A Proof of Theorem 4

We now turn to the proof of the theorem. The strategy is to construct a sequence of large economies with finitely many firms such that in every case there is a stable matching. Then, by a limit argument we obtain a stable matching for the original economy. In doing so, we structure the proof in three parts: First we define a sequence of truncated versions of the original economy. Second we show that each of them has a stable matching and third we study the limit point of the ensuing sequence of stable matchings and show it to be a stable matching of the original economy.

First part: Specification of a sequence of economies with finitely many firms.

For every natural number n, we define the set Fn:=f1,f2,...,fn and F˜n:=Fn∪∅. It is clear that F˜n⊂F˜n+1⊂F˜n+2⊂..., F˜=⋃n=1∞F˜n and |F˜n|=n+1. The correspondence Cf:χ⟶χ is that of Section 2 for f∈F˜n.

The preference of the workers are given by the restricted preference P|F˜n:{1,...,n+1}↦F˜n. That is to say P|F˜n and P order the firms in F˜n in the same way.

For every f∈F˜n the measures Dn,⪰fM and Dn,⪯fM are defined as

and

Hence, the truncated large economy is specified by (G,F˜n,(ΘP)P∈P,(Cf)f∈F˜n) and we obtain a sequence of large economies with finite sets of firms directed by inclusion {G,F˜n,(ΘP)P∈P,(Cf)f∈F˜n}n≥1.

Second part: Each of the terms of the sequence has a stable matching.

It is easy to check that Assumptions 1 and 2 hold in every economy (G,F˜n,(ΘP)P∈P,(Cf)f∈F˜n). Consequently, by Theorem 2 of Che, Kim, and Kojima (forthcoming), there exists a stable matching Mn=Mfnf∈F˜n∈χn+1 in every (G,F˜n,(ΘP)P∈P,(Cf)f∈F˜n)^[4]. We expand it to the product space χ|F˜| while still writing it as Mn=Mfnf∈F˜.^[5] Let us recall that χ|F˜| is a subset of the topological dual of C(Θ)|F˜| given by M|F˜|.

Let us consider the set Nff∈F˜∈M|F˜|:∑f∈F˜Nf=G. It can be easily checked that it is norm-bounded and by Alaoglu’s Theorem (Rudin 1991) it is weak∗-compact. Since Mn belongs to the above set  for every n, there exists a subsequence Mnkk∈K, where K is a subset of the natural numbers, converging to M in the weak∗-topology. Since a sequence in a product space converges if and only if the projection of each component converges (Kelly 1955, p. 91), we have a limit M=Mff∈F˜.

Third part: M is a stable matching.

Since for every k ∈ K, ∑f∈F˜Mfnk=G and χ is weak∗-closed, we have that ∑f∈F˜Mf=G, hence M is a matching. We now show that it is stable:

1. Let f∈F˜ and P∈P such that ∅≻Pf. There exists n0 such that f∈F˜n0. There exists k0∈K such that for all k≥k0, F˜n0⊂F˜nk. Since Mnk=Mfnkf∈F˜ is a stable matching we have that MfnkΘP=0 for all k≥k0. Hence, MfΘP=0.

2. Let f∈F˜. There exists n0 such that f∈F˜n0. There exists k0∈K such that for all k≥k0, F˜n0⊂F˜nk. Since Mnk=Mfnkf∈F˜ is a stable matching we have that Mfnk∈CfMfnk for all k ∈ K. By Assumption 2., Mf∈CfMf.

We have proven that the matching M satisfies individual rationality. It only remains to show that there is no blocking coalition. We proceed by contradiction. Suppose that there exists f ∈ F and Mf′∈χ such that Mf′≤D⪯fM, Mf′∈CfMf∨Mf′ and Mf∉CfMf∨Mf′. It is clear that Mf≤Mf∨Mf′, whence Mf∈χMf∨Mf′. By (ii) above, Mf∈CfMf and hence, Mf∈CfMf∩χMf∨Mf′. Consequently, by the revealed preference condition, Mf∈CfMf∨Mf′, a contradiction.

B Proof of Proposition 5

First of all, we claim that^[6]

for all S ∈ σ(Θ) and all fˉ∈F˜ is equivalent to saying that

for all S ∈ σ(Θ), for all fˉ∈F˜ and all P∈P. Indeed, for S ∈ σ(Θ), one has that S∩ΘP belongs to σ(Θ) since ΘP is assumed measurable. Then, since S∩ΘP does not meet ΘP′ for P^ʹ≠P it follows that (1) implies (2). The converse is immediate.

We proceed by contradiction. Let us suppose that there exists M′  such that M′≻FM and M′≻ΘM. Consequently there exists f ∈ F such that Mf′≻Mf. For a given preference P∈P we denote the immediate predecessor of f as f−P.^[7]. Since M′≻ΘM then, for each fˉ∈F˜, one has

for all S ∈ σ(Θ) and P∈P. In particular for f−P

for all S ∈ σ(Θ)

Which is equivalent to

for all S ∈ σ(Θ)

Given P∈P we have

which can be rewritten as

for all S ∈ σ(Θ).

We claim that ∑f∈F˜,f′⪯fMf′′(ΘP∩S)≤∑f′∈F˜,f′⪯fMf′(ΘP∩S) for all S ∈ σ(Θ). Otherwise, from the previous equality one deduces that ∑f′∈F˜,f′≻fMf′′(ΘP∩S)<∑f′∈F˜,f′≻fMf′(ΘP∩S) for some S ∈ σ(Θ) which contradicts (3). Summing over P,

for all S ∈ σ(Θ)

Since Mf′≻Mf and Mf′≤D⪯f(M′) we have a contradiction with the fact that M is a stable matching. Consequently, M is weakly Pareto efficient.

Now, if |Cf(X)|=1 for all X ∈ χ and we assume that M is not Pareto efficient, then there is another matching M′  and F⊂F˜ such that M′≠M, M′⪰FM and M′⪰ΘM. Consequently, there exists f ∈ F such that Mf′⪰Mf which means that Mf′=Cf(Mf′∨Mf). Then Mf≠Mf′, whence Mf′≻Mf which contradicts the fact that M is a stable matching.