This document summarizes a research paper on stable matching with incomplete information. The paper defines a model of matching between firms and workers where: 1) Workers' productivity types are private information, though their distribution is common knowledge. 2) A matching is considered stable if there is no incentive for a worker-firm pair to deviate given firms' beliefs about worker types. 3) Stability is defined through an iterative belief formation process where unstable beliefs are eliminated. 4) The paper shows that a stable matching always exists under this definition and characterizes stable matchings through a fixed point argument. It also shows stable matchings are efficient if productivity is supermodular in types.

1. Stable Matching with Incomplete Information
Liu, Q., G. J. Mailath, A. Postlewaite, and L. Samuelson; Econometrica (2014)
Takuya Irie
October 11, 2016
1 / 30

2. Introduction (1/4)
▶ Matching market with firms and workers, along with payments
▶ A matching is said to be stable if there is no blocking pair.
2 / 30

3. Introduction (2/4)
For example, if Taro preferred Softbank to Google, then
Taro-Softbank pair could not increase both their payoffs through
any payment.
But...
3 / 30

4. Introduction (3/4)
▶ In general, workers’ “qualities” are not commonly known.
▶ How does Softbank estimate Softbank’s payoff when deviating
to the match with Taro?
4 / 30

5. Introduction (4/4)
Goals:
▶ Define an appropriate stability concept.
▶ What are the properties of stable outcomes?
▶ To what extent does the introduction of asymmetric
information in a matching problem alter equilibrium
outcomes?
Method:
▶ Iterative belief-formation process (reminiscent of
rationalizability)
5 / 30

6. Model (1/4)
▶ I: a finite set of workers
▶ J: a finite set of firms
▶ Productive characteristics:
▶ W ⊂ R: the finite set of possible worker types
▶ F ⊂ R: the finite set of possible firm types
▶ f : J → F: the function mapping each firm to her type
▶ w : I → W: the function mapping each worker to his type
6 / 30

7. Model (2/4)
▶ νwf ∈ R: the worker’s value to which a match between worker
type w ∈ W and firm type f ∈ F gives rise
▶ ϕwf ∈ R: the firm’s value to which a match between worker
type w ∈ W and firm type f ∈ F gives rise
▶ νwf + ϕwf : the surplus of the match
7 / 30

8. Model (3/4)
▶ Each firm’s type and the functions ν : W × F → R and
ϕ : W × F → R are common knowledge, but the function w
will not be known (though workers will know their own types).
▶ Assume that the worker type assignment w is drawn from
some distribution with support Ω ⊂ WI.
▶ Given a match between worker i (of type w(i)) and firm j (of
type f(j)),
▶ the worker’s payoff is πw
i := νw(i),f(j) + p;
▶ the firm’s payoff is πf
j := ϕw(i),f(j) − p,
where p ∈ R is the payment made to worker i by firm j.
8 / 30

9. Model (4/4)
▶ µ : I → J ∪ {∅}: a matching function that is one-to-one on
µ−1(J) and assigns worker i to firm µ(i)
▶ µ(i) = ∅ and µ−1(j) = ∅ mean that worker i is unemployed
and firm j does not hire a worker respectively.
▶ p: a payment scheme (or, vector) associated with µ that
specifies a payment pi,µ(i) ∈ R for each i ∈ I and
pµ−1(j),j ∈ R for each j ∈ J
9 / 30

10. Allocation and Outcome
Definition 1: An allocation (µ, p) consists of a matching function
µ and a payment scheme p associated with µ. An outcome of the
matching game (µ, p, w, f) specifies a realized type assignment
(w, f) and an allocation (µ, p).
10 / 30

11. An Example
worker indices: a b c
worker payoffs, πw
i : πw
a πw
b πw
c
worker types, w: 1 3 2
payments, p: paa′ pbb′ pcc′
firm types, f: 2 4 5
firm payoffs, πf
j : πf
a′ πf
b′ πf
c′
firm indices: a′ b′ c′
Figure 1.— An example of an outcome. The matching of types is
specified by µ(i) = i′ for i ∈ {a, b, c}.
11 / 30

12. Individual Rationality (IR)
Definition 2: An outcome (µ, p, w, f) is individually rational if
νw(i),f(µ(i)) + pi,µ(i) ≥ 0 for all i ∈ I and (1)
ϕw(µ−1(j)),f(j) − pµ−1(j),j ≥ 0 for all j ∈ J. (2)
12 / 30

