1. Applied Game Theory - A Matching
Problem
Treesha Sarkar
July 27, 2014
Abstract
This paper deals with the matching of doctors with hospitals. Every
doctors have their individual preferences as well as couple preferences (if
they are couples), where for couples, couple preferences are strictly greater
than the individual preferences. All the hospitals are assumed to have the
same preference orderings over the doctors. In this paper, we consider two
hospitals, two single doctors and a couple (who have their single preference
as well as joint preference). We check for all possible preference orderings
of the hospitals, where we find that in some preference profiles there exist
no stable matching. This is because there always exist a hospital and a
doctor who have an incentive to deviate from their original matching (we
call this a blocking pair). In the later part of this paper, we introduce the
concept of credible blocking and credible stability, where we find that in
every preference profile, there exist one credibly stable matching.
1 Introduction
In this paper we are dealing with problem of allocating employees with their
employers. Employees can be categorized into married (here we call couple)
and unmarried. A married man would like to be employed where his wife is
employed rather than being employed where his wife is not (and this also holds
good for his wife). Most often, it is the government who deals with the problem
of allocating employers to employees. In our present democratic structure, both
the employers and the employees give their respective preferences, and the gov-
ernment has to come up with a mechanism to allocate these two groups so that
there is a minimum incentive for anyone in any group to change their position.
This situation can hold for doctors and hospitals, professors and university, and
in many other practical cases.Let’s take an example of two hospitals(hospital 1
and hospital 2) and a couple(say, male and female). The male prefers hospital 1
and the female prefers hospital 2; but they both prefer to stay together than to
stay apart. Assuming that both hospital 1 and hospital 2 prefer male doctor to
female doctor, then both the doctors would prefer to stay in the same hospital
than to stay to two different hospitals. This phenomenon in vividly explained
in this paper.
1

2. This paper deals with the matching for doctors with hospitals, where both
doctors and hospitals have their respective preferences. For simplicity, we con-
sider hospitals to have a common preference ordering, where as, doctors can
be singles(unmarried) or couples. The couples have their preferences which
are different from the single doctors. In other words, couples, though have
their preferences as single individuals, have their joint preferences where cou-
ples ranks their joint preferences higher than their single preferences . We also
consider that hospitals prefer a couple over a single doctor. Now, matching
specifies which doctors are assigned or matched to which hospitals; or rather
which hospitals gets which doctors. Both the hospitals or doctors have a chance
to get their preferred positions filled. If a hospital get a male for a couple but
not the respective female, then there is a blocking pair where the female as well
as the hospital would like to be assigned under one another. This constitutes a
blocking pair. A matching is stable at a particular preference profile structure
if there exist no blocking pair.
Blocking itself is meaningful if there is no counter block that affects one of
the candidates in the former blocking party. From here we move into the idea of
credible blocking which ensures that if a couple blocks with a hospital, then the
hospital cannot subsequently change its assignment so as to make the couple
worse off compared to the original match. Consequently, a matching is credibly
stable if there exist no credible blocking. It is taken into consideration that the
hospitals offer positions to as many doctors as the number of vacancy. We show
that a stable matching may not exist but a credibly stable matching always
exists.
2 The Model
In this section we consider a matching market consisting of a set of hospitalsH,
a set of doctors, and their preferences. Hospital h ∈ H has a capacity of kh,
where kh ≥ 1. S is the set of single doctors, while C = M ∪ F is the set of
couples of doctors. Here, M and F denote respectively the set of male and
female members constituting C. So, the whole set of doctors is D, where,
D = M ∪ F ∪ S. For every m ∈ M, there is an unique f ∈ F such that the pair
(m, f) forms a couple.
Each doctor s ∈ S has a preference ordering (a strict linear order) over the
set of hospitals in (H ∪ ∅), which we denote as s. Similarly, each male doctor
m ∈ M and each female doctor f ∈ F have preference orderings m and f .
For any couple c ∈ C, where c = (m, f), its preference ordering is a strict linear
order over (H ∪ ∅) × (H ∪ ∅), and is denoted by (mf). (m.f) is required
to satisfy the following conditions (with respect to m and f ). We use the
notation (k) to denote the k−th ranked alternative in the preference ordering
.
1. { (m,f)(1), (m,f)(2)} = {( m(1), m(1)),( f (1), f (1))}.
2. (h, h ) (m.f) (ˆh, h ) if h m
ˆh and (ˆh, h ) = ( f (1), f (1)).
2

