[PPT] - Cores of Convex Games and Pascals Triangle Julio Gonz alez-D az PowerPoint Presentation

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Cores of Convex Games and Pascal’s Triangle

Julio Gonz´ alez-D´ ıaz

Kellogg School of Management (CMS-EMS) Northwestern University and Research Group in Economic Analysis Universidad de Vigo ........................ (joint with Estela S´ anchez-Rodr´ ıguez)

July 4th, 2007

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Definitions

Cores of Convex Games and Pascal’s Triangle Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 1/7

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Definitions

Convex

Cores of Convex Games and Pascal’s Triangle Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 1/7

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Definitions

Convex

for each i ∈ N and each S and T such that S ⊆ T ⊆ N\{i},

Cores of Convex Games and Pascal’s Triangle Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 1/7

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Definitions

Convex

for each i ∈ N and each S and T such that S ⊆ T ⊆ N\{i}, v(S ∪ i) − v(S) ≤ v(T ∪ i) − v(T)

Cores of Convex Games and Pascal’s Triangle Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 1/7

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Definitions

Convex

for each i ∈ N and each S and T such that S ⊆ T ⊆ N\{i}, v(S ∪ i) − v(S) ≤ v(T ∪ i) − v(T)

Strictly Convex

for each i ∈ N and each S and T such that S T ⊆ N\ {i}, v(S ∪ i) − v(S) < v(T ∪ i) − v(T)

Cores of Convex Games and Pascal’s Triangle Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 1/7

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Definitions

Convex

for each i ∈ N and each S and T such that S ⊆ T ⊆ N\{i}, v(S ∪ i) − v(S) ≤ v(T ∪ i) − v(T)

Strictly Convex

for each i ∈ N and each S and T such that S T ⊆ N\ {i}, v(S ∪ i) − v(S) < v(T ∪ i) − v(T)

The core of a strictly convex n-player game. . .

Cores of Convex Games and Pascal’s Triangle Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 1/7

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Definitions

Convex

for each i ∈ N and each S and T such that S ⊆ T ⊆ N\{i}, v(S ∪ i) − v(S) ≤ v(T ∪ i) − v(T)

Strictly Convex

for each i ∈ N and each S and T such that S T ⊆ N\ {i}, v(S ∪ i) − v(S) < v(T ∪ i) − v(T)

The core of a strictly convex n-player game. . .

has n! extreme points

Cores of Convex Games and Pascal’s Triangle Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 1/7

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Definitions

Convex

for each i ∈ N and each S and T such that S ⊆ T ⊆ N\{i}, v(S ∪ i) − v(S) ≤ v(T ∪ i) − v(T)

Strictly Convex

for each i ∈ N and each S and T such that S T ⊆ N\ {i}, v(S ∪ i) − v(S) < v(T ∪ i) − v(T)

The core of a strictly convex n-player game. . .

has n! extreme points

(one for each vector of marginal contributions)

Cores of Convex Games and Pascal’s Triangle Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 1/7

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Definitions

Convex

for each i ∈ N and each S and T such that S ⊆ T ⊆ N\{i}, v(S ∪ i) − v(S) ≤ v(T ∪ i) − v(T)

Strictly Convex

for each i ∈ N and each S and T such that S T ⊆ N\ {i}, v(S ∪ i) − v(S) < v(T ∪ i) − v(T)

The core of a strictly convex n-player game. . .

has n! extreme points

(one for each vector of marginal contributions)

and its core is full dimensional

Cores of Convex Games and Pascal’s Triangle Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 1/7

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Definitions

Convex

for each i ∈ N and each S and T such that S ⊆ T ⊆ N\{i}, v(S ∪ i) − v(S) ≤ v(T ∪ i) − v(T)

Strictly Convex

for each i ∈ N and each S and T such that S T ⊆ N\ {i}, v(S ∪ i) − v(S) < v(T ∪ i) − v(T)

The core of a strictly convex n-player game. . .

has n! extreme points

(one for each vector of marginal contributions)

and its core is full dimensional

(an (n − 1)-dimensional polytope inside the set of imputations)

Cores of Convex Games and Pascal’s Triangle Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 1/7

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The Core and its Faces

Cores of Convex Games and Pascal’s Triangle Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 2/7

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The Core and its Faces

Cores of Convex Games and Pascal’s Triangle Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 2/7

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The Core and its Faces

Cores of Convex Games and Pascal’s Triangle Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 2/7

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The Core and its Faces

HT := {x ∈ Rn :

i∈T xi = v(T)}

Cores of Convex Games and Pascal’s Triangle Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 2/7

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The Core and its Faces

HT := {x ∈ Rn :

i∈T xi = v(T)}

FT := C(N, v) ∩ HN\T

Cores of Convex Games and Pascal’s Triangle Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 2/7

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The Core and its Faces

HT := {x ∈ Rn :

i∈T xi = v(T)}

FT := C(N, v) ∩ HN\T

1 2 3

Cores of Convex Games and Pascal’s Triangle Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 2/7

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The Core and its Faces

HT := {x ∈ Rn :

i∈T xi = v(T)}

FT := C(N, v) ∩ HN\T

1 2 3 F1

Cores of Convex Games and Pascal’s Triangle Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 2/7

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The Core and its Faces

HT := {x ∈ Rn :

i∈T xi = v(T)}

FT := C(N, v) ∩ HN\T

1 2 3 x2 + x3 = v(23) F1

Cores of Convex Games and Pascal’s Triangle Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 2/7

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The Core and its Faces

HT := {x ∈ Rn :

i∈T xi = v(T)}

FT := C(N, v) ∩ HN\T

1 2 3 x2 + x3 = v(23) F1 x1 = v(1) F23

Cores of Convex Games and Pascal’s Triangle Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 2/7

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The Core and its Faces

HT := {x ∈ Rn :

i∈T xi = v(T)}

FT := C(N, v) ∩ HN\T

1 2 3 x2 + x3 = v(23) F1 x1 = v(1) F23

Cores of Convex Games and Pascal’s Triangle Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 2/7

