Cooperative Games I
Ignacio García-Jurado
Departamento de Matemáticas
Universidade da Coruña
Doc-course: The Mathematics of Games, Strategies, Cooperation
and Fair Division
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Cooperative Games I
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Introduction to Cooperative Games
Preliminaries
What is Game Theory?
Game theory can be defined as the mathematical theory of
interactive decision situations.
It officially started in 1944, when the first edition of the book
“Theory of Games and Economic Behavior” by John von
Neumann and Oskar Morgenstern was launched.
From its very beginning, game theory is the result of the
collaboration between mathematicians and economists.
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Introduction to Cooperative Games
Preliminaries
Non-Cooperative versus Cooperative Game Theory
Non-Cooperative Game Theory assumes that players’ possibilities
for interacting and collaborating can be fully modelled. It analyses
how players should strategically behave within the rules of the
game.
Cooperative Game Theory assumes that players’ possibilities for
interacting and collaborating are too complex to be formally
modelled. It just aims to allocate among players the estimated
benefits of their cooperation.
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Introduction to Cooperative Games
Models of Cooperative Game Theory
NTU-games
Definition
A non-transferable utility game (NTU-game) is a pair (N, V ), where N
is the finite set of players and V is a function that assigns, to each
coalition S ⊂ N, a set V (S) ⊂ RS . By convention, V (∅) := {0}.
Moreover, for each S ⊂ N, S 6= ∅:
1
2
3
V (S) is a non-empty and closed subset of RS .
V (S) is comprehensive (i.e., for each pair x, y ∈ RS such that
x ∈ V (S) and y ≤ x, we have that y ∈ V (S)). Moreover, for each
i ∈ N, V ({i}) 6= R, i.e., there is vi ∈ R such that V ({i}) = (−∞, vi ].
The set V (S) ∩ {y ∈ RS | for each i ∈ S, yi ≥ vi } is bounded.
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Introduction to Cooperative Games
Models of Cooperative Game Theory
Bargaining problems
Definition
A bargaining problem with finite set of players N is a pair (F , d) whose
elements are the following:
Feasible set F is a closed, convex, and comprehensive subset of RN
such that Fd := {x ∈ F | x ≥ d} is compact.
Disagreement point d is an allocation in F . It is assumed that there is
x ∈ F such that x > d.
Remark
(F , d), a bargaining problem with finite set of players N, can be seen
as an NTU-game (N, V ), where V (N) := F and, for each non-empty
coalition S 6= N, V (S) := {y ∈ RS : for each i ∈ S, yi ≤ di }.
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Introduction to Cooperative Games
Models of Cooperative Game Theory
TU-games
Definition
A TU-game is a pair (N, v ), where N is the finite set of players, and
v : 2N → R is the characteristic function of the game, which satisfies
v (∅) = 0. In general, we interpret v (S) as the benefit that S can
generate. We will often refer to (N, v ) simply as v , and will denote by
G(N) the class of TU-games with set of players N.
Remark
A TU-game (N, v ) can be seen as an NTU-game (N,P
V ) by defining, for
each non-empty coalition S ⊂ N, V (S) := {y ∈ RS : i∈S yi ≤ v (S)}.
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Introduction to Cooperative Games
Introduction to TU-games
Example
Divide a million. A rich person dies and leave one million euros to
three nephews, with the condition that at least two of them must agree
on how to divide this amount among them; otherwise, the million of
euros will be given to other person. This situation can be modelled as
the TU-game (N, v ), where N = {1, 2, 3}, and v (1) = v (2) = v (3) = 0,
v (12) = v (13) = v (23) = v (N) = 1.
Example
The glove game. Three players are willing to divide the benefits of
selling a pair of gloves. Player one has a left glove, and players two
and three have one right glove each. A pair of gloves can be sold for
one hundred euros. This situation can be modelled as the TU-game
(N, v ), where N = {1, 2, 3}, and v (1) = v (2) = v (3) = v (23) = 0,
v (12) = v (13) = v (N) = 1.
