Computational Game Theory
Lecture 9
P2/2015
György Dán
Laboratory for Communication Networks
Computational Game Theory – P2/2015
1
György Dán, http://www.ee.kth.se/~gyuri
Example: Facility location game
•
•
Set of cities N
Set of facilities F
•
•
Opening facility j costs fj
City i can produce goods worth k using the facility
•
Distance of facility j from city i
•
Cities can cooperate to decrease costs
•
dij:N FR
• SN
• c( S )  min F ' F {

a
2
Computational Game Theory – P2/2015
f j  iS min jF ' d ij }
b
6
8
f1=1
jF '
2
1
2
c
4
1
5
f2=2
György Dán, http://www.ee.kth.se/~gyuri
f3=2
2
Coalitional games
•
N player non-cooperative games
•
•
•
•
N player coalitional games
•
•
•
•
Actions, strategies, payoffs
Individual payoff maximization
• Actions chosen individually
No negotiation allowed
• Correlated NE…
Players seek to maximize their payoffs
Players form coalitions
• If they benefit from it
• Payoff assigned to the members of the coalition
Negotiation is the game
What coalitions will form?
•
What are the plausible payoff allocations between players
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3
Example: Majority game
•
Three player majority game
•
Payoffs to the players
•
•
•
•
•
Payoff to single player 0
Payoff to coalition of two players: 0.6
Payoff to grand coalition: 1
Formal model of the problem?
What would be a reasonable solution?
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4
Coalitional game with
transferable payoff
•
Coalitional game with transferable payoff <N,v>
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•
•
v(S) payoff to coalition S
•
•
•
Finite set N of players
Function v:SR SN, S (characteristic function)
arbitrary division possible between coalition members
independent of the decisions of N\S
Coalitional game with transferable payoff <N,V> is
cohesive
Sk
v ( N )   v ( S k ) partition{S1 ,..., S k } of N
k 1
•
Grand coalition has the highest payoff
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Example: Majority game
•
•
Three player majority game (|N|=3)
Payoffs
v(S ) ||S |1  0
v(S ) ||S | 2  0.6
v( N )  1
•
What would be a reasonable solution?
•
Pareto efficient?
x  (0.38,0.31,0.31)
x  (0.3,0.3,0.4)
x  (0.4,0.3,0.3)
x  (0.2,0.3,0.5)
x  (0,0.5,0.5)
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Feasible payoff profile
•
•
<N,v> coalitional game with TP
Payoff vector and its value for S
( xi ) iN
x( S )   xi
iS
•
S-feasible payoff vector
x( S )  v ( S )
•
Feasible payoff profile is an N-feasible payoff vector
x( N )  v ( N )
•
•
Allocation of payoff to the players
Feasible payoff profiles for majority game
x  (0.3,0.3,0.4)
x  (0.38,0.31,0.31)
x  (0.4,0.3,0.3)
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Individual rationality, imputation
•
Payoff profile (xi)iN is individually rational if
xi  v({i})
•
An imputation is a feasible payoff profile that is
individually rational
•
Payoff profile (xi)iN is coalitionally rational for SN if
x ( S )  v (S )
•
Example for the majority game – are these rational?
x  (0,0.5,0.5)
x  (0.1,0.5,0.4)
x  (0.2,0.4,0.4)
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Dummy and Symmetry
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Player i is a dummy player if
v ( S  {i})  v ( S )  v({i}) S  N , i  S
•
A payoff profile x satisfies the dummy property if for
every dummy player i
x ({i})  v({i})
•
Players i and j are interchangeable if
v ( S \ {i})  { j})  v ( S ) S  i, j  S
•
A payoff profile x satisfies the symmetry property if for
every pair of interchangeable players i and j
x({i})  x({ j})
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Desirable solution properties
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Existence
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Uniqueness
•
Might be several plausible solutions
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Feasibility (Efficiency)
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Individual rationality
•
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Symmetry
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Coalitional rationality?
All players do not have to be equal
Dummy player
•
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Non-contributors should get something?
György Dán, http://www.ee.kth.se/~gyuri
10
The Core
•
The core of a coalitional game with transferable payoff
<N,v> is the set of feasible payoff profiles (xi)iN for
which 
( S , ( yi ) iS )  yi  v ( S ),
•
•
yi  xi
i  S
iS
no S-feasible payoff profile offers more
The core of a coalitional game with transferable payoff
<N,v> is the set of feasible payoff profiles (xi)iN for
which
v (S )  x( S ) S  N
•
•
Coalitionally rational for all coalitions
The core is a convex, closed set
•
nonempty?
D.B. Gillies, “Solutions to General Non-zero Sum Games”,
in Contributions to the Theory of Games, Vol 4, Annals of
Math. Studies, 40, pp. 47-85, 1959
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11
Example: Majority game
•
•
Three player majority game (|N|=3)
Payoffs
v(S ) ||S |1  0
v(S ) ||S | 2  
v( N )  1
•
What is the core of the game?
x1  x2  x3  1
x1  x2  
x2  x3  
x1  x3  