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The Core and its Faces

HT := {x ∈ Rn :

i∈T xi = v(T)}

FT := C(N, v) ∩ HN\T

1 2 3 x2 + x3 = v(23) F1 x1 = v(1) F23

T-face game: (N, vFT )

vFT (S) :=

Cores of Convex Games and Pascal’s Triangle Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 2/7

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The Core and its Faces

HT := {x ∈ Rn :

i∈T xi = v(T)}

FT := C(N, v) ∩ HN\T

1 2 3 x2 + x3 = v(23) F1 x1 = v(1) F23

T-face game: (N, vFT )

vFT (S) := v(S ∩ (N\T))

Cores of Convex Games and Pascal’s Triangle Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 2/7

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The Core and its Faces

HT := {x ∈ Rn :

i∈T xi = v(T)}

FT := C(N, v) ∩ HN\T

1 2 3 x2 + x3 = v(23) F1 x1 = v(1) F23

T-face game: (N, vFT )

vFT (S) :=v((S ∩ T) ∪ (N\T)) − v(N\T) + v(S ∩ (N\T))

Cores of Convex Games and Pascal’s Triangle Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 2/7

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The Core and its Faces

1 2 3 F1 F23 Cores of Convex Games and Pascal’s Triangle Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 3/7

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The Core and its Faces

1 2 3 F1 F23

If v is convex, then the face games are convex (not strictly convex)

Cores of Convex Games and Pascal’s Triangle Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 3/7

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The Core and its Faces

1 2 3 F1 F23

If v is convex, then the face games are convex (not strictly convex)

Proposition

Let (N, v) be a convex game and T ⊆ N.

Cores of Convex Games and Pascal’s Triangle Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 3/7

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The Core and its Faces

1 2 3 F1 F23 F1 = C(N, vF1) F23 = C(N, vF23)

If v is convex, then the face games are convex (not strictly convex)

Proposition

Let (N, v) be a convex game and T ⊆ N. Then, C(N, vFT ) = FT .

Cores of Convex Games and Pascal’s Triangle Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 3/7

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The Core and its Faces

1 2 3 F1 F23 F1 = C(N, vF1) F23 = C(N, vF23)

If v is convex, then the face games are convex (not strictly convex)

Proposition

Let (N, v) be a convex game and T ⊆ N. Then, C(N, vFT ) = FT . Therefore, C(N, v) = co{C(N, vFT ) : ∅ = T N}

Cores of Convex Games and Pascal’s Triangle Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 3/7

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Decomposable Games

Cores of Convex Games and Pascal’s Triangle Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 4/7

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Decomposable Games

Let P = {N1, . . . , Np} be a partition of N, with p ≥ 2

Cores of Convex Games and Pascal’s Triangle Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 4/7

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Decomposable Games

Let P = {N1, . . . , Np} be a partition of N, with p ≥ 2 (N, v) is decomposable with respect to P if, for each S ⊆ N,

Cores of Convex Games and Pascal’s Triangle Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 4/7

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Decomposable Games

Let P = {N1, . . . , Np} be a partition of N, with p ≥ 2 (N, v) is decomposable with respect to P if, for each S ⊆ N, v(S) = v(S ∩ N1) + . . . + v(S ∩ Np)

Cores of Convex Games and Pascal’s Triangle Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 4/7

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Decomposable Games

Let P = {N1, . . . , Np} be a partition of N, with p ≥ 2 (N, v) is decomposable with respect to P if, for each S ⊆ N, v(S) = v(S ∩ N1) + . . . + v(S ∩ Np) =

Ni∈P

v(S ∩ Ni)

Cores of Convex Games and Pascal’s Triangle Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 4/7

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Decomposable Games

Let P = {N1, . . . , Np} be a partition of N, with p ≥ 2 (N, v) is decomposable with respect to P if, for each S ⊆ N, v(S) = v(S ∩ N1) + . . . + v(S ∩ Np) =

Ni∈P

v(S ∩ Ni)

Lemma

Cores of Convex Games and Pascal’s Triangle Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 4/7

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Decomposable Games

Let P = {N1, . . . , Np} be a partition of N, with p ≥ 2 (N, v) is decomposable with respect to P if, for each S ⊆ N, v(S) = v(S ∩ N1) + . . . + v(S ∩ Np) =

Ni∈P

v(S ∩ Ni)

Lemma

The core of a decomposable convex game is the cartesian product of the cores of the components of any decomposition

Cores of Convex Games and Pascal’s Triangle Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 4/7

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Decomposable Games

Let P = {N1, . . . , Np} be a partition of N, with p ≥ 2 (N, v) is decomposable with respect to P if, for each S ⊆ N, v(S) = v(S ∩ N1) + . . . + v(S ∩ Np) =

Ni∈P

v(S ∩ Ni)

Lemma

The core of a decomposable convex game is the cartesian product of the cores of the components of any decomposition A strictly convex game is indecomposable

Cores of Convex Games and Pascal’s Triangle Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 4/7

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Decomposable Games

Let P = {N1, . . . , Np} be a partition of N, with p ≥ 2 (N, v) is decomposable with respect to P if, for each S ⊆ N, v(S) = v(S ∩ N1) + . . . + v(S ∩ Np) =

Ni∈P

v(S ∩ Ni)

Lemma

The core of a decomposable convex game is the cartesian product of the cores of the components of any decomposition A strictly convex game is indecomposable (N, vFT ) is decomposable with respect to P = {T, N\T}

Cores of Convex Games and Pascal’s Triangle Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 4/7

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Decomposable Games

Let P = {N1, . . . , Np} be a partition of N, with p ≥ 2 (N, v) is decomposable with respect to P if, for each S ⊆ N, v(S) = v(S ∩ N1) + . . . + v(S ∩ Np) =

Ni∈P

v(S ∩ Ni)

