Von Neumann-Morgenstern Stable Sets in ... - Semantic Scholar
Von Neumann-Morgenstern Stable Sets in Matching Problems Lars Ehlersy February 2005
Abstract The following properties of the core of a one-to-one matching problem are well-known: (i) the core is non-empty; (ii) the core is a lattice; and (iii) the set of unmatched agents is identical for any two matchings belonging to the core. The literature on two-sided matching focuses almost exclusively on the core and studies extensively its properties. Our main result is the following characterization of (von Neumann-Morgenstern) stable sets in one-to-one matching problems.
We show that a set of matchings is a stable set of a one-to-one
matching problem only if it is a maximal set satisfying the following properties: (a) the core is a subset of the set; (b) the set is a lattice; and (c) the set of unmatched agents is identical for any two matchings belonging to the set. Furthermore, a set is a stable set if it is the unique maximal set satisfying properties (a), (b), and (c). We also show that our main result does not extend from one-to-one matching problems to many-to-one matching problems.
JEL Classi cation: Keywords:
C78, J41, J44.
Matching Problem, Von Neumann-Morgenstern Stable Sets.
First version: November 2002. I thank Joseph Greenberg for drawing my attention to this question and Utku Unver for helpful comments. y Departement de Sciences Economiques and CIREQ, Universite de Montreal, Montreal, Quebec H3C 3J7, Canada; e-mail: [email protected]
1
1
Introduction
Von Neumann and Morgenstern (1944) introduced the notion of a stable set of a cooperative game.1 The idea behind a stable set is the following (Myerson, 1991; Osborne and Rubinstein, 1994): suppose the players consider a certain set of allocations of the cooperative game to be the possible outcomes (or proposals) of the game, without knowing which one will be ultimately chosen. Then any stable set of the game is a set of possible outcomes having the following properties: (i) for any allocation in the stable set there does not exist any coalition which prefers a certain other possible (attainable) outcome to this allocation, i.e. no coalition has a credible objection to any stable outcome; and (ii) for any allocation outside of the stable set there exists a coalition which prefers a certain other possible (attainable) outcome to this allocation, i.e. any unstable outcome is credibly objected by a coalition through a stable outcome. Conditions (i) and (ii) are robustness conditions of stable sets. (i) is referred to as internal stability of a set and (ii) as external stability of a set. The core of a cooperative game is always internally stable but it may violate external stability. Von Neumann and Morgenstern believed that stable sets should be the main solution concept for cooperative games in economic environments. Unfortunately, there is no general theory for stable sets. The theory has been prevented from being successful because it is very dicult working with it, which Aumann (1987) explains as follows: \Finding stable sets involves a new tour de force of mathematical reasoning for each game or class of games that is considered. Other than a small number of elementary truisms (e.g. that the core is contained in every stable set), there is no theory, no tools, certainly no algorithms." These facts helped the core to become the dominant multi-valued solution concept of cooperative games. The core of a game is extensively studied and well understood 1
Stable sets are called \solutions" in their book. We follow the convention of most of the recent
literature and refer to \solutions" as stable sets.
2
by the literature. This led a number of papers to identify classes of games where the core is the unique stable set of the game (e.g., Shapley (1971), Peleg (1986a), Einy, Holzman, Monderer, and Shitovitz (1997) and Biswas, Parthasarathy, and Ravindran (2001)).2 This paper is the rst study of stable sets in matching markets. In a matching market there are two disjoint sets of agents, usually called men and women or workers and rms, and we face the problem of matching agents from one side of the market with agents from the other side where each individual has the possibility of remaining unmatched. Matching problems arise in a number of important economic environments such as entry-level labor markets, college admissions, or school choice. The literature on two-sided matching problems focuses almost exclusively on the core.3 However, the core may violate external stability, i.e. there may be matchings outside the core which are not blocked (or objected) by a coalition through a core matching. Those matchings are only blocked through some \hypothetical matching" which does not belong to the core. Once such a matching proposed it is not clear why it will be replaced by an element in the core. We show that a sucient condition for this is that at the core matching, which is optimal for one side of the market, the agents of that side can gain by reallocating their partners. Here our purpose is not to investigate when the core is the unique stable set for a matching problem. However, the answer to this question will be a straightforward corollary of our main result. We nd that any stable set shares a number of wellknown and extensively studied properties of the core of a matching problem. Our main result shows that for one-to-one matching problems any stable set is a maximal set satisfying the following properties: (a) the core is a subset of the set; (b) the set is a lattice; and (c) the set of unmatched agents is identical for any two matchings belonging to the set. The converse also holds (i.e. a set is a stable set for a one-to-one 2 3
Note that the core of a cooperative game is always unique. One of the few exceptions is Klijn and Masso (2003) who apply the bargaining set of Zhou (1994)
to one-to-one matching problems.
3
matching problem) if a set is the unique maximal set satisfying the properties (a), (b), and (c). The literature on matching studied extensively when the core is a lattice and when the set of unmatched agents is identical. However, there is no such result saying that if a set possesses certain properties, then it coincides with the core. From our main result it is immediate that the core is the unique stable set if and only if it is a maximal set satisfying (b) the set is a lattice and (c) the set of unmatched agents is identical for all matchings belonging to the set. Furthermore, our main result facilitates considerably the search for stable sets in one-to-one matching problems: we just need to look at maximal sets satisfying (a), (b), and (c) (and if the maximal set is unique, then it is a stable set). We also show that the main result does not extend to many-to-one matching problems. Two papers in the literature on stable sets contain some similar features as our paper. One is Einy, Holzman, Monderer, and Shitovitz (1996) who study (non-atomic) glove games with a continuum of agents. They show that the core is the unique stable set of any glove game where the mass of agents holding left hand and right hand gloves is identical. Glove games are a special case of assignment games where there are two disjoint sets of buyers and sellers and each buyer-seller pair obtains a certain surplus from exchanging the good owned by the seller. Note that their result requires a continuum of agents, an equal mass of sellers and traders, and each seller's good has the same value for all buyers. Our main result does not impose any restriction on the one-to-one matching problem under consideration. The other paper is Einy and Shitovitz (2003) who study neoclassical pure exchange economies with a nite set of agents or with a continuum of agents. They show that the set of symmetric and Pareto-optimal allocations is the unique symmetric stable set. Their result holds in the continuum case without any restriction and in the nite case with the restriction that any endowment is owned by an identical number of agents and the agents owning the same endowment have identical preferences. The spirit of their result is similar as ours in the sense of determining properties of stable sets and showing that any set 4
satisfying these properties is a stable set. Note, however, that they focus on symmetric stable sets only and for the nite case the result only holds if any endowment is owned by an identical number of agents who have identical preferences.4 Symmetry is not meaningful in matching problems because no pair of agents is identical. The paper is organized as follows. Section 2 introduces one-to-one matching problems. Section 3 de nes stable sets and states some helpful insights. Section 4 contains the main result for one-to-one matching problems. It characterizes stable sets in terms of well-known properties of the core. Section 5 shows that this characterization does not extend to many-to-one matching problems. Section 6 concludes. 2
One-To-One Matching Problems
A one-to-one matching problem is a triple (M; W; R) where M is a nite set of men,
W is a nite set of women, and R is a preference pro le specifying for each man m
2M
a strict preference relation Rm over W
a strict preference relation Rw over M
[ fwg.
[ fmg and for each woman w 2 W
Then vRi v means that v is weakly 0
preferred to v under Ri , and vPi v means v is strictly preferred to v under Ri . 0
0
0
Strictness of a preference relation Ri means that vRi v implies v = v or vPi v . We 0
0
0
will keep M and W xed and thus, a matching problem is completely described by
R denote the set of all pro les. We will call N = M [ W the set of agents. Given Rm and S W , let Rm jS denote the restriction of Rm to S . Furthermore,
R. Let
let A(Rm ) denote the set of women who are acceptable for man m under Rm , i.e.
A(Rm ) = fw 2 W jwPm mg. Similarly we de ne Rw jS (where S M ) and A(Rw ). A matching is a function : N
all m 2 M , (m) 2 W
i 2 N , ( (i)) = i.
!N
satisfying the following properties: (i) for
[ fmg; (ii) for all w 2 W , (w) 2 M [ fwg; and (iii) for all Let M denote the set of all matchings. We say that an agent
i is unmatched at matching if (i) = i. Given the matching , let U () denote 4
This assumption is similar to the one of Einy, Holzman, Monderer, and Shitovitz (1996) that an
equal mass of agents holds left hand and right hand gloves.
5
the set of agents who are unmatched at . Given a pro le R, a matching is called
individually rational if for all i 2 N , (i)Ri i; and is called Pareto-optimal if there is
no matching = 6 such that (i)Ri(i) for all i 2 N with strict preference holding 0
0
N , we say that is Pareto-optimal 6 such that (i)Ri(i) for all i 2 S with strict =
for at least one agent. Given a coalition S for S if there is no matching
0
0
preference holding for at least one agent in S . Furthermore we say that matching is attainable for coalition S if (S ) = S (where (S ) = f(i)ji 2 S g).
Let R be a pro le. Given two matchings ; and a coalition S N , we say that 0
dominates via S under R, denoted by RS , if (i) (S ) = S and (ii) for all 0
0
i 2 S , (i)Pi (i). We say that dominates under R, denoted by R , if there 0
0
0
N such that RS . We will omit the superscript when the pro le R is unambiguous and write S and .
exists S
0
The core of a matching problem contains all matchings which are not blocked by some coalition through a matching attainable for that coalition. Given a pro le R, let C (R) denote the core of R,5 i.e.
C (R) = f 2 Mj for all ; = 6 S N and all
0
2 M we have 6 RS g: 0
The core of a matching problem is always non-empty (Gale and Shapley, 1962). Furthermore, the set of unmatched agents is identical for all matchings belonging to the core (McVitie and Wilson, 1970; Roth, 1984). We also consider the core where blocking is only allowed by a certain set of coalitions (instead of all coalitions). Given a set of coalitions
T,
let C (R) denote the T
T -core of R (Kalai,
Postlewaite, and
Roberts, 1979), i.e.
C (R) = f 2 Mj for all S 2 T and all T
0
2 M we have 6 RS g: 0
It is well-known that the core of a matching problem is a complete lattice (Knuth 5
The core of a one-to-one matching problem is often referred to as the set of stable matchings.
In avoiding any confusion with stable sets we will not use this terminology.
6
(1976) attributes this result to John Conway).6 Therefore, the core contains two matchings, called the M -optimal matching and W -optimal matching (the two extremes of the lattice), such that the M -optimal matching is the matching which is both most preferred by the men and least preferred by the women in the core (similar for the W -optimal matching). Given a pro le R, let M (R) denote the M -optimal matching and W (R) the W optimal matching in C (R). We will suppress the pro le R and instead write M and
2 M, let _ denote the mapping _ : N ! N such that (i) for all m 2 M , ( _ )(m) = (m) if (m)Rm (m), and otherwise ( _ )(m) = (m), and (ii) for all w 2 W , ( _ )(m) = (w) if (w)Rw (w), and otherwise ( _ )(w) = (w). Note that _ does not need to be a matching. Similarly we de ne ^ . Given a pro le R and V M, we say that V is a lattice (under R) if for all ; 2 V we have _ 2 V and ^ 2 V .
W . Given two matchings ;
0
0
0
0
0
0
0
0
0
0
0
0
0
0
3
0
0
Stable Sets
A set of matchings is a stable set for a matching problem if it satis es the following two robustness conditions: (i) no matching belonging to the set is blocked by a coalition through a matching which both belongs to the set and is attainable for this coalition; and (ii) any matching outside of the set is blocked by a coalition through a matching which both belongs to the set and is attainable for this coalition.
De nition 1 Let R
2 R and V M.
Then V is called a stable set for R if the
following two properties hold: (i) (Internal stability) For all ;
0
(ii) (External stability) For all
0
6
2 V , 6 . 0
2 MnV
there exists 2 V such that . 0
Many papers study the lattice structure of the strong core and the set of stable matchings in
matching problems; see for example, Blair (1988), Alkan (2001), and Alkan and Gale (2003).
