345
i
S
ŵ(S ) 600
480
480
360
180
60
0
S 123 124 125 134 135 145 234 235 245 345
ŵ(S ) 300 240 240 120 120 120 40 40 40 40
12
S
ŵ(S ) 120
13
60
14
60
15
60
23
20
24
20
25
20
34
20
35
20
45
20
Definition. A PDG ( N ,C, w) is zero-monotonic if it has a
zero-monotonic extension.
4
Definitions. The allocation method  is
 efficient if
iN i (N ,C, w)  w(N ) for all PDGs (N ,C, w) .
 symmetric if  (i) ( N , C, w) i ( N , C, w) for all PDGs
( N , C, w) , permutations  of N , and players i N .
 dummy subsidy-free if i ( N ,C, w)  0 for all PDGs ( N ,C, w)
and players i N satisfying wˆ (S )  wˆ (S {i}) for all
coalitions S  N and zero-monotonic extensions ( N , wˆ ) .
 additive if i ( N ,C,v  w) i ( N , C, v) i ( N , C, w) for all
PDGs ( N ,C, v) and ( N ,C, w) .
Theorem (Housman, 2001). The Shapley value (with 0s
substituted for the unknown coalitional worths) is the unique
allocation method on zero-monotonic PDGs that is efficient,
symmetric, dummy subsidy-free, and additive.
S
u (S )
u (S )
u (S )
u (S )
u (S )
N 1234 1235 1245 1345 2345 1
2
3
4
5
15
0
90
120
120
15
45
0
120
120
15
45
90
0
120
15
45
90
120
0
60
0
180 0
360 0
480 0
480 480
0
0
0
480
0
0
0
360
0
0
0
180
0
0
0
60
0
0
0
0
0
45
90
120
120
u (S ) 960 0
w(S ) 600 480
0
480
0
360
0
180
0
60
192 192 192 192 192
183 153 108 78 78
1
2
3
4
5
0
5
Why not use the Shapley Value?
Five-player PDG:
12345 1234
S
w(S )
5
4
1235
3
1245
2
1345
1
2345
0
i
0
Any zero-monotonic extension satisfies
S 123 124 125 134 135 145 234 235 245 345
ŵ(S ) 3 2 2 1 1 1 0
0
0
0
12
S
ŵ(S ) 0
13
0
14
0
15
0
23
0
24
0
25
0
34
0
35
0
45
0
0  wˆ (ij)  wˆ (ijk )  w(ijkl)
The Shapley Value of w equals the Shapley Value of the
extension ŵ for which wˆ (ij)  wˆ (ijk )  0 .
This is the apex of the “pyramid” of extensions.
It gives player 1 the minimum payoff among all possible
extension Shapley values.
Additivity is too strong of a property because we ask for the
allocations to add even when the sets of extensions do not.
6
Weakly Additive Values
Definitions. The allocation method  is
 weakly additive if i ( N ,C,v  w) i ( N , C, v) i ( N , C, w)
whenever ext( N ,C,v)  ext( N , C, w)  ext( N, C, v  w) .
 proportional if i ( N ,C, aw)  ai ( N , C, w) for all real
numbers a and PDGs ( N ,C, w) .
Theorem. Suppose C {S  N :| S |{1, n 1, n}} and ext(w) is
the set of zero-monotonic extensions of the PDG ( N ,C, w) .
Then ext(v)  ext(w)  ext(v  w) if and only if there exists a
permutation  of N such that
v(N { (1)})  v(N { (2)})   v(N { (n)}) and
w(N { (1)})  w( N { (2)})   w( N { (n)}) .
12345
1234
1235
1245
1345
2345
Suppose w(1234)  w(1235)  w(1245)  w(1345)  w(2345) .
1
1
1
1
1
1
1
1
1
1
1
1
0
1
1
1
1
0
0
1
1
1
0
0
0
1
1
0
0
0
0
1
0
0
0
0
0
1
5
1 a4
1 (1 a )
3
2
1 (1 a )
2
3
1
4
1
5
2
3
4
5
1
5
1
5
1
5
1
5
1a
4 4
1a
4 4
1a
3 3
1a
4 4
1a
3 3
1a
2 2
1
4
1
5
1a
4 4
1a
3 3
1a
2 2
1 (1 a )
3
2
1 (1 a )
2
3
1
4
1
5
1 (1 a )
2
3
1
4
1
5
0
1
5
7
In the following, assume that C {S  N :| S |{1, n 1, n}} and
all extensions are zero-monotonic.
Definition. Suppose M  N . Define the PDG vM by
vM (N )  vM (N {i}) 1 if iM and vM (S )  0 otherwise.
Theorem. If  is efficient and symmetric, then there are
a0, a1, , an satisfying
1
a ,
M
i (v )   m1 m
 nm (1 am ),

if iM
if iM
where a0  0 and an 1. If  is also dummy subsidy-free,
1)
then a1  0 . If  is also additive, then am  mn((m
for all m .
n1)
Theorem. The allocation method  is efficient, symmetric,
dummy subsidy-free, proportional, and weakly additive if
and only if
 (i) (w)  1n w( N { (1)})
i
  n1j1 an j1[w( N { ( j)})  w( N { ( j 1)})]