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Introduction to Cooperative Games
Introduction to TU-games
Example
The Parliament of Aragón. This example illustrates that TU-games can also be used
to model coalitional bargaining situations in which players negotiate with something
more abstract than money. In this case we consider the Parliament of Aragón, one of
the regions in which Spain is divided. After the elections which took place in May
1991, its composition was: PSOE (Socialist Party) had 30 seats, PP (Conservative
Party) had 17 seats, PAR (Regional Party of Aragón) had 17 seats, and IU (a coalition
mainly formed by communists) had 3 seats. In a Parliament, the most relevant
decisions are taken making use of the simple majority rule. Measuring the power of
the different parties in a Parliament, which is an interesting problem, can be thought
as “dividing” the power among them. A coalition is said to have the power if it collects
more than half of the seats of the Parliament, thirty-four seats in this example. So, this
situation can be modelled as the TU-game (N, v ), where N = {1, 2, 3, 4} (1=PSOE,
2=PP, 3=PAR, 4=IU), and v (S) = 1 if there is T ∈ {{1, 2}, {1, 3}, {2, 3}} with T ⊂ S,
v (S) = 0 otherwise. Notice that we indicate that S has more than half of the seats
making v (S) = 1.
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Introduction to Cooperative Games
Introduction to TU-games
Example
The visiting professor. Three research groups belonging to the universities of Milano
(group one), Genova (group two) and Santiago de Compostela (group three) plan to
invite a Japanese professor to give a course on game theory. To minimize the cost,
they coordinate the courses, so that the professor makes a tour visiting Milano,
Genova and Santiago de Compostela. Then the groups want to allocate the cost of
the tour among them. For that purpose they have estimated the travel cost (in euros)
of the visit for all the possible coalitions of groups: c(1) = 1500, c(2) = 1600,
c(3) = 1900, c(12) = 1600, c(13) = 2900, c(23) = 3000, c(N) = 3000. Take
N = {1, 2, 3}. Notice that (N, c) is a TU-game; however, it is what we usually call a
cost game, in the sense that c(S) (for every S) does not represent the benefits that S
can generate, but the costs it must support. The saving game associated to this
situation (displaying the benefits generated by each coalition) is (N, v ) where, for
every S ⊂ N,
X
v (S) =
c(i) − c(S).
i∈S
Thus, v (1) = v (2) = v (3) = 0, v (12) = 1500, v (13) = 500, v (23) = 500,
v (N) = 2000.
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Introduction to Cooperative Games
Introduction to TU-games
Superadditive Games
Definition
Take a TU-game v ∈ G(N). We say that v is superadditive if, for all
S, T ⊂ N, with S ∩ T = ∅,
v (S ∪ T ) ≥ v (S) + v (T ).
We denote by SG(N) the set of superadditive TU-games with set of
players N.
Note that a game is superadditive when players have true incentives
for cooperation, in the sense that the union of any two disjoint groups
of players never diminishes the total benefits. In fact, most part of the
theory of TU-games is really developed for superadditive games,
although it is formally presented, for simplicity, for the whole class of
TU-games. Notice that all the games in the examples above, with the
only exception of (N, c) in the visiting professor game, are
superadditive.
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Introduction to Cooperative Games
Introduction to TU-games
Solutions for TU-Games
The main goal of the theory of TU-games is to select, for every
TU-game, an allocation, or a set of allocations, which is admissible for
the players. In making this, there are two main approaches.
The first approach is based on stability: it aims to find a set of
allocations which is stable, in the sense that it can be expected
that the final agreement adopted by the players lies within this set;
this is the approach underlying, for instance, the core (Gillies
(1953)), the stable sets (von Neumann and Morgenstern (1944))
and the bargaining set (Aumann and Maschler (1964)).
