Two-sided Matching with Incomplete Information
Sushil Bikhchandani∗
March 2014
Preliminary Draft
Abstract
Stability in a two-sided matching model with non-transferrable utility (NTU)
and with incomplete information is investigated. Each agent has interdependent preferences which depend on his own type and on the possibly unknown
types of agents on the other side of the market. Agents’ utilities are increasing
in types. The notion of incomplete-information stability I use was introduced
by Liu, Mailath, Postlewaite, and Samuelson (2013) in a matching model with
transferrable utility and side payments. Initially, I examine a model with anonymous preferences, i.e., utility depends only on the type and not on the identity
of the matched agent. With anonymous preferences, all individually rational
matching outcomes are incomplete-information stable under one-sided incomplete information. Thus, in a positive assortative matching all matchings are
stable including the negative assortative matching. When preferences are not
anonymous, I identify a necessary and sufficient condition for stability. An ex
post incentive compatible mechanism exists. This mechanism implements a
complete information stable matching for the side of the market with private
information. Extensions to two-sided incomplete information are investigated.
∗
UCLA Anderson School of Management ([email protected])
1
Introduction
The concept of stability in two-sided matching was introduced by Gale and Shapley [4]. A matching is stable if there does not exist a blocking pair, i.e., two agents,
one on each side of the market, who prefer each other to their respective partners
under the matching. Agents do not have any private information about their own
types that might affect their own preferences or the preferences of other agents. Much
of the subsequent literature makes the same complete information assumption – see
Roth and Sotomayor [9]. A few exceptions are discussed below.
Recently, Liu, Mailath, Postlewaite, and Samuelson [6] introduced a notion of
stability that is appropriate for matching models with interdependent values and
one-sided asymmetric information. They consider a matching model with workers
and firms in which each worker’s type is private information but each firm’s type
is common knowledge and side payments between matched workers and firms are
permitted. The utility of an agent, say a worker, depends on the worker’s type and
the type of the firm it is matched with, less any payment from the worker to the firm.
Once a matching prevails, a worker’s type becomes known to the firm it is matched
with but not to other firms. Thus, workers are at an informational advantage. Firms
compensate by being conservative in their participation in potential block. A contract
is a matching and side payments between workers and firms. A worker-firm pair
blocks a contract if there is a side payment between the blocking worker-firm pair
which makes the worker better off (than in the status quo contract) and the firm
better off for all “reasonable” types of workers who prefer the blocking contract to
the status quo. The set of reasonable worker types of evolves sequentially, with firms
drawing inferences about worker types from blocks that do not materialize. Thus, as
a contract persists over time, it becomes common knowledge among agents that there
are no blocking opportunities. Liu et al. [6] point out that this is similar in spirit to
Holmstrom and Myerson [5]’s notion of durable mechanisms.
In this paper, I consider a two-sided matching model with non-transferrable utility. Side payments between worker and firms are not possible. Workers have private
information about their types. The utility functions of a worker and the matched firm
increase in the worker’s type. I investigate the restrictions imposed by stability on the
set of matching outcomes when there is incomplete information and there are no side
payments. I also investigate the existence a “good” incentive compatible mechanism.
1
Stable matching outcomes exist under incomplete information and are a super1
While the model in this paper is of one-to-one matching, it is easily generalized to a many-to-one
matching model. For instance, all the results presented here hold for a model in which firms employ
more than one worker and each firm’s utility is additive over workers.
1
set of complete-information stable matchings. Thus, the set of stable matchings is
larger than under complete information and, in a sense, too large: if preferences are
anonymous, that is if each agent’s utility depends on the type but not on the identity
of the matched agent, then every strictly individually rational matching outcome is
stable. Under anonymous preferences, stability is essentially equivalent to individual
rationality. When preferences are not anonymous, I identify a necessary and sufficient
condition for stability.
Next I consider blocking coalitions of size greater than two. It is necessary to make
assumptions about what members of a coalition know. In a worker-firm blocking pair,
it is assumed that the firm draws inferences about the worker’s type from the worker’s
willingness to block. That is, the firm participates in blocking only if it is better off
with all types of workers who benefit from participating. Similarly, a firm j in a larger
coalition too draw inferences about the type of worker i, its proposed match in the
block, from worker i’s willingness to block; further, if the firm currently matched to
worker i is also part of the coalition then that conveys additional information about
worker i’s type. It is shown that larger coalitions do not restrict the set of stable
outcomes: a blocking pair exists whenever a blocking coalition of size greater than
two exists.
An mechanism that is ex post incentive compatible for almost all worker types
exists. In this mechanism, workers reveal their types to the mechanism designer.
The mechanism designer computes the complete-information stable matching (for
the stated types) that is optimal for workers. The mechanism designer reveals the
reported types of workers and implements the computed matching. If all agents report
truthfully, then the resulting matching is stable. If any one agent lies, then either the
implemented matching is unstable and agents settle on another stable matching or it
is stable. In either event, the agent does not benefit by lying.
Next, a notion of stability for two-sided incomplete information is proposed. The
set of two-sided incomplete information matching outcomes is a superset of the set of
one-sided incomplete information stable matching outcomes. With two-sided incomplete information, an efficient, ex post incentive compatible mechanism exists if there
is a unique complete-information stable matching in each state of the world.
Overall, the set of incomplete-information stable-matching outcomes is large. If, as
seems plausible in many applications, there is incomplete-information among participants then the absence of instability should not lead to the conclusion that the status
quo matching cannot be improved for all participants. Pareto dominated matching
outcomes can be incomplete-information stable.
In addition to Liu et al. [6], there are several recent papers on matching with
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incomplete information. Chakraborty, Citana, and Ostrovsky [1] investigate the
incomplete-information stability of a mechanism, rather than of a matching, in a
college-admissions model. Students’ preferences are known but not their quality is
unknown. Preferences of colleges depend on quality. Chakraborty et al. [1] show that
stable mechanisms do not usually exist.