2
 continuum of solutions
3

2
1 1 1
 x( , , )
3
3 3 3
2( x1  x2  x3 )  3
2

3
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
2
 core is empty
3
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12
Non-emptiness of the core
•
Characteristic vector of S
1 i  S
(1S ) i  
0 i  S
•
(S)SC is a balanced collection of weights if
 1
S S
 1N
S C
•
Example for majority game
S ||S|2  0.5 S ||S|2  0
S ||S|1  1 S ||S|1  0
•
Players allocate S portion of their time to coalition S
•
•
They have unit amount of time to spend
Coalitional game TP <N,v> is balanced if for every
balanced collection of weights
  v ( S )  v( N )
S
SC
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13
Bondareva-Shapley Theorem
•
A coalitional game TP <N,v> has a nonempty core iff it
is balanced
•
Necessary
• (xi)iN feasible payoff profile in the core
• ( S)SC balanced
  v( S )    x ( S )   x     x
S
S C
•
•
S
S C
i
iN
S
S i
i
 v( N )
iN
Sufficient
• Separating hyperplane theorem
Example for majority game
2
1
1

S ||S | 2 
N    S v ( S )  1
3
3
3
SC
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14
Shapley value
•
Marginal contribution of player i to SN (iS) in the
game <N,v>
 i (S )  v(S  {i})  v(S )
•
All possible ordering of players (permutations)
•
•
•

Set of all orderings
Players preceding player i in ordering
R   is S i (R )
Shapley value of player i in the game <N,v> is
 i ( N , v) 
•
•
•
1
1
(| N |  | S | 1)!| S |! i ( S )
  i ( Si ( R ))  | N |! S 
| N |! R
N \{i }
Average marginal contribution of player i
Assigns unique feasible payoff profile to every game <N,v>
Note that
  ( S (R))  v( N )
i
i
R  
iN
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Example
•
Three player majority game N={A,B,C}
•
Possible permutations
•
=1 (core empty)
  {( A, B, C ), ( A, C , B), ( B, A, C ), ( B, C , A), (C , A, B), (C , B, A)}
•
Shapley value
 i ( N , v)   A ( N , v ) 
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1
0  0  1  0  1  0  1
6
3
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Properties of the Shapley value
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A value  is additive if for any two games <N,v> and
<N,w> we have
 i (v  w)   i (v)   i (w) (v  w)( S )  v( S )  w(S ) S  N
•
The Shapley value is the only value that satisfies
•
•
•
•
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Symmetry
Dummy player
Additivity
Efficiency
A value  satisfies the balanced contributions property
if for every coalitional game with TP <N,v> and i,jN
 i ( N , v)   i ( N \ { j}, v N \{ j} )   j ( N , v)  j ( N \ {i}, v N \{i} )
•
•
Reciprocity of loss if other player leaves
The unique value that satisfies the balanced
contributions property is the Shapley value
R.B. Myerson, “Conference structures and fair allocation rules,” International
Journal of Game Theory vol. 9, pp. 169–182, 1980
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Convex games
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A coalitional game with TP is convex if
v( S )  v(T )  v( S  T )  v( S  T ) S , T  N
•
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Supermodularity of v(S)
Practical consequence
v( S '{i})  v ( S ' )  v( S ' '{i})  v ( S ' ' ) i  N , S ' '  S '  N \ {i}
•
•
Marginal contribution of the player increases with |S|
Proof:
• S=S’ and T=S’’{i}
L.S. Shapley, “Cores of Convex Games”, International
Journal of Game Theory 1(1), 11-26, 1971
M. Maschler, B. Peleg, L.S. Shapley, “The Kernel and
Bargaining Set for Convex Games”, International
Journal of Game Theory 1(1), pp. 73-93, 1971
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18
Example: Markets w. TU
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Set of players N={1,2,3,4}
Two kinds of resources owned by players (endowment)
e1  e 2  (1,0) e 3  e 4  (0,1)
•
Player i’s utility function of the resources
U i ( x i )  x1i x2i
•
Players can form coalitions to increase their utilities
•
Coalitional game with TP
•
Combine available resources to produce goods