The second approach is based on fairness: it aims to propose for
every TU-game one allocation which represents a fair compromise
for the players; this is the approach underlying, for instance, the
Shapley value (Shapley (1953)), the nucleolus (Schmeidler
(1969)) and the Tijs value (Tijs (1981)).
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The core of a TU-game and related concepts
The core
Imputations and the core
Definition
Let v ∈ G(N) be a TU-game. An imputation of v is an x ∈ RN such
that:
P
i∈N xi = v (N),
xi ≥ v (i), for all i ∈ N.
We denote by I(v ) the set of imputations of v .
Definition
Let v ∈ G(N) be a TU-game. The core of v , that we denote by C(v ), is
the following set:
X
xi ≥ v (S) for all S ⊂ N}.
C(v ) = {x ∈ I(v ) |
i∈S
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The core of a TU-game and related concepts
The core
Example
Consider the TU-game in the visiting professor game, which is given
by: N = {1, 2, 3}, and v (1) = v (2) = v (3) = 0, v (12) = 1500,
v (13) = 500, v (23) = 500, v (N) = 2000. Its set of imputations is
depicted below.
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The core of a TU-game and related concepts
The core
Example
Consider the TU-game in the visiting professor game, which is given
by: N = {1, 2, 3}, and v (1) = v (2) = v (3) = 0, v (12) = 1500,
v (13) = 500, v (23) = 500, v (N) = 2000. Now the core of this game is
displayed.
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The core of a TU-game and related concepts
The core
Example
Consider the glove game, which is given by: N = {1, 2, 3}, and
v (1) = v (2) = v (3) = v (23) = 0, v (12) = v (13) = v (N) = 1. It is easy
to check that its core is the set {(1, 0, 0)}.
Example
Consider the divide a million game, which is given by: N = {1, 2, 3},
and v (1) = v (2) = v (3) = 0, v (12) = v (13) = v (23) = v (N) = 1. It is
easy to check that its core is empty.
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The core of a TU-game and related concepts
Balancedness and the core
Balancedness
Definition
A family of coalitions F ⊂ 2N \ {∅} is said to be balanced if there exists
a corresponding family of positive real numbers (called balancing
coefficients) {yS | S ∈ F} such that, for all i ∈ N,
X
yS = 1.
S∈F ,i∈S
Definition
A game v ∈ G(N) is said to be balanced if, for every balanced family F
with balancing coefficients {yS | S ∈ F}, it holds that
X
yS v (S) ≤ v (N).
S∈F
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The core of a TU-game and related concepts
Balancedness and the core
The Bondareva-Shapley theorem
Definition
A game v ∈ G(N) is said to be balanced if, for every balanced family F
with balancing coefficients {yS | S ∈ F}, it holds that
X
yS v (S) ≤ v (N).
S∈F
The fact that v is balanced can be roughly interpreted as that its
intermediate coalitions are not too powerful. Hence, the balancedness
of a game should be, in some sense, related to the stability of the
coalitional bargaining situation modelled by it. This is precisely what
the Bondareva-Shapley theorem asserts.
Theorem
(Bondareva-Shapley). Let v ∈ G(N) be a TU-game. Then C(v ) 6= ∅ if
and only if v is balanced.
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The core of a TU-game and related concepts
Balancedness and the core
Bondareva-Shapley theorem: a sketch of the proof
Suppose that v is balanced and consider the following linear
programming problem (P):
P
Minimise
i∈N xi ,
P
N
provided that
i∈S xi ≥ v (S), ∀S ∈ 2 \ {∅}.
Clearly,
P C(v ) 6= ∅ if and only if there exists x̄ an optimal solution of (P)
with i∈N x̄i = v (N). The dual of (P) is the following linear
programming problem (D):
P
Maximise
yS v (S),
N
P S∈2 \{∅}
provided that
S∈2N \{∅},i∈S yS = 1, ∀i ∈ N,
yS ≥ 0,
∀S ∈ 2N \ {∅}.