Starting with Roth [8], several papers study incomplete-information matching
models where agent’s know their own preferences but these are not common knowledge. Roth [8] established that there exists no mechanism in which it is a dominant
strategy for all agents to truthfully reveal their preferences. Ehlers and Masso [3]
showed that in a matching model with two-sided incomplete information, an ordinally Bayesian incentive compatible mechanism exists if and only if there is exactly
one stable matching at every state of the world. A related negative result was obtained
by Majumdar [7].
The rest of the paper is organized as follows. The basic model and results for
one-sided uncertainty are presented in Section 2. Blocking coalitions larger than two
are considered in Section 2.1 and the incentive compatible mechanism is presented in
Section 2.2. Section 2.3 compares this model with a TU model. Section 3 extends
the results to two-sided incomplete information. Section 4 concludes.
2
One-sided Uncertainty
There are i = 1, 2, . . . , n workers and j = 1, 2, . . . , m firms. Each worker has private
information about his own type. Worker i’s type, wi , is in the interval [w, w].2 If
worker-firm (i, j) are matched then their respective utilities are:
ui (wi , j) and vj (wi , i)
(1)
A dummy worker is indexed i = 0 with type w0 and a dummy firm is indexed j = 0.
An unmatched worker (firm) is matched to the dummy firm (worker). The utility of
an unmatched worker or firm is normalized to zero: ui (wi , 0) = 0 and vj (w0 , 0) = 0.
The following assumption is made throughout the paper.
Increasing and continuous utility: The utility functions ui (wi , j) and vj (wi , i)
are strictly increasing and continuous in wi for all i and j.
A matching is a function µ : {1, 2, . . . , n} → {0, 1, 2, . . . , m}, where µ(i) is the firm
2
Assuming the same interval of types for all workers is without loss of generality as agents’ utility
functions are different.
3
that worker i is matched to. If µ(i) = 0 then worker i is unmatched and if µ(i) = µ(î)
then µ(i) = 0. For any j ∈ {1, 2, . . . , m},
i, if there is an i ∈ {1, 2, . . . , n} s.t. µ(i) = j,
−1
µ (j) =
0, otherwise.
Let w = (w1 , w2 , . . . , wn ) be the type vector of the workers. A matching outcome
is a matching together with a worker type vector: (µ, w). The matching outcome
(µ, w) is individually rational if
ui (wi , µ(i)) ≥ 0,
∀i
vj (wµ−1 (j) , µ−1 (j)) ≥ 0,
∀j
Let Σ0 be the set of individually rational matching outcomes. By monotonicity, if
(µ, w) ∈ Σ0 then (µ, w0 ) ∈ Σ0 for all w0 ≥ w. A (µ, w) is strictly individually rational
if the above inequalities are satisfied strictly whenever µ(i) 6= 0 or µ−1 (j) 6= 0. Let
int(Σ0 ) denote the set of strictly individually rational matching outcomes.
It is useful to state the definition of complete-information stability before defining
incomplete-information stability.
A matching outcome (µ, w) ∈ Σ0 , where w = (w1 , w2 , . . . , wn ), is complete information stable if there is no worker-firm pair (i, j) such that
ui (wi , j) > ui (wi , µ(i)) and vj (wi , i) > vj (wµ−1 (j) , µ−1 (j))
(2)
If there exists an (i, j) satisfying (2), then (i, j) blocks (µ, w).
A matching outcome (µ, w) ∈ A, where w = (w1 , w2 , . . . , wn ), is A-blocked if there
is a worker-firm pair (i, j) satisfying
ui (wi , j)
>
ui (wi , µ(i))
(3)
and for all (µ, w0 ) ∈ A s.t. wµ0 −1 (j) = wµ−1 (j) ,
ui (wi0 , j)
>
ui (wi0 , µ(i))
(4)
then vj (wi0 , i)
>
vj (wµ−1 (j) , µ−1 (j))
(5)
If
As (µ, w) ∈ A, if (3) is satisfied then (4) is also satisfied with wi0 = wi (and possibly
with other wi0 as well).
Inequality (3) requires that worker i is better off in the potential block. The
corresponding condition for firm j is more complicated. Firm j knows only that
worker µ−1 (j)’s type is wµ−1 (j) , but does not know worker i’s type. Therefore, in
order to participate in the block, firm j should be better off with any worker i type
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wi0 (in A at a worker type vector w0 that is consistent with firm j’s knowledge about
worker µ−1 (j)) that is better off in the match. If whenever (4) is satisfied, (5) is
satisfied then the pair (i, j) blocks (µ, w) in A.
The matching outcome (µ, w) ∈ A is A-stable if it is not A-blocked by any firmworker pair. The set A is self-stabilizing if every (µ, w) ∈ A is A-stable.
If the set {(µ, w)} is self-stabilizing then, by definition, µ is a complete-information
matching at w. Additional results are gathered in the following lemma, where A and
B are sets of matching outcomes.
Lemma 1
(i) Suppose that B ⊂ A. If (µ, w) ∈ B is B-stable then it is A-stable.
(ii) Let µ be a complete-information stable matching at w. Then, (µ, w) is A-stable
for any A such that (µ, w) ∈ A.
Proof: (i) It follows directly from (4) and (5) that if (µ, w) is A-blocked then it is
also B-blocked.
(ii) Let B = {(µ, w)}. As µ is complete-information stable matching at w, (µ, w) is
B-stable. Therefore, by (i), (µ, w) is A-stable for any A ⊃ B.
The next assumption states that preferences depend only on the types of agents
and not on their identity.
