Further Applications of Core Theory to Market Exchange

Abstract

The set of imputations in the core satisfy a number of inequalities of the type

$$\sum\limits_{i\;in\;S} {{x_i} \geqslant v(S)} ,$$

(1)

where x_i is the imputation of individual i and v(S) denotes the value of the characteristic function for the coalition S. The term v(S) gives the largest amount that the members of S can guarantee themselves under the most adverse conditions. Therefore, v(S) gives the largest amount that the members of S can obtain by their own resources. If the individuals may join any coalition, then their imputations must satisfy the system of inequalities given by (1). In addition it must be true that

$$\sum\limits_I {{x_i} \leqslant v(I)} ,$$

(2)

where I is the set of all individuals. It is natural, therefore, to study the linear programming problem

$$\min \sum {{x_i}} subject\;to\;(1), for\;all S \subset I and S \ne \;I.$$

(3)

The solution of this problem gives the smallest amount that the individuals can obtain while satisfying all of the core constraints given by (1). Hence for the core to be nonempty it is necessary that

$$\min \sum {{x_i} \leqslant v(I)} .$$

(4)