3. 3. (h , h) (m,f) (h , ˆh) if h f
ˆh and (h , ˆh) = ( m(1), m(1)).
Similarly, we assume that all the hospitals have a common preference order-
ing over the set of doctors D. Without loss of generality, we assume that all
doctors are acceptable, i.e. we assume a common preference orderings. We de-
note this preference orderings as PH. Further, hospitals like to have the couple
over two single doctors, i.e., {m.f}PH{s1, s2}, where si ∈ S.
A matching specifies which doctors are assigned or matched to which hos-
pitals. Formally, a matching µ is a function defined on the set H ∪ D such
that:
• µ(h) ⊆ D with |µ(h)| ≤ kh for all h ∈ H
• µ(d) = h if and only if d ∈ µ(h).
We now define a blocking pair. Let h be a hospital which gets m but not
f. Then it can be said that (h, f) is a blocking pair. This follows from the fact
that h like f to any single doctor and (m, f) like to be together than separate.
It can also be the case that (h, s) form a blocking pair where s likes h than
assigned h and h also like s than any other individual s , m, f, where s, s ∈ S,
h, h ∈ H, m ∈ M and f ∈ F.
A matching µ is stable if there is no blocking pair. In other words, a matching
is individually stable if no hospital or doctor can be better off by unilaterally
rejecting some of the existing partners.
3 An Example:
Let s1, s2 ∈ S be the two single doctors, m ∈ M and f ∈ F be the corresponding
male and female in the couple. Here we consider only one couple c = (m, f).
Again, let h1, h2 ∈ H be the two hospital where kh1
= kh2
= 2. The preferences
of m and f are considered to be symmetric. It is assumed that both the hos-
pitals have same preference profile. We will check for stability for all possible
preference profiles of the hospitals and the doctors.
3.1 Preference profiles with no stable matching :
Firstly, we deal with those preference profile where in all the matching, there
exist a blocking pair. In other words, in these profiles none of the matching are
stable.
Profile 1
s1 s2 m f (m, f) h1 h2
h2 h1 h1 h2 (h1, h1) f f
h1 h2 h2 h1 (h2, h2) s2 s2
(h1, h2) s1 s1
(h2, h1) m m
3

4. We consider all possible matching.
• µh1
= {m, f} and µh2
= {s1, s2}. Then {h1, s2} is a blocking pair.
• µh1
= {s1, s2} and µh2
= {m, f}. Then {h2, s1} is a blocking pair.
• µh1
= {m, s1} and µh2
= {f, s2}. Then {h1, s2} is a blocking pair.
• µh1 = {m, s2} and µh2 = {f, s1}. Then {h1, f} is a blocking pair.
• µh1 = {f, s1} and µh2 = {m, s2}. Then {h2, f} is a blocking pair.
• µh1
= {f, s2} and µh2
= {m, s1}. Then {h2, f} is a blocking pair.
So we can conclude that in this profile none of the matching is stable.
Profile 2
s1 s2 m f (m, f) h1 h2
h2 h1 h1 h2 (h1, h1) f f
h1 h2 h2 h1 (h2, h2) s1 s1
(h1, h2) s2 s2
(h2, h1) m m
We consider all possible matching.
• µh1
= {m, f} and µh2
= {s1, s2}. Then {h1, s2} is a blocking pair.
• µh1 = {s1, s2} and µh2 = {m, f}. Then {h2, s1} is a blocking pair.
• µh1 = {m, s1} and µh2 = {f, s2}. Then {h1, s2} is a blocking pair.
• µh1
= {m, s2} and µh2
= {f, s1}. Then {h1, f} is a blocking pair.
• µh1
= {f, s1} and µh2
= {m, s2}. Then {h2, f} is a blocking pair.
• µh1
= {f, s2} and µh2
= {m, s1}. Then {h2, f} is a blocking pair.
So we can conclude that in this profile none of the matching is stable.
Profile 3
s1 s2 m f (m, f) h1 h2
h2 h1 h1 h2 (h1, h1) m m
h1 h2 h2 h1 (h2, h2) s2 s2
(h1, h2) s1 s1
(h2, h1) f f
We consider all possible matching.
• µh1
= {m, f} and µh2
= {s1, s2}. Then {h1, s2} is a blocking pair.
• µh1
= {s1, s2} and µh2
= {m, f}. Then {h2, s1} is a blocking pair.
• µh1
= {m, s1} and µh2
= {f, s2}. Then {h1, s2} is a blocking pair.
• µh1
= {m, s2} and µh2
= {f, s1}. Then {h2, m} is a blocking pair.
• µh1 = {f, s1} and µh2 = {m, s2}. Then {h1, m} is a blocking pair.
• µh1 = {f, s2} and µh2 = {m, s1}. Then {h1, m} is a blocking pair.
So we can conclude that in this profile none of the matching is stable.
4

