Cooperative Game TheorySolutionThe set S is a superset of al

Cooperative Game Theory

Solution

The set S is a superset of all the numbers obtained from 2^(N)

since j belongs to domain of natural numbers hence j will belong to S and j will also be the subset of S

Hence we can write

S U j = S + j - S(int)j

S(int)j = j (since j is a subset of S)

S U j = S + j - j = S

Hence the statement v(S) = v(S U j) holds true for this case.

If the core is non-empty implies that there exists at least one element in the core, the lowest positive integer present in the group S will be 1

2^(j) = 1

Hence j = 0, for 2^(j) to be equal to 1

Hence for every non-empty core the jth component of the elements in the core is 0