i i
i
2
i
i
v ( S )  max
 x1 x2 : x  R ,  x   e 
( x i )iS
 iS
iS
iS

v ({i})  0
v({i, i  2})  1
v ( S ) ||S |  3  2 v ( N )  4
•
v ({i, i  3})  1
Solution?
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19
Properties of convex games
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The core of a convex coalitional game with TP <N,v> is
non-empty
•
Greedy allocation: arbitrary ordering of players
xi  v (Si  i )  v (S i ) Si  {1, , i  1}
•
The stable set of a convex game is unique and
coincides with the core
•
The kernel is a singleton  coincides with the nucleolus
•
The Shapley value of a convex game is within its core,
it is the center of gravity of the vertices
L.S. Shapley, “Cores of Convex Games”, International
Journal of Game Theory 1(1), 11-26, 1971
M. Maschler, B. Peleg, L.S. Shapley, “The Kernel and
Bargaining Set for Convex Games”, International
Journal of Game Theory 1(1), pp. 73-93, 1971
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20
Cost Games
•
Coalitional game with divisible cost <N,c>
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•
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c(S) cost of coalition S
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•
•
Finite set N of players
Function c:SR SN, S
Arbitrary division possible between coalition members
Independent of the decisions of N\S
Cohesive
Sk
c ( N )   c( S k ) partition{S1 ,..., S k } of N
k 1
•
Grand coalition has the smallest cost
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21
György Dán, http://www.ee.kth.se/~gyuri
Example: Facility location
•
•
Given set of cities N
Set of facilities F
•
•
Opening facility j costs fj
City i can produce goods worth k using the facility
•
Distance of facility j from city i
•
Cities can cooperate to decrease costs
•
dij:N FR cost function
• S N
• c( S )  min F ' F {