5. Profile 4
s1 s2 m f (m, f) h1 h2
h2 h1 h1 h2 (h1, h1) m m
h1 h2 h2 h1 (h2, h2) s1 s1
(h1, h2) s2 s2
(h2, h1) f f
We consider all possible matching.
• µh1
= {m, f} and µh2
= {s1, s2}. Then {h1, s2} is a blocking pair.
• µh1
= {s1, s2} and µh2
= {m, f}. Then {h2, s1} is a blocking pair.
• µh1 = {m, s1} and µh2 = {f, s2}. Then {h2, s1} is a blocking pair.
• µh1 = {m, s2} and µh2 = {f, s1}. Then {h2, m} is a blocking pair.
• µh1
= {f, s1} and µh2
= {m, s2}. Then {h1, s2} is a blocking pair.
• µh1
= {f, s2} and µh2
= {m, s1}. Then {h1, m} is a blocking pair.
So we can conclude that in this profile none of the matching is stable.
Profile 5
s1 s2 m f (m, f) h1 h2
h2 h1 h1 h2 (h1, h1) s2 s2
h1 h2 h2 h1 (h2, h2) s1 s1
(h1, h2) m m
(h2, h1) f f
We consider all possible matching.
• µh1
= {m, f} and µh2
= {s1, s2}. Then {h1, s2} is a blocking pair.
• µh1
= {s1, s2} and µh2
= {m, f}. Then {h2, s1} is a blocking pair.
• µh1
= {m, s1} and µh2
= {f, s2}. Then {h2, s1} is a blocking pair.
• µh1 = {m, s2} and µh2 = {f, s1}. Then {h2, m} is a blocking pair.
• µh1 = {f, s1} and µh2 = {m, s2}. Then {h1, s2} is a blocking pair.
• µh1
= {f, s2} and µh2
= {m, s1}. Then {h1, m} is a blocking pair.
So we can conclude that in this profile none of the matching is stable.
Profile 6
s1 s2 m f (m, f) h1 h2
h2 h1 h1 h2 (h1, h1) s2 s2
h1 h2 h2 h1 (h2, h2) s1 s1
(h1, h2) f f
(h2, h1) m m
5