Anonymous Preferences: For any i, î ∈ {1, 2, . . . , n} and j, ĵ ∈ {1, 2, . . . , m}
and wi , wi0 ∈ (w, w],
vj (wi , i)
=
vj (wi , î)
If ui (wi , j) > ui (wi , ĵ) then ui (wi0 , j) > ui (wi0 , ĵ)
The condition of anonymous preferences is satisfied whenever the utility of firms
(workers) over workers (firms) depends only on the type and not on the identity of
the workers (firms). The above equation states that firms’ preferences over workers
depend only on workers’ types and not on their identities. The if-then statement
above requires that if worker i of type wi > w prefers firm j to ĵ then worker i of
any other type wi0 > w prefers firm j to ĵ. Essentially, this also states that workers’
preferences over firms depend only on firms’ “types” and not on their identities.
To see this, assume that firms also have types and that these types are common
5
knowledge. The underlying utility functions of workers and firms, Ui and Vj , depend
on the agent’s own type and the type but not the identity of the matched agent. The
utility functions ui and vj are obtained from Ui and Vj by replacing firm types fj and
fĵ with firm identities j and ĵ:
ui (wi , j) ≡ Ui (wi , fj ),
ui (wi , ĵ) ≡ Ui (wi , fĵ )
vj (wi , i) ≡ Vj (wi , fj )
(6)
(7)
Assuming that worker utility increases with firm types, the hypothesis of the if statement, ui (wi , fj ) > ui (wi , fĵ ) implies that fj > fĵ , which in turn implies ui (wi0 , fj ) >
ui (wi0 , fĵ ).
The next result says that if preferences are anonymous and any matching is strictly
preferred to being unmatched by a worker or a firm (except when the worker’s type
is the lowest), then every strictly IR matching outcome is stable.
Proposition 1 Suppose that ui (w, j) = 0, vj (w, i) = 0 for all i, j. Then, under
anonymous preferences the set of strictly individually rational matching outcomes is
self-stabilizing.
Proof: Take any matching µ. As ui (w, j) = 0, vj (w, i) = 0 for all i, j, increasing
utility implies that for any w ∈ (w, w]n , (µ, w) ∈ int(Σ0 ). Suppose that the workerfirm pair (i, j) satisfies (3) at w, where (µ, w) ∈ int(Σ0 ). Then by anonymity, (3)
is satisfied for all wi0 ∈ (w, w]. Because vj (wµ−1 (j) , µ−1 (j)) > 0 by strict individual
rationality, and vj (w, µ−1 (j)) = 0, we have wµ−1 (j) > w as utility is increasing. Select
wi0 ∈ (w, wµ−1 (j) ]. Now, for (µ, w0 ), where w0 = (w−i , wi0 ), (4) is satisfied but, by
increasing utility and anonymity, (5) is not satisfied. Hence, (µ, w) is not int(Σ0 )blocked.
If, instead, there does not exist a worker-firm pair (i, j) that satisfies (3) at any
w, where (µ, w) ∈ int(Σ0 ), then again (µ, w) is not int(Σ0 )-blocked.
Remark: The condition that ui (w, j) = 0, vj (w, i) = 0 for all i, j in the hypothesis of
Proposition 1 can be dropped if, instead of anonymous preferences, equations (6) and
(7) are assumed. Note that the Anonymous Preferences assumption is more general
than (6) and (7). It is possible that ui (w, j) > ui (w, ĵ) and uî (w, j) < uî (w, ĵ) under
the anonymous preferences but not under (6) and (7).
Thus, under the assumptions of Proposition 1, stability provides no restrictions
on matching outcomes. In particular, consider the following example which satisfies
the hypothesis of Proposition 1.
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Example 1: (Assortative Matching)
Let the number of workers equal the number of firms, n = m. The utility functions
of workers and firms are
ui (wi , j) = vj (wi , i) = jwi
Then, with 0 < wi1 ≤ wi2 ≤ . . . ≤ win , the unique complete-information stable matching pairs worker ij with firms j, the positive assortative matching. But Proposition 1
implies that all matchings are incomplete-information stable.
To see this directly, consider the negative assortative matching where worker in is
matched to firm 1, worker in−1 is matched to firm 2, etc. Firm n, who is matched to
the lowest type worker i1 but does not know that i1 has the lowest type, will reject a
blocking proposal from all other workers, including worker in . This is because types
wi0n < wi1 would also do better by matching with firm n than with their current match
firm 1.
Similarly, every firm j ≥ 2 will reject a blocking proposal from any other worker.
No worker will propose a block with firm 1.
The difference between an NTU model and a TU model is stark. In a TU model
with side payments, Proposition 3 of Liu et al. [6] implies that in any positive assortative matching model, only the efficient matching is incomplete information stable.
In the sequel, I do not assume either anonymous preferences or that any firm is
indifferent between being matched to a lowest type worker and remaining unmatched.
Before proceeding, I define the notion of incomplete-information stability. As in Liu et
al. [6], this definition calls for an iterative elimination of unstable matching outcomes
from the beliefs of firms. The idea is that if a matching persists, then it is reasonable
to conclude that the worker types are not in a subset that would block the matching.
With Σ0 = Σ, the set of individually rational matching outcomes, define
k
k−1 Σ = {(µ, w) ∈ Σ (µ, w) is not Σk−1 -blocked}
Then
∗
Σ ≡
∞
\
(8)
Σk
k=0
is the set of incomplete-information stable-matching outcomes.
It is clear from the definition that Σk ⊆ Σk+1 and that Σ∗ is self-stabilizing. The
next result shows that Σ∗ is the largest self-stabilizing set.
Lemma 2 If A be a self-stabilizing set then A ⊆ Σ∗ .
7
Proof: If A is self-stabilizing then A ⊆ Σ0 . Let k ≥ 0 be such that A ⊆ Σk . As every
(µ, w) ∈ A is A-stable, it is also Σk -stable by Lemma 1(i). Therefore, A ⊆ Σk+1 .