Lemma

The core of a decomposable convex game is the cartesian product of the cores of the components of any decomposition A strictly convex game is indecomposable (N, vFT ) is decomposable with respect to P = {T, N\T} vFT (S) := v((S ∩ T) ∪ (N\T)) − v(N\T)

+ v(S ∩ (N\T))
Cores of Convex Games and Pascal’s Triangle

Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 4/7

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Decomposable Games

Let P = {N1, . . . , Np} be a partition of N, with p ≥ 2 (N, v) is decomposable with respect to P if, for each S ⊆ N, v(S) = v(S ∩ N1) + . . . + v(S ∩ Np) =

Ni∈P

v(S ∩ Ni)

Lemma

The core of a decomposable convex game is the cartesian product of the cores of the components of any decomposition A strictly convex game is indecomposable (N, vFT ) is decomposable with respect to P = {T, N\T} vFT (S) := v((S ∩ T) ∪ (N\T)) − v(N\T)

+ v(S ∩ (N\T))
vFT (S∩(N\T))

Cores of Convex Games and Pascal’s Triangle Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 4/7

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Decomposable Games

Let P = {N1, . . . , Np} be a partition of N, with p ≥ 2 (N, v) is decomposable with respect to P if, for each S ⊆ N, v(S) = v(S ∩ N1) + . . . + v(S ∩ Np) =

Ni∈P

v(S ∩ Ni)

Lemma

The core of a decomposable convex game is the cartesian product of the cores of the components of any decomposition A strictly convex game is indecomposable (N, vFT ) is decomposable with respect to P = {T, N\T} vFT (S) := v((S ∩ T) ∪ (N\T)) − v(N\T)

vFT (S∩T)

+ v(S ∩ (N\T))

vFT (S∩(N\T))

Cores of Convex Games and Pascal’s Triangle Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 4/7

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Decomposable Games

Let P = {N1, . . . , Np} be a partition of N, with p ≥ 2 (N, v) is decomposable with respect to P if, for each S ⊆ N, v(S) = v(S ∩ N1) + . . . + v(S ∩ Np) =

Ni∈P

v(S ∩ Ni)

Lemma

The core of a decomposable convex game is the cartesian product of the cores of the components of any decomposition A strictly convex game is indecomposable (N, vFT ) is decomposable with respect to P = {T, N\T} vFT (S) := v((S ∩ T) ∪ (N\T)) − v(N\T)

vFT (S∩T)

+ v(S ∩ (N\T))

vFT (S∩(N\T))

Cores of Convex Games and Pascal’s Triangle Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 4/7

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Decomposable Games

Cores of Convex Games and Pascal’s Triangle Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 5/7

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Decomposable Games

v(N) = 5

Cores of Convex Games and Pascal’s Triangle Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 5/7

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Decomposable Games

v(N) = 5 v(1) = v(2) = v(3) = 0

Cores of Convex Games and Pascal’s Triangle Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 5/7

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Decomposable Games

v(N) = 5 v(1) = v(2) = v(3) = 0

v(3) = 0 v(1) = 0 v(2) = 0

Cores of Convex Games and Pascal’s Triangle Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 5/7

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Decomposable Games

v(N) = 5 v(1) = v(2) = v(3) = 0

v(3) = 0 v(1) = 0 v(2) = 0

1 2 3 Cores of Convex Games and Pascal’s Triangle Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 5/7

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Decomposable Games

v(N) = 5 v(1) = v(2) = v(3) = 0 v(12) = 2 ; v(23) = v(13) = 1

v(3) = 0 v(1) = 0 v(2) = 0

1 2 3 Cores of Convex Games and Pascal’s Triangle Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 5/7

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Decomposable Games

v(N) = 5 v(1) = v(2) = v(3) = 0 v(12) = 2 ; v(23) = v(13) = 1

v(3) = 0 v(1) = 0 v(2) = 0 v(12) = 2 v(23) = 1 v(13) = 1

1 2 3 Cores of Convex Games and Pascal’s Triangle Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 5/7

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Decomposable Games

v(N) = 5 v(1) = v(2) = v(3) = 0 v(12) = 2 ; v(23) = v(13) = 1

v(3) = 0 v(1) = 0 v(2) = 0 v(12) = 2 v(23) = 1 v(13) = 1

1 2 3 Cores of Convex Games and Pascal’s Triangle Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 5/7

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Decomposable Games

v(N) = 5 v(1) = v(2) = v(3) = 0 v(12) = 2 ; v(23) = v(13) = 1

1 2 3 Cores of Convex Games and Pascal’s Triangle Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 5/7

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Decomposable Games

v(N) = 5 v(1) = v(2) = v(3) = 0 v(12) = 2 ; v(23) = v(13) = 1

1 2 3 F23 Cores of Convex Games and Pascal’s Triangle Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 5/7

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Decomposable Games

v(N) = 5 v(1) = v(2) = v(3) = 0 v(12) = 2 ; v(23) = v(13) = 1

1 2 3 F23 x2 + x3 = 5 x1 = 0 Cores of Convex Games and Pascal’s Triangle Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 5/7

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Decomposable Games

v(N) = 5 v(1) = v(2) = v(3) = 0 v(12) = 2 ; v(23) = v(13) = 1

1 2 3 F23 x2 + x3 = 5 x1 = 0 F23 = Cores of Convex Games and Pascal’s Triangle Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 5/7