7
Since the core consists of all undominated matchings, the core is always contained in any stable set. However, the core is not necessarily a stable set. A sucient condition for the core not to be a stable set is that at the M -optimal matching the men can gain by reallocating their partners (and thus, the M -optimal matching is not Pareto-optimal for the men) or at the W -optimal matching the women can gain by reallocating their partners (and thus, the W -optimal matching is not Pareto-optimal for the women). To see this, let R be a pro le such that at M the men can gain by reallocating their partners. Then there is a matching such that (M ) \ W =
M (M ) \ W and for all m 2 M , (m)Rm M (m), with strict preference holding for at least one man. Obviously, by the individual rationality of M , this implies that any man who is matched to a woman at M must be also matched to a woman at , i.e.
(m) 2 W for all m 2 M such that M (m) 6= m. Therefore, by the assumption that the matched men reallocate their partners ((M ) \ W = M (M ) \ W ), the set of unmatched agents is identical for both and M . Since M is the matching which is most preferred by all men in C (R) and the set of unmatched agents is the same at any two matchings belonging to the core (noting U () = U (M )), there is no matching
0
2 C (R) such that . Hence, C (R) is not externally stable. 0
Furthermore, stable sets are not necessarily individually rational (the same is true for other cooperative games).
Example 1 Let M = fm1 ; m2 ; m3 g and W = fw1 ; w2 ; w3 g. Let R 2 R be such that Rm1 Rm2 Rm3 Rw1 Rw2 Rw3
0 m1 Let = @
m2
w1
w2
w3
m2
m3
m1
w2
w3
w1
m3
m1
m2
m1
m2
m3
w1
w2
w3
w3
w1
w2
m1
m2
m3
1 0 m3 A m1 , =@ 0
m2
1 m3 A , and
00
0 m1 =@
m2
1 m3 A .
w1 w2 w3 w2 w3 w1 w3 w1 w2 Then C (R) = f g. It is a direct consequence of our main result (we will state 0
8
Theorem 2 in the next section) that V = f; ; 0
g is the unique stable set for R.
00
Because of the bilateral structure of one-to-one matching problems the essential blocking coalitions are man-woman pairs and individuals. Therefore, with any set of matchings we may associate the man-woman pairs which are matched by some element belonging to this set and the individuals who are unmatched under some element belonging to this set. Given V
M, let
T (V ) ffi; (i)g j i 2 N and 2 V g: The following is a simple and useful characterization of stable sets.
M. Then V
Theorem 1 Let R be a pro le and V if V = C
T
(V )
is a stable set for R if and only
(R).
Proof. (Only if) Let V be a stable set for R. By internal stability of V we have
C (V )(R). Suppose V 6 C (V )(R). Let ~ 2 C (V )(R)nV . But then by de nition of T (V ), there is no 2 V such that ~, which contradicts external stability of V
T
T
T
V. (If) Let V = C
T
(V )
(R). By de nition of
MnV . Then there is 2 V
T (V ), V
is internally stable. Let ~
such that ~ and V is externally stable.
2
Remark 1 By Theorem 1, given a pro le R, any stable set for R is a xed point of the mapping
C
T
()
(R ) : 2
M
V
! 7!
2
M
C
T
(V )
(R):
Theorem 1 suggests a tool for nding stable sets. Unfortunately, it is not clear which algorithm one may apply.7 Note that a similar result holds for stable sets of any cooperative game. However, then the corresponding mapping has to take into 7
The following is an obvious way to de ne an iterative procedure for nding stable sets. Let
R 2 R.
9
account both the blocking coalition and the attainable allocation through which this coalition is allowed to block.8 4
The Main Result
First, we show the following useful insight: if a matching is not dominated by any matching belonging to the core, then the set of unmatched agents is identical for this matching and any matching belonging to the core. Because any stable set contains the core and is internally stable, Proposition 1 implies that the set of unmatched agents is identical for any two matchings belonging to a stable set.
Proposition 1 Let R be a pro le and ~ 2 MnC (R). If for all 2 C (R), 6 ~, then the set of agents who are unmatched is identical for ~ and for all matchings belonging to C (R).
Proof. Since the set of unmatched agents is identical for any two matchings belonging to C (R), it suces to show U (~) = U (W ). First, suppose that there is m
2M
such that ~(m) = m and W (m) = 6 m. Let
W (m) = w. Since W is individually rational, we have W (m)Pm m. Thus, from W
6 m;w f
~ we obtain ~(w)Pw W (w). Hence, ~(w) = 6 w. Let ~(w) = m . If 0
g
W (m ) = m , then by W 0
0
6 m f
0
~ we must have ~(m )Pm m . But then (m ; w) 0
g
0
0
0
ALGORITHM
SET V0 M and STEP k: Let Vk
T0 T (V0 ). C T 1 (R). If V k
k
= Vk 1 , then STOP. Otherwise let
T T (V k
k + 1.
k
) and goto STEP
Note the following facts: (i) for any even k the set Vk is externally stable and for any odd k the set Vk is internally stable; (ii) for any even k we have Vk
V
k
V +2 and for any odd k we have k
V +2 ; (iii) when the ALGORITHM stops for some k, we have V k
k
= Vk
1
and Vk = C Tk 1 (R).
Thus, Vk = C T (Vk ) (R) and by Theorem 1, Vk is a stable set for R. Unfortunately, we are not able to show whether the ALGORITHM stops or not. 8 This was already noted by von Neumann and Morgenstern (1944).
10
blocks W , i.e. ~
2 C (R). Therefore, we must have W (m ) 6= m . Thus, by m Pw W (w), ~(w) = m , and W 2 C (R), we have W (m )Pm ~(m ). Let W (m ) = w . Then again by W 6 m ;w ~ we must have m0 ;wg
f
0
0
0
W , which is a contradiction to W
0
0
0
0
0
0
f
0
0
g
~(w )Pw W (w ). Continuing this way we nd an in nite sequence of men and women 0
0
0
which contradicts the niteness of M
[ W.
Hence, we have shown that if a man is
unmatched at ~, then he is also unmatched at all matchings belonging to C (R). Since the same argumentation is also valid for women, we obtain U (~) U (W ).
Second, suppose that there is m 2 M such that ~(m) 6= m and W (m) = m. Be-
cause W
have ~ 6
6 m f
m;wg
f
g
2 C (R) we Because W 6 w ~,
~ we must have ~(m)Pm m. Let ~(m) = w. Then by W
W . Thus, by ~(m)Pm m, we have W (w)Pw m.
f
g
w cannot be unmatched at W . Thus, W (w) = 6 w. Let W (w) = m . Again from 0
W
6 m ;w f
0
~ and W (w)Pw ~(w) we obtain ~(m )Pm W (m ). Thus, ~(m ) 6= m . Let 0
g
~(m ) = w . Then by W 0
0
0
0
0
0
2 C (R) and ~(m )Pm W (m ), we have W (w )Pw ~(w ). 0
0
0
0
0
0
Continuing this way we nd an in nite sequence of men and women which contradicts the niteness of M
[ W.
Hence, we have shown that if a man is unmatched
at all matchings belonging to C (R), then he is also unmatched at ~. Since the same argumentation is also valid for women, we obtain U (~) U (W ).
Next we show that if V is a stable set for R, then it is also a stable set for the pro le where all agents in the opposite set become acceptable (without changing any preferences between them) for any agent who is matched under the core and no agent is acceptable for all agents who are unmatched under the core. Therefore, the individual rationality constraint is irrelevant for the matched agents and when investigating stable sets we may constrain ourselves to one-to-one matching problems which contain the same number of men and women and any agent ranks all members belonging to the opposite set acceptable.
Proposition 2 Let R be a pro le, 2 C (R), and V
M. Let R be such that (i) for all i 2 U (), A(R i ) = ;, (ii) for all m 2 M nU (), A(R m ) = W and R m jW = Rm jW , 11
and (iii) for all w 2 W nU (), A(R w ) = M and R w jM = Rw jM . Then V is a stable set for R if and only if V is a stable set for R .
Proof. (Only if) Let V be a stable set for R. By Theorem 1, it suces to show V =C
T
(V )
(R ).
V . Thus, by internal stability of under R and Proposition 1, we have for all 2 V , U ( ) = U (). Note that the Since V is a stable set for R, we have C (R)
V
0
0
construction of R does not change any preferences of Ri between any partners for any
agent i 2 N nU (). Hence, by internal stability of V under R and U ( ) = U () for all 2 V , we have V C (V ) (R ). Suppose V 6 C (V ) (R ). Let 2 C (V ) (R )nV . 0
0
T
Then for all
0
T
2 V , 6 R . 0
T
Thus, by C (R) V and Proposition 1, U () = U ().
Hence, by construction of R , we have for all
0
2 V , 6 R , and V 0
is not externally
stable under R, a contradiction. (If) Let V be a stable set for R . By Theorem 1, it suces to show V = C (V ) (R). By the stability of V under R , C (R ) V . Let 2 C (R ). By construction of R , we T
have C (R) C (R ). Since the set of unmatched agents is identical for all matchings belonging to C (R ) and 2 C (R) C (R ), we have U () = U (). By internal stability
of V under R and Proposition 1, we have for all 2 V , U ( ) = U () = U (). Hence, by construction of R from R and internal stability of V under R , V C (V ) (R). Sup0
0
T
pose V
6 C
T
(V )
(R). Let ^
2C
T
(V )
(R)nV . Since C (R)
V , we then have for all
2 C (R), 6 R ^, and by Proposition 1, U (^) = U (). But then by construction of R from R, 6 R ^, and V is not externally stable under R , a contradiction.
0
0
0
Proposition 2 is a strategic equivalence result in the sense that any stable set for a pro le R is also a stable set for the pro le R where all core-unmatched agents rank all partners unacceptable and all core-matched agents rank all possible partners acceptable. This fact also implies that the core of R must be contained in any stable set for R.9 9
However, the core of R is not necessarily a stable set for R.
12
Our main result is the following characterization of stable sets.
Theorem 2 Let R be a pro le and V
M.
Then V is a stable set for R only if V
is a maximal set satisfying the following properties: (a) C (R) V . (b) V is a lattice. (c) The set of unmatched agents is identical for all matchings belonging to V . Furthermore, V is a stable set for R if V is the unique maximal set satisfying properties (a), (b), and (c).
