Material for “ Large Matching Markets as Two - Sided Demand Systems

TO VERIFY THAT THE STATEMENT IN LEMMA 2.1 is indeed equivalent to the usual definition of pairwise stability, notice that if μ is not pairwise stable, there exists a pair i j = μw(i) such that Uij > Uiμw(i) and Vji > Vjμm(j). In particular, j is available to i under μ, that is, j ∈Mi[μ], so that U∗ i (Mi[μ]) ≥ Uij > Uiμw(i), which violates the first part of the condition. Conversely, if the condition in the lemma does not hold for woman i, then there exists j ∈ W ∗ i [μ] such that U∗ i (Mi[μ]) ≥ Uij > Uiμw(i). On the other hand, j ∈ Mi[μ] implies that Vji ≥ Vjμm(j), and that inequality is strict in the absence of ties since, by assumption, i = μm(j). On the other hand, if the condition is violated for a man j, we can find a blocking pair consisting of j and a woman i ∈Wj[μ] using an analogous argument. Q.E.D.