Next, a necessary and sufficient condition for self-stabilizing sets is presented. Let
βj (i, wµ−1 (j) ; µ, A) = inf{wi (µ, (w−i ,µ−1 (j) , wi , wµ−1 (j) )) ∈ A and vj (wi , i)−vj (wµ−1 (j) , µ−1 (j)) > 0},
if the inf is over a non-empty set; otherwise
βj (i, wµ−1 (j) ; µ, A) = inf{wi (µ, (w−i ,µ−1 (j) , wi , wµ−1 (j) )) ∈ A}
Thus, firm j strictly prefers its current match in µ to any worker i of type wi if
and only if wi < βj (i, wµ−1 (j) ; µ, A), where it is understood that wi is consistent with
firm j’s information, i.e., (µ, w) ∈ A such that w = (w−i ,µ−1 (j) , wi , wµ−1 (j) ).
Thus,
βj (i, w
µ−1 (j)
; µ, A) ≥ inf{wi (µ, (w−i ,µ−1 (j) , wi , wµ−1 (j) )) ∈ A}
If the above inequality is strict then βj is strictly increasing in wµ−1 (j) . If, instead, it
is an equality then βj does not vary with wµ−1 (j) .
Further, let
n o
αi (j; µ, A) = inf wi (µ, (w−i , wi )) ∈ A and ui (wi , j) − ui (wi , µ(i)) > 0 ,
if the inf is over a non-empty set; otherwise
αi (j; µ, A) = inf{wi (µ, (w−i , wi )) ∈ A}
Thus, any worker i of type wi < αi (j; µ, A) such that (µ, (w−i , wi )) ∈ A prefers its
current match in µ to firm j.
Proposition 2 A set A is self-stabilizing if and only if for every matching outcome
(µ, w) ∈ A,
βj (i, wµ−1 (j) ; µ, A) > αi (j; µ, A)
for each (i, j) such that µ(i) 6= j.
Proof: Sufficiency. Let(µ, w) ∈ A. By the continuity of worker utility functions, (4)
is satisfied by wi0 = αi (j; µ, A)+ for all small > 0. Suppose that βj (i, wµ−1 (j) ; µ, A) >
αi (j; µ, A). Then we can take small enough such that βj (i, wµ−1 (j) ; µ, A) > wi0 . Thus,
(4) is satisfied but (5) is not by wi0 and (i, j) does not block (µ, w) in A.
8
Necessity. Suppose that βj (i, wµ−1 (j) ; µ, A) ≤ αi (j; µ, A). From the definition of
αi , there exist wi0 > αi (j; µ, A) such that (4) is satisfied by worker i of type wi0 . For
any such wi0 we have wi0 > βj (i, wµ−1 (j) ; µ, A), and monotonicity implies that (5) is
satisfied by wi0 . Hence, (i, j) blocks (µ, w) in A.
Thus, (i, j) blocks µ if and only if the lowest type of i in A that prefers j to µ(i)
is preferred by j to µ−1 (j). The following examples are illustrative.
Example 2: There are two firms and two workers. The workers’ types lie in the
interval [0, 2]. The utility functions of workers are in the table below:
Worker utility
Firm 1
Firm 2
u1 (w1 , j)
u2 (w2 , j)
w1 + 1
w2 + 1
2w1
2w2
The utility of worker 1 of type w1 , when matched with firm 1 is w1 + 1, etc. Note
that worker i prefers firm 1 [firm 2] if wi < 1 [wi > 1].
Firm’s utility functions are displayed below:
Firm utility Worker 1 Worker 2
v1 (wi , i)
v2 (wi , i)
w1
w1
w2 + 1
w2
The utility of firm 1, when matched with worker 1 of type w1 , is w1 , etc. Firm 1 has
a preference for worker 2 while firm 2 always prefers the worker with the higher type.
Suppose that the workers’ types are w1 = 0.5 and w2 = 2, and each worker’s type
is privately known to that worker. In any matching, the worker’s type also becomes
known to the matched firm.
At (w1 , w2 ) = (0.5, 2), any matching in which both workers are matched is (strictly)
individually rational. This set consists of the following matchings:
µ1 = {(W1 , F1 ), (W2 , F2 )}
µ2 = {(W1 , F2 ), (W2 , F1 )}
That is, at µ1 worker 1 is matched to firm 1, worker 2 is matched to firm 2.
Of these two matchings, µ1 is Σ0 -stable at (w1 , w2 ) = (0.5, 2). However, µ2 is
Σ0 -blocked by (W2 , F2 ) at (w1 , w2 ) = (0.5, 2) as all W2 types w2 > 1 prefer F2 to F1
and F2 prefers all W2 types w2 > 1 to W1 with w1 = 0.5.
The unique complete-information stable matching at (w1 , w2 ) = (0.5, 2) is µ1 .
9
(This matching is also maximizes the sum of agents’ utilities, but it is not the only
Pareto-efficient matching.)
Next, observe that (µ1 , (w1 , w2 )) is Σ0 -blocked when w2 < 1. To see this, note that
if w1 > 1 and w2 < 1 then (W1 , F2 ) Σ0 -blocks (µ1 , (w1 , w2 )). If, instead, w1 ≤ 1 and
w2 < 1, then (W2 , F1 ) Σ0 -blocks (µ1 , (w1 , w2 )). Thus, (µ1 , (w1 , w2 )) is not Σ0 -stable
for w2 < 1.
Finally, (µ2 , (w1 , w2 )) is Σ0 -blocked by (W2 , F2 ) if w1 < 1 and w2 > 1.
This information is summarized in the table below.
Incomplete-Information Stable-Matching Outcomes Set, Σ∗
Matching
µ1 = {(W1 , F1 ), (W2 , F2 )}
µ2 = {(W1 , F2 ), (W2 , F1 )}
Worker types
w2 ≥ 1
(w1 ≥ 1) or (w2 ≤ 1)
The matching outcomes in the above table constitute Σ1 and all are Σ1 -stable.
That is, Σ1 is self-stabilizing and therefore Σ1 = Σ∗ .
The next example adds one more firm to the previous example and now Σ1 ( Σ2 .