Proof. (Only if) Let V be a stable set for R. First, we show that V satis es (a), (b), and (c). By external stability of V , we have C (R)
and V satis es (a).
ffi; (i)g j i 2 N and 2 V g. By Theorem 1, V = C (R). For all i 2 N , let T (i) = f(i) j 2 V gnfig. Let R 2 R be such that (i) for all m 2 M , R m jT (m) = Rm jT (m), and for all w 2 T (m) and all w 2 W nT (m), wPm mPm w , and (ii) for all w 2 W , R w jT (w) = Rw jT (w), and for all m 2 T (w) and all m 2 M nT (w), Let
T
V
=
T
0
0
0
mPw wPw m . 0
We show that from the construction of R it follows that C (R ) = C (R). Let 2 C (R ). If 2= C (R), then there exists some S 2 T and 2 M such that T
T
RS . Since T = T (V ), we may assume 2 V = C (R). Because is individually rational under R , is also individually rational under R. Thus, if RS , then T
S = fm; wg for some man-woman pair. Then, by the construction of R and 2 V , we also have RS which contradicts 2 C (R ). Hence, we have C (R ) C (R). T
2 C (R). If 2= C (R ), then there exists some ; 6= S N and 2 M such that RS . Since the matching problem In showing the reverse inclusion relation, let
T
is one-to-one, the essential blocking coalitions are only individuals and man-woman pairs. Thus, we may assume that S is a singleton or a man-woman pair. By the construction of R , there exists a matching ^ 2 V such that ^(S ) = (S ). From the 13
construction of R , then we also have ^ RS . This contradicts the internal stability of V because ; ^ 2 V . Hence, we have C (R ) C (R). T
We know that C (R ) is a lattice. Because the preferences restricted to C (R ) are identical under R and R and V = C (R ), we have that V is a lattice under R and V satis es (b). Furthermore, the set of unmatched agents is identical for all matchings belonging to C (R ). Since V = C (R ), V satis es (c). Second, we show that V is a maximal set satisfying (a), (b), and (c). Suppose not. Because V satis es (a), (b), and (c), then there exists a set V (a), (b), and (c) such that V
V
0
0
M satisfying
and V = 6 V . Let ~ 2 V nV . By external stability 0
0
of V there exists 2 V such that ~. Because V
V
0
and V satis es (c), the 0
~, we must have m;w ~ for some man-woman pair (m; w). Hence, (m) = w, ~(m) = 6 m, ~(w) =6 w, wPm ~(m), and mPw ~(w). Now when calculating _ ~ we obtain ( _ ~)(m) = w and ( _ ~)(w) = ~(w). By ~(w) 6= m, _ ~ is not a matching which is a contradiction to V M and V being a lattice. set of unmatched agents is identical for and ~. Thus, by f
g
0
0
(If) Let V be the unique maximal set satisfying (a), (b), and (c). We prove that V
is a stable set for R. First, we show that V is internally stable. Let ; ~ 2 V . By (c),
the set of unmatched agents is identical for and ~. Thus, if ~, then
m;wg
f
~
for some man-woman pair (m; w), i.e. (m) = w, wPm ~(m) and mPw ~(w). Then similarly as above it follows that _ ~ is not a matching, a contradiction to V being a lattice. Second, we show that V is externally stable. Suppose not. Then there is some
~ 2 MnV such that for all 2 V , 6 ~. By C (R) V , (c) and Proposition 1, the set of unmatched agents is identical for ~ and all matchings belonging to V .
T = ffi; (i)g j i 2 N and 2 C (R) [ f~gg. For all i 2 N , let T (i) = f(i) j 2 C (R) [ f~ggnfig. Let R 2 R be such that (i) for all m 2 M , RmjT (m) = Rm jT (m), and for all w 2 T (m) and all w 2 W nT (m), wPm mPm w , and (ii) for all w 2 W , R w jT (w) = Rw jT (w), and for all m 2 T (w) and all m 2 M nT (w), Let
0
0
0
14
mPw wPw m . By construction, C (R) [ f~g C (R ). We know that C (R ) is a lattice under R . Furthermore, for all 2 C (R ) and all i 2 N , (i) 2 T (i) [fig. Because the 0
preferences restricted to C (R ) are identical under R and R, it follows that C (R ) is a lattice for R. By construction, C (R ) C (R). Furthermore, the set of unmatched agents is identical for all matchings belonging to C (R ). Hence, C (R ) is a set of matchings satisfying (a), (b), and (c).
Because M is nite, there exists a maximal set V
0
C (R ) such that V
0
satis es
(a), (b), and (c). Then V is a maximal set satisfying (a), (b), and (c) and ~ 2 V nV , 0
0
which contradicts the fact that V is the unique maximal set satisfying (a), (b), and
(c). Hence, V must be externally stable.
Remark 2 It is straightforward to check that properties (a), (b), and (c) are mutually independent in Theorem 2. Let M = fm1 ; m2 g and W = fw1 ; w2 g.
(b) & (c)
6) (a):
Let R be a pro le and I denote the matching such that
2 N . Then fI g is a (maximal) set satisfying (b) and (c). Whenever C (R) 6= fI g, the set fI g violates (a). I (i) = i for all i
(a) & (c) 6) (b): Let R be the pro le such that w01 Pm1 w2 Pm1 1 m1 , w2 Pm2 w0 1 Pm2 m2 , m1 m2 A m1 m2 m1 Pw1 m2 Pw1 w1 , and m2 Pw2 m1 Pw2 w2 . Let = @ and = @ w1 w2 w2 w1 Then C (R) = fg and f; g is a (maximal) set satisfying (a) and (c). Since 0
0
_ is not a matching, the set f; g violates (b). 0
0
(a) & (b) 6) (c): Let R be the pro le such that m1 P0m1 w1 Pm1 w21 , m2 Pm2 w1 Pm2 w2 , m1 m2 A m1 Pw1 w1 Pw1 m2 , and w2 Pw2 m2 Pw2 m1 . Let = @ . Then C (R) = w1 m2 fI g and fI ; g is a (maximal) set satisfying (a) and (b) (where I is de ned 00
00
as above). Obviously, the set fI ; g violates (c).
An important consequence of Theorem 2 is that any stable set contains a matching which is both most preferred by the men and least preferred by the women in the 15
1 A.
stable set. This is due to the fact that by (b), any stable set is a lattice, i.e. the preferences of men and women are opposed for any two matchings belonging to a stable set.
Corollary 1 Let R be a pro le. Then any stable set V for R contains a matching which is both most preferred by the men and least preferred by the women in V , namely
_
V ,
2
and V contains a matching which is both least preferred by the men and most
preferred by the women in V , namely
^
V .
2
Note that if V is the unique maximal set satisfying (a), (b), and (c) of Theorem 2, then V is the unique stable set for R. An immediate corollary of our main result is the answer to the question when the core is the unique stable set for a one-to-one matching problem.
Corollary 2 Let R be a pro le. The core C (R) is the unique stable set for R if and only if C (R) is a maximal set satisfying (b) the set is a lattice and (c) the set of unmatched agents is identical for all matchings belonging to the set.
Proof. (Only if) If C (R) is a stable set for R, then by Theorem 2, C (R) is a maximal set satisfying (a), (b), and (c). Hence, C (R) is a maximal set satisfying (b) and (c). (If) If C (R) is a maximal set satisfying (b) and (c) of Theorem 2, then C (R) is the unique maximal set satisfying (a), (b), and (c) of Theorem 2. Hence, by Theorem
2, C (R) is the unique stable set for R.
The following example shows that for the stability of a set V it is not sucient for V to be a maximal set satisfying properties (a), (b), and (c) in Theorem 2. Furthermore, for the stability of a set V it is not necessary for V to be the unique maximal set satisfying properties (a), (b), and (c) in Theorem 2.
Example 2 Let M = fm1 ; m2 ; m3 ; m4 g and W = fw1 ; w2 ; w3 ; w4 g. Let R 16
2 R be
such that
Let
00
Rm1 Rm2 Rm3 Rm4 Rw1 Rw2 Rw3 Rw4
=
0 m1 =@
w2
w1
w1
w1
m1
m2
m3
m4
w3
w2
w3
w4
m4
m1
m1
m1
w1
w3
w2
w2
m2
m3
m2
m2
w4
w4
w4
w3
m3
m4
m4
m3
m1
m2
m3
m4
w1
w2
w3
w4
0 @ m1
m2 m3 m4
w1 w2 w3 w4
m2 m3
1 A,
=
0
0 @ m1
m2 m3 m4
w2 w1 w3 w4
1 A,
and
1 m4 A . Then C (R) = fg. Let V f; g and V f; g. 0
0
00
w3 w2 w1 w4 Note that is Pareto dominated for the men via both and and that no other 0
00
2 Mnf; ; g is dominated by , i.e. .10 Furthermore, m ;w and 6 . Hence, we have (i) V is a stable set for R because and for all 2 Mnf; ; g, and (ii) V is a maximal set satisfying properties (a), (b), and (c) in Theorem 2 but V is not a stable set for R because 6 and 6 . matching Pareto dominates for the men . Thus, any matching
000
000
0
f
0
000
2
00
1g
00
0
00
000
00
0
0
00
0
0
0
00
0
Remark 3 The if-part of Theorem 2 is one of very few results saying that if a set possesses certain properties, then it is a stable set or the core. Characterizations of the core as a solution for all problems have been obtained via properties relating di erent problems. For example, \consistency" plays the important role in the characterizations of the core of Sasaki and Toda (1992) for one-to-one matching problems and of Peleg (1986b) for cooperative games.11 In Theorem 2 all properties apply only to a single problem. 10
Since 000 6= 0 and 000 6= 00 , 000 cannot Pareto dominate for the men. Thus, by 000 6= , there
is some mi 2 M such that 000 (mi ) 6= wi and wi Pmi 000 (mi ). Then dominates 000 via fmi ; wi g. 11 In these contexts, Demange (1986) nds a certain \strong stability" condition of the core which
is sucient for the core to be non-manipulable by agents who evaluate any set of outcomes in terms of its most preferred element.
17
Remark 4 Some literature (see for instance Einy and Shitovitz (2003)) studies only stable sets which are individually rational. Then the de nition of stable sets needs to be adjusted by requiring external stability for individually rational matchings only. It can be checked that Theorem 2 remains unchanged if we restrict ourselves to individually rational matchings. 5
Many-To-One Matching Problems
It is a typical feature that results for one-to-one matching problems do not extend to many-to-one matching problems.12 We will show that this also applies to most of our results. Instead of introducing the formal many-to-one matching model we will use the reverse version of the ingenious trick by Gale and Sotomayor (1985) and only consider one-to-one matching problems and associate with it (if possible) a many-to-one matching problem with responsive preferences. For all our examples it suces to consider the possibility of merging two men, say m1 and m2 , to one agent. Given a one-to-one matching problem (M; W; R), we say that (M; W; R) corresponds
to a many-to-one matching problem where m1 and m2 are merged to fm1 ; m2 g if (i)
Rm1 jW = Rm2 jW and A(Rm1 ) = A(Rm2 ) (the preferences of m1 and m2 are identical) and (ii) for all w 2 W , m1 Pw m2 and there is no v 2 M [ fwg such that m1 Pw vPw m2
(each woman ranks m1 above m2 and the positions of m1 and m2 in the woman's
ranking are adjacent to each other). In the corresponding problem, fm1 ; m2 g can be matched with up to two women and their preference R
m1 ;m2 g
f
is responsive to
Rm1 over the sets containing fewer than or equal to two women, i.e. for all distinct w; w ; w 0
00
2 W,
fw; w gP m ;m fw; w g , w Pm w : 0
f
12
00
1
2g
0
1
00
This has been shown already for manipulation issues and that with substitutable preferences
the set of unmatched agents may change for matchings in the strong core (Martinez, Masso, Neme, and Oviedo, 2000).
18
It is easy to see that Theorem 1 remains true for many-to-one matching problems and that in general the following implications hold: (i) V is internally stable in the
one-to-one matching problem ) V is internally stable in the corresponding many-to-
one matching problem and (ii) V is externally stable in the corresponding many-to-
one matching problem ) V is externally stable in the one-to-one matching problem. However, the reverse directions of these statements are not true in general. There does not need to be any relationship between the stable sets of the one-to-one matching problem and its corresponding many-to-one matching problem, i.e. (i) V is a stable set in the one-to-one matching problem
6) V
is a stable set in the corresponding
many-to-one matching problem and (ii) V is a stable set in the corresponding manyto-one matching problem
6) V
is a stable set in the one-to-one matching problem.
Furthermore, in the corresponding many-to-one matching problem a stable set may not be a lattice and the set of unmatched agents may not be identical for all matchings belonging to a stable set. Thus, Proposition 1 and Theorem 2 do not carry over to many-to-one matching problems. The following example establishes these facts.