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Decomposable Games

v(N) = 5 v(1) = v(2) = v(3) = 0 v(12) = 2 ; v(23) = v(13) = 1

1 2 3 F23 x2 + x3 = 5 x1 = 0 F23 = ×

Cores of Convex Games and Pascal’s Triangle

Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 5/7

SLIDE 56

Decomposable Games

v(N) = 5 v(1) = v(2) = v(3) = 0 v(12) = 2 ; v(23) = v(13) = 1

1 2 3 F23 x2 + x3 = 5 x1 = 0 F23 = ×

Cores of Convex Games and Pascal’s Triangle

Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 5/7

SLIDE 57

Decomposable Games

v(N) = 5 v(1) = v(2) = v(3) = 0 v(12) = 2 ; v(23) = v(13) = 1

1 2 3 F23 x2 + x3 = 5 x1 = 0 F23 = ×

Cores of Convex Games and Pascal’s Triangle

Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 5/7

SLIDE 58

Decomposable Games

In FT the “negotiations” between T and N\T have been decided in favor of T

v(N) = 5 v(1) = v(2) = v(3) = 0 v(12) = 2 ; v(23) = v(13) = 1

1 2 3 F23 x2 + x3 = 5 x1 = 0 F23 = ×

Cores of Convex Games and Pascal’s Triangle

Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 5/7

SLIDE 59

Decomposable Games

In FT the “negotiations” between T and N\T have been decided in favor of T Thus, vFT is decomposable with respect to T and N\T.

v(N) = 5 v(1) = v(2) = v(3) = 0 v(12) = 2 ; v(23) = v(13) = 1

1 2 3 F23 x2 + x3 = 5 x1 = 0 F23 = ×

Cores of Convex Games and Pascal’s Triangle

Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 5/7

SLIDE 60

Decomposable Games

In FT the “negotiations” between T and N\T have been decided in favor of T Thus, vFT is decomposable with respect to T and N\T. Denote by (T, vT ) and (N\T, vN\T ) the games in the decomposition

v(N) = 5 v(1) = v(2) = v(3) = 0 v(12) = 2 ; v(23) = v(13) = 1

1 2 3 F23 x2 + x3 = 5 x1 = 0 F23 = ×

Cores of Convex Games and Pascal’s Triangle

Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 5/7

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Decomposable Games

In FT the “negotiations” between T and N\T have been decided in favor of T Thus, vFT is decomposable with respect to T and N\T. Denote by (T, vT ) and (N\T, vN\T ) the games in the decomposition If |T| > 1, the players in T still have to “negotiate”

(similarly in (N\T, vN\T ))

v(N) = 5 v(1) = v(2) = v(3) = 0 v(12) = 2 ; v(23) = v(13) = 1

1 2 3 F23 x2 + x3 = 5 x1 = 0 F23 = ×

Cores of Convex Games and Pascal’s Triangle

Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 5/7

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Results

Cores of Convex Games and Pascal’s Triangle Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 6/7

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Results

The combinatorial complexity of the core of a game is the number

f “different” kinds of polytopes there are among its facets

Cores of Convex Games and Pascal’s Triangle Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 6/7

SLIDE 64

Results

The combinatorial complexity of the core of a game is the number

f “different” kinds of polytopes there are among its facets

Cores of Convex Games and Pascal’s Triangle Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 6/7

SLIDE 65

Results

The combinatorial complexity of the core of a game is the number

f “different” kinds of polytopes there are among its facets

Cores of Convex Games and Pascal’s Triangle Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 6/7

SLIDE 66

Results

The combinatorial complexity of the core of a game is the number

f “different” kinds of polytopes there are among its facets

Let (N, v) be a strictly convex game and ∅ = T N

Cores of Convex Games and Pascal’s Triangle Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 6/7

SLIDE 67

Results

The combinatorial complexity of the core of a game is the number

f “different” kinds of polytopes there are among its facets

Let (N, v) be a strictly convex game and ∅ = T N

Result 1 Result 2 Result 3

Cores of Convex Games and Pascal’s Triangle Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 6/7

SLIDE 68

Results

The combinatorial complexity of the core of a game is the number

f “different” kinds of polytopes there are among its facets

Let (N, v) be a strictly convex game and ∅ = T N

Result 1

(T, vT ) and (N\T, vN\T ) are strictly convex

Result 2 Result 3

Cores of Convex Games and Pascal’s Triangle Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 6/7

SLIDE 69

Results

The combinatorial complexity of the core of a game is the number

f “different” kinds of polytopes there are among its facets

Let (N, v) be a strictly convex game and ∅ = T N

Result 1

(T, vT ) and (N\T, vN\T ) are strictly convex and C(N, vFT ) = C(T, vT ) × C(N\T, vN\T )

Result 2 Result 3

Cores of Convex Games and Pascal’s Triangle Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 6/7

SLIDE 70

Results

The combinatorial complexity of the core of a game is the number

f “different” kinds of polytopes there are among its facets

Let (N, v) be a strictly convex game and ∅ = T N

Result 1

(T, vT ) and (N\T, vN\T ) are strictly convex and C(N, vFT ) = C(T, vT ) × C(N\T, vN\T )

Result 2

For each t ∈ {1, . . . , n − 1}, C(N, v) has 2 n

t

“equal” facets

Result 3

Cores of Convex Games and Pascal’s Triangle Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 6/7

SLIDE 71

Results

The combinatorial complexity of the core of a game is the number

f “different” kinds of polytopes there are among its facets

Let (N, v) be a strictly convex game and ∅ = T N

Result 1

(T, vT ) and (N\T, vN\T ) are strictly convex and C(N, vFT ) = C(T, vT ) × C(N\T, vN\T )

Result 2

For each t ∈ {1, . . . , n − 1}, C(N, v) has 2 n

t

“equal” facets

(decomposable as the product of the cores of two strictly convex games with t and n − t players, respectively)

Result 3

Cores of Convex Games and Pascal’s Triangle Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 6/7

SLIDE 72

Results

The combinatorial complexity of the core of a game is the number

f “different” kinds of polytopes there are among its facets

Let (N, v) be a strictly convex game and ∅ = T N

Result 1

(T, vT ) and (N\T, vN\T ) are strictly convex and C(N, vFT ) = C(T, vT ) × C(N\T, vN\T )

Result 2

For each t ∈ {1, . . . , n − 1}, C(N, v) has 2 n

t

“equal” facets

(decomposable as the product of the cores of two strictly convex games with t and n − t players, respectively)