13. Complete-Information Stability
Definition 3:
▶ A matching outcome (µ, p, w, f) is complete-information
stable if it is individually rational, and there is no worker-firm
combination (i, j) and payment p ∈ R from j to i such that
νw(i),f(j) + p > νw(i),f(µ(i)) + pi,µ(i) and (3)
ϕw(i),f(j) − p > ϕw(µ−1(j)),f(j) − pµ−1(j),j. (4)
▶ If (µ, p, w, f) is a complete-information stable outcome, the
allocation (µ, p) is a complete-information stable allocation at
(w, f).
13 / 30

14. Incomplete-Information (1/2)
What a firm can observe:
▶ the types of all firms (f)
▶ the distribution from which the function assigning workers’
types is drawn (in particular, support Ω)
▶ the type of the firm’s current worker (w(µ−1(j)))
▶ which worker is matched with which firm at which payment
(µ and p)
14 / 30

15. Incomplete-Information (2/2)
For example, Softbank knows:
▶ the types of Google and Toyota
▶ possible types of Taro and Naoki
▶ Kenta’s type
▶ Taro works at Google with 10$ per day and Naoki works at
Toyota with 1$ per day
15 / 30

16. Iterative Belief-Formation Process
How to check whether a matching outcome (µ, p, w, f) in a set of
IR matching outcomes Σ is stable (or, rationalizable):
1. Check whether there is no blocking pair, for some reasonable
beliefs consistent with Σ.
2. Eliminate any beliefs in which there is a blocking pair at some
payment, and restrict Σ to the new set of IR matching
outcomes Σ′.
3. Check whether there is no blocking pair, for some reasonable
beliefs consistent with Σ′.
4. Eliminate any beliefs in which there is a blocking pair at some
payment, and restrict Σ to the new set of IR matching
outcomes Σ′′.
5. Continue this process until no further beliefs are eliminated.
Formally...
16 / 30

17. Σ-Stability
Definition 4:
▶ Fix a nonempty set of individually rational matching
outcomes, Σ.
▶ A matching outcome (µ, p, w, f) ∈ Σ is Σ-blocked if there is
a worker-firm pair (i, j) and payment p ∈ R satisfying
νw(i),f(j) + p > νw(i),f(µ(i)) + pi,µ(i) and (5)
ϕw′(i),f(j) − p > ϕw′(µ−1(j)),f(j) − pµ−1(j),j (6)
for all w′ ∈ Ω satisfying
(µ, p, w′
, f) ∈ Σ, (7)
w′
(µ−1
(j)) = w(µ−1
(j)), and (8)
νw′(i),f(j) + p > νw′(i),f(µ(i)) + pi,µ(i). (9)
▶ A matching outcome (µ, p, w, f) ∈ Σ is Σ-stable if it is not
Σ-blocked.
17 / 30

18. Incomplete-Information Stability
Definition 5:
▶ Let Σ0 be the set of all individually rational outcomes.
▶ For k ≥ 1, define
Σk
:= {(µ, p, w, f) ∈ Σk−1
: (µ, p, w, f) is Σk−1
-stable}.
(10)
▶ The set of incomplete-information stable outcomes is given by
Σ∞
:=
∞∩
k=1
Σk
. (11)
▶ If (µ, p, w, f) is a incomplete-information stable outcome, the
allocation (µ, p) is a incomplete-information stable allocation
at (w, f).
18 / 30

19. An Example of Iterative Belief-Formation Process (1/4)
worker indices: a b c
worker payoffs, πw
i : 2 16 6
worker types, w: 1 3 2
payments, p: 0 4 −4
firm types, f: 2 4 5
firm payoffs, πf
j : 2 8 14
firm indices: a′ b′ c′
Figure 2.— A matching outcome that is not Σ1-stable but
Σ0-stable. The matching of types is specified by µ(i) = i′ for
i ∈ {a, b, c}. νwf = ϕwf = wf. Firms believe that the set Ω of
possible vectors (w(a), w(b), w(c)) is the set of permutations of
(1, 2, 3).
19 / 30

20. An Example of Iterative Belief-Formation Process (2/4)
This matching outcome is Σ0-stable because
▶ There is a candidate blocking pair (c, b′) with some payment
˜p ∈ (−2, 0).
▶ However, firm b′ does not know whether worker c is of type 1
or 2.
▶ Furthermore, firm b′’s belief that worker c is of type 1 is
reasonable because if worker c were type 1, his current payoff
would be 1, while he would receive a payoff of 4 + ˜p > 1 in
the candidate blocking pair.
20 / 30