Example 3 Let M = fm1 ; m2 ; m3 g and W = fw1 ; w2 ; w3 ; w4 g. Let R 2 R be such that
Rm1 Rm2 Rm3 Rw1 Rw2 Rw3 Rw4
0 m1 Let = @
w1
w1
w2
m3
m1
m1
m1
w2
w2
w1
m1
m2
m2
m2
w3
w3
w3
m2
m3
m3
m3
w4
w4
w4
w1
w2
w3
w4
m1
m2
m
m2 m3
1 3 0 w4 A m1 and = @ 0
m2 m3
1 w4 A . Then C (R) = fg
w2 w3 w1 w4 w1 w3 w2 w4 and f; g is the unique maximal set satisfying properties (a), (b), and (c) of Theorem 0
2. Hence, f; g is the unique stable set for the one-to-one matching problem. 0
Now consider the corresponding many-to-one matching problem where we merge
m1 and m2 to one agent, denoted by fm1 ; m2 g (note that this is possible since Rm1 and 19
Rm2 agree over the set0 of women and each woman 1 ranks 0 m1 and m2 adjacent1 and in the m1 m2 m3 w3 A m1 m2 m3 w3 A , and ~ = , ~ = @ same order). Let ~ = @ w1 w4 w2 w3 w2 w4 w1 w3 00
0
0 @ m1
1 w1 A . Then in the corresponding many-to-one matching problem,
m2 m3
w3 w4 w2 w1 6 m1 ;m2 ;w2 ;w3 ~ (since m1 and m2 are merged to one agent and w2 is indi erent ff
g
g
6
between and ~) and
0
m1 ;m2 g;w1 ;w3 g
ff
~ (since w1 prefers m3 to m1 ). Thus, 6 ~
6 ~, which implies that f; g is not externally stable in the corresponding many-to-one matching problem. Let V = f; ; ~; ~ ; ~ g. Without loss of generality, let fw3 ; w4 gP m ;m w1 P m ;m w2 . It is easy to check that V is a stable set for R in the corresponding many-to-one matching problem if fw1 ; w4 gR m ;m fw2 ; w3 g (if fw2; w3gP m ;m fw1; w4g, then m ;m ;w ;w ~ and f; ; ~; ~ g is a stable set and
0
0
0
0
00
1
f
2g
f
1
2g
f
f
1
ff
2g
1
2g
2
3g
0
0
1
2g
00
for R).13 Hence, we have established the following facts: (i) In the one-to-one matching problem, f; g is a stable set for R and V is not 0
a stable set for R because V is not internally stable. (ii) In the corresponding many-to-one matching problem, V is a stable set for R and f; g is not a stable set for R because f; g is not externally stable. 0
0
(iii) In the corresponding many-to-one matching problem, V is a stable set for R. The set of unmatched agents is not identical for any two matchings belonging
to V since U () = fw4 g 6= fw3 g = U (~). Furthermore, all women strictly prefer being matched with any man to being unmatched. Since under ^ ~ all
To see this, let ^
2 MnV .
If ^(m3 )
2 fm3 ; w3 ; w4 g, then f
g ^. If ^(w2 ) = w2 , then 0 f 3 2 g ^. If two or more women are unmatched under ^, then by ^(w2 ) 6= w2 , two women out of fw1 ; w3 ; w4 g are unmatched. Since ^(m3 ) 2 fw1 ; w2 g, the merged agent fm1 ; m2 g is matched to at most one woman and by fw3 ; w4 gPf 1 2 g w1 , it follows that ^ is dominated by a matching in V (because 0 (fm1 ; m2 g) = fw1 ; w3 g, ~0 (fm1 ; m2 g) = fw1 ; w4 g, and ~00 (fm1 ; m2 g) = fw3 ; w4 g). Now it follows that (i) if ^(m3 ) = w2 , then ^ 2 f0 ; ~0 ; ~00 g V ; and (ii) if ^(m3 ) = w1 , then ^ 2 f; ~g V or ^(fm1 ; m2 g) = fw3 ; w4 g (which is not possible because we cannot have ^(w2 ) = w2 ). Hence, V is externally stable. It is straightforward that V is internally stable. 13
m ;w
m ;m
20
m3 ;w1
women choose their most preferred partner from and ~ and U () \ U (~) = ;,
under ^ ~ no woman can be unmatched, which is impossible because there are only three man. Hence, ^ ~ is not a matching and V is not a lattice. Thus, Proposition 1 and Theorem 2 do not carry over to many-to-one matching problems.
It is straightforward to check that the conclusions of Example 3 are independent
of which responsive extension we choose for fm1 ; m2 g.
Recall that a matching is dominated by another matching via a coalition only if all members of the coalition strictly prefer the other matching to the initial matching. The literature also refers to
as the strong dominance relation among matchings.
The weak dominance relation allows some members of the blocking coalition to be indi erent between the initial and the new matching. It is well known that results for many-to-one matching problems change when considering weak dominance instead of strong dominance. For one-to-one matching problems this distinction is irrelevant since agents' preferences are strict and the problem is one-to-one. Therefore, all results remain identical under either dominance relation. The same is true for most cooperative games like games with transferable utility or with non-transferable utility. Since for one-to-one matching problems it is irrelevant which dominance relation we use, one may wonder whether the conclusions of Example 3 remain true when considering weak dominance. To be more precise, we introduce the weak dominance relation. Let R be a pro le. Given two matchings ; and a coalition S N , we say 0
that weakly dominates via S (under R), denoted by wS , if (i) (S ) = S , (ii) 0
0
for all i 2 S , (i)Ri (i), and (iii) for some i 2 S , (i)Pi (i). We say that weakly 0
0
dominates (under R), denoted by w , if there exists S 0
0
N such that wS . 0
We say that a set V is a strongly stable set for R if it satis es conditions (i) and (ii) of
De nition 1 when is replaced by w . We will refer to (i) as internal strong stability and to (ii) as external strong stability. It is easy to see that in Example 3, the set f; g is a strongly stable set for R in the corresponding many-to-one matching 0
21
problem. Nevertheless, as the following example shows, there does not need to be any relationship between the strongly stable sets of the one-to-one matching problem and its associated many-to-one matching problem.
Example 4 Let M = fm1 ; m2 ; m3 ; m4 g and W = fw1 ; w2 ; w3 g. Let R 2 R be such that
0 m1 Let = @
Rm1 Rm2 Rm3 Rm4 Rw1 Rw2 Rw3 w1
w1
w1
w1
m1
m3
m3
w2
w2
w2
w2
m2
m4
m4
w3
w3
w3
w3
m3
m1
m1
m1
m2
m3
m4
m4
m2
m2
w1
w2
w3
m2 m3 m4
1 A. Then C (R) = fg and fg is the unique maximal
w1 m2 w2 w3 set satisfying properties (a), (b), and (c) of Theorem 2. Hence, C (R) is the unique strongly stable set for the one-to-one matching problem.
Now consider the corresponding many-to-one matching problem where we merge
m1 and m2 to one agent fm1 ; m2 g and m3 and m4 to one agent fm3 ; m4 g (note that this is possible since the men's preferences agree over the set of women and each woman ranks m1 and m2 adjacent and in the same order and m3 and m4 adjacent and in the same order). Let fw2 ; w3 gP
m1 ;m2 g w1
and w1 P
m3 ;m4 g
0 m fw2; w3g. Let = @ 1 0
m2 m3 m4
1 A.
w2 w3 w1 m4 Then in the corresponding many-to-one matching problem, 6 , which implies that f
f
0
fg is not externally strongly stable in the corresponding many-to-one matching problem. Let V = f; g. It is easy to check that V is a strongly stable set for R in 0
the corresponding many-to-one matching problem.14 Hence, we have established the 14
To see this, let ^
2 MnV .
If ^(w1 ) = w1 , then 0
ff
2 ^(fm1 ; m2 g), then either ^ = or ff 3 4 g 2 fw1 g = ^(fm3 ; m4 g), then either ^ = 0 or 0 ff 1 2 g 2
If w1
w
m ;m
;w ;w
w
m ;m
22
g g ^. Let ^(w1 ) 6= w1 . ^. Let w1 2 ^(fm3 ; m4 g). If 3g ^. If fw1 g ( ^(fm3 ; m4 g), then 3g w
;w ;w
m3 ;m4 ;w1
following facts: (i) In the one-to-one matching problem, fg is a strongly stable set for R and V is not a strongly stable set for R because V is not internally strongly stable. (ii) In the corresponding many-to-one matching problem, V is a strongly stable set for R and fg is not a strongly stable set for R because fg is not externally
strongly stable.15 (iii) In the corresponding many-to-one matching problem, V is a strongly stable set for R. The set of unmatched agents is not identical for any two matchings belonging to V since U () = fm2 g 6= fm4 g = U ( ). Thus, Proposition 1 0
and Theorem 2 do not carry over to many-to-one matching problems when considering the weak dominance relation w . 6
Conclusion
In general both the core and stable sets may not exist for cooperative games (Lucas (1969) and Einy and Shitovitz (1996) for stable sets). Here it is still an open question whether stable sets always exist for one-to-one matching problems and whether there is a unique stable set if they exist. Since the core of one-to-one matching problems is always non-empty, one may wonder why we should be interested in stable sets. Such a judgement would be based on properties of solution concepts, i.e. such reasoning is a posteriori after having de ned a solution concept. However, more importantly any
judgement of any solution concept should be a priori based on the economic meaning of its de nition. Any core matching is unblocked, i.e., no coalition has any incentive to ff w
g g ^. Hence, V is externally strongly stable. It is straightforward that V is internally
m1 ;m2 ;w1
0 m Note that V is not a stable set for R since for ~ = @
strongly stable. 15
1
w1
and 0 6 ~.
23
m2 m3 m4 w2 w3 m4
1 A we have both 6 ~
deviate from it. However, the core as a set does not possess any additional appealing property other than the stability of any single core matching. This is not true for a stable set since as a whole set it satis es internal stability and external stability. If we select a single matching from a stable set, then this matching is unlikely to be externally stable as a set (unless all agents unanimously agree which matching is most preferred among all matchings). Stable sets should be truly understood as a multi-valued solution concept. They are appealing for situations where agents agree to choose a set of possible outcomes and the nally chosen outcome is enforced.16 Then it may be questionable to rule out matchings which are not blocked by any possible enforceable outcome. Of course, on a practical level the success of a solution concept depends on its properties and its applicability. For one-to-one matching problems we found that the core and stable sets share a number of well-known properties. Our main result did not impose any restriction on the matching problem under consideration (other than it is one-to-one). Most results on stable sets in the literature apply to speci c classes of games with certain restrictions. References
Alkan, A. (2001): \On Preferences over Subsets and the Lattice Structure of Stable Matchings," Review of Economic Design 6:99{111. Alkan, A., and D. Gale (2003): \Stable Schedule Matching under Revealed Preference," Journal of Economic Theory 112:289{306. Aumann, R.J. (1987): \What is game theory trying to accomplish?," in Frontiers of Economics (K.J. Arrow and S. Honkapohja, Eds.), Blackwell, Oxford, 1987. 16
For example, in school choice agents may be prohibited from changing their partners chosen by
the possible outcome.
24
Biswas, A.K., T. Parthasarathy, and G. Ravindran (2001): \Stability and Largeness of the Core," Games and Economic Behavior 34:227{237. Blair, C. (1988): \The Lattice Structure of the Set of Stable Matchings with Multiple Partners," Mathematics of Operations Research 13:619{628. Demange, G. (1987): \Nonmanipulable Cores," Econometrica 55:1057{1074. Einy, E., and B. Shitovitz (1996): \Convex Games and Stable Sets," Games and Economic Behavior 16:192{201.
Einy, E., and B. Shitovitz (2003): \Symmetric von NeumannMorgenstern stable sets in pure exchange economies," Games and Economic Behavior 43:28{43. Einy, E., R. Holzman, D. Monderer, and B. Shitovitz (1996): \Core and Stable Sets of Large Games Arising in Economics," Journal of Economic Theory 16:192{ 201. Einy, E., R. Holzman, D. Monderer and B. Shitovitz (1997): \Core Equivalence Theorems for In nite Convex Games," Journal of Economic Theory 76:1{12. Gale, D., and L.S. Shapley (1962): \College Admissions and the Stability of Marriage," American Mathematical Monthly 69:9{15. Greenberg, J., X. Luo, R. Oladi, and B. Shitovitz (2002): \(Sophisticated) Stable Sets in Exchange Economies," Games and Economic Behavior 39:54{70. Kalai, E., A. Postlewaite, and J. Roberts (1979): \A Group Incentive Compatible Mechanism Yielding Core Allocations, Journal of Economic Theory 20:13{22. Klijn, F., and J. Masso (2003): \Weak stability and a bargaining set for the one-toone matching model," Games and Economic Behavior 42:91{100. Knuth, D., 1976, Mariages Stables, Montreal: Les Presses de l'Universite de Montreal. 25
Lucas, W.F. (1969): \The Proof That a Game May Not Have a Solution," Trans. Am. Math. Soc. 136:219{229.