6. We consider all possible matching.
• µh1
= {m, f} and µh2
= {s1, s2}. Then {h1, s2} is a blocking pair.
• µh1
= {s1, s2} and µh2
= {m, f}. Then {h2, s1} is a blocking pair.
• µh1
= {m, s1} and µh2
= {f, s2}. Then {h1, s2} is a blocking pair.
• µh1 = {m, s2} and µh2 = {f, s1}. Then {h1, f} is a blocking pair.
• µh1 = {f, s1} and µh2 = {m, s2}. Then {h1, s2} is a blocking pair.
• µh1
= {f, s2} and µh2
= {m, s1}. Then {h2, f} is a blocking pair.
So we can conclude that in this profile none of the matching is stable.
Profile 7
s1 s2 m f (m, f) h1 h2
h2 h1 h1 h2 (h1, h1) s1 s1
h1 h2 h2 h1 (h2, h2) s2 s2
(h1, h2) m m
(h2, h1) f f
We consider all possible matching.
• µh1
= {m, f} and µh2
= {s1, s2}. Then {h1, s2} is a blocking pair.
• µh1 = {s1, s2} and µh2 = {m, f}. Then {h2, s1} is a blocking pair.
• µh1 = {m, s1} and µh2 = {f, s2}. Then {h2, s1} is a blocking pair.
• µh1
= {m, s2} and µh2
= {f, s1}. Then {h2, m} is a blocking pair.
• µh1
= {f, s1} and µh2
= {m, s2}. Then {h2, s1} is a blocking pair.
• µh1
= {f, s2} and µh2
= {m, s1}. Then {h1, m} is a blocking pair.
So we can conclude that in this profile none of the matching is stable.
Profile 8
s1 s2 m f (m, f) h1 h2
h2 h1 h1 h2 (h1, h1) s1 s1
h1 h2 h2 h1 (h2, h2) s2 s2
(h1, h2) f f
(h2, h1) m m
We consider all possible matching.
• µh1
= {m, f} and µh2
= {s1, s2}. Then {h1, s2} is a blocking pair.
• µh1
= {s1, s2} and µh2
= {m, f}. Then {h2, s1} is a blocking pair.
• µh1
= {m, s1} and µh2
= {f, s2}. Then {h2, s1} is a blocking pair.
• µh1
= {m, s2} and µh2
= {f, s1}. Then {h1, f} is a blocking pair.
• µh1 = {f, s1} and µh2 = {m, s2}. Then {h2, s1} is a blocking pair.
• µh1 = {f, s2} and µh2 = {m, s1}. Then {h2, f} is a blocking pair.
So we can conclude that in this profile none of the matching is stable.
6

7. Profile 9
s1 s2 m f (m, f) h1 h2
h2 h1 h1 h2 (h1, h1) s1 s1
h1 h2 h2 h1 (h2, h2) m m
(h1, h2) s2 s2
(h2, h1) f f
We consider all possible matching.
• µh1
= {m, f} and µh2
= {s1, s2}. Then {h1, s2} is a blocking pair.
• µh1
= {s1, s2} and µh2
= {m, f}. Then {h2, s1} is a blocking pair.
• µh1 = {m, s1} and µh2 = {f, s2}. Then {h2, s1} is a blocking pair.
• µh1 = {m, s2} and µh2 = {f, s1}. Then {h2, m} is a blocking pair.
• µh1
= {f, s1} and µh2
= {m, s2}. Then {h2, s1} is a blocking pair.
• µh1
= {f, s2} and µh2
= {m, s1}. Then {h1, m} is a blocking pair.
So we can conclude that in this profile none of the matching is stable.
Profile 10
s1 s2 m f (m, f) h1 h2
h2 h1 h1 h2 (h1, h1) s2 s2
h1 h2 h2 h1 (h2, h2) m m
(h1, h2) s1 s1
(h2, h1) f f
We consider all possible matching.
• µh1
= {m, f} and µh2
= {s1, s2}. Then {h1, s2} is a blocking pair.
• µh1
= {s1, s2} and µh2
= {m, f}. Then {h2, s1} is a blocking pair.
• µh1
= {m, s1} and µh2
= {f, s2}. Then {h1, s2} is a blocking pair.
• µh1 = {m, s2} and µh2 = {f, s1}. Then {h2, m} is a blocking pair.
• µh1 = {f, s1} and µh2 = {m, s2}. Then {h2, s1} is a blocking pair.
• µh1
= {f, s2} and µh2
= {m, s1}. Then {h1, m} is a blocking pair.
So we can conclude that in this profile none of the matching is stable.
Profile 11
s1 s2 m f (m, f) h1 h2
h2 h1 h1 h2 (h1, h1) s2 s2
h1 h2 h2 h1 (h2, h2) f f
(h1, h2) s1 s1
(h2, h1) m m
7