The balancedness of v implies that C(v ) 6= ∅.
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The core of a TU-game and related concepts
Domination and the core
Domination
Definition
Let v ∈ G(N) be a TU-game and take a non-empty S ⊂ N and
x, y ∈ I(v ). We say that x dominates y through S if:
xi > yi for all i ∈ S,
P
i∈S xi ≤ v (S).
We say that x dominates y if there exists a non-empty S ⊂ N such that
x dominates y through S. Finally, y is said to be an undominated
imputation of v if there does not exist another imputation x ∈ I(v ) such
that x dominates y .
Proposition
Let v ∈ G(N) be a TU-game.
1
If x ∈ C(v ), then x is undominated.
2
If, moreover, v ∈ SG(N), then
C(v ) = {x ∈ I(v ) | x is undominated }.
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The core of a TU-game and related concepts
Stable sets and the core
Stable sets
Definition
(von Neumann and Morgenstern). Let v ∈ G(N) be a TU-game. A
subset M of I(v ) is said to be a stable set if it satisfies the following two
properties:
Internal stability. For every x, y ∈ M, x does not dominate y and y
does not dominate x.
External stability. For every x ∈ I(v ) \ M, there exists y ∈ M such
that y dominates x.
If M is a stable set of v , then C(v ) ⊂ M.
If C(v ) is a stable set of v , then it is the unique stable set of v .
A game can have many stable sets. In 1968 W. Lucas constructed
a 10-person game without stable sets.
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The Shapley value of a TU-game
Definition and axiomatic characterization
Values for TU-games
Now we study the Shapley value, the most important solution
concept for TU-games dealing with fairness.
Shapley wants to propose, for every TU-game, one allocation
which is a fair compromise for the players.
To that aim, Shapley gives the concept of value of a TU-game, a
value being a map φ which associates to every v ∈ G(N) an
element φ(v ) ∈ RN ; he also introduces some properties that a fair
value should satisfy.
Finally, Shapley proves that his properties characterize a unique
value and finds an explicit expression for it. This value is what we
call the Shapley value.
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The Shapley value of a TU-game
Definition and axiomatic characterization
Fairness and values
Definition
Let v ∈ G(N) be a TU-game.
i ∈ N is said to be a null player of v if, for every S ⊂ N,
v (S ∪ {i}) = v (S).
Two players i, j ∈ N are said to be interchangeable in v if, for every
coalition S ⊂ N \ {i, j}, v (S ∪ {i}) = v (S ∪ {j}).
Efficiency
(EFF). φ satisfies EFF if, for all v ∈ G(N),
P
i∈N φi (v ) = v (N).
The Null Player Property (NPP). φ satisfies NPP if, for every
v ∈ G(N) and every null player i ∈ N, φi (v ) = 0.
Anonimity (AN). φ satisfies AN if, for every v ∈ G(N) and every
i, j ∈ N, interchangeable players in v , it holds that φi (v ) = φj (v ).
Additivity (ADD). φ satisfies ADD if, for every v , w ∈ G(N), it holds
that φ(v + w) = φ(v ) + φ(w).
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The Shapley value of a TU-game
Definition and axiomatic characterization
The Shapley value
Theorem
There exists a unique value Φ satisfying EFF, NPP, AN and ADD. This
value, that is called the Shapley value, is given by
Φi (v ) =
X
S⊂N\{i}
s!(n − s − 1)!
(v (S ∪ {i}) − v (S))
n!
(1)
for all v ∈ G(N) and all i ∈ N, n and s denoting the cardinalities of N
and S, respectively.
Sketch of the proof. For every non-empty S ⊂ N, define u S ∈ G(N)
(the unanimity game of S) by u S (T ) = 1 if S ⊂ T , and u S (T ) = 0
otherwise. Now take into account that every v ∈ G(N) can be seen as
n
n
a vector in R2 −1 and that the set {u S }S∈2N \∅ is a basis of R2 −1 .