Result 3

The combinatorial complexity of C(N, v) is ⌊ n

2 ⌋

Cores of Convex Games and Pascal’s Triangle Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 6/7

SLIDE 73

Results

The combinatorial complexity of the core of a game is the number

f “different” kinds of polytopes there are among its facets

Let (N, v) be a strictly convex game and ∅ = T N

Result 1

(T, vT ) and (N\T, vN\T ) are strictly convex and C(N, vFT ) = C(T, vT ) × C(N\T, vN\T )

Result 2

For each t ∈ {1, . . . , n − 1}, C(N, v) has 2 n

t

“equal” facets

(decomposable as the product of the cores of two strictly convex games with t and n − t players, respectively)

Result 3

The combinatorial complexity of C(N, v) is ⌊ n

2 ⌋

Cores of Convex Games and Pascal’s Triangle Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 6/7

SLIDE 74

Results

The combinatorial complexity of the core of a game is the number

f “different” kinds of polytopes there are among its facets

Let (N, v) be a strictly convex game and ∅ = T N

Result 1

(T, vT ) and (N\T, vN\T ) are strictly convex and C(N, vFT ) = C(T, vT ) × C(N\T, vN\T )

Result 2

For each t ∈ {1, . . . , n − 1}, C(N, v) has 2 n

t

“equal” facets

(decomposable as the product of the cores of two strictly convex games with t and n − t players, respectively)

Result 3

The combinatorial complexity of C(N, v) is ⌊ n

2 ⌋

Cores of Convex Games and Pascal’s Triangle Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 6/7

SLIDE 75

Result 1: (T, vT ), (N\T, vN\T ) strictly convex. C(N, vFT ) = C(T, vT ) × C(N\T, vN\T ) Result 2: For each t ∈ {0, . . . , n}, C(N, v) has 2 n

t

“equal” facets

Result 3: The combinatorial complexity of C(N, v) is ⌊ n

2 ⌋

SLIDE 76

Result 1: (T, vT ), (N\T, vN\T ) strictly convex. C(N, vFT ) = C(T, vT ) × C(N\T, vN\T ) Result 2: For each t ∈ {0, . . . , n}, C(N, v) has 2 n

t

“equal” facets

Result 3: The combinatorial complexity of C(N, v) is ⌊ n

2 ⌋ 2

1

3

1

3

2

4

1

4

2

4

3

5

1

5

2

5

3

5

4

6

1

6

2

6

3

6

4

6

5

1
1

1

2
2

2

3
3

3

4
4

4

5
5

5

6
6

6

.

. .

SLIDE 77

Result 1: (T, vT ), (N\T, vN\T ) strictly convex. C(N, vFT ) = C(T, vT ) × C(N\T, vN\T ) Result 2: For each t ∈ {0, . . . , n}, C(N, v) has 2 n

t

“equal” facets

Result 3: The combinatorial complexity of C(N, v) is ⌊ n

2 ⌋ 2

1

3

1

3

2

4

1

4

2

4

3

5

1

5

2

5

3

5

4

6

1

6

2

6

3

6

4

6

5

1
1

1

2
2

2

3
3

3

4
4

4

5
5

5

6
6

6

0-players

1-player 2-players 3-players 4-players 5-players 6-players . . . . . .

SLIDE 78

Result 1: (T, vT ), (N\T, vN\T ) strictly convex. C(N, vFT ) = C(T, vT ) × C(N\T, vN\T ) Result 2: For each t ∈ {0, . . . , n}, C(N, v) has 2 n

t

“equal” facets

Result 3: The combinatorial complexity of C(N, v) is ⌊ n

2 ⌋

and their facets (cores of face games) cores of strictly convex games

2

1

3

1

3

2

4

1

4

2

4

3

5

1

5

2

5

3

5

4

6

1

6

2

6

3

6

4

6

5

1
1

1

2
2

2

3
3

3

4
4

4

5
5

5

6
6

6

0-players

1-player 2-players 3-players 4-players 5-players 6-players . . . . . .

SLIDE 79

Result 1: (T, vT ), (N\T, vN\T ) strictly convex. C(N, vFT ) = C(T, vT ) × C(N\T, vN\T ) Result 2: For each t ∈ {0, . . . , n}, C(N, v) has 2 n

t

“equal” facets

Result 3: The combinatorial complexity of C(N, v) is ⌊ n

2 ⌋

and their facets (cores of face games) cores of strictly convex games

2

1

3

1

3

2

4

1

4

2

4

3

5

1

5

2

5

3

5

4

6

1

6

2

6

3

6

4

6

5

1
1

1

2
2

2

3
3

3

4
4

4

5
5

5

6
6

6

∅

0-players 1-player 2-players 3-players 4-players 5-players 6-players . . . . . .

SLIDE 80

Result 1: (T, vT ), (N\T, vN\T ) strictly convex. C(N, vFT ) = C(T, vT ) × C(N\T, vN\T ) Result 2: For each t ∈ {0, . . . , n}, C(N, v) has 2 n

t

“equal” facets

Result 3: The combinatorial complexity of C(N, v) is ⌊ n

2 ⌋

and their facets (cores of face games) cores of strictly convex games

2

1

3

1

3

2

4

1

4

2

4

3

5

1

5

2

5

3

5

4

6

1

6

2

6

3

6

4

6

5

1
1

1

2
2

2

3
3

3

4
4

4

5
5

5

6
6

6

∅
0-players

1-player 2-players 3-players 4-players 5-players 6-players . . . . . .

SLIDE 81

Result 1: (T, vT ), (N\T, vN\T ) strictly convex. C(N, vFT ) = C(T, vT ) × C(N\T, vN\T ) Result 2: For each t ∈ {0, . . . , n}, C(N, v) has 2 n

t

“equal” facets

Result 3: The combinatorial complexity of C(N, v) is ⌊ n

2 ⌋

and their facets (cores of face games) cores of strictly convex games

2

1

3

1

3

2

4

1

4

2

4

3

5

1

5

2

5

3

5

4

6

1

6

2

6

3

6

4

6

5

1
1

1

2
2

2

3
3

3

4
4

4

5
5

5

6
6

6

∅
0-players

1-player 2-players 3-players 4-players 5-players 6-players . . . . . .