Martinez, R., J. Masso, A. Neme, and J. Oviedo (2000): \Single Agents and the Set of Many-to-One Stable Matchings," Journal of Economic Theory 91:91{105. McVitie, D.G., and L.B. Wilson (1970): \Stable Marriage Assignments for Unequal Sets," BIT 10:295{309. Myerson R.B., 1991, Game Theory: Analysis of Con ict, Cambridge/London: Harvard University Press. Osborne, M.J., and A. Rubinstein, 1994, A Course in Game Theory, Cambridge/London: The MIT Press. Owen G., 1995, Game Theory, 3rd Edition, United States: Academic Press. Peleg, B. (1986a): \A Proof that the Core of an Ordinal Convex Game Is a von Neumann-Morgenstern Solution," Mathematical Social Sciences 11:83{87. Peleg, B. (1986b): \On the Reduced Game Property and its Converse," International Journal of Game Theory 15:187{200.
Roth, A.E. (1984): \The Evolution of the Labor Market for Medical Interns and Residents: a Case Study in Game Theory," Journal of Political Economy 92:991{ 1016. Roth, A.E., and M. Sotomayor, 1990, Two-Sided Matching: A Study in Game Theoretic Modeling and Analysis, London/New York: Cambridge University Press. Sasaki, H., and M. Toda (1992): \Consistency and Characterization of the Core of the Two-Sided Matching Problems," Journal of Economic Theory 56:218{227.
26
Shapley, L.S. (1971): \Cores of Convex Games," International Journal of Game Theory 1:11{26.
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27
Abstract The following properties of the core of a one-to-one matching problem are well-known: (i) the core is non-empty; (ii) the core is a lattice; and (iii) the set of unmatched agents is identical for any two matchings belonging to the core. The literature on two-sided matching focuses almost exclusively on the core and studies extensively its properties. Our main result is the following characterization of (von Neumann-Morgenstern) stable sets in one-to-one matching problems.
We show that a set of matchings is a stable set of a one-to-one
matching problem only if it is a maximal set satisfying the following properties: (a) the core is a subset of the set; (b) the set is a lattice; and (c) the set of unmatched agents is identical for any two matchings belonging to the set. Furthermore, a set is a stable set if it is the unique maximal set satisfying properties (a), (b), and (c). We also show that our main result does not extend from one-to-one matching problems to many-to-one matching problems.
JEL Classi cation: Keywords:
C78, J41, J44.
Matching Problem, Von Neumann-Morgenstern Stable Sets.
First version: November 2002. I thank Joseph Greenberg for drawing my attention to this question and Utku Unver for helpful comments. y Departement de Sciences Economiques and CIREQ, Universite de Montreal, Montreal, Quebec H3C 3J7, Canada; e-mail: [email protected]
1
1
Introduction
Von Neumann and Morgenstern (1944) introduced the notion of a stable set of a cooperative game.1 The idea behind a stable set is the following (Myerson, 1991; Osborne and Rubinstein, 1994): suppose the players consider a certain set of allocations of the cooperative game to be the possible outcomes (or proposals) of the game, without knowing which one will be ultimately chosen. Then any stable set of the game is a set of possible outcomes having the following properties: (i) for any allocation in the stable set there does not exist any coalition which prefers a certain other possible (attainable) outcome to this allocation, i.e. no coalition has a credible objection to any stable outcome; and (ii) for any allocation outside of the stable set there exists a coalition which prefers a certain other possible (attainable) outcome to this allocation, i.e. any unstable outcome is credibly objected by a coalition through a stable outcome. Conditions (i) and (ii) are robustness conditions of stable sets. (i) is referred to as internal stability of a set and (ii) as external stability of a set. The core of a cooperative game is always internally stable but it may violate external stability. Von Neumann and Morgenstern believed that stable sets should be the main solution concept for cooperative games in economic environments. Unfortunately, there is no general theory for stable sets. The theory has been prevented from being successful because it is very dicult working with it, which Aumann (1987) explains as follows: \Finding stable sets involves a new tour de force of mathematical reasoning for each game or class of games that is considered. Other than a small number of elementary truisms (e.g. that the core is contained in every stable set), there is no theory, no tools, certainly no algorithms." These facts helped the core to become the dominant multi-valued solution concept of cooperative games. The core of a game is extensively studied and well understood 1
Stable sets are called \solutions" in their book. We follow the convention of most of the recent
literature and refer to \solutions" as stable sets.
2
by the literature. This led a number of papers to identify classes of games where the core is the unique stable set of the game (e.g., Shapley (1971), Peleg (1986a), Einy, Holzman, Monderer, and Shitovitz (1997) and Biswas, Parthasarathy, and Ravindran (2001)).2 This paper is the rst study of stable sets in matching markets. In a matching market there are two disjoint sets of agents, usually called men and women or workers and rms, and we face the problem of matching agents from one side of the market with agents from the other side where each individual has the possibility of remaining unmatched. Matching problems arise in a number of important economic environments such as entry-level labor markets, college admissions, or school choice. The literature on two-sided matching problems focuses almost exclusively on the core.3 However, the core may violate external stability, i.e. there may be matchings outside the core which are not blocked (or objected) by a coalition through a core matching. Those matchings are only blocked through some \hypothetical matching" which does not belong to the core. Once such a matching proposed it is not clear why it will be replaced by an element in the core. We show that a sucient condition for this is that at the core matching, which is optimal for one side of the market, the agents of that side can gain by reallocating their partners. Here our purpose is not to investigate when the core is the unique stable set for a matching problem. However, the answer to this question will be a straightforward corollary of our main result. We nd that any stable set shares a number of wellknown and extensively studied properties of the core of a matching problem. Our main result shows that for one-to-one matching problems any stable set is a maximal set satisfying the following properties: (a) the core is a subset of the set; (b) the set is a lattice; and (c) the set of unmatched agents is identical for any two matchings belonging to the set. The converse also holds (i.e. a set is a stable set for a one-to-one 2 3
Note that the core of a cooperative game is always unique. One of the few exceptions is Klijn and Masso (2003) who apply the bargaining set of Zhou (1994)
to one-to-one matching problems.
3
matching problem) if a set is the unique maximal set satisfying the properties (a), (b), and (c). The literature on matching studied extensively when the core is a lattice and when the set of unmatched agents is identical. However, there is no such result saying that if a set possesses certain properties, then it coincides with the core. From our main result it is immediate that the core is the unique stable set if and only if it is a maximal set satisfying (b) the set is a lattice and (c) the set of unmatched agents is identical for all matchings belonging to the set. Furthermore, our main result facilitates considerably the search for stable sets in one-to-one matching problems: we just need to look at maximal sets satisfying (a), (b), and (c) (and if the maximal set is unique, then it is a stable set). We also show that the main result does not extend to many-to-one matching problems. Two papers in the literature on stable sets contain some similar features as our paper. One is Einy, Holzman, Monderer, and Shitovitz (1996) who study (non-atomic) glove games with a continuum of agents. They show that the core is the unique stable set of any glove game where the mass of agents holding left hand and right hand gloves is identical. Glove games are a special case of assignment games where there are two disjoint sets of buyers and sellers and each buyer-seller pair obtains a certain surplus from exchanging the good owned by the seller. Note that their result requires a continuum of agents, an equal mass of sellers and traders, and each seller's good has the same value for all buyers. Our main result does not impose any restriction on the one-to-one matching problem under consideration. The other paper is Einy and Shitovitz (2003) who study neoclassical pure exchange economies with a nite set of agents or with a continuum of agents. They show that the set of symmetric and Pareto-optimal allocations is the unique symmetric stable set. Their result holds in the continuum case without any restriction and in the nite case with the restriction that any endowment is owned by an identical number of agents and the agents owning the same endowment have identical preferences. The spirit of their result is similar as ours in the sense of determining properties of stable sets and showing that any set 4
satisfying these properties is a stable set. Note, however, that they focus on symmetric stable sets only and for the nite case the result only holds if any endowment is owned by an identical number of agents who have identical preferences.4 Symmetry is not meaningful in matching problems because no pair of agents is identical. The paper is organized as follows. Section 2 introduces one-to-one matching problems. Section 3 de nes stable sets and states some helpful insights. Section 4 contains the main result for one-to-one matching problems. It characterizes stable sets in terms of well-known properties of the core. Section 5 shows that this characterization does not extend to many-to-one matching problems. Section 6 concludes. 2
One-To-One Matching Problems
A one-to-one matching problem is a triple (M; W; R) where M is a nite set of men,
W is a nite set of women, and R is a preference pro le specifying for each man m
2M
a strict preference relation Rm over W
a strict preference relation Rw over M
[ fwg.
[ fmg and for each woman w 2 W
Then vRi v means that v is weakly 0
preferred to v under Ri , and vPi v means v is strictly preferred to v under Ri . 0
0
0
Strictness of a preference relation Ri means that vRi v implies v = v or vPi v . We 0
0
0
will keep M and W xed and thus, a matching problem is completely described by
R denote the set of all pro les. We will call N = M [ W the set of agents. Given Rm and S W , let Rm jS denote the restriction of Rm to S . Furthermore,
R. Let
let A(Rm ) denote the set of women who are acceptable for man m under Rm , i.e.
A(Rm ) = fw 2 W jwPm mg. Similarly we de ne Rw jS (where S M ) and A(Rw ). A matching is a function : N
all m 2 M , (m) 2 W
i 2 N , ( (i)) = i.
!N
satisfying the following properties: (i) for
[ fmg; (ii) for all w 2 W , (w) 2 M [ fwg; and (iii) for all Let M denote the set of all matchings. We say that an agent
i is unmatched at matching if (i) = i. Given the matching , let U () denote 4
This assumption is similar to the one of Einy, Holzman, Monderer, and Shitovitz (1996) that an
equal mass of agents holds left hand and right hand gloves.
5
the set of agents who are unmatched at . Given a pro le R, a matching is called
individually rational if for all i 2 N , (i)Ri i; and is called Pareto-optimal if there is
no matching = 6 such that (i)Ri(i) for all i 2 N with strict preference holding 0
0
N , we say that is Pareto-optimal 6 such that (i)Ri(i) for all i 2 S with strict =
for at least one agent. Given a coalition S for S if there is no matching
0
0
preference holding for at least one agent in S . Furthermore we say that matching is attainable for coalition S if (S ) = S (where (S ) = f(i)ji 2 S g).
Let R be a pro le. Given two matchings ; and a coalition S N , we say that 0
dominates via S under R, denoted by RS , if (i) (S ) = S and (ii) for all 0
0
i 2 S , (i)Pi (i). We say that dominates under R, denoted by R , if there 0
0
0
N such that RS . We will omit the superscript when the pro le R is unambiguous and write S and .
exists S
0
The core of a matching problem contains all matchings which are not blocked by some coalition through a matching attainable for that coalition. Given a pro le R, let C (R) denote the core of R,5 i.e.
C (R) = f 2 Mj for all ; = 6 S N and all
0
2 M we have 6 RS g: 0
The core of a matching problem is always non-empty (Gale and Shapley, 1962). Furthermore, the set of unmatched agents is identical for all matchings belonging to the core (McVitie and Wilson, 1970; Roth, 1984). We also consider the core where blocking is only allowed by a certain set of coalitions (instead of all coalitions). Given a set of coalitions
T,
let C (R) denote the T
T -core of R (Kalai,
Postlewaite, and
Roberts, 1979), i.e.