Finally, it is clear that EFF, NPP and AN characterize a unique solution
for every unanimity game.
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The Shapley value of a TU-game
Some comments on the Shapley value
Heuristic interpretation
The Shapley value can be seen as the expected reward vector in
the following situation: a) players agree to meet at a certain
location, b) all possible arrival orderings are equally probable, and
c) on arrival every player receives a reward equal to its
contribution to the coalition of players who arrived before.
In other words, the proposal of the Shapley value for game
v ∈ G(N) can also be written as
Φi (v ) =
1 X
v (B π (i) ∪ {i}) − v (B π (i)),
n!
π∈Π(N)
for all i ∈ N, where Π(N) is the set of permutations of N, and B π (i)
is the set {j ∈ N | π(j) < π(i)} (for all π ∈ Π(N) and all i ∈ N).
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The Shapley value of a TU-game
Some comments on the Shapley value
Example
The proposal of the Shapley value for the visiting professor game is
Φ(v ) = (5000/6, 5000/6, 2000/6).
123
1
2
3
0
1500
500
132
0
1500
500
213
1500
0
500
231
1500
0
500
312
500
1500
0
321
1500
500
0
5000
5000
2000
According to this allocation of the savings, the players have to pay
(4000/6, 4600/6, 9400/6). Note that the latter vector is precisely Φ(c).
Finally, observe that Φ(v ) ∈ C(v ).
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The Shapley value of a TU-game
Some comments on the Shapley value
Example
The proposal of the Shapley value for the divide a million game is
(1/3, 1/3, 1/3). Note that, although the core of this game is empty, the
Shapley value proposes an allocation for it. Remember that the core
dealt with stability and that the Shapley value deals with fairness.
Example
The proposal of the Shapley value for the glove game is the vector
(2/3, 1/6, 1/6). Remember that the core of this game is {(1, 0, 0)}.
Example
The proposal of the Shapley value for the Parliament of Aragón is
(1/3, 1/3, 1/3, 0). This is a measure of the power of the four political
parties in this Parliament. Note that IU is a null player and that the
other three parties are interchangeable if we only take into account
their voting power.
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The Shapley value of a TU-game
Some comments on the Shapley value
The Shapley value and the core
Definition
Let v ∈ G(N) a TU-game. We say that v is convex if, for every
S, T ⊂ N \ {i} with S ⊂ T ,
v (T ∪ {i}) − v (T ) ≥ v (S ∪ {i}) − v (S).
It can be checked that every convex game is superadditive. Moreover,
Shapley demonstrated the following result.
Theorem
Let v ∈ G(N) be a TU-game.
If v is superadditive then Φ(v ) ∈ I(v ).
If v is convex then Φ(v ) ∈ C(v ).
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The nucleolus of a TU-game
An introduction to the nucleolus
The definition of excess
The nucleolus is a value for TU-games based on the idea of minimizing
the dissatisfaction of the most dissatisfied groups. To that aim, the
concept of excess is defined.
Definition
Let v ∈ G(N) be a TU-game and take x ∈ RN and S ∈ 2N \ ∅. The
excess of x with respect to S, denoted by e(S, x), is given by
X
e(S, x) =
xi − v (S).
i∈S
n
The excess of x, denoted by e(x), is the vector in R2 −1 containing the
excesses of x with respect to all the non-empty coalitions disposed in
non-decreasing order. More precisely, if i ∈ {1, . . . , 2n − 2} and ei (x)
denotes the i-th component of e(x), then ei (x) ≤ ei+1 (x).