SLIDE 82

Result 1: (T, vT ), (N\T, vN\T ) strictly convex. C(N, vFT ) = C(T, vT ) × C(N\T, vN\T ) Result 2: For each t ∈ {0, . . . , n}, C(N, v) has 2 n

t

“equal” facets

Result 3: The combinatorial complexity of C(N, v) is ⌊ n

2 ⌋

and their facets (cores of face games) cores of strictly convex games

2

1

3

1

3

2

4

1

4

2

4

3

5

1

5

2

5

3

5

4

6

1

6

2

6

3

6

4

6

5

1
1

1

2
2

2

3
3

3

4
4

4

5
5

5

6
6

6

∅
0-players

1-player 2-players 3-players 4-players 5-players 6-players . . . . . .

SLIDE 83

Result 1: (T, vT ), (N\T, vN\T ) strictly convex. C(N, vFT ) = C(T, vT ) × C(N\T, vN\T ) Result 2: For each t ∈ {0, . . . , n}, C(N, v) has 2 n

t

“equal” facets

Result 3: The combinatorial complexity of C(N, v) is ⌊ n

2 ⌋

and their facets (cores of face games) cores of strictly convex games

2

1

3

1

3

2

4

1

4

2

4

3

5

1

5

2

5

3

5

4

6

1

6

2

6

3

6

4

6

5

1
1

1

2
2

2

3
3

3

4
4

4

5
5

5

6
6

6

∅
0-players

1-player 2-players 3-players 4-players 5-players 6-players . . . . . .

SLIDE 84

Result 1: (T, vT ), (N\T, vN\T ) strictly convex. C(N, vFT ) = C(T, vT ) × C(N\T, vN\T ) Result 2: For each t ∈ {0, . . . , n}, C(N, v) has 2 n

t

“equal” facets

Result 3: The combinatorial complexity of C(N, v) is ⌊ n

2 ⌋

and their facets (cores of face games) cores of strictly convex games

2

1

3

1

3

2

4

1

4

2

4

3

5

1

5

2

5

3

5

4

6

1

6

2

6

3

6

4

6

5

1
1

1

2
2

2

3
3

3

4
4

4

5
5

5

6
6

6

∅
Core of a

5-player game Core of a 5-player game Core of a 6-player game Core of a 6-player game 0-players 1-player 2-players 3-players 4-players 5-players 6-players . . . . . .

SLIDE 85

Result 1: (T, vT ), (N\T, vN\T ) strictly convex. C(N, vFT ) = C(T, vT ) × C(N\T, vN\T ) Result 2: For each t ∈ {0, . . . , n}, C(N, v) has 2 n

t

“equal” facets

Result 3: The combinatorial complexity of C(N, v) is ⌊ n

2 ⌋

and their facets (cores of face games) cores of strictly convex games

2

1

3

1

3

2

4

1

4

2

4

3

5

1

5

2

5

3

5

4

6

1

6

2

6

3

6

4

6

5

1
1

1

2
2

2

3
3

3

4
4

4

5
5

5

6
6

6

∅
Core of a

5-player game Core of a 5-player game Core of a 6-player game Core of a 6-player game 0-players 1-player 2-players 3-players 4-players 5-players 6-players . . . . . .

SLIDE 86

Result 1: (T, vT ), (N\T, vN\T ) strictly convex. C(N, vFT ) = C(T, vT ) × C(N\T, vN\T ) Result 2: For each t ∈ {0, . . . , n}, C(N, v) has 2 n

t

“equal” facets

Result 3: The combinatorial complexity of C(N, v) is ⌊ n

2 ⌋

and their facets (cores of face games) cores of strictly convex games

2

1

3

1

3

2

4

1

4

2

4

3

5

1

5

2

5

3

5

4

6

1

6

2

6

3

6

4

6

5

1
1

1

2
2

2

3
3

3

4
4

4

5
5

5

6
6

6

∅
Core of a

5-player game Core of a 5-player game Core of a 6-player game Core of a 6-player game 0-players 1-player 2-players 3-players 4-players 5-players 6-players . . . . . .

2

1

→
=
×

SLIDE 87

Result 1: (T, vT ), (N\T, vN\T ) strictly convex. C(N, vFT ) = C(T, vT ) × C(N\T, vN\T ) Result 2: For each t ∈ {0, . . . , n}, C(N, v) has 2 n

t

“equal” facets

Result 3: The combinatorial complexity of C(N, v) is ⌊ n

2 ⌋

and their facets (cores of face games) cores of strictly convex games

2

1

3

1

3

2

4

1

4

2

4

3

5

1

5

2

5

3

5

4

6

1

6

2

6

3

6

4

6

5

1
1

1

2
2

2

3
3

3

4
4

4

5
5

5

6
6

6

∅
Core of a

5-player game Core of a 5-player game Core of a 6-player game Core of a 6-player game 0-players 1-player 2-players 3-players 4-players 5-players 6-players . . . . . .

2

1

→
=
×

SLIDE 88

Result 1: (T, vT ), (N\T, vN\T ) strictly convex. C(N, vFT ) = C(T, vT ) × C(N\T, vN\T ) Result 2: For each t ∈ {0, . . . , n}, C(N, v) has 2 n

t

“equal” facets

Result 3: The combinatorial complexity of C(N, v) is ⌊ n

2 ⌋

and their facets (cores of face games) cores of strictly convex games

2

1

3

1

3

2

4

1

4

2

4

3

5

1

5

2

5

3

5

4

6

1

6

2

6

3

6

4

6

5

1
1

1

2
2

2

3
3

3

4
4

4

5
5

5

6
6

6

∅
Core of a

5-player game Core of a 5-player game Core of a 6-player game Core of a 6-player game 0-players 1-player 2-players 3-players 4-players 5-players 6-players . . . . . .

