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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

The Core and its Faces H T := { x ∈ R n : � i ∈ T x i = v ( T ) } Cores of Convex Games and Pascal’s Triangle Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 2/7

The Core and its Faces H T := { x ∈ R n : � i ∈ T x i = v ( T ) } F T := C ( N, v ) ∩ H N \ T Cores of Convex Games and Pascal’s Triangle Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 2/7

The Core and its Faces H T := { x ∈ R n : � i ∈ T x i = v ( T ) } F T := C ( N, v ) ∩ H N \ T 3 1 2 Cores of Convex Games and Pascal’s Triangle Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 2/7

The Core and its Faces H T := { x ∈ R n : � i ∈ T x i = v ( T ) } F T := C ( N, v ) ∩ H N \ T 3 F 1 1 2 Cores of Convex Games and Pascal’s Triangle Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 2/7

The Core and its Faces H T := { x ∈ R n : � i ∈ T x i = v ( T ) } F T := C ( N, v ) ∩ H N \ T 3 x 2 + x 3 = v (23) F 1 1 2 Cores of Convex Games and Pascal’s Triangle Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 2/7

The Core and its Faces H T := { x ∈ R n : � i ∈ T x i = v ( T ) } F T := C ( N, v ) ∩ H N \ T 3 x 1 = v (1) F 23 x 2 + x 3 = v (23) F 1 1 2 Cores of Convex Games and Pascal’s Triangle Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 2/7

The Core and its Faces H T := { x ∈ R n : � i ∈ T x i = v ( T ) } F T := C ( N, v ) ∩ H N \ T 3 x 1 = v (1) F 23 x 2 + x 3 = v (23) F 1 1 2 Cores of Convex Games and Pascal’s Triangle Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 2/7

The Core and its Faces H T := { x ∈ R n : � i ∈ T x i = v ( T ) } F T := C ( N, v ) ∩ H N \ T 3 x 1 = v (1) F 23 x 2 + x 3 = v (23) F 1 1 2 T -face game: ( N, v F T ) v F T ( S ) := Cores of Convex Games and Pascal’s Triangle Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 2/7

The Core and its Faces H T := { x ∈ R n : � i ∈ T x i = v ( T ) } F T := C ( N, v ) ∩ H N \ T 3 x 1 = v (1) F 23 x 2 + x 3 = v (23) F 1 1 2 T -face game: ( N, v F T ) v F T ( S ) := v ( S ∩ ( N \ T )) Cores of Convex Games and Pascal’s Triangle Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 2/7

The Core and its Faces H T := { x ∈ R n : � i ∈ T x i = v ( T ) } F T := C ( N, v ) ∩ H N \ T 3 x 1 = v (1) F 23 x 2 + x 3 = v (23) F 1 1 2 T -face game: ( N, v F T ) v F T ( 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

The Core and its Faces 3 F 23 F 1 1 2 Cores of Convex Games and Pascal’s Triangle Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 3/7

The Core and its Faces 3 F 23 F 1 1 2 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

The Core and its Faces 3 F 23 F 1 1 2 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

The Core and its Faces 3 F 23 = C ( N, v F 23 ) F 23 F 1 = C ( N, v F 1 ) F 1 1 2 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, v F T ) = F T . Cores of Convex Games and Pascal’s Triangle Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 3/7

The Core and its Faces 3 F 23 = C ( N, v F 23 ) F 23 F 1 = C ( N, v F 1 ) F 1 1 2 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, v F T ) = F T . Therefore, C ( N, v ) = co { C ( N, v F T ) : ∅ � = T � N } Cores of Convex Games and Pascal’s Triangle Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 3/7

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

Decomposable Games Let P = { N 1 , . . . , N p } 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

Decomposable Games Let P = { N 1 , . . . , N p } 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

Decomposable Games Let P = { N 1 , . . . , N p } 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 ∩ N 1 ) + . . . + v ( S ∩ N p ) Cores of Convex Games and Pascal’s Triangle Gonz´ alez-D´ ıaz and S´ anchez-Rodr´ ıguez 4/7