8. We consider all possible matching.
• µh1
= {m, f} and µh2
= {s1, s2}. Then {h1, s2} is a blocking pair.
• µh1
= {s1, s2} and µh2
= {m, f}. Then {h2, s1} is a blocking pair.
• µh1
= {m, s1} and µh2
= {f, s2}. Then {h1, s2} is a blocking pair.
• µh1 = {m, s2} and µh2 = {f, s1}. Then {h1, f} is a blocking pair.
• µh1 = {f, s1} and µh2 = {m, s2}. Then {h1, s2} is a blocking pair.
• µh1
= {f, s2} and µh2
= {m, s1}. Then {h2, f} is a blocking pair.
So we can conclude that in this profile none of the matching is stable.
Profile 12
s1 s2 m f (m, f) h1 h2
h2 h1 h1 h2 (h1, h1) s1 s1
h1 h2 h2 h1 (h2, h2) f f
(h1, h2) s2 s2
(h2, h1) m m
We consider all possible matching.
• µh1
= {m, f} and µh2
= {s1, s2}. Then {h1, s2} is a blocking pair.
• µh1
= {s1, s2} and µh2
= {m, f}. Then {h2, s1} is a blocking pair.
• µh1
= {m, s1} and µh2
= {f, s2}. Then {h2, s1} is a blocking pair.
• µh1 = {m, s2} and µh2 = {f, s1}. Then {h1, f} is a blocking pair.
• µh1 = {f, s1} and µh2 = {m, s2}. Then {h2, s1} is a blocking pair.
• µh1
= {f, s2} and µh2
= {m, s1}. Then {h2, f} is a blocking pair.
So we can conclude that in this profile none of the matching is stable.
All the matching in above 12 profiles have a blocking pair, which means that
there is no stable matching in the above 12 profiles.
TO PROVE : To show that there is no stable matching in the above profiles.
PROOF : To show this, first we show that a matching where m and f are
separated cannot be a stable one.
Suppose, without loss of generality, we take µm = h1 and µf = h2.
Then acting according to the couple preference, m would apply to h2 and h2
would happily accept m by rejecting s1 or s2. So, this means that h2 and m
blocks this matching.
Now consider matching where m and f are together at some hospital, say h1.
Then, as s2’s most preferred hospital is h1, s2 would apply to h1 and h1 would
accept s2. So, h1 and s2 blocks and hence the proof.
8

9. 3.2 Preference profiles with atleast one stable matching :
Now we are dealing with those preference profiles where their exist atleast one
stable matching. In other words, in atleast one matching in every profile, there
is an absence of a blocking pair.
Profile 13
s1 s2 m f (m, f) h1 h2
h2 h1 h1 h2 (h1, h1) m m
h1 h2 h2 h1 (h2, h2) f f
(h1, h2) s1 s1
(h2, h1) s2 s2
In this preference profile, µh1
= {m, f} and µh2
= {s1, s2} is a stable matching.
Profile 14
s1 s2 m f (m, f) h1 h2
h2 h1 h1 h2 (h1, h1) m m
h1 h2 h2 h1 (h2, h2) f f
(h1, h2) s2 s2
(h2, h1) s1 s1
In this preference profile, µh1
= {m, f} and µh2
= {s1, s2} is a stable matching.
Profile 15
s1 s2 m f (m, f) h1 h2
h2 h1 h1 h2 (h1, h1) f f
h1 h2 h2 h1 (h2, h2) m m
(h1, h2) s1 s1
(h2, h1) s2 s2
In this preference profile, µh1 = {m, f} and µh2 = {s1, s2} is a stable matching.
Profile 16
9