Example 3: Add a third firm to Example 2, with utilities as specified below:
Worker utility
Firm 1
Firm 2
Firm 3
u1 (w1 , j)
u2 (w2 , j)
w1 + 1
w2 + 1
2w1
2w2
4w1
4w2
Firm utility Worker 1 Worker 2
v1 (wi , i)
v2 (wi , i)
v3 (wi , i)
w1
w1
w1 − 0.8
w2 + 1
w2
w2 − 0.8
Firm 3 would rather stay unmatched than be with a worker of type less than 0.8.
In the previous example, we saw that µ1 was the unique incomplete-information
stable-matching at w1 = 0.5 and w2 = 2. We show that it is no longer incompleteinformation stable-matching as it is not Σ1 -stable.
The unique complete-information stable matching at (w1 , w2 ) = (0.5, 2) is {(W1 , F1 ),
(W2 , F3 )} – worker 1 is matched to firm 1, worker 2 is matched to firm 3, and firm 2
is unmatched.
10
At (w1 , w2 ) = (0.5, 2), any matching in which both workers are matched and
worker 1 is not matched to firm 3 is (strictly) individually rational. This set consists
of the following matchings: µ1 = {(W1 , F1 ), (W2 , F2 )}, µ2 = {(W1 , F2 ), (W2 , F1 )},
µ3 = {(W1 , F1 ), (W2 , F3 )}, and µ4 = {(W1 , F2 ), (W2 , F3 )}.
Of these four matchings, µ2 continues to be Σ0 -blocked by (W2 , F2 ) at (w1 , w2 ) =
(0.5, 2) for the same reason as in Example 2: all W2 types w2 > 1 prefer F2 to F1 and
F2 prefers all W2 types w2 > 1 to W1 with w1 = 0.5.
Further, µ1 is not Σ0 -blocked at w1 = 0.5, w2 = 2. But note that (µ1 , (w1 , w2 )) ∈
Σ1 only if w2 ≥ 1. Therefore, µ1 is not Σ1 -stable at w1 = 0.5, w2 = 2 as (W2 , F3 ) is a
blocking coalition in Σ1 : all W2 types with w2 ≥ 1 prefer F3 to F2 and F3 prefers all
such types of W2 to being unmatched. Therefore, (µ1 , (0.5, 2)) is not Σ1 -stable.
The unique Σ1 -stable matching at w1 = 0.5, w2 = 2 is µ3 .
There are stable matchings at different values of w. For instance, consider matching µ2 for w2 ≤ 1. First, F3 cannot be part of any blocking pair as W1 of any type
prefers F3 to the current match F2 , and W2 of types w2 > 1/3 prefer F3 to its current
match F1 ; F3 , on the other hand, prefers to be unmatched rather than be matched
with a worker of type wi < 0.8. Next, (W1 , F1 ) is not a blocking pair because W1 of
types w1 ≤ 1 prefer F1 to F2 but F1 prefers w2 > 0 to any w1 ≤ 1. Finally, (W2 , F2 )
is not a blocking pair because W2 prefers F1 to F2 .
Observe that (µ3 , (w1 , w2 )) and (µ4 , (w1 , w2 )) are in Σ0 only if w2 > 0.8. Further,
(µ3 , (w1 , w2 )) is Σ0 -blocked by (W1 , F1 ) if w1 > 1 and (µ4 , (w1 , w2 )) is Σ0 -blocked by
(W1 , F2 ) if w1 < 1. The incomplete-information stable matchings are:
Incomplete-Information Stable-Matching Outcomes Set, Σ∗
µ2
µ3
µ4
µ5
µ6
Matching
= {(W1 , F2 ), (W2 , F1 )}
= {(W1 , F1 ), (W2 , F3 )}
= {(W1 , F2 ), (W2 , F3 )}
= {(W2 , F2 ), (W1 , F3 )}
= {(W2 , F1 ), (W1 , F3 )}
Worker types
w2 ≤ 1
w1 ≤ 1, w2 ≥ 0.8
w1 ≥ 1, w2 ≥ 0.8
w1 ≥ 0.8, w2 ≥ 1
w1 ≥ 0.8, w2 ≤ 1
Of these, only µ3 is supported at w1 = 0.5, w2 = 2. This is also the unique completeinformation stable matching at w1 = 0.5, w2 = 2.
Note that an incomplete-information stable matching outcome need not be a
complete-information stable matching. For example, µ1 , µ2 ∈ Σ∗ at w1 = 2, w2 = 0.9
even though µ4 is the unique complete-information stable matching.
11
As the preceding examples show, the set of incomplete-information matching outcomes can be quite large even when preferences are not anonymous. The next example demonstrates that even a Pareto-dominated matching outcome can be incomplete
information stable.
Example 4: Pareto-dominated stable matching outcome
There are two firms and two workers. The workers’ types lie in the interval [0, 2].
The utility functions of workers are in the table below:
Worker utility
Firm 1
Firm 2
u1 (w1 , j)
u2 (w2 , j)
w1 + 1
w2 + 1
2w1
2w2
Firm utility Worker 1 Worker 2
v1 (wi , i)
v2 (wi , i)
w1 + 0.5
w1 + 0.1
w2
w2
Consider the following matchings:
µ1 = {(W1 , F1 ), (W2 , F2 )}
µ2 = {(W1 , F2 ), (W2 , F1 )}
Of these two matchings, (µ1 , (w1 , w2 )) is Σ-blocked by (W1 , F2 ) when w1 > 1 and
w2 < 1.1. Next, (µ2 , (w1 , w2 )) is Σ-blocked by (W1 , F1 ) if w1 < 1 and w2 < 0.5. And
(µ2 , (w1 , w2 )) is Σ-blocked by (W2 , F2 ) if w1 < 0.9 and w2 > 1.