a
2
Computational Game Theory – P2/2015
f j  iS min jF ' d ij }
b
6
8
f1=1
jF '
2
1
2
c
4
1
5
f2=2
György Dán, http://www.ee.kth.se/~gyuri
f3=2
22
Duality of cost and profit games
•
Payoff function defined by
v(S)  c(N) - c(N \ S)
v ({a})  3
v ({b})  1
v ({c})  1
v ({a, b})  4
v ({b, c})  4
v ({a, c})  4
v ({a, b, c})  7
c ({a})  3
c ({b})  3
c ({c})  3
c ({a, b})  6
c ({b, c})  4
c ({a, c})  6
c ({a, b, c})  7
•
The core of dual games is equivalent
x( S )  c( S )  x ( S )  v ( S )
•
•
“Submodular” cost games vs. “supermodular” profit games
“Supermodular” profit games vs. “submodular” cost games
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23
Example: Minimum Spanning Tree
•
Multicast cost sharing
•
•
•
•
Graph G(N,E)
Service provided by one node (server)
Nodes on the graph want to use the service
• Either directly connected
• Or connected via another node
Coalitional game <N,v>
•
Set of players N the customers
•
v(S)=c(grand coalition)
- c(minimum spanning tree for S and server)
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24
Coalitional game with nontransferable payoffs
•
Coalitional game with non-transferable payoff <N,X,V,
•
•
•
•
•
f
~i >
f
~
Coalition
•
•
Finite set of players N
Set of consequences X
Function V:S2X,  SN, S
Preference relation
on X
i
Subset SN
From consequences to utilities
•
V(S)=(xi)R|S| assigns utility to each member of the coalition S
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25
Final words
•
Large number of solution concepts for TP games
•
•
•
•
•
•
•
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Core, stable sets, kernel, nucleolus, Shapley value
Aumann-Shapley value
Bargaining set
Prenucleolus
Prekernel
-core
Owen–Banzhaf coalitional value
Solution concepts can be adapted to games with nontransferable payoffs
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27
Literature
•
•
M. Osborne, A. Rubinstein, “A Course in Game Theory”, MIT press, 1994
Nisan, Roughgarden, Tardos, Vazirani (eds.), “Algorithmic Game Theory”,
Cambridge UP, 2007
•
L.S. Shapley, “Cores of Convex Games”, International Journal of Game
Theory 1(1), 11-26, 1971
D.B. Gillies, “Solutions to General Non-zero Sum Games”, in Contributions
to the Theory of Games, Vol 4, Annals of Math. Studies, 40, pp. 47-85,
1959
J. von Neumann, O. Morgenstern, “Theory of Games and Economic
Behavior,” Princeton University Press, 1944
W. F. Lucas, M. Rabie, “Games with No Solutions and Empty Cores”,
Mathematics of Operations Research 7(4), 1982, pp. 491-500
M. Davis, M. Maschler, "The kernel of a cooperative game", Naval
Research Logistics Quarterly, vol 12, pp. 223–259, 1965
D. Schmeidler,"The nucleolus of a characteristic function game,” SIAM
Journal of Applied Math. vol. 17, pp. 1163–1170,1969
M.D. Davis, “Game Theory: A Nontechnical Introduction”, Dover, 1997
•
•
•
•
•
•
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Other solution concepts
•
•
•
Stable sets (Von-Neumann and Morgenstern)
Kernel (Davis and Maschler)
Nucleolus (Schmeidler)
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Stable sets and objections
•
Imputation y dominates the imputation x for some
coalition S
y  S x  y i  x i i  S ,
•
•
y ( S )  v( S )
y dominates x via S
Set of imputations dominated by Y
D(Y )  {z  X | S  N , y  Y y  S z}
•
The subset Y of the imputations X of the coalitional
game with TP <N,v> is a stable set if
•
Y is internally stable
•
Y is externally stable
y  Y
z  S y, z  Y
 S  N
z  X \ Y
Y  X \ D(Y )
Y  X \ D(Y )
S  N , y  Y
y S z
Y  X \ D(Y )
J. von Neumann, O. Morgenstern, “Theory of Games and
Economic Behavior,” Princeton University Press, 1944
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Example: Majority game
•
•
Set of Players N={1,2,3}
Payoff function
v ( S ) ||S |2  1 v ( S ) ||S |2  0
•
Stable sets
•
Core is empty (>2/3)
Y  {( 1 , 1 ,0), ( 1 ,0, 1 ), (0, 1 , 1 )}
2 2
2
2
2 2
•
•
Internally stable
• No two players prefer any profile to the other
Externally stable
• For any zY there are two players with zi<0.5
Yi ,c  {x  X : xi  c}
•
•
z  X \ Y3,c
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c  [0,0.5)
Internally stable
• No two players prefer any profile to the other
Externally stable
 z3  c  z1  z 2  1  c  y  Y3, c y  {1, 2} z

