E L S E V I E R European Journal of Operational Research 88 (1996) 348-357
EUROPEAN JOURNAL
OF OPERATIONAL RESEARCH
T h e o r y a n d M e t h o d o l o g y
Arbitrated matching: Formulation and protocol
W . - Y . N g *, K . - W . C h o i , K . - H . S h u m
Department of Information Engineering, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong
Received: May 1993
Abstract
This paper presents an arbitration approach to bipartite matching in which a set of actors are matched up with a set of mates in a one-to-one manner. The arbitration is composed of two stages, viz. stage I in which ordinal preference is elicited on a need-to basis until all stable matchings are obtained. Stage II is sequential bargaining in which risk preference information is extracted by eliciting certainty equivalents of default randomizations declared by the arbitrator. Stage II comes out with a particular stable subset, randomization of which is preferable for all, to that of the entire stable set. Stage II is particularly effective when actors and mates are risk-averse, whence reducing uncertainty with a smaller randomization set proves to be advantageous to all. We also prove that it is superior to the traditional one-off approach in terms of the elicitation effort required as well as strategyproofness.
Real-life examples of assignment problems, like college admission [1], the labor market for medi- cal interns [3], and sorority rush [4], can be formu- lated as bipartite matching, in which a set of actors are matched up with a set of mates, in a one-to-one manner, into ac tor -mate pairs. Each mate exhibit ordinal preference over actors and each actor likewise exhibits ordinal preference over mates. As far as the matching is concerned, the two sets of actors and mates are symmetrical. However, we distinguish mates as the potentially less active side, from actors who tend to interact more actively with the matchmaking authority during the matching process. In the example of
* Corresponding author.
college admission, the applicant prefers more competent colleges and each college prefers more qualified applicants. A good match should pair up applicants and college places for their mutual satisfaction in a demonstrably fair manner. Appli- cants who fill in and submit their choice forms in the beginning and passively wait for replies from the authority may be considered the mates, while the colleges who tend to compete for good stu- dents may be considered actors.
A central authority is often responsible for conducting the matching process. Procedures are set up to elicit preference from both actors and mates. The simplest way to matching is a one-off approach, in which the central authority elicits preference information from both actors and mates, once and for all, often in the form of priority lists. A predefined algorithm, invariably
W.-Y. Ng et al. / European Journal of Operational Research 88 (1996) 248-357 349
heuristic in nature, is then used to produce a final matching. For instance, some affinity mea- sure may be used to rank the mates (actors) according to how much they are liked by the actors (mates) in general. The mates (actors) are then sequentially assigned according to their most preferred and still available actors (mates). The interaction between the central authority and the actors and mates is relatively simple for such a one-off approach.
In principle, a one-off matching procedure se- lects one solution among the set of all possible matchings based on preferences of all mates and actors. Due to preference mismatches - for in- stance, the mate an actor likes most may prefer some other actor more - there is obviously no win-win solution that appeals to all in general. Compromise and trade-off among all mates ' and actors' degrees of satisfaction is inevitable. Ar- row's Impossibility Theorem [9] that proves the non-existence of an ideal form of rational, fair and democratic aggregation of ordinal prefer- ences poses a limit on the general existence of ideal matchings.
The need for elicitation also poses a practical problem in that it is too demanding or even impractical to request complete ordinal prefer- ence information. Very often, mates and actors supply and rank only their top few choices. An important and novel result we prove is that a one-off procedure that uses only partial ordinal information may generate matches that are un-
stable. It is also not s trategyproof as mates and actors may refrain from supplying true top choices that they somehow judge unattainable. This is a consequence of the somewhat arbitrary par tner that the matchmaking authority has to find for an unfortunate candidate, whose supplied choices have all been used up.
We consider an alternative to the one-off ap- proach, namely, the arbitration approach in which rounds of arbitration are conducted to elicit pref- erence information on a need-to basis, and ac- t o r - m a t e pairs are matched in a progressive man- ner. The arbitration is conducted according to a two-stage protocol, in which either stage is di- vided into rounds of information exchange be- tween an arbitrator on one side, and the mates
and actors on the other side. The first stage extracts ordinal preference information incre- mentally on a need-to basis, until all stable
matchings [1] are generated. The second stage treats the problem as bargaining and strives to generate a choice set of stable matchings. The protocol is similar to a voting correspondence [2] in social choice theory that nominates a set rather than a specific member in general. For the sake of equity, a matching may then be randomly chosen from the choice set.
The organization of the paper is as follows. The marriage model of bipartite matching is for- mulated in Section 2. The classical notion of stability is then discussed in Section 3. In Section 4, matching is analysed using social choice theory. In Section 5, the arbitration approach is proposed and described in detail. Sections 6 and 7 finish off with discussions and final conclusions.
2. Formulation
The bipartite matching of actors and mates is an instance of the marriage model [5]. It differs from the basic marriage model in that the two sides are distinguished by their different modes of communication with the central authority. While the two sides are indeed symmetrical in the marriage model, they are differently named men and women for clarity of discussion. Never- theless, all results in the marriage model apply to the matching problem.