10. s1 s2 m f (m, f) h1 h2
h2 h1 h1 h2 (h1, h1) f f
h1 h2 h2 h1 (h2, h2) m m
(h1, h2) s2 s2
(h2, h1) s1 s1
In this preference profile, µh1
= {m, f} and µh2
= {s1, s2} is a stable matching.
Profile 17
s1 s2 m f (m, f) h1 h2
h2 h1 h1 h2 (h1, h1) m m
h1 h2 h2 h1 (h2, h2) s1 s1
(h1, h2) f f
(h2, h1) s2 s2
In this preference profile, µh1 = {m, f} and µh2 = {s1, s2} is a stable matching.
Profile 18
s1 s2 m f (m, f) h1 h2
h2 h1 h1 h2 (h1, h1) f f
h1 h2 h2 h1 (h2, h2) s1 s1
(h1, h2) m m
(h2, h1) s2 s2
In this preference profile, µh1
= {m, f} and µh2
= {s1, s2} is a stable matching.
Profile 19
s1 s2 m f (m, f) h1 h2
h2 h1 h1 h2 (h1, h1) m m
h1 h2 h2 h1 (h2, h2) s2 s2
(h1, h2) f f
(h2, h1) s1 s1
In this preference profile, µh1 = {s1, s2} and µh2 = {m, f} is a stable matching.
10

11. Profile 20
s1 s2 m f (m, f) h1 h2
h2 h1 h1 h2 (h1, h1) f f
h1 h2 h2 h1 (h2, h2) s2 s2
(h1, h2) m m
(h2, h1) s1 s1
In this preference profile, µh1 = {s1, s2} and µh2 = {m, f} is a stable matching.
Profile 21
s1 s2 m f (m, f) h1 h2
h2 h1 h1 h2 (h1, h1) s1 s1
h1 h2 h2 h1 (h2, h2) m m
(h1, h2) f f
(h2, h1) s2 s2
In this preference profile, µh1 = {m, f} and µh2 = {s1, s2} is a stable matching.
Profile 22
s1 s2 m f (m, f) h1 h2
h2 h1 h1 h2 (h1, h1) s1 s1
h1 h2 h2 h1 (h2, h2) f f
(h1, h2) m m
(h2, h1) s2 s2
In this preference profile, µh1 = {m, f} and µh2 = {s1, s2} is a stable matching.
Profile 23
s1 s2 m f (m, f) h1 h2
h2 h1 h1 h2 (h1, h1) s2 s2
h1 h2 h2 h1 (h2, h2) m m
(h1, h2) f f
(h2, h1) s1 s1
11

12. In this preference profile, µh1
= {s1, s2} and µh2
= {m, f} is a stable matching.
Profile 24
s1 s2 m f (m, f) h1 h2
h2 h1 h1 h2 (h1, h1) s2 s2
h1 h2 h2 h1 (h2, h2) f f
(h1, h2) m m
(h2, h1) s1 s1
In this preference profile, µh1
= {s1, s2} and µh2
= {m, f} is a stable matching.
3.3 The notion of Credible Blocking
We define a block (h, c, µ ) to µ to be credible if -
• There is no block (h, d, µ ) to µ such that µ(c)Pcµ (c), or,
• No block (h , c, µ ) to µ such that µ(h)Phµ (h).
Credible blocking ensures that if a couple blocks µ with hospital h, then the
hospital cannot subsequently change its assignment so as to make the couple
worse off compared to the original matching µ. It is sufficient to define credible
blocking only for hospital-couple blocking pairs.
It can be shown that all the above preference profiles are credible stable, i.e., in
none of the above preference profiles, there is no scope of credible blocking. We
take into consideration of the preference profile where there exist none stable
matching.
Profile 1
s1 s2 m f (m, f) h1 h2
h2 h1 h1 h2 (h1, h1) f f
h1 h2 h2 h1 (h2, h2) s2 s2
(h1, h2) s1 s1
(h2, h1) m m
Taking consideration of a typical matching -
µh1 = {f, s2} and µh2 = {m, s1}. Then {h2, f} used to be a blocking pair.
Now h2 prefers s1 than m; so it will reject m and take f. But f then would
cheat h2 as she prefers to be with m. So this blocking is not credible and the
matching is credibly stable.
12