 z3  c  zi  z j  y  { yi  1  c, y3  c}  {i , 3} z
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Example
v({A,B,C})=12
v({A})=1
•
•
Three players: A,B,C
Core empty
•
v({A,C})=11
v({A,B})=11
xi  x j  11
  24  33
x( N )  12 
v({B})=1
v({B,C})=11
Stable set
•
Sum of distances from edges of interior point is constant
•
Y={(5.5,5.5,1),
(1,5.5,5.5),
(5.5,1,5.5)}
v({C})=1
y=(4,4,4)
x  BC y
Player C
Player A
1=v({i})
M.D. Davis, “Game Theory: A Nontechnical
Introduction”, Dover, 1997
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Player B
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Properties of the stable set
•
The core is the set of imputations x that is not
dominated
•
The core is a subset of every stable set
{ y  X :  S  N , x  X
•
•
Consequence of external stability: every member is an
imputation, and is not dominated by any other imputation
No stable set is a proper subset of any other
•
•
x  S y}
Consequence of external stability: if Y1 Y2 then for some
yY2\Y1, zY1 z dominates y for some S.
But zY2, so this is a contradiction
If the core is a stable set then it is the only stable set
•
Follows from the above two
•
The stable set satisfies the dummy property
•
Not all games have stable sets
•
There exist games with 14 or more players…
W. F. Lucas, M. Rabie, “Games with No Solutions
and Empty Cores”, Mathematics of Operations
Research 7(4), 1982, pp. 491-500
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33
The Kernel
•
The excess of coalition S under imputation x is
e( S , x)  v ( S )  x ( S )
•
Maximum excess of player i compared to player j
•
Sacrifice of coalition members
sij ( x )  max{e( S , x ) : i  S , j  N \ S }
S C
•
•
sij(x)-sji(x) bargaining power of i against j
The kernel of the coalitional game with TP is the set of
imputations xX such that for every pair (i,j) of players
[ sij ( x)  s ji ( x)] [ x({ j})  v({ j})]  0
[ s ji ( x)  sij ( x)] [ x({i})  v({i})]  0
0
M. Davis, M. Maschler, "The kernel of a cooperative game",
Naval Research Logistics Quarterly, vol 12, pp. 223–259, 1965
Computational Game Theory – P2/2015
György Dán, http://www.ee.kth.se/~gyuri
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Example
•
•
Three player majority game
Kernel
x (
•
1 1 1
, , )
3 3 3
Assume kernel x=(x1,x2,x3)
s31(x)  1  (x2  x3 )
s13(x)  1  (x2  x1 )
•
Assume x1x2x3 with at least one strict inequality (x1>x3)
s31 ( x )  s13 ( x )
•
Furthermore x1>0
x ({1})  v ({1})
•
Computational Game Theory – P2/2015
 s31 ( x )  s13 ( x )  0
 x({1})  v({1})  0
x is not in the kernel
György Dán, http://www.ee.kth.se/~gyuri
35
The Nucleolus
•
Ordering of the coalitions S for imputation x wrt. e(S,x)
e( S l , x )  e( S l 1 , x) l  1, ,2| N |  2
•
Vector of excesses
E ( x)  [ E1 ( x ), , E2|N | 2 ( x )], El ( x )  e( Sl , x )
•
•
Excesses in decreasing order
Partition B(x)=(B1(x),…,BK(x)) of the coalitions SN
•
Coalitions with equal excess
S , S ' Bk ( x )  e(S , x )  e( S ' , x)  ek ( x )
e1 ( x )    eK ( x)
Computational Game Theory – P2/2015
György Dán, http://www.ee.kth.se/~gyuri
36
The Nucleolus
•
Lexicographical ordering on E(x)
E ( x)  E ( y )  El ( x)  El ( y )
for l  min[ k {1,...,2|N |  2} : Ek ( x )  Ek ( y )]
k * | Bk ( x ) || Bk ( y ) | , ek ( x )  ek ( y ) k  k *
and
e k * ( x )  e k * ( y ) or e k * ( x)  e k * ( y ) and | Bk ( x) || Bk ( y ) |
•
The nucleolus is the set xX of imputations for which
E(x)  E ( y )
•
y  X
Lexicographical minimization (of maximum excess)
D. Schmeidler,"The nucleolus of a characteristic function game,”
SIAM Journal of Applied Math. vol. 17, pp. 1163–1170,1969
Computational Game Theory – P2/2015
György Dán, http://www.ee.kth.se/~gyuri
37
Example
•
•
Three player majority game
Nucleolus
1 1 1
x( , , )
3 3 3
•
Excess
e( S , x) ||S |1  
•
Ordered coalitions
•
Excess vector
1
1
e( S , x ) ||S | 2 
3
3
[{ A, B}, { A, C}, {B, C}, { A}, {B}, {C}]
1 1 1 1 1 1
E ( x)  [ , , , , , ]
3 3 3 3 3 3
•
Partitions
1
B1 ( x)  {{ A, B}, { A, C}, {B, C}} e1 ( x ) 
3
1
B2 ( x)  {{ A}, {B}, {C}}
e2 ( x)  
3
Computational Game Theory – P2/2015
György Dán, http://www.ee.kth.se/~gyuri
38
Properties of the nucleolus
•
The nucleolus of any coalitional game with TP is
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non-empty
E k ( x) 
•
min
{max{e( S , x)}}
2 |N | ,|T |  k 1 S C \
•
a singleton
•
a subset of the kernel
• Consequence: the kernel is non-empty
If the core is non-empty, the nucleolus is in the core
D. Schmeidler,"The nucleolus of a characteristic function game,”
SIAM Journal of Applied Math. vol. 17, pp. 1163–1170,1969
Computational Game Theory – P2/2015
György Dán, http://www.ee.kth.se/~gyuri
39