Incomplete-information Stable Matching Outcomes
Matching
Worker types
µ1
w1 ≤ 1 or w2 ≥ 1.1
µ2
(w1 ≥ 1) or (w1 ∈ [0.9, 1] ∩ w2 ≥ 0.5) or (w1 ∈ [0, 0.9] ∩ w2 ∈ [0.5, 1])
However, note that an incomplete-information stable matching outcome need not
be a complete-information stable matching. For example, µ1 , µ2 ∈ Σ∞ at w1 =
0.95, w2 = 1.1 even though µ1 is the unique complete-information stable matching
(as µ2 is complete-information blocked by (W1 , F1 )).
This example also shows that an incomplete-information stable matching outcome
need not be Pareto-efficient. At w1 = 0.95, w2 = 1.1, µ2 is strongly Pareto-dominated
by µ1 . That is, under complete information about worker types, if matching µ2 were
proposed then each worker and each firm would be part of a blocking pair. Under
incomplete information, however, it is not common knowledge among workers and
firms that a blocking pair exists.
12
The next lemma shows that if at worker types wi , wµ−1 (j) the matching µ is [not]
A-blocked by the pair (i, j), then at worker µ−1 (j) types more [less] favorable to the
block the matching µ is [not] A-blocked by the pair (i, j). The types of workers other
than i and µ−1 (j) could be anything.
Lemma 3
(i) If (µ, (w−i ,µ−1 (j) , wi , wµ−1 (j) )) ∈ A is A-blocked by (i, j) then for any wµb −1 (j) ≤
b
b
b
b
wµ−1 (j) and any w−i
,µ−1 (j) , (µ, (w−i ,µ−1 (j) , wi , wµ−1 (j) )) ∈ A also is A-blocked.
(ii) If (µ, (w−i ,µ−1 (j) , wi , wµ−1 (j) )) ∈ A is not A-blocked by (i, j) then for any wµnb−1 (j) ≥
nb
nb
nb
nb
wµ−1 (j) and any w−i
,µ−1 (j) , (µ, (w−i ,µ−1 (j) , wi , wµ−1 (j) )) ∈ A also is not Ablocked.
Proof: Suppose that (µ, (w−i ,µ−1 (j) , wi , wµ−1 (j) )) ∈ A is A-blocked by (i, j). That is,
0
0
ui (wi , j) > ui (wi , µ(i)) and for all (µ, (w−i
,µ−1 (j) , wi , wµ−1 (j) )) ∈ A we have
If
ui (wi0 , j) > ui (wi0 , µ(i)) then vj (wi0 , i) > vj (wµ−1 (j) , µ−1 (j))
(9)
b
b
b
Consider any (µ, (w−i
,µ−1 (j) , wi , wµ−1 (j) )) ∈ A with wµ−1 (j) ≤ wµ−1 (j) . By increasing
0
0
b
utility and (9), we have for all (µ, (w−i
,µ−1 (j) , wi , wµ−1 (j) )) ∈ A
If
ui (wi0 , j) > ui (wi0 , µ(i)) then vj (wi0 , i) > vj (wµb −1 (j) , µ−1 (j))
b
b
Therefore, (µ, (w−i
,µ−1 (j) , wi , wµ−1 (j) )) is A-blocked by (i, j).
Next, suppose that (µ, (w−i ,µ−1 (j) , wi , wµ−1 (j) )) ∈ A is not A-blocked by (i, j).
Then either
(i) ui (wi , j) ≤ ui (wi , µ(i)), or
(ii) there exists wi0 such that ui (wi0 , j) ≥ ui (wi0 , µ(i)) and vj (wi0 , i) ≤ vj (wµ−1 (j) , µ−1 (j)).
nb
nb
nb
Consider any (µ, (w−i
,µ−1 (j) , wi , wµ−1 (j) )) ∈ A with wµ−1 (j) ≥ wµ−1 (j) . If (i) then
nb
nb
(µ, (w−i
,µ−1 (j) , wi , wµ−1 (j) )) is not A-blocked by (i, j). Suppose, instead, that (ii).
nb
Then, by increasing utility, (ii) holds with wµnb−1 (j) replacing wµ−1 (j) . Thus (µ, (w−i
,µ−1 (j) ,
nb
wi , wµ−1 (j) )) is not A-blocked by (i, j).
2.1
Blocking Coalitions with Multiple Agents
In a complete-information model, if a matching is not blocked by a worker-firm pair
then it is not blocked by a larger coalition. In an incomplete-information setting, it
13
is possible that a larger coalition blocks a matching that is unblocked by any twoagent coalition, depending on the information-sharing assumptions within coalitions.
One assumption is that coalition members share their private information and the
incentive constraints to do so are disregarded. The opposite case is the assumption
that only what is common knowledge between coalition members is shared. These
two cases were examined in Wilson [10].An intermediate case, considered in Dutta
and Vohra [2], only information that can be inferred from a planned block is available
to coalition members. I make this intermediate assumption as that is consistent with
the earlier definition of a two-person block.
A coalition blocks a matching if each member is strictly better off in any state
of the world, consistent with the coalition-member’s beliefs, under which the block
might be contemplated. One immediate conclusion is that a blocking coalition must
have equal numbers of workers and firms because an unmatched worker (or firm)
cannot be strictly better off. Thus, let (i1 , j1 ), . . . , (ik , jk ) be a blocking coalition,
where each pair (i` , j` ), ` = 1, 2, . . . , k are matched together in the proposed blocking
coalition.
A matching outcome (µ, w) ∈ A is A-blocked by a coalition (i1 , j1 ), . . . , (ik , jk ) if
for all ` = 1, . . . , k,
ui` (wi` , j` )
>
ui` (wi` , µ(i` )),
and for all (µ, w0 ) ∈ A s.t. wµ0 −1 (j` ) = wµ−1 (j` ) ,
If ui` (wi0` , j` )
then
vj` (wi0` , i` )
>
>
ui` (wi0` , µ(i` ))
(10)
wi0` ≤ wi∗` ,
(11)
−1
vj` (wµ−1 (j` ) , µ (j` ))
(12)
where if i` 6= µ−1 (j`ˆ) for any `ˆ = 1, . . . k then wi∗` = w; that is, no additional restriction
is imposed on wi0` than is implied by (11).