13. Profile 2
s1 s2 m f (m, f) h1 h2
h2 h1 h1 h2 (h1, h1) f f
h1 h2 h2 h1 (h2, h2) s1 s1
(h1, h2) s2 s2
(h2, h1) m m
Taking consideration of a typical matching -
µh1
= {f, s2} and µh2
= {m, s1}. Then {h2, f} used to be a blocking pair.
Now h2 prefers s1 than m; so it will reject m and take f. But f then would
cheat h2 as she prefers to be with m. So this blocking is not credible and the
matching is credibly stable.
Profile 3
s1 s2 m f (m, f) h1 h2
h2 h1 h1 h2 (h1, h1) m m
h1 h2 h2 h1 (h2, h2) s2 s2
(h1, h2) s1 s1
(h2, h1) f f
Taking consideration of a typical matching -
µh1 = {f, s2} and µh2 = {m, s1}. Then {h1, m} used to be a blocking pair.
Now h1 prefers s2 than f; so it will reject f and take m. But m then would
cheat h1 as she prefers to be with f. So this blocking is not credible and the
matching is credibly stable.
Profile 4
s1 s2 m f (m, f) h1 h2
h2 h1 h1 h2 (h1, h1) m m
h1 h2 h2 h1 (h2, h2) s1 s1
(h1, h2) s2 s2
(h2, h1) f f
Taking consideration of a typical matching -
µh1
= {f, s2} and µh2
= {m, s1}. Then {h1, m} used to be a blocking pair.
Now h1 prefers s2 than f; so it will reject f and take m. But m then would
cheat h1 as she prefers to be with f. So this blocking is not credible and the
matching is credibly stable.
13

14. Profile 5
s1 s2 m f (m, f) h1 h2
h2 h1 h1 h2 (h1, h1) s2 s2
h1 h2 h2 h1 (h2, h2) s1 s1
(h1, h2) m m
(h2, h1) f f
Taking consideration of a typical matching -
µh1
= {m, s2} and µh2
= {f, s1}. Then {h2, m} used to be a blocking pair.
Now m prefers h2 as h2 have taken f. But h2, after taking m would reject f as
it prefers s1 than f. So m will not be blocked by h2 in the first place and that
makes this matching credibly stable.
Profile 6
s1 s2 m f (m, f) h1 h2
h2 h1 h1 h2 (h1, h1) s2 s2
h1 h2 h2 h1 (h2, h2) s1 s1
(h1, h2) f f
(h2, h1) m m
Taking consideration of a typical matching -
µh1 = {m, s2} and µh2 = {f, s1}. Then {h1, f} used to be a blocking pair.
Now f prefers h1 as h1 have taken m. But h1, after taking f would reject m as
it prefers s2 than m. So f will not be blocked by h1 in the first place and that
makes this matching credibly stable.
Profile 7
s1 s2 m f (m, f) h1 h2
h2 h1 h1 h2 (h1, h1) s1 s1
h1 h2 h2 h1 (h2, h2) s2 s2
(h1, h2) m m
(h2, h1) f f
Taking consideration of a typical matching -
µh1
= {m, s2} and µh2
= {f, s1}. Then {h2, m} used to be a blocking pair.
Now m prefers h2 as h2 have taken f. But h2, after taking m would reject f as
it prefers s1 than f. So m will not be blocked by h2 in the first place and that
makes this matching credibly stable.
14

