f 1 f 2 j 1 i 3 j 2 i 1 j 3 i 2 O u j 3 u j 1 u j 2 a f 1 f 2 f 3 j 1 i 3 j 2 i

F 1 f 2 j 1 i 3 j 2 i 1 j 3 i 2 o u j 3 u j 1 u j 2 a

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f 1 f 2 j 1 i 3 j 2 i 1 j 3 i 2 O u j 3 u j 1 u j 2 (a) f 1 f 2 f 3 j 1 i 3 j 2 i 1 j 3 i 2 O u j 3 u j 1 u j 2 (b) f 1 f 2 f 3 f 4 j 1 i 3 j 2 i 1 j 3 i 2 O u j 3 u j 1 u j 2 (c) f 1 f 2 f 3 f 4 j 1 i 3 j 2 i 1 j 3 i 2 O u j 3 u j 1 u j 2 (d) f 1 f 2 f 3 f 4 f 5 j 1 i 3 j 2 i 1 j 3 i 2 O u j 3 u j 1 u j 2 (e) f 1 f 2 f 3 f 4 f 5 f 6 j 1 i 3 j 2 i 1 j 3 i 2 O u j 3 u j 1 u j 2 (f) Figure 5: Sets of surplus and family partitions We observe that in family partitions for which all alternative husbands for the daugh- ters are in the same family, as in (a), (b) and (c), the shares of surplus are independent of each other. By contrast, when alternative husbands are scattered among different families (as in (d), (e) and (f)), we observe not only lower bounds for women’s shares, but also a functional relationship between the shares of surplus. Moreover, we observe that the more competition (i.e. the more families for the same number of males and partitions (b) and (d). To graphically represent the sets in Figure 5, we choose π 12 = π 23 = π 31 . 20
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females), the smaller the set of shares: the set shrinks when we go from (b) to (c), and when we go from (d) to (e). Finally, the family partitions for which inefficient match- ings can be family-stable, (b), (c), (d) and (e), are characterized by families having same-gender children and are also heterogenous in terms of family size, as opposed to (a) and (f). In particular, we find that for the family partition such that each family is composed of one son and one daughter, familial stability implies individual stability. Therefore for this family partition, the only family-stable assignments are the efficient ones and the sets of the shares of surplus that support the efficient assignments as stable are the same for individuals and families. We state our result formally in Theorem 2. Theorem 2 For the family partition such that each family is composed of one son and one daughter, a family-stable matching must be stable for individuals. Proof. Consider the family partition such that each family is composed of one son and one daughter. Consider a matching ( μ * ij , u * ij ) . This matching is family-stable if there is no pair of families who would like to deviate from it together (see Definition 1). We need to consider all possible deviations from this matching, which should cover families that are linked and families that are not linked. First consider any pair of linked families, f k and f k . If these two families are already linked in terms of all four of their children, then they cannot deviate together. This is because this family partition is such that each family is composed of one son and one daughter, so two families already linked through two marriages cannot deviate by swapping the marriages of their children. If these two families are linked only in terms of their children i k and j k , they could deviate together if they chose a marriage between their two other children, j k and i k . Conditions on the sharing of surplus of linked families for ( μ * ij , u * ij ) to be a family-stable matching are: u * f k + u * f k > u f k + u f k π k,k + u * i k + u * j k > π k,k + π k ,k u * i k + u * j k > π k ,k
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  • Spring '10
  • JAMES
  • J2, family dimension

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