C (R) = f 2 Mj for all S 2 T and all T
0
2 M we have 6 RS g: 0
It is well-known that the core of a matching problem is a complete lattice (Knuth 5
The core of a one-to-one matching problem is often referred to as the set of stable matchings.
In avoiding any confusion with stable sets we will not use this terminology.
6
(1976) attributes this result to John Conway).6 Therefore, the core contains two matchings, called the M -optimal matching and W -optimal matching (the two extremes of the lattice), such that the M -optimal matching is the matching which is both most preferred by the men and least preferred by the women in the core (similar for the W -optimal matching). Given a pro le R, let M (R) denote the M -optimal matching and W (R) the W optimal matching in C (R). We will suppress the pro le R and instead write M and
2 M, let _ denote the mapping _ : N ! N such that (i) for all m 2 M , ( _ )(m) = (m) if (m)Rm (m), and otherwise ( _ )(m) = (m), and (ii) for all w 2 W , ( _ )(m) = (w) if (w)Rw (w), and otherwise ( _ )(w) = (w). Note that _ does not need to be a matching. Similarly we de ne ^ . Given a pro le R and V M, we say that V is a lattice (under R) if for all ; 2 V we have _ 2 V and ^ 2 V .
W . Given two matchings ;
0
0
0
0
0
0
0
0
0
0
0
0
0
0
3
0
0
Stable Sets
A set of matchings is a stable set for a matching problem if it satis es the following two robustness conditions: (i) no matching belonging to the set is blocked by a coalition through a matching which both belongs to the set and is attainable for this coalition; and (ii) any matching outside of the set is blocked by a coalition through a matching which both belongs to the set and is attainable for this coalition.
De nition 1 Let R
2 R and V M.
Then V is called a stable set for R if the
following two properties hold: (i) (Internal stability) For all ;
0
(ii) (External stability) For all
0
6
2 V , 6 . 0
2 MnV
there exists 2 V such that . 0
Many papers study the lattice structure of the strong core and the set of stable matchings in
matching problems; see for example, Blair (1988), Alkan (2001), and Alkan and Gale (2003).
7
Since the core consists of all undominated matchings, the core is always contained in any stable set. However, the core is not necessarily a stable set. A sucient condition for the core not to be a stable set is that at the M -optimal matching the men can gain by reallocating their partners (and thus, the M -optimal matching is not Pareto-optimal for the men) or at the W -optimal matching the women can gain by reallocating their partners (and thus, the W -optimal matching is not Pareto-optimal for the women). To see this, let R be a pro le such that at M the men can gain by reallocating their partners. Then there is a matching such that (M ) \ W =
M (M ) \ W and for all m 2 M , (m)Rm M (m), with strict preference holding for at least one man. Obviously, by the individual rationality of M , this implies that any man who is matched to a woman at M must be also matched to a woman at , i.e.
(m) 2 W for all m 2 M such that M (m) 6= m. Therefore, by the assumption that the matched men reallocate their partners ((M ) \ W = M (M ) \ W ), the set of unmatched agents is identical for both and M . Since M is the matching which is most preferred by all men in C (R) and the set of unmatched agents is the same at any two matchings belonging to the core (noting U () = U (M )), there is no matching
0
2 C (R) such that . Hence, C (R) is not externally stable. 0
Furthermore, stable sets are not necessarily individually rational (the same is true for other cooperative games).
Example 1 Let M = fm1 ; m2 ; m3 g and W = fw1 ; w2 ; w3 g. Let R 2 R be such that Rm1 Rm2 Rm3 Rw1 Rw2 Rw3
0 m1 Let = @
m2
w1
w2
w3
m2
m3
m1
w2
w3
w1
m3
m1
m2
m1
m2
m3
w1
w2
w3
w3
w1
w2
m1
m2
m3
1 0 m3 A m1 , =@ 0
m2
1 m3 A , and
00
0 m1 =@
m2
1 m3 A .
w1 w2 w3 w2 w3 w1 w3 w1 w2 Then C (R) = f g. It is a direct consequence of our main result (we will state 0
8
Theorem 2 in the next section) that V = f; ; 0
g is the unique stable set for R.
00
Because of the bilateral structure of one-to-one matching problems the essential blocking coalitions are man-woman pairs and individuals. Therefore, with any set of matchings we may associate the man-woman pairs which are matched by some element belonging to this set and the individuals who are unmatched under some element belonging to this set. Given V
M, let
T (V ) ffi; (i)g j i 2 N and 2 V g: The following is a simple and useful characterization of stable sets.
M. Then V
Theorem 1 Let R be a pro le and V if V = C
T
(V )
is a stable set for R if and only
(R).
Proof. (Only if) Let V be a stable set for R. By internal stability of V we have
C (V )(R). Suppose V 6 C (V )(R). Let ~ 2 C (V )(R)nV . But then by de nition of T (V ), there is no 2 V such that ~, which contradicts external stability of V
T
T
T
V. (If) Let V = C
T
(V )
(R). By de nition of
MnV . Then there is 2 V
T (V ), V
is internally stable. Let ~
such that ~ and V is externally stable.
2
Remark 1 By Theorem 1, given a pro le R, any stable set for R is a xed point of the mapping
C
T
()
(R ) : 2
M
V
! 7!
2
M
C
T
(V )
(R):
Theorem 1 suggests a tool for nding stable sets. Unfortunately, it is not clear which algorithm one may apply.7 Note that a similar result holds for stable sets of any cooperative game. However, then the corresponding mapping has to take into 7
The following is an obvious way to de ne an iterative procedure for nding stable sets. Let
R 2 R.
9
account both the blocking coalition and the attainable allocation through which this coalition is allowed to block.8 4
The Main Result
First, we show the following useful insight: if a matching is not dominated by any matching belonging to the core, then the set of unmatched agents is identical for this matching and any matching belonging to the core. Because any stable set contains the core and is internally stable, Proposition 1 implies that the set of unmatched agents is identical for any two matchings belonging to a stable set.
Proposition 1 Let R be a pro le and ~ 2 MnC (R). If for all 2 C (R), 6 ~, then the set of agents who are unmatched is identical for ~ and for all matchings belonging to C (R).
Proof. Since the set of unmatched agents is identical for any two matchings belonging to C (R), it suces to show U (~) = U (W ). First, suppose that there is m
2M
such that ~(m) = m and W (m) = 6 m. Let
W (m) = w. Since W is individually rational, we have W (m)Pm m. Thus, from W
6 m;w f
~ we obtain ~(w)Pw W (w). Hence, ~(w) = 6 w. Let ~(w) = m . If 0
g
W (m ) = m , then by W 0
0
6 m f
0
~ we must have ~(m )Pm m . But then (m ; w) 0
g
0
0
0
ALGORITHM
SET V0 M and STEP k: Let Vk
T0 T (V0 ). C T 1 (R). If V k
k
= Vk 1 , then STOP. Otherwise let
T T (V k
k + 1.
k
) and goto STEP
Note the following facts: (i) for any even k the set Vk is externally stable and for any odd k the set Vk is internally stable; (ii) for any even k we have Vk
V
k
V +2 and for any odd k we have k
V +2 ; (iii) when the ALGORITHM stops for some k, we have V k
k
= Vk
1
and Vk = C Tk 1 (R).
Thus, Vk = C T (Vk ) (R) and by Theorem 1, Vk is a stable set for R. Unfortunately, we are not able to show whether the ALGORITHM stops or not. 8 This was already noted by von Neumann and Morgenstern (1944).
10
blocks W , i.e. ~
2 C (R). Therefore, we must have W (m ) 6= m . Thus, by m Pw W (w), ~(w) = m , and W 2 C (R), we have W (m )Pm ~(m ). Let W (m ) = w . Then again by W 6 m ;w ~ we must have m0 ;wg
f
0
0
0
W , which is a contradiction to W
0
0
0
0
0
0
f
0
0
g
~(w )Pw W (w ). Continuing this way we nd an in nite sequence of men and women 0
0
0
which contradicts the niteness of M
[ W.
Hence, we have shown that if a man is
unmatched at ~, then he is also unmatched at all matchings belonging to C (R). Since the same argumentation is also valid for women, we obtain U (~) U (W ).
Second, suppose that there is m 2 M such that ~(m) 6= m and W (m) = m. Be-
cause W
have ~ 6
6 m f
m;wg
f
g
2 C (R) we Because W 6 w ~,
~ we must have ~(m)Pm m. Let ~(m) = w. Then by W
W . Thus, by ~(m)Pm m, we have W (w)Pw m.
f
g
w cannot be unmatched at W . Thus, W (w) = 6 w. Let W (w) = m . Again from 0
W
6 m ;w f
0
~ and W (w)Pw ~(w) we obtain ~(m )Pm W (m ). Thus, ~(m ) 6= m . Let 0
g
~(m ) = w . Then by W 0
0
0
0
0
0
2 C (R) and ~(m )Pm W (m ), we have W (w )Pw ~(w ). 0
0
0
0
0
0
Continuing this way we nd an in nite sequence of men and women which contradicts the niteness of M
[ W.
Hence, we have shown that if a man is unmatched
at all matchings belonging to C (R), then he is also unmatched at ~. Since the same argumentation is also valid for women, we obtain U (~) U (W ).
Next we show that if V is a stable set for R, then it is also a stable set for the pro le where all agents in the opposite set become acceptable (without changing any preferences between them) for any agent who is matched under the core and no agent is acceptable for all agents who are unmatched under the core. Therefore, the individual rationality constraint is irrelevant for the matched agents and when investigating stable sets we may constrain ourselves to one-to-one matching problems which contain the same number of men and women and any agent ranks all members belonging to the opposite set acceptable.
Proposition 2 Let R be a pro le, 2 C (R), and V
M. Let R be such that (i) for all i 2 U (), A(R i ) = ;, (ii) for all m 2 M nU (), A(R m ) = W and R m jW = Rm jW , 11
and (iii) for all w 2 W nU (), A(R w ) = M and R w jM = Rw jM . Then V is a stable set for R if and only if V is a stable set for R .
Proof. (Only if) Let V be a stable set for R. By Theorem 1, it suces to show V =C
T
(V )
(R ).
V . Thus, by internal stability of under R and Proposition 1, we have for all 2 V , U ( ) = U (). Note that the Since V is a stable set for R, we have C (R)
V
0
0
construction of R does not change any preferences of Ri between any partners for any
agent i 2 N nU (). Hence, by internal stability of V under R and U ( ) = U () for all 2 V , we have V C (V ) (R ). Suppose V 6 C (V ) (R ). Let 2 C (V ) (R )nV . 0
0
T
Then for all
0
T
2 V , 6 R . 0
T
Thus, by C (R) V and Proposition 1, U () = U ().
Hence, by construction of R , we have for all
0
2 V , 6 R , and V 0
is not externally
stable under R, a contradiction. (If) Let V be a stable set for R . By Theorem 1, it suces to show V = C (V ) (R). By the stability of V under R , C (R ) V . Let 2 C (R ). By construction of R , we T
have C (R) C (R ). Since the set of unmatched agents is identical for all matchings belonging to C (R ) and 2 C (R) C (R ), we have U () = U (). By internal stability
of V under R and Proposition 1, we have for all 2 V , U ( ) = U () = U (). Hence, by construction of R from R and internal stability of V under R , V C (V ) (R). Sup0
0
T
pose V
6 C
T
(V )
(R). Let ^
2C
T
(V )
(R)nV . Since C (R)
V , we then have for all
2 C (R), 6 R ^, and by Proposition 1, U (^) = U (). But then by construction of R from R, 6 R ^, and V is not externally stable under R , a contradiction.
0
0
0
Proposition 2 is a strategic equivalence result in the sense that any stable set for a pro le R is also a stable set for the pro le R where all core-unmatched agents rank all partners unacceptable and all core-matched agents rank all possible partners acceptable. This fact also implies that the core of R must be contained in any stable set for R.9 9
However, the core of R is not necessarily a stable set for R.
