same reasoning for the second deviation involving f 1 and f 3 and find that the

Same reasoning for the second deviation involving f 1

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same reasoning for the second deviation involving f 1 and f 3 and find that the upper bound on u j 1 is: u j 1 π 11 + π 22 - π 12 . We find that all pairs ( u j 1 , u j 2 ) satisfying inequalities u j 1 π 22 + π 11 - π 12 and u j 2 π 22 + π 11 - π 21 with 0 u j 1 π 11 and 0 u j 2 π 22 yield imputations u j 1 , u j 2 , u i 1 = π 11 - u j 1 and u i 2 = π 22 - u j 2 that support the inefficient assignment as family-stable. To represent this set graphically, assume π 22 > π 21 , π 12 > π 11 , and π 12 = π 21 16 . This example shows that considering families generates some coordination problems which may translate into inefficient outcomes. The coordination problem emerges here because deviations are only allowed for pairs of families. It is interesting to note that, in our example, if we allowed families to deviate in triples, the three families could coordinate their deviations to reach the efficient assignment. This means that 15 π 22 - u j 2 + u i 1 + u j 1 = u i 2 + u j 1 π 22 - u j 2 + π 11 = π 21 u j 2 = π 22 + π 11 - π 12 , which is the higher bound on u j 2 . 16 This is the same assumption as that made by Browning et al. (2014) in Chapter 8. We can use a numerical example to explore this result. For instance with π 22 = 8 , π 21 = π 12 = 6 , π 11 = 2 we have that the shares u j 1 = 2 , u j 2 = 3 , u i 1 = 0 , u i 2 = 5 support the inefficient assignment as family-stable, but obviously not as individual-stable. 14
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u j 2 π 22 u j 1 π 11 π 22 + π 11 - π 21 π 22 + π 11 - π 12 Figure 3: Inefficient family-stable matching the three families could obtain a higher aggregate surplus to share, and could find a sharing mode that would benefit all three. This is in sharp contrast with classical matching models on marriage in which individual-stable matchings are located in the core, defined in Shapley and Shubik (1971) as “the set of outcomes that no coalition can improve upon”. In our setting, the core is the set of the shares of surplus that support efficient matchings as family-stable, as no coalition of families would be able to improve upon it without making other families worse off. In other words, either the coalition would reach an inefficient matching which would destroy the surplus for other families, or the coalition would modify their shares of surplus within the efficient matching, reducing the surplus for other families. Our result here is that family- stable matchings exceed the core, as some inefficient family-stable outcomes could be improved upon by a coalition of a subset of families. This result comes from our definition of familial stability, which considers deviations by pairs of families. However, we could also choose an alternative definition which considers deviations of a subset of families. We could assume that families negotiate the marriage of their children multilaterally and commit through betrothal contracts. This alternative definition would resolve some situations where families are stuck in an inefficient matching. In reality, however, deviations by k > 2 families should generate coordination costs that 15
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may offset this positive result. In any case, as long as the number of families who can deviate together is bounded, inefficient outcomes are still likely to arise.
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