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The nucleolus of a TU-game
An introduction to the nucleolus
The nucleolus
Definition
Let v ∈ G(N) be a TU-game such that I(v ) 6= ∅. The nucleolus maps v
to N (v ) ∈ RN , where N (v ) is the unique point of the set
{x ∈ I(v ) | e(y ) ≤L e(x) for all y ∈ I(v )},
n −1
≤L denoting the lexicographic order on R2
nucleolus of v .
(2)
. N (v ) is said to be the
It can be check that N is well-defined, in the sense that the set in (2)
contains in fact a unique point for those v ∈ G(N) with I(v ) 6= ∅.
Moreover, it can be easily proved that if C(v ) 6= ∅ then N (v ) ∈ C(v ).
There are several procedures to compute the nucleolus of a TU-game,
but its computation can be quite hard.
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Bibliography
Bibliography
R.J. Aumann and M. Maschler (1964). “The bargaining set for cooperative games”. In M. Dresher, L.S. Shapley and A.W. Tucker
(eds.), Advances in game theory. Princeton University Press.
O. Bondareva (1963). “Certain applications of the methods of linear programming to the theory of cooperative games”. Problemy
Kibernetiki 10, 119-139.
D.B. Gillies (1953). “Some theorems on n-person games”. PhD dissertation. Princeton University.
J. González-Díaz, I. García-Jurado, M.G. Fiestras Janeiro (2010). “An introductory course on mathematical game theory”.
Graduate Studies in Mathematics 115. American Mathematical Society.
Y. Kannai (1992). “The Core and balancedness”. In R.J. Aumann and S. Hart (eds.), Handbook of Game Theory Vol. I.
North-Holland.
J. von Neumann and O. Morgenstern (1944). “Theory of games and economic behavior”. Princeton University Press.
M.J. Osborne and A. Rubinstein (1994). “A course on game theory”. The MIT Press.
G. Owen (1995). “Game theory”. Academic Press.
M. Maschler (1992). “The bargaining set, kernel, and nucleolus”. In R.J. Aumann and S. Hart (eds.), Handbook of Game Theory
Vol. I. North-Holland.
S. Moretti and F. Patrone (2008). “Transversality of the Shapley value”. Top 16, 1-41.
D. Schmeidler (1969). “The nucleolus of a characteristic function game”. SIAM Journal on Applied Mathematics 17, 1163-1170.
L.S. Shapley (1953). “A value for n-person games”. In H.W. Kuhn and A.W. Tucker (eds.), Contributions to the theory of games II.
Princeton University Press.
L.S. Shapley (1967). “On balanced sets and cores”, Naval Research Logistics Quarterly 14, 453-460.
L.S. Shapley (1971). “Cores of convex games”, International Journal of Game Theory 1, 11-26.
S.H. Tijs (1981). “Bounds for the core and the τ -value”. In O. Moeschlin and D. Pallaschke (eds.), Game theory and
mathematical economics. North Holland.
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Exercises
Exercises
1
Compute the Shapley value of the TU-game (N, v ) given by:
N = {1, 2, 3}, v (1) = v (2) = v (3) = 0, v (12) = 2,
v (13) = v (23) = 4, v (N) = 8. Depict C(v ) and obtain its extreme
points. Show that Φ(v ) ∈ C(v ).
2
Compute the Shapley value of the TU-game (N, v ) given by:
N = {1, 2, 3}, v (1) = v (2) = v (3) = 0, v (12) = 4, v (13) = 8,
v (23) = 10, v (N) = 10. Give an imputation of v which dominates
(2, 4, 4). Use the Bondareva-Shapley theorem to prove that C(v )
is an empty set. If w(13) = w(23) = 4 and w(S) = v (S) for all
other coalition S, depict C(w) and obtain its extreme points.
Prove that a TU-game (N, v ) which satisfies that
3
I
I
v (S) < 0P
for all S ⊂ N, and
v (S)2 = i∈S v (i)2 for all S ⊂ N,
has a non-empty core.
I. García-Jurado (UDC)
Cooperative Games I
Sevilla, March 16, 2011
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