12
Our main result is the following characterization of stable sets.
Theorem 2 Let R be a pro le and V
M.
Then V is a stable set for R only if V
is a maximal set satisfying the following properties: (a) C (R) V . (b) V is a lattice. (c) The set of unmatched agents is identical for all matchings belonging to V . Furthermore, V is a stable set for R if V is the unique maximal set satisfying properties (a), (b), and (c).
Proof. (Only if) Let V be a stable set for R. First, we show that V satis es (a), (b), and (c). By external stability of V , we have C (R)
and V satis es (a).
ffi; (i)g j i 2 N and 2 V g. By Theorem 1, V = C (R). For all i 2 N , let T (i) = f(i) j 2 V gnfig. Let R 2 R be such that (i) for all m 2 M , R m jT (m) = Rm jT (m), and for all w 2 T (m) and all w 2 W nT (m), wPm mPm w , and (ii) for all w 2 W , R w jT (w) = Rw jT (w), and for all m 2 T (w) and all m 2 M nT (w), Let
T
V
=
T
0
0
0
mPw wPw m . 0
We show that from the construction of R it follows that C (R ) = C (R). Let 2 C (R ). If 2= C (R), then there exists some S 2 T and 2 M such that T
T
RS . Since T = T (V ), we may assume 2 V = C (R). Because is individually rational under R , is also individually rational under R. Thus, if RS , then T
S = fm; wg for some man-woman pair. Then, by the construction of R and 2 V , we also have RS which contradicts 2 C (R ). Hence, we have C (R ) C (R). T
2 C (R). If 2= C (R ), then there exists some ; 6= S N and 2 M such that RS . Since the matching problem In showing the reverse inclusion relation, let
T
is one-to-one, the essential blocking coalitions are only individuals and man-woman pairs. Thus, we may assume that S is a singleton or a man-woman pair. By the construction of R , there exists a matching ^ 2 V such that ^(S ) = (S ). From the 13
construction of R , then we also have ^ RS . This contradicts the internal stability of V because ; ^ 2 V . Hence, we have C (R ) C (R). T
We know that C (R ) is a lattice. Because the preferences restricted to C (R ) are identical under R and R and V = C (R ), we have that V is a lattice under R and V satis es (b). Furthermore, the set of unmatched agents is identical for all matchings belonging to C (R ). Since V = C (R ), V satis es (c). Second, we show that V is a maximal set satisfying (a), (b), and (c). Suppose not. Because V satis es (a), (b), and (c), then there exists a set V (a), (b), and (c) such that V
V
0
0
M satisfying
and V = 6 V . Let ~ 2 V nV . By external stability 0
0
of V there exists 2 V such that ~. Because V
V
0
and V satis es (c), the 0
~, we must have m;w ~ for some man-woman pair (m; w). Hence, (m) = w, ~(m) = 6 m, ~(w) =6 w, wPm ~(m), and mPw ~(w). Now when calculating _ ~ we obtain ( _ ~)(m) = w and ( _ ~)(w) = ~(w). By ~(w) 6= m, _ ~ is not a matching which is a contradiction to V M and V being a lattice. set of unmatched agents is identical for and ~. Thus, by f
g
0
0
(If) Let V be the unique maximal set satisfying (a), (b), and (c). We prove that V
is a stable set for R. First, we show that V is internally stable. Let ; ~ 2 V . By (c),
the set of unmatched agents is identical for and ~. Thus, if ~, then
m;wg
f
~
for some man-woman pair (m; w), i.e. (m) = w, wPm ~(m) and mPw ~(w). Then similarly as above it follows that _ ~ is not a matching, a contradiction to V being a lattice. Second, we show that V is externally stable. Suppose not. Then there is some
~ 2 MnV such that for all 2 V , 6 ~. By C (R) V , (c) and Proposition 1, the set of unmatched agents is identical for ~ and all matchings belonging to V .
T = ffi; (i)g j i 2 N and 2 C (R) [ f~gg. For all i 2 N , let T (i) = f(i) j 2 C (R) [ f~ggnfig. Let R 2 R be such that (i) for all m 2 M , RmjT (m) = Rm jT (m), and for all w 2 T (m) and all w 2 W nT (m), wPm mPm w , and (ii) for all w 2 W , R w jT (w) = Rw jT (w), and for all m 2 T (w) and all m 2 M nT (w), Let
0
0
0
14
mPw wPw m . By construction, C (R) [ f~g C (R ). We know that C (R ) is a lattice under R . Furthermore, for all 2 C (R ) and all i 2 N , (i) 2 T (i) [fig. Because the 0
preferences restricted to C (R ) are identical under R and R, it follows that C (R ) is a lattice for R. By construction, C (R ) C (R). Furthermore, the set of unmatched agents is identical for all matchings belonging to C (R ). Hence, C (R ) is a set of matchings satisfying (a), (b), and (c).
Because M is nite, there exists a maximal set V
0
C (R ) such that V
0
satis es
(a), (b), and (c). Then V is a maximal set satisfying (a), (b), and (c) and ~ 2 V nV , 0
0
which contradicts the fact that V is the unique maximal set satisfying (a), (b), and
(c). Hence, V must be externally stable.
Remark 2 It is straightforward to check that properties (a), (b), and (c) are mutually independent in Theorem 2. Let M = fm1 ; m2 g and W = fw1 ; w2 g.
(b) & (c)
6) (a):
Let R be a pro le and I denote the matching such that
2 N . Then fI g is a (maximal) set satisfying (b) and (c). Whenever C (R) 6= fI g, the set fI g violates (a). I (i) = i for all i
(a) & (c) 6) (b): Let R be the pro le such that w01 Pm1 w2 Pm1 1 m1 , w2 Pm2 w0 1 Pm2 m2 , m1 m2 A m1 m2 m1 Pw1 m2 Pw1 w1 , and m2 Pw2 m1 Pw2 w2 . Let = @ and = @ w1 w2 w2 w1 Then C (R) = fg and f; g is a (maximal) set satisfying (a) and (c). Since 0
0
_ is not a matching, the set f; g violates (b). 0
0
(a) & (b) 6) (c): Let R be the pro le such that m1 P0m1 w1 Pm1 w21 , m2 Pm2 w1 Pm2 w2 , m1 m2 A m1 Pw1 w1 Pw1 m2 , and w2 Pw2 m2 Pw2 m1 . Let = @ . Then C (R) = w1 m2 fI g and fI ; g is a (maximal) set satisfying (a) and (b) (where I is de ned 00
00
as above). Obviously, the set fI ; g violates (c).
An important consequence of Theorem 2 is that any stable set contains a matching which is both most preferred by the men and least preferred by the women in the 15
1 A.
stable set. This is due to the fact that by (b), any stable set is a lattice, i.e. the preferences of men and women are opposed for any two matchings belonging to a stable set.
Corollary 1 Let R be a pro le. Then any stable set V for R contains a matching which is both most preferred by the men and least preferred by the women in V , namely
_
V ,
2
and V contains a matching which is both least preferred by the men and most
preferred by the women in V , namely
^
V .
2
Note that if V is the unique maximal set satisfying (a), (b), and (c) of Theorem 2, then V is the unique stable set for R. An immediate corollary of our main result is the answer to the question when the core is the unique stable set for a one-to-one matching problem.
Corollary 2 Let R be a pro le. The core C (R) is the unique stable set for R if and only if C (R) is a maximal set satisfying (b) the set is a lattice and (c) the set of unmatched agents is identical for all matchings belonging to the set.
Proof. (Only if) If C (R) is a stable set for R, then by Theorem 2, C (R) is a maximal set satisfying (a), (b), and (c). Hence, C (R) is a maximal set satisfying (b) and (c). (If) If C (R) is a maximal set satisfying (b) and (c) of Theorem 2, then C (R) is the unique maximal set satisfying (a), (b), and (c) of Theorem 2. Hence, by Theorem
2, C (R) is the unique stable set for R.
The following example shows that for the stability of a set V it is not sucient for V to be a maximal set satisfying properties (a), (b), and (c) in Theorem 2. Furthermore, for the stability of a set V it is not necessary for V to be the unique maximal set satisfying properties (a), (b), and (c) in Theorem 2.
Example 2 Let M = fm1 ; m2 ; m3 ; m4 g and W = fw1 ; w2 ; w3 ; w4 g. Let R 16
2 R be
such that
Let
00
Rm1 Rm2 Rm3 Rm4 Rw1 Rw2 Rw3 Rw4
=
0 m1 =@
w2
w1
w1
w1
m1
m2
m3
m4
w3
w2
w3
w4
m4
m1
m1
m1
w1
w3
w2
w2
m2
m3
m2
m2
w4
w4
w4
w3
m3
m4
m4
m3
m1
m2
m3
m4
w1
w2
w3
w4
0 @ m1
m2 m3 m4
w1 w2 w3 w4
m2 m3
1 A,
=
0
0 @ m1
m2 m3 m4
w2 w1 w3 w4
1 A,
and
1 m4 A . Then C (R) = fg. Let V f; g and V f; g. 0
0
00
w3 w2 w1 w4 Note that is Pareto dominated for the men via both and and that no other 0
00
2 Mnf; ; g is dominated by , i.e. .10 Furthermore, m ;w and 6 . Hence, we have (i) V is a stable set for R because and for all 2 Mnf; ; g, and (ii) V is a maximal set satisfying properties (a), (b), and (c) in Theorem 2 but V is not a stable set for R because 6 and 6 . matching Pareto dominates for the men . Thus, any matching
000
000
0
f
0
000
2
00
1g
00
0
00
000
00
0
0
00
0
0
0
00
0
Remark 3 The if-part of Theorem 2 is one of very few results saying that if a set possesses certain properties, then it is a stable set or the core. Characterizations of the core as a solution for all problems have been obtained via properties relating di erent problems. For example, \consistency" plays the important role in the characterizations of the core of Sasaki and Toda (1992) for one-to-one matching problems and of Peleg (1986b) for cooperative games.11 In Theorem 2 all properties apply only to a single problem. 10
Since 000 6= 0 and 000 6= 00 , 000 cannot Pareto dominate for the men. Thus, by 000 6= , there
is some mi 2 M such that 000 (mi ) 6= wi and wi Pmi 000 (mi ). Then dominates 000 via fmi ; wi g. 11 In these contexts, Demange (1986) nds a certain \strong stability" condition of the core which
is sucient for the core to be non-manipulable by agents who evaluate any set of outcomes in terms of its most preferred element.
17
Remark 4 Some literature (see for instance Einy and Shitovitz (2003)) studies only stable sets which are individually rational. Then the de nition of stable sets needs to be adjusted by requiring external stability for individually rational matchings only. It can be checked that Theorem 2 remains unchanged if we restrict ourselves to individually rational matchings. 5
Many-To-One Matching Problems
It is a typical feature that results for one-to-one matching problems do not extend to many-to-one matching problems.12 We will show that this also applies to most of our results. Instead of introducing the formal many-to-one matching model we will use the reverse version of the ingenious trick by Gale and Sotomayor (1985) and only consider one-to-one matching problems and associate with it (if possible) a many-to-one matching problem with responsive preferences. For all our examples it suces to consider the possibility of merging two men, say m1 and m2 , to one agent. Given a one-to-one matching problem (M; W; R), we say that (M; W; R) corresponds
to a many-to-one matching problem where m1 and m2 are merged to fm1 ; m2 g if (i)
Rm1 jW = Rm2 jW and A(Rm1 ) = A(Rm2 ) (the preferences of m1 and m2 are identical) and (ii) for all w 2 W , m1 Pw m2 and there is no v 2 M [ fwg such that m1 Pw vPw m2
(each woman ranks m1 above m2 and the positions of m1 and m2 in the woman's
ranking are adjacent to each other). In the corresponding problem, fm1 ; m2 g can be matched with up to two women and their preference R
m1 ;m2 g
f
is responsive to
Rm1 over the sets containing fewer than or equal to two women, i.e. for all distinct w; w ; w 0
00
2 W,
fw; w gP m ;m fw; w g , w Pm w : 0
f
12
00
1
2g
0
1
00
This has been shown already for manipulation issues and that with substitutable preferences
the set of unmatched agents may change for matchings in the strong core (Martinez, Masso, Neme, and Oviedo, 2000).