This definition is similar to the earlier definition of a blocking pair except for the
possible restrictions wi01 ≤ wi∗1 , wi02 ≤ wi∗2 , etc. If i1 6= µ−1 (j` ) for any ` = 2, . . . , k then
j1 learns nothing from the fact that (i2 , j2 ), . . . , (ik , jk ) are in the blocking coalition
and we have wi∗1 = w. Therefore, (11) is similar to (4). If, instead, i1 = µ−1 (j` ) for
some ` ≥ 2, then an upper bound on wµ−1 (j` ) , that is on wi1 , is implied by Lemma 3;
that is, from the fact that j` will be better off with all possible types of i` that are
“reasonable” than by staying with µ−1 (j` ). Hence, the restriction wi01 ≤ wi∗1 . The
exact value of wi∗1 is unimportant for the next result.
Proposition 3 If a matching outcome is blocked by a coalition then each matched
worker-firm pair within the coalition constitutes a blocking pair.
Proof: Let (i1 , j1 ), . . . , (ik , jk ) be a blocking coalition.
14
Suppose that i1 6= µ−1 (j` ) for any ` ∈ {2, . . . , k}. Then wi∗1 = w and for the pair
(i1 , j1 ), (11) implies (12) if and only if (4) implies (5). Therefore, (i1 , j1 ) is a blocking
pair.
Next, suppose that i1 = µ−1 (j` ) for some ` ∈ {2, . . . , k} and wi∗1 < w. Then any
wi01 ≤ wi∗1 that satisfies (11) also satisfies (12). By increasing utility, (12) implies
vj1 (wi001 , i1 ) > vj` (wµ−1 (j1 ) , µ−1 (j1 )) for all wi001 > wi01 , thus (5) is true for all wi00 > wi∗1
including wi00 that satisfy (4). Hence, (i1 , j1 ) is a blocking pair.
2.2
An Ex Post Incentive-Compatible Mechanism
Properties of the set of complete-information stable matchings can be exploited to
construct an efficient, ex post incentive-compatible mechanism. For almost all worker
types w, this mechanism implements the worker-optimal complete-information stable
matching.
I assume that the mechanism designer knows the utility functions of the workers
and the firms. The rules of the mechanism are as follows. Workers report their types
to the mechanism designer. The mechanism designer computes the worker-optimal
complete-information stable matching (if one exists) for the reported worker types,
pairs the workers and firms according to this stable matching, and reveals the reported
worker types to all firms. If a worker-optimal matching does not exist then the mechanism designer picks an arbitrary complete-information stable matching. As before,
after the matching is implemented each firm will learn its matched worker’s type and,
in particular, will find out whether it is different from the worker type revealed by
the mechanism designer. Call this mechanism the worker-optimal mechanism.
Proposition 4 For almost all worker types, truthful reporting of types is an ex post
equilibrium for workers in the worker-optimal mechanism.
Proof: Consider a vector of worker types, w = (w1 , w2 , . . . , wn ), at which all agents
have strict preference. (As utility is increasing, this holds for almost all type vectors.)
Therefore, a worker-optimal mechanism exists at w; call it µw . Suppose that worker 1
reports a type w10 6= w1 . Let µ0 be the matching implemented at (w10 , w−1 ). If
µ0 (1) = µw (1), then worker 1 does not benefit from this deviation. Therefore, suppose
that µ0 (1) 6= µw (1) and that u1 (w1 , µ0 (1)) > u1 (w1 , µw (1)). But then, firm µ0 (1) is
unattainable for worker 1 at w. After the matching µ0 is implemented, firm µ0 (1)
learns that worker 1’s type is w1 and not w10 . Therefore, there exists a worker î that
together with firm µ0 (1) blocks matching µ0 . Hence, worker 1 does not profit from
the deviation.
15
Observe that increasing utility is used in the proof only in so far as it implies that
for almost all worker types there is strict preference.
2.3
Comparison with a TU model
In this section, the previous NTU model is compared to a TU model with quasilinear
utility in which side payments between matched agents are allowed. Thus, a matching
outcome is a (µ, w, p) where p = (p1 , p2 , . . . , pn ) is the vector of payments received
by the workers from their matched firms, i.e., worker i receives pi from firm µ(i) at
the matching outcome (µ, w, p). If µ(i) = 0 then pi = 0. This is the model of Liu et
al. (2013).
The impact of allowing side-payments on the size of self-stabilizing set is not
immediately clear. On the one hand, each matching could potentially be stabilized
by one of many side-payments, increasing the possibility that a matching is in a selfstabilizing set. On the other hand, the number of potential blocking contracts is also
much larger. It turns out, however, that the self-stabilizing set is smaller.
A matching outcome (µ, w, p) is A-blocked if there is a worker-firm pair (i, j) and
a payment p satisfying
ui (wi , j) + p
>
ui (wi , µ(i)) + pi
(13)
and for all (µ, w0 ) ∈ A s.t. wµ0 −1 (j) = wµ−1 (j) ,
ui (wi0 , j) + p
>
ui (wi0 , µ(i)) + pi
(14)
then vj (wi0 , i) − p
>
vj (wµ−1 (j) , µ−1 (j)) − pi
(15)
If
As (µ, w) ∈ A, if (3) is satisfied then (4) is also satisfied with wi0 = wi (and possibly
with other wi0 as well).
Proposition 5 If (µ, w) is A-blocked in the NTU game then, for any side-payment
vector p, (µ, w, p) is A-blocked in the TU game.
Proof: Suppose that (µ, w) ∈ A is A-blocked by (i, j) in the NTU game. Then, the
definitions of blocking in the NTU and TU games imply that for any side payments
p, (µ, w) in the TU game is A-blocked by (i, j) with the same side payments p.