j 2
n

1 (1 a
)[w( N { ( j)})  w( N { ( j 1)})]
n j 1
j 1
j i1
 1n [w( N )  w( N { (n)})]
8
Geometric Approach
Definition. Let eT be the game satisfying eT (T ) 1 and
eT (S )  0 for all S  T .
Definition. Suppose w is a PDG. The extension ŵ is a
coordinate center of ext(w) if ŵ is the midpoint of the line
segment {wˆ  eT :  }ext(w) for all coalitions T  N .
Theorem (Brutt, 1994). For each zero-monotonic PDG, a
coordinate center exists and is unique.
Example. The PDG
12345
S
w(S ) 600
1234
480
1235
480
1245
360
1345
180
2345
60
i
0
has coordinate center
S 123 124 125 134 135 145 234 235 245 345
ŵ(S ) 300 240 240 120 120 120 40 40 40 40
12
S
ŵ(S ) 120
13
60
14
60
15
60
23
20
24
20
25
20
34
20
35
20
45
20
9
Coordinate Center Value
Definition. The coordinate center value  is defined by
 (w) is the Shapley value of the coordinate center of ext(w) .
Theorem (Brutt, 1994). The coordinate center value is
given by the formula
 (i) (w)  1n w( N { (1)})
i
  n1j1 an j1[w( N { ( j)})  w( N { ( j 1)})]

j 2
n

1 (1 a
)[w( N { ( j)})  w( N { ( j 1)})]
n j 1
j 1
j i1
 1n [w( N )  w( N { (n)})]
m 1 for m 1,2, , n  2 and a  n 1.
n1 2n
n(n  m 1)
Thus, it is an efficient, symmetric, dummy subsidy-free,
proportional, and weakly additive allocation method.
where am 
Example. The PDG
12345
S
w(S ) 600
1234
480
1235
480
1245
360
1345
180
2345
60
i
0
has the coordinate center value (224, 164, 94, 59, 59).
Compare to the Shapley value (183, 153, 108, 78, 78).
10
Other Centers
Centroid:
o The center of mass of ext(w) .
o Not weakly additive and difficult to compute.
Coordinate Extrema Center:
o wˆ (S )  1 min{uˆ(S ):uˆ ext(w)} max{uˆ(S ):uˆ ext(w)}

2
o Theorem (Brutt, 1994). The coordinate extrema center
ŵ of ext(w) satisfies wˆ (S )  1 min{w(T ): S  T}.
2
o So, the Shapley value of the coordinate extrema center
is an efficient, symmetric, subsidy free, proportional,
and weakly additive allocation method.
Chebyshev Center:
o The center of the smallest hypersphere containing
ext(w) .
o Theorem (Engelsone, 1999). The Chebyshev center of
ext(w) is the coordinate extrema center of ext(w) .
11
Comparison
1
0
0
0
0
0
0
0
0
0
Max for 1
1
1
0
0
0
Coordinate
1
1
1
0
0
Chebyshev
1
1
1
1
0
Centroid
1
1
1
1
1
Shapley
N
1234
1235
1245
1345
2345
Payoff to Strong Players
.400
.350
.300
.250
.200
.583
.420
.317
.250
.200
.600
.425
.317
.250
.200
.600
.433
.317
.250
.200
.800
.500
.333
.250
.200
1
1
0
0
0
1
0
0
0
0
Max for 1
1
1
1
0
0
Coordinate
1
1
1
1
0
Chebyshev
1
1
1
1
1
Centroid
1
1
1
1
1
Shapley
N
1234
1235
1245
1345
2345
Payoff to Weak Players
.200
.150
.100
.050
.000
.200
.104
.053
.025
.000
.200
.100
.050
.025
.000
.200
.100
.045
.025
.000
.200
.050
.000
.000
.000
12
2345
1
1
0
0
0
1
0
0
0
0
.20
.40
.35
.30
.25
1
1
0
0
0
1
0
0
0
0
.200
.600
.433
.317
.250
2345
1
1
1
0
0
1345
1
1
1
1
0
1245
1
1
1
1
1
1235
1
1
1
1
1
1234
1
N
2345
1345
1
1
1
0
0
1345
1245
1
1
1
1
0
1245
1235
1
1
1
1
1
1235
1234
1
1
1
1
1
1234
1
N
N
Comparison
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
1
1
1
0
0
1
1
0
0
0
1
0
0
0
0
Shapley Value
2
3
4
5
.20
.15
.35
.30
.25
.20
.15
.10
.30
.25
.20
.15
.10
.05
.25
.20
.15
.10
.05
.00
Coordinate Center Value
2
3
4
5
.200
.100
.433
.317
.250
.200
.100
.044
.317
.250
.200
.100
.044
.025
.250
.200
.100
.044
.025
.000
Shapley
Coord Center
strong weak strong weak
.40
.35
.30
.25
.20
.15
.10
.05
.00
.600
.433
.317
.250
.200
.100
.044
.025
.000
13
Other Past Results and Future Work
Generalize zero-monotonic PDGs on {S  N :| S |{1, n 1, n}}
to arbitrary C .
Axiomatically characterize the coordinate center value.
Consider other types of extensions, including superadditive
and convex games.
Consider other types of centers, including centroid, extreme
coordinate center, and Chebyshev center.
14