2

1

→
=
×
3

1

,

3

2

→

= ×

SLIDE 89

Result 1: (T, vT ), (N\T, vN\T ) strictly convex. C(N, vFT ) = C(T, vT ) × C(N\T, vN\T ) Result 2: For each t ∈ {0, . . . , n}, C(N, v) has 2 n

t

“equal” facets

Result 3: The combinatorial complexity of C(N, v) is ⌊ n

2 ⌋

and their facets (cores of face games) cores of strictly convex games

2

1

3

1

3

2

4

1

4

2

4

3

5

1

5

2

5

3

5

4

6

1

6

2

6

3

6

4

6

5

1
1

1

2
2

2

3
3

3

4
4

4

5
5

5

6
6

6

∅
Core of a

5-player game Core of a 5-player game Core of a 6-player game Core of a 6-player game 0-players 1-player 2-players 3-players 4-players 5-players 6-players . . . . . .

2

1

→
=
×
3

1

,

3

2

→

= ×

SLIDE 90

Result 1: (T, vT ), (N\T, vN\T ) strictly convex. C(N, vFT ) = C(T, vT ) × C(N\T, vN\T ) Result 2: For each t ∈ {0, . . . , n}, C(N, v) has 2 n

t

“equal” facets

Result 3: The combinatorial complexity of C(N, v) is ⌊ n

2 ⌋

and their facets (cores of face games) cores of strictly convex games

2

1

3

1

3

2

4

1

4

2

4

3

5

1

5

2

5

3

5

4

6

1

6

2

6

3

6

4

6

5

1
1

1

2
2

2

3
3

3

4
4

4

5
5

5

6
6

6

∅
Core of a

5-player game Core of a 5-player game Core of a 6-player game Core of a 6-player game 0-players 1-player 2-players 3-players 4-players 5-players 6-players . . . . . .

2

1

→
=
×
3

1

,

3

2

→

= ×

4

1

,

4

3

→

=

×

SLIDE 91

Result 1: (T, vT ), (N\T, vN\T ) strictly convex. C(N, vFT ) = C(T, vT ) × C(N\T, vN\T ) Result 2: For each t ∈ {0, . . . , n}, C(N, v) has 2 n

t

“equal” facets

Result 3: The combinatorial complexity of C(N, v) is ⌊ n

2 ⌋

and their facets (cores of face games) cores of strictly convex games

2

1

3

1

3

2

4

1

4

2

4

3

5

1

5

2

5

3

5

4

6

1

6

2

6

3

6

4

6

5

1
1

1

2
2

2

3
3

3

4
4

4

5
5

5

6
6

6

∅
Core of a

5-player game Core of a 5-player game Core of a 6-player game Core of a 6-player game 0-players 1-player 2-players 3-players 4-players 5-players 6-players . . . . . .

2

1

→
=
×
3

1

,

3

2

→

= ×

4

1

,

4

3

→

=

×

SLIDE 92

Result 1: (T, vT ), (N\T, vN\T ) strictly convex. C(N, vFT ) = C(T, vT ) × C(N\T, vN\T ) Result 2: For each t ∈ {0, . . . , n}, C(N, v) has 2 n

t

“equal” facets

Result 3: The combinatorial complexity of C(N, v) is ⌊ n

2 ⌋

and their facets (cores of face games) cores of strictly convex games

2

1

3

1

3

2

4

1

4

2

4

3

5

1

5

2

5

3

5

4

6

1

6

2

6

3

6

4

6

5

1
1

1

2
2

2

3
3

3

4
4

4

5
5

5

6
6

6

∅
Core of a

5-player game Core of a 5-player game Core of a 6-player game Core of a 6-player game 0-players 1-player 2-players 3-players 4-players 5-players 6-players . . . . . .

2

1

→
=
×
3

1

,

3

2

→

= ×

4

1

,

4

3

→

=

×

4

2

→

= ×

SLIDE 93

Result 1: (T, vT ), (N\T, vN\T ) strictly convex. C(N, vFT ) = C(T, vT ) × C(N\T, vN\T ) Result 2: For each t ∈ {0, . . . , n}, C(N, v) has 2 n

t

“equal” facets

Result 3: The combinatorial complexity of C(N, v) is ⌊ n

2 ⌋

and their facets (cores of face games) cores of strictly convex games

2

1

3

1

3

2

4

1

4

2

4

3

5

1

5

2

5

3

5

4

6

1

6

2

6

3

6

4

6

5

1
1

1

2
2

2

3
3

3

4
4

4

5
5

5

6
6

6

∅
Core of a

5-player game Core of a 5-player game Core of a 6-player game Core of a 6-player game 0-players 1-player 2-players 3-players 4-players 5-players 6-players . . . . . .

2

1

→
=
×
3

1

,

3

2

→

= ×

4

1

,

4

3

→

=

×

4

2

→

= ×

SLIDE 94

Result 1: (T, vT ), (N\T, vN\T ) strictly convex. C(N, vFT ) = C(T, vT ) × C(N\T, vN\T ) Result 2: For each t ∈ {0, . . . , n}, C(N, v) has 2 n

t

“equal” facets

Result 3: The combinatorial complexity of C(N, v) is ⌊ n

2 ⌋

and their facets (cores of face games) cores of strictly convex games

2

1

3

1

3

2

4

1

4

2

4

3

5

1

5

2

5

3

5

4

6

1

6

2

6

3

6

4

6

5

1
1

1

2
2

2

3
3

3

4
4

4

5
5

5

6
6

6

∅
Core of a

5-player game Core of a 5-player game Core of a 6-player game Core of a 6-player game 0-players 1-player 2-players 3-players 4-players 5-players 6-players . . . . . .