15. Profile 8
s1 s2 m f (m, f) h1 h2
h2 h1 h1 h2 (h1, h1) s1 s1
h1 h2 h2 h1 (h2, h2) s2 s2
(h1, h2) f f
(h2, h1) m m
Taking consideration of a typical matching -
µh1
= {m, s2} and µh2
= {f, s1}. Then {h1, f} used to be a blocking pair.
Now f prefers h1 as h1 have taken m. But h1, after taking f would reject m as
it prefers s2 than m. So f will not be blocked by h1 in the first place and that
makes this matching credibly stable.
Profile 9
s1 s2 m f (m, f) h1 h2
h2 h1 h1 h2 (h1, h1) s1 s1
h1 h2 h2 h1 (h2, h2) m m
(h1, h2) s2 s2
(h2, h1) f f
Taking consideration of a typical matching -
µh1 = {m, s2} and µh2 = {f, s1}. Then {h2, m} used to be a blocking pair.
Now m prefers h2 as long as h2 have f. But h2, after taking m would reject f
as it prefers s1 than f. So m will not be blocked by h2 in the first place and
that makes this matching credibly stable.
Profile 10
s1 s2 m f (m, f) h1 h2
h2 h1 h1 h2 (h1, h1) s2 s2
h1 h2 h2 h1 (h2, h2) m m
(h1, h2) s1 s1
(h2, h1) f f
Taking consideration of a typical matching -
µh1
= {m, s2} and µh2
= {f, s1}. Then {h2, m} used to be a blocking pair.
Now m prefers h2 as long as h2 have f. But h2, after taking m would reject f
as it prefers s1 than f. So m will not be blocked by h2 in the first place and
that makes this matching credibly stable.
15

16. Profile 11
s1 s2 m f (m, f) h1 h2
h2 h1 h1 h2 (h1, h1) s2 s2
h1 h2 h2 h1 (h2, h2) f f
(h1, h2) s1 s1
(h2, h1) m m
Taking consideration of a typical matching -
µh1
= {m, s2} and µh2
= {f, s1}. Then {h1, f} used to be a blocking pair.
Now f prefers h1 as long as h1 have m. But h1, after taking f would reject m
as it prefers s2 than m. So f will not be blocked by h1 in the first place and
that makes this matching credibly stable.
Profile 12
s1 s2 m f (m, f) h1 h2
h2 h1 h1 h2 (h1, h1) s2 s2
h1 h2 h2 h1 (h2, h2) f f
(h1, h2) s1 s1
(h2, h1) m m
Taking consideration of a typical matching -
µh1 = {m, s2} and µh2 = {f, s1}. Then {h1, f} used to be a blocking pair.
Now f prefers h1 as long as h1 have m. But h1, after taking f would reject m
as it prefers s2 than m. So f will not be blocked by h1 in the first place and
that makes this matching credibly stable.
4 Conclusion
From the whole analysis, one can conclude that in a typical matching problem,
it is not always the case that one can get a stable matching. This is because after
the first allocation, a particular hospital prefers atleast one other doctor than
its allocated doctors and simultaneously, that doctor also prefers that particular
hospital than the one he is employed in.
Now, under credible blocking, the couple doctor(let’s say the male doctor)
check if after moving into the newly preferred hospital, whether the hospital
have any tendency to reject his female counterpart. If so, then the male doctor
will never block in the first place. Similarly, the hospital checks that after
accepting a member of a couple, if there is any incentive for that member for
16

17. the couple to leave the newly employed hospital. And this will happen if the
other member of the couple is not present in that particular hospital. So, the
hospital will not block in the first place.
With this notion, we find out that after the initial matching, none of the
hospitals and couples block and thus we get a credibly stable matching.
5 References
• Klaus, B., and Klijn, F. (2007). Fair and efficient student placement with
couples. International Journal of Game Theory, 36, 177-207.
• B. Klaus and F. Klijn. Stable Matchings and Preferences of Married Couples,
Journal of Economic Theory 121, 75-106 (2005).
• L. Ehlers. In Search of Advice for Participants in Matching Markets which
Use the Deferred-Acceptance Algorithm, Games and Economic Behavior 48,
249-270 (2004).
• D. Cantala. Matching Markets: The Particular Case of Couples, Mimeo,
University of Edinburgh (2002).
• J. Eeckhout. On the Uniqueness of Stable Marriage Matchings, Economics
Letters 69, 1-8 (2000).
• B. Dutta and J. Masso. Stability of Matchings When Individuals Have Pref-
erences over Colleagues, Journal of Economic Theory 75, 464-475 (1997).
• A.E. Roth. The Economics of Matching: Stability and Incentives, Mathematics
of Operations Research 7, 617-628 (1982).
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