18
It is easy to see that Theorem 1 remains true for many-to-one matching problems and that in general the following implications hold: (i) V is internally stable in the
one-to-one matching problem ) V is internally stable in the corresponding many-to-
one matching problem and (ii) V is externally stable in the corresponding many-to-
one matching problem ) V is externally stable in the one-to-one matching problem. However, the reverse directions of these statements are not true in general. There does not need to be any relationship between the stable sets of the one-to-one matching problem and its corresponding many-to-one matching problem, i.e. (i) V is a stable set in the one-to-one matching problem
6) V
is a stable set in the corresponding
many-to-one matching problem and (ii) V is a stable set in the corresponding manyto-one matching problem
6) V
is a stable set in the one-to-one matching problem.
Furthermore, in the corresponding many-to-one matching problem a stable set may not be a lattice and the set of unmatched agents may not be identical for all matchings belonging to a stable set. Thus, Proposition 1 and Theorem 2 do not carry over to many-to-one matching problems. The following example establishes these facts.
Example 3 Let M = fm1 ; m2 ; m3 g and W = fw1 ; w2 ; w3 ; w4 g. Let R 2 R be such that
Rm1 Rm2 Rm3 Rw1 Rw2 Rw3 Rw4
0 m1 Let = @
w1
w1
w2
m3
m1
m1
m1
w2
w2
w1
m1
m2
m2
m2
w3
w3
w3
m2
m3
m3
m3
w4
w4
w4
w1
w2
w3
w4
m1
m2
m
m2 m3
1 3 0 w4 A m1 and = @ 0
m2 m3
1 w4 A . Then C (R) = fg
w2 w3 w1 w4 w1 w3 w2 w4 and f; g is the unique maximal set satisfying properties (a), (b), and (c) of Theorem 0
2. Hence, f; g is the unique stable set for the one-to-one matching problem. 0
Now consider the corresponding many-to-one matching problem where we merge
m1 and m2 to one agent, denoted by fm1 ; m2 g (note that this is possible since Rm1 and 19
Rm2 agree over the set0 of women and each woman 1 ranks 0 m1 and m2 adjacent1 and in the m1 m2 m3 w3 A m1 m2 m3 w3 A , and ~ = , ~ = @ same order). Let ~ = @ w1 w4 w2 w3 w2 w4 w1 w3 00
0
0 @ m1
1 w1 A . Then in the corresponding many-to-one matching problem,
m2 m3
w3 w4 w2 w1 6 m1 ;m2 ;w2 ;w3 ~ (since m1 and m2 are merged to one agent and w2 is indi erent ff
g
g
6
between and ~) and
0
m1 ;m2 g;w1 ;w3 g
ff
~ (since w1 prefers m3 to m1 ). Thus, 6 ~
6 ~, which implies that f; g is not externally stable in the corresponding many-to-one matching problem. Let V = f; ; ~; ~ ; ~ g. Without loss of generality, let fw3 ; w4 gP m ;m w1 P m ;m w2 . It is easy to check that V is a stable set for R in the corresponding many-to-one matching problem if fw1 ; w4 gR m ;m fw2 ; w3 g (if fw2; w3gP m ;m fw1; w4g, then m ;m ;w ;w ~ and f; ; ~; ~ g is a stable set and
0
0
0
0
00
1
f
2g
f
1
2g
f
f
1
ff
2g
1
2g
2
3g
0
0
1
2g
00
for R).13 Hence, we have established the following facts: (i) In the one-to-one matching problem, f; g is a stable set for R and V is not 0
a stable set for R because V is not internally stable. (ii) In the corresponding many-to-one matching problem, V is a stable set for R and f; g is not a stable set for R because f; g is not externally stable. 0
0
(iii) In the corresponding many-to-one matching problem, V is a stable set for R. The set of unmatched agents is not identical for any two matchings belonging
to V since U () = fw4 g 6= fw3 g = U (~). Furthermore, all women strictly prefer being matched with any man to being unmatched. Since under ^ ~ all
To see this, let ^
2 MnV .
If ^(m3 )
2 fm3 ; w3 ; w4 g, then f
g ^. If ^(w2 ) = w2 , then 0 f 3 2 g ^. If two or more women are unmatched under ^, then by ^(w2 ) 6= w2 , two women out of fw1 ; w3 ; w4 g are unmatched. Since ^(m3 ) 2 fw1 ; w2 g, the merged agent fm1 ; m2 g is matched to at most one woman and by fw3 ; w4 gPf 1 2 g w1 , it follows that ^ is dominated by a matching in V (because 0 (fm1 ; m2 g) = fw1 ; w3 g, ~0 (fm1 ; m2 g) = fw1 ; w4 g, and ~00 (fm1 ; m2 g) = fw3 ; w4 g). Now it follows that (i) if ^(m3 ) = w2 , then ^ 2 f0 ; ~0 ; ~00 g V ; and (ii) if ^(m3 ) = w1 , then ^ 2 f; ~g V or ^(fm1 ; m2 g) = fw3 ; w4 g (which is not possible because we cannot have ^(w2 ) = w2 ). Hence, V is externally stable. It is straightforward that V is internally stable. 13
m ;w
m ;m
20
m3 ;w1
women choose their most preferred partner from and ~ and U () \ U (~) = ;,
under ^ ~ no woman can be unmatched, which is impossible because there are only three man. Hence, ^ ~ is not a matching and V is not a lattice. Thus, Proposition 1 and Theorem 2 do not carry over to many-to-one matching problems.
It is straightforward to check that the conclusions of Example 3 are independent
of which responsive extension we choose for fm1 ; m2 g.
Recall that a matching is dominated by another matching via a coalition only if all members of the coalition strictly prefer the other matching to the initial matching. The literature also refers to
as the strong dominance relation among matchings.
The weak dominance relation allows some members of the blocking coalition to be indi erent between the initial and the new matching. It is well known that results for many-to-one matching problems change when considering weak dominance instead of strong dominance. For one-to-one matching problems this distinction is irrelevant since agents' preferences are strict and the problem is one-to-one. Therefore, all results remain identical under either dominance relation. The same is true for most cooperative games like games with transferable utility or with non-transferable utility. Since for one-to-one matching problems it is irrelevant which dominance relation we use, one may wonder whether the conclusions of Example 3 remain true when considering weak dominance. To be more precise, we introduce the weak dominance relation. Let R be a pro le. Given two matchings ; and a coalition S N , we say 0
that weakly dominates via S (under R), denoted by wS , if (i) (S ) = S , (ii) 0
0
for all i 2 S , (i)Ri (i), and (iii) for some i 2 S , (i)Pi (i). We say that weakly 0
0
dominates (under R), denoted by w , if there exists S 0
0
N such that wS . 0
We say that a set V is a strongly stable set for R if it satis es conditions (i) and (ii) of
De nition 1 when is replaced by w . We will refer to (i) as internal strong stability and to (ii) as external strong stability. It is easy to see that in Example 3, the set f; g is a strongly stable set for R in the corresponding many-to-one matching 0
21
problem. Nevertheless, as the following example shows, there does not need to be any relationship between the strongly stable sets of the one-to-one matching problem and its associated many-to-one matching problem.
Example 4 Let M = fm1 ; m2 ; m3 ; m4 g and W = fw1 ; w2 ; w3 g. Let R 2 R be such that
0 m1 Let = @
Rm1 Rm2 Rm3 Rm4 Rw1 Rw2 Rw3 w1
w1
w1
w1
m1
m3
m3
w2
w2
w2
w2
m2
m4
m4
w3
w3
w3
w3
m3
m1
m1
m1
m2
m3
m4
m4
m2
m2
w1
w2
w3
m2 m3 m4
1 A. Then C (R) = fg and fg is the unique maximal
w1 m2 w2 w3 set satisfying properties (a), (b), and (c) of Theorem 2. Hence, C (R) is the unique strongly stable set for the one-to-one matching problem.
Now consider the corresponding many-to-one matching problem where we merge
m1 and m2 to one agent fm1 ; m2 g and m3 and m4 to one agent fm3 ; m4 g (note that this is possible since the men's preferences agree over the set of women and each woman ranks m1 and m2 adjacent and in the same order and m3 and m4 adjacent and in the same order). Let fw2 ; w3 gP
m1 ;m2 g w1
and w1 P
m3 ;m4 g
0 m fw2; w3g. Let = @ 1 0
m2 m3 m4
1 A.
w2 w3 w1 m4 Then in the corresponding many-to-one matching problem, 6 , which implies that f
f
0
fg is not externally strongly stable in the corresponding many-to-one matching problem. Let V = f; g. It is easy to check that V is a strongly stable set for R in 0
the corresponding many-to-one matching problem.14 Hence, we have established the 14
To see this, let ^
2 MnV .
If ^(w1 ) = w1 , then 0
ff
2 ^(fm1 ; m2 g), then either ^ = or ff 3 4 g 2 fw1 g = ^(fm3 ; m4 g), then either ^ = 0 or 0 ff 1 2 g 2
If w1
w
m ;m
;w ;w
w
m ;m
22
g g ^. Let ^(w1 ) 6= w1 . ^. Let w1 2 ^(fm3 ; m4 g). If 3g ^. If fw1 g ( ^(fm3 ; m4 g), then 3g w
;w ;w
m3 ;m4 ;w1
following facts: (i) In the one-to-one matching problem, fg is a strongly stable set for R and V is not a strongly stable set for R because V is not internally strongly stable. (ii) In the corresponding many-to-one matching problem, V is a strongly stable set for R and fg is not a strongly stable set for R because fg is not externally
strongly stable.15 (iii) In the corresponding many-to-one matching problem, V is a strongly stable set for R. The set of unmatched agents is not identical for any two matchings belonging to V since U () = fm2 g 6= fm4 g = U ( ). Thus, Proposition 1 0
and Theorem 2 do not carry over to many-to-one matching problems when considering the weak dominance relation w . 6
Conclusion
In general both the core and stable sets may not exist for cooperative games (Lucas (1969) and Einy and Shitovitz (1996) for stable sets). Here it is still an open question whether stable sets always exist for one-to-one matching problems and whether there is a unique stable set if they exist. Since the core of one-to-one matching problems is always non-empty, one may wonder why we should be interested in stable sets. Such a judgement would be based on properties of solution concepts, i.e. such reasoning is a posteriori after having de ned a solution concept. However, more importantly any
judgement of any solution concept should be a priori based on the economic meaning of its de nition. Any core matching is unblocked, i.e., no coalition has any incentive to ff w
g g ^. Hence, V is externally strongly stable. It is straightforward that V is internally
m1 ;m2 ;w1
0 m Note that V is not a stable set for R since for ~ = @
strongly stable. 15
1
w1
and 0 6 ~.
23
m2 m3 m4 w2 w3 m4
1 A we have both 6 ~
deviate from it. However, the core as a set does not possess any additional appealing property other than the stability of any single core matching. This is not true for a stable set since as a whole set it satis es internal stability and external stability. If we select a single matching from a stable set, then this matching is unlikely to be externally stable as a set (unless all agents unanimously agree which matching is most preferred among all matchings). Stable sets should be truly understood as a multi-valued solution concept. They are appealing for situations where agents agree to choose a set of possible outcomes and the nally chosen outcome is enforced.16 Then it may be questionable to rule out matchings which are not blocked by any possible enforceable outcome. Of course, on a practical level the success of a solution concept depends on its properties and its applicability. For one-to-one matching problems we found that the core and stable sets share a number of well-known properties. Our main result did not impose any restriction on the matching problem under consideration (other than it is one-to-one). Most results on stable sets in the literature apply to speci c classes of games with certain restrictions. References
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27