2

1

→
=
×
3

1

,

3

2

→

= ×

5

1

,

5

4

→

=

×

4

1

,

4

3

→

=

×

4

2

→

= ×

SLIDE 95

Result 1: (T, vT ), (N\T, vN\T ) strictly convex. C(N, vFT ) = C(T, vT ) × C(N\T, vN\T ) Result 2: For each t ∈ {0, . . . , n}, C(N, v) has 2 n

t

“equal” facets

Result 3: The combinatorial complexity of C(N, v) is ⌊ n

2 ⌋

and their facets (cores of face games) cores of strictly convex games

2

1

3

1

3

2

4

1

4

2

4

3

5

1

5

2

5

3

5

4

6

1

6

2

6

3

6

4

6

5

1
1

1

2
2

2

3
3

3

4
4

4

5
5

5

6
6

6

∅
Core of a

5-player game Core of a 5-player game Core of a 6-player game Core of a 6-player game 0-players 1-player 2-players 3-players 4-players 5-players 6-players . . . . . .

2

1

→
=
×
3

1

,

3

2

→

= ×

5

1

,

5

4

→

=

×

4

1

,

4

3

→

=

×

4

2

→

= ×

SLIDE 96

Result 1: (T, vT ), (N\T, vN\T ) strictly convex. C(N, vFT ) = C(T, vT ) × C(N\T, vN\T ) Result 2: For each t ∈ {0, . . . , n}, C(N, v) has 2 n

t

“equal” facets

Result 3: The combinatorial complexity of C(N, v) is ⌊ n

2 ⌋

and their facets (cores of face games) cores of strictly convex games

2

1

3

1

3

2

4

1

4

2

4

3

5

1

5

2

5

3

5

4

6

1

6

2

6

3

6

4

6

5

1
1

1

2
2

2

3
3

3

4
4

4

5
5

5

6
6

6

∅
Core of a

5-player game Core of a 5-player game Core of a 6-player game Core of a 6-player game 0-players 1-player 2-players 3-players 4-players 5-players 6-players . . . . . .

2

1

→
=
×
3

1

,

3

2

→

= ×

5

1

,

5

4

→

=

×

5

2

,

5

3

→

= × 4

1

,

4

3

→

=

×

4

2

→

= ×

SLIDE 97

Result 1: (T, vT ), (N\T, vN\T ) strictly convex. C(N, vFT ) = C(T, vT ) × C(N\T, vN\T ) Result 2: For each t ∈ {0, . . . , n}, C(N, v) has 2 n

t

“equal” facets

Result 3: The combinatorial complexity of C(N, v) is ⌊ n

2 ⌋

and their facets (cores of face games) cores of strictly convex games

2

1

3

1

3

2

4

1

4

2

4

3

5

1

5

2

5

3

5

4

6

1

6

2

6

3

6

4

6

5

×

Core of a 5-player game

×

Core of a 5-player game

1
1

1

2
2

2

3
3

3

4
4

4

5
5

5

6
6

6

∅
Core of a

5-player game Core of a 5-player game Core of a 6-player game Core of a 6-player game 0-players 1-player 2-players 3-players 4-players 5-players 6-players . . . . . .

2

1

→
=
×
3

1

,

3

2

→

= ×

5

1

,

5

4

→

=

×

5

2

,

5

3

→

= × 4

1

,

4

3

→

=

×

4

2

→

= ×

SLIDE 98

Result 1: (T, vT ), (N\T, vN\T ) strictly convex. C(N, vFT ) = C(T, vT ) × C(N\T, vN\T ) Result 2: For each t ∈ {0, . . . , n}, C(N, v) has 2 n

t

“equal” facets

Result 3: The combinatorial complexity of C(N, v) is ⌊ n

2 ⌋

and their facets (cores of face games) cores of strictly convex games

2

1

3

1

3

2

4

1

4

2

4

3

5

1

5

2

5

3

5

4

6

1

6

2

6

3

6

4

6

5

×

Core of a 5-player game

× ×

×

Core of a 5-player game

1
1

1

2
2

2

3
3

3

4
4

4

5
5

5

6
6

6

∅
Core of a

5-player game Core of a 5-player game Core of a 6-player game Core of a 6-player game 0-players 1-player 2-players 3-players 4-players 5-players 6-players . . . . . .

2

1

→
=
×
3

1

,

3

2

→

= ×

5

1

,

5

4

→

=

×

5

2

,

5

3

→

= × 4

1

,

4

3

→

=

×

4

2

→

= ×

SLIDE 99

Result 1: (T, vT ), (N\T, vN\T ) strictly convex. C(N, vFT ) = C(T, vT ) × C(N\T, vN\T ) Result 2: For each t ∈ {0, . . . , n}, C(N, v) has 2 n

t

“equal” facets

Result 3: The combinatorial complexity of C(N, v) is ⌊ n

2 ⌋

and their facets (cores of face games) cores of strictly convex games

2

1

3

1

3

2

4

1

4

2

4

3

5

1

5

2

5

3

5

4

6

1

6

2

6

3

6

4

6

5

×

Core of a 5-player game

× × ×

×

Core of a 5-player game

1
1

1

2
2

2

3
3

3

4
4

4

5
5

5

6
6

6

∅
Core of a

5-player game Core of a 5-player game Core of a 6-player game Core of a 6-player game 0-players 1-player 2-players 3-players 4-players 5-players 6-players . . . . . .

2

1

→
=
×
3

1

,

3

2

→

= ×

5

1

,

5

4

→

=

×

5

2

,

5

3

→

= × 4

1

,

4

3

→

=

×

4

2

→

= ×

SLIDE 100

Cores of Convex Games and Pascal’s Triangle

Julio Gonz´ alez-D´ ıaz

Kellogg School of Management (CMS-EMS) Northwestern University and Research Group in Economic Analysis Universidad de Vigo ........................ (joint with Estela S´ anchez-Rodr´ ıguez)