21. An Example of Iterative Belief-Formation Process (3/4)
▶ This matching outcome is not Σ1-stable because outcomes
with w′(c) = 1 (such as the one displayed in Figure 3) are not
contained in Σ1.
▶ (µ, p, w′, f) /∈ Σ1 (or, is Σ0-blocked) because
▶ Firm c′
knows that worker a is of type at least 2.
▶ Regardless of the type of worker a being 2 or 3, firm c′
can
form a blocking pair with worker a.
21 / 30

22. An Example of Iterative Belief-Formation Process (4/4)
worker indices: a b c
worker payoffs, πw
i : 4 16 1
worker types, w′: 2 3 1
payments, p: 0 4 −4
firm types, f: 2 4 5
firm payoffs, πf
j : 4 8 9
firm indices: a′ b′ c′
Figure 3.— The payments and matching from Figure 2 with a
different worker type realization.
22 / 30

23. Existence
Proposition 1: For each type assignment (w, f), there is an
incomplete-information stable outcome (µ, p, w, f), and so the set
of incomplete-information stable allocations is nonempty.
Proof: If (µ, p) is a complete-information stable allocation at
(w, f), then, by definition, (µ, p, w, f) ∈ Σk for each k ≥ 0. 2
23 / 30

24. Fixed-Point Characterization (1/3)
Definition 6:
▶ A nonempty set of individually rational matching outcomes E
is self-stabilizing if every (µ, p, w, f) ∈ E is E-stable.
▶ The set E stabilizes a given matching outcome (µ, p, w, f) if
(µ, p, w, f) ∈ E and E is self-stabilizing.
▶ A set of worker type assignments Ω∗ ⊂ Ω stabilizes an
allocation (µ, p) if {(µ, p, w, f) : w ∈ Ω∗} is a self-stabilizing
set.
24 / 30

25. Fixed-Point Characterization (2/3)
Lemma 1:
1. The singleton set {(µ, p, w, f)} is self-stabilizing if and only if
(µ, p, w, f) is a complete-information stable outcome.
2. If both E1 and E2 are self-stabilizing, then E1 ∪ E2 is
self-stabilizing.
3. If E is self-stabilizing, then its closure ¯E is self-stabilizing.
4. If E is a self-stabilizing set and (µ, p, w, f) ∈ E, then
E ∩ {(µ, p, w′, f) : w′ ∈ Ω} is a self-stabilizing.
25 / 30

26. Fixed-Point Characterization (3/3)
Proposition 2:
1. If E is a self-stabilizing set, then E ⊂ Σ∞.
2. The set of incomplete-information stable outcomes, Σ∞ is a
self-stabilizing set, and hence the largest self-stabilizing set.
3. The set Σ∞ is closed.
Implication: to show (µ, p, w, f) is a stable outcome, it suffices to
construct a subset Ω∗ containing w stabilizing the allocation
(µ, p).
26 / 30

27. Payoff Assumptions
Assumption 1 (Monotonicity): The worker premuneration values
νwf and firm premuneration values ϕwf are increasing in w and f,
with νwf strictly increasing in w and ϕwf strictly increasing in f.
Assumption 2 (Supermodularity): The worker premuneration
value νwf and the match surplus νwf + ϕwf are strictly
supermodular in w and f.
27 / 30

28. Efficiency under Supermodularity
Proposition 3: Under Assumptions 1 and 2, every
incomplete-information stable outcome is efficient (i.e. maximizes
total surplus).
28 / 30

29. Restrictions of Workers’ Types
For the result of Proposition 6, we focus on the case where
|I| = |J| and assume that νwf > 0 and ϕwf > 0 for any w ∈ W
and f ∈ F.
Definition 7: The support Ω is a set of permutations if, for any
w, w′ ∈ Ω, there exists a one-to-one mapping ι : I → I such that
w(i) = w′(ι(i)).
29 / 30

30. Relation to Complete-Information Stability II
Proposition 6: Suppose Assumptions 1 and 2 hold, and assume Ω
is a set of permutations. Incomplete-information stability coincides
with complete-information stability if either
1. different firms have different types, or
2. different workers have different types.
30 / 30