Thus, the set of incomplete-information stable-matching outcomes is at least as
large in the NTU model as it is in the TU model. That the converse of Proposition 5
is not true follows from Example 1 of Section 2, where it is shown that every strictly
individually rational matching is incomplete-information stable in the NTU model,
16
and Liu et al.’s Proposition 3, which shows that only efficient matchings are stable
under assortative matching in a TU model.
3
Two-sided Uncertainty
As before, worker i’s type, wi , is in the interval [w, w]. In addition, firm j’s type, fj ,
is in the interval [f , f ]. If worker-firm (i, j) are matched then their respective utilities
are:
ui (wi , fj , j) and vj (wi , fj , i)
(16)
As before, the utility of an unmatched worker or firm is normalized to zero. Further,
it is common knowledge that agents know types on own side and know type of the
agent they are matched with, if any.
Increasing and Continuous Utility: The utility functions ui (wi , fj , j) and
vj (wi , fj , i) are strictly increasing in wi and fj , for all i and j.
Let w = (w1 , w2 , . . . , wn ) and f = (f1 , f2 , . . . , fm ) be the type vectors of the
workers and agents. A matching outcome is a matching function together with type
vectors: (µ, w, f ). The matching outcome (µ, w, f ) is individually rational if
ui (wi , fµ(i) , µ(i)) ≥ 0,
∀i
vj (wµ−1 (j) , fj , µ−1 (j)) ≥ 0,
∀j
Let Σ0 be the set of individually rational matching outcomes.
A matching outcome (µ, w, f ) ∈ A is A-blocked if there is a worker-firm pair (i, j)
satisfying
vj (wi , fj , i) > vj (wµ−1 (j) , fj , µ−1 (j))(17)
ui (wi , fj , j) > ui (wi , fµ(i) , µ(i)),
and for all (µ, w0 , f 0 ) ∈ A s.t. wµ0 −1 (j) = wµ−1 (j) ,
ui (wi0 , fj0 , j)
>
ui (wi0 , fµ(i) , µ(i))
(18)
then vj (wi0 , fj , i)
>
vj (wµ−1 (j) , fj , µ−1 (j))
(19)
If
0
and for all (µ, w0 , f 0 ) ∈ A s.t. fµ(i)
= fµ(i) ,
vj (wi0 , fj0 , i)
>
vj (wµ−1 (j) , fj0 , µ−1 (j))
(20)
then ui (wi , fj0 , j)
>
ui (wi , fµ(i) , µ(i))
(21)
If
We can define Σk and Σ∗ as earlier. The following result follows from arguments
used previously.
17
Lemma 4
(i) If B ⊂ A then if (µ, w, f ) is B-stable then it is A-stable.
(ii) The set of two-sided incomplete-information stable matching outcomes “includes”
the set of one-sided incomplete-information stable matching outcomes.
Proof: (i) Follows directly from the definition.
(ii) Suppose that it is common knowledge that the firms’ types vector is f . Let
B(f ) be the set of one-sided incomplete-information stable matching outcomes at f
and let A be the set of two-sided incomplete-information stable matching outcomes.
Then B(f ) ⊂ A from (i).
The following example shows that the set of incomplete information stable outcomes can be very large.
Example 5: Let wi , fj ∈ [0, 1] and
ui (wi , fj , j) = vj (wi , fj , i) = 1 − (wi − fj )2 .
Each agent wants to be matched with an agent of the closest type and would rather
be matched with an agent than be single.
Suppose that w1 = f1 = x < 21 , w2 = f2 = 1 − x. Let µ = {(W1 , F2 ), (W2 , F1 )},
an inefficient matching. This matching is one-sided incomplete-information unstable
for < 31 . But it is stable under two-sided incomplete information.
We know that (W1 , F1 ) satisfies (17). Can µ be blocked by (W1 , F1 )? That is,
does (18) imply (19)? The answer is no as w10 = f10 = 1 − 0.5x satisfies (18) but not
(19). A symmetric argument shows that (W2 , F2 ) is not a blocking pair.
It is assumed above that each agent’s type is common knowledge on their side
of the market but is not known to agents on the other side with whom they are
not matched. If this assumption is dropped and each agent’s type is known only to
him self and the matched agent, the definition of blocking needs to be changed with
0
fµ(i)
replacing fµ(i) in (18) and wµ0 −1 (j) replacing wµ−1 (j) in (20). Thus, the set of
incomplete-information stable-matching outcomes becomes larger.
Using the argument in the proof of Proposition 4, it can be shown that an ex
post incentive compatible mechanism exists if there exists exactly one (though not
necessarily the same one) matching for almost all worker type and firm type vectors.
18
4
Discussion
The preceding results suggest that under incomplete information, stability does not
provide much restriction on the observed matching outcomes. In particular, under
anonymous preferences stability provides no restriction at all. Therefore, in a matching environment with substantial incomplete information, the absence of signs of
instability, such as attempts by workers and firms to change their matches, need not
imply that the market is at a desirable outcome. Matching outcomes that make some,
or perhaps all, participants better off may exist but cannot be reached because of a
lack of information. Therefore, in such environments the process by which a matching
outcome is reached is important. An efficient incentive compatible stable matching
mechanism exists when there is one-sided incomplete information.
A natural question is whether that definition of incomplete-information blocking
used here is too strong. If, for instance, the firm in a potential blocking pair is required
to do better in expectation only rather than for all reasonable worker types, then it
becomes easier to block a matching outcome, thereby decreasing the set of stable
matching outcomes. Observe that a Bayesian notion of blocking assumes an ability
to commit to binding contracts. Suppose that a firm agrees to join with a worker to
form a blocking pair (to a status quo matching) because the firm’s expected utility
will increase. Later, after learning the type of the matched worker, the firm may find
that its utility in fact decreased and then it may wish to return to the status quo in
the absence of a binding contract. But if a binding contract is feasible in the blocking
pair, then it ought to be feasible in the status quo matching as well, rendering a
potential block to the status quo infeasible.
19
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[3] Ehlers, L., and J. Masso, 2007. “Incomplete Information and Singleton Cores in
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