Las Tecnologías de la Información y la Comunicación (TIC)

Proof Let S ∈ . By Lemma 2.47 there exist Weber strings in(S) as well as _{(N\S), say (R}

i)i∈Sand (Rj)j∈N\S, respectively. Then (Ri)i∈Nis a Weber string in where

Ri=

R_i for i∈S

{i} ∪(N\R_i) for i∈/S

Clearly if i∈S, Ri∈; furthermore, i∈R_i =Riand for i such that Ri=Ri= {i} there is a player j ∈S with R−i\ {i} =R_i \ {i} =R_j =Rj. Now if i ∈/ S either Ri = {i} ∪N\R_i =N\(R_i \ {i})=N\R_j for a player j∈/S, or Ri =N\∅=N and, since (_{)=for every lattice, we conclude that Ri∈.}

Furthermore, i∈Riand

• if R_i =N\S, then Ri\ {i} =N\R_i =S=R_j=Rjfor some j∈S;

• if R_i =N\S then there is a player k∈/ S with R_i=R_k\ {k}. Hence, Ri\ {i} = N\R_i =N\(R_k\ {k})= {k} ∪N\R_k =Rk.

We conclude that for all players i ∈ N we have Ri ∈ and if Ri = {i}there is a

player j such that Ri\ {i} =Rj.

Corollary 2.49 Let S1,. . ., SK be elements insuch that S1 ⊂ S2 ⊂ · · · ⊂ SK. Then there exists a Weber string inof which S1,. . ., SKare members.

Proof of 2.41(a)

Sinceis discerning by Lemma 2.39 there exists at least one Weber string in. Now suppose to the contrary that there exists a Core allocation x∈C(, v) such that x ∈/ W(, v)+. By Theorem 2.34 the set W(, v)+ is polyhedral and since is discerning/non-degenerate, it does not contain a nontrivial linear subspace. Thus x is an extreme point of

Conv [{x} ∪(W(, v)+)] .

The normals of all supporting hyperplanes in an extreme point of a polyhedral set are well known to form a full dimensional cone. Therefore, there exists a normal vector, say p∈RN, with non-equal coefficients such that for each y∈W(, v) and y∈:

p·x<p·(y+y).

Now by Theorem 2.34 for every i ∈N, j ∈ ∂iand M 0 it holds that for every y∈W(, v):

p·x<p·y+M p·(ej−ei)=p·y+M (pj−pi) (2.36) implying that for every i∈N and j∈∂i: pjpi. Now label the players in N such that p1> p2 >· · · >pn. Consider for every k∈ N the coalition Sk = {1,. . ., k}. If i ∈ Sk, then pi pk, in turn implying that pj pi pkfor all j ∈∂i. Hence, ∂i⊂Skfor all i∈Skimplying that

Sk =( i_∈Sk

∂i∈.

Hence, (Sk)k∈Ndetermines a Weber string in. The corresponding marginal vector y for v now belongs toW(, v). Thus,

p·x<p·y= k∈N % v(Sk)−v(Sk−1)& =v(N) pn+ k∈N v(Sk) (pk−pk+1).

Since x∈C(, v) and Sk ∈(k∈N), we conclude that x(N)=v(N) and x(Sk) v(Sk), k=n. Furthermore, pk−pk₊1 0 for k =n. Therefore, the above may be rewritten as

p·x<v(N) pn+ k∈N v(Sk) (pk−pk+1) x(N) pn+ n−1 k=1 k j=1 xj(pk−pk+1) = n k=1 k j=1 xjpk− n k=2 k−1 j=1 xjpk =x1p1+ n k=2 pkxk=p·x. This constitutes a contradiction, proving the desired inclusion. Proof of 2.41(b)

Only if:

Suppose that v ∈ GN. Given the already established facts, we only have to show that every Weber allocation x ∈ W(, v) belongs to C(, v). So, let x ∈ W(, v) correspond to some Weber string (Ri)i∈Nwith for every i∈N

xi=v(Ri)−v(Ri\ {i}). Since v is convex onfor every i∈S∈we have that

v(Ri)+v(S∩(Ri\ {i}))v(S∩Ri)+v(Ri\ {i}). Thus, for every i∈S:

xiv(S∩Ri)−v(S∩Ri\ {i}).

Without loss of generality we may assume that N is labelled such that the players in S are labelled first and{1,. . ., k} ⊂Rkfor all k∈S. Then

i∈S xi |S_| k=1 xk |S_| k=1 v(S∩Rk)−v(S∩Rk\ {k}) = |S| k=1 v({1,. . ., k})− |S|−1 k=1 v({1,. . ., k}) =v({1,. . .,|S|})=v(S).

If:

Suppose that C(, v)=W(, v)+. Take S, T∈. Sinceis discerning by Corollary 2.49 there exists a Weber string in, say (Ri)i∈N, such that it contains the coalitions S∪T and S∩T.

Let x ∈ W(, v) be the corresponding Weber allocation—or marginal payoff vector. It is evident that x(S∩T) =v(S∩T) as well as x(S∪T) =v(S∪T). By hypothesis x∈C(, v), which leads to the conclusion that

v(S)+v(T)x(S)+x(T)= =x(S∪T)+x(S∩T)= =v(S∪T)+v(S∩T). This proves the convexity of v.

2.5 Problems

Problem 2.1 Let N = {1, 2, 3}and consider the game v∈GN_{given by the following} table:

S ∅ 1 2 3 12 13 23 123

v(S) 0 0 5 6 15 0 10 20

(a) Give the (0, 1)-normalization vof this game v.

(b) Compute the Core C(v) ⊂ I(v) of the game v and of its (0, 1)-normalization v. Show that these two Cores are essentially identical. Formulate this identity precisely.

(c) Draw the Core C(v) of the (0, 1)-normalization v of the game v in the two- dimensional simplexS2.

Problem 2.2 Consider the so-called bridge game, introduced by Kaneko and Wood- ers (1982, Example 2.3). Let N = {1,. . ., n}and define v∈GNby

v(S)=

1 if|S| =4 0 otherwise. In the bridge game only quartets can generate benefits.

Show that C(v)=∅if and only if n=4m for some m∈N.

Problem 2.3 Construct a non-essential three player game that has an empty Core.