We describe the marriage model as follows. There are two finite and disjoint sets M and W:
M = {ml, m 2 . . . . , m,}
is a set of n men, and
W = {wl, w2 . . . . . Wr}
is a set of r women. The two sets may be of unequal sizes. Men are women may be commonly addressed as candidates.
We assume that the ordinal preference of each man over all women and that of each woman over all men are strict, complete and transitive. Let >" r n and >- w be the resulting binary strict prefer- ence relations of man m and woman w respec- tively. Being strict, the preference relation may
350 w.-Y. Ng et al. / European Journal of Operational Research 88 (1996) 248-357
be concisely described by a preference list. For instance, man m's preference list being
e ( m ) = [Wl, w3, w2, w4 . . . . ]
means w 1 >'m we > ' m ' ' ' , etc. Further, any candidate prefers being matched to remaining single.
Let ~M, ~ w be the sets of all possible man and woman preference lists respectively, and PM, W e ~ × ~ v is a preference profile - an in- stance of all preference lists. The marriage prob- lem is characterized by the triple (M, W; PM, W)"
Definition 1. A matching I~ is a one-to-one corre- spondence from the set M U W onto itself of order 2 (that is,/zE(x) = x ) such that 1) i f /z(m) ¢ m, m e M, then ~(m) e W, and 2) if ~(w) ~ w, w e W, then / x (w )eM. /z(x) is called the /~- partner of x.
A matching may also be described by the re- sulting set of man-woman pairs:
i x = { ( m , w) J m e M , w e W,
/x(m) = w and / x ( w ) = m } .
The set of all possible matchings is denoted by .¢t'.
Definition 2. A matching function is a mapping ~pf that maps any preference profile to a unique matching. A matching correspondence is a set-val- ued mapping tp¢ that maps any preference profile to a non-empty of matchings.
Arrow's Impossibility Theorem sets a theoreti- cal limit to the ideal of any deterministic match- ing function. A practicable resolution is to ran- domize the final choice among a choice set of matchings produced by a matching correspon- dence that circumscribes all candidates' interest. Consequently, our goal is to derive properties of a somewhat optimal matching correspondence ~*, or equivalently those of the particular choice set ~ * it produces.
3. Stability
The fundamental notion of stability in the marriage problem is introduced by Gale and Shapley [8] in 1962.
Definition 3. A man-woman pair (m, w) blocks the matching/~ if and only if
W>'ml.t(m), m>'wtz(w ).
We call (m, w) a blocking pair of /z . A matching /~ is stable if and only if there is no blocking pair in/~, and is otherwise unstable.
A somewhat surprising result is that stable matchings always exist. Further, the number of stable matchings in general grow exponentially with the problem size [1]. Although the blocking pair (m, w) may not know each others' prefer- ence, or the rule of the game may not allow them to defect the matching, instability is undesirable in principle. In this paper, we regard stability as necessary for an acceptable matching. Denoting the set of all stable matchings by I , ' s c.~r, we therefore opine that ~" * c ~ .
However, derivation of stable matchings re- quires full knowledge of the preference profile. A fundamental limitation of the one-off approach results:
Theorem 1. There exist preference profiles for which any one-off procedure cannot guarantee matching stability with only partial preference information.
Proof. We prove by constructing such a prefer- ence profile. Let P be a preference profile with a unique stable matching ~. Further, some man m exists such that w >'m ]./,(m) but whether /~(w) >'w m or not is unknown. That is, whether/x is stable is unknown. []
The full preference profile must be elicited in a one-off approach if stability is to be guaranteed. That is, each woman /man must give her /h is full preference list. Such elicitation rapidly becomes impracticable with increasing problem size.
4. Matching as a social choice problem
Social choice theory is concerned with the collective choice problem, namely, how prefer- ences of individual decision makers may be ag- gregated most effectively to produce a single
W..-Y. Ng et al. / European Journal of Operational Research 88 (1996) 248-357 351
group preference. The decision makers are called voters for the apparent analogy to voting.
Consider n voters
D = { d l , d 2 . . . . . dn}
who are jointly responsible for choosing an alter- native from a set of alternatives
A = { a l , a2 , . . . ,an}.
Each individual d i exhibits a complete transitive weak preference represented by a binary relation
i over A. Weak preference allows indifference. Let
~o:= [~ , , ~2 . . . . . ~.] be the preference profile. A social choice prob- lem is then characterized by (D; ~ o ) .
For the general social choice problem, Arrow [9] constructs a set of axioms to represent an ideal of rationality, equity and democracy and subsequently proves that they are mutually con- tradictory. Consequently, an ideal deterministic single-valued social choice function does not ex- ist. Later on, Plott [7] shows that even relaxing the single value requirement does not help; it is just as impossible to construct an ideal set-valued social choice correspondence.
A marriage problem (M, W; PM,w) can be modeled as a social choice problem (D; Po) when all candidates are considered the voters, i.e. M w W as D, and the set of all possible matchings as the alternative set, i.e. ~tr as A. Assume all candidates are selfish in the sense that any candi- date's affinity for a matching/X depends solely on he r /h i s /x-partner, the candidates' complete transitive strict preferences over their respective partner sets induces complete transitive weak preferences over .av. In particular, denote the binary weak preference relation of a man m over
p t K by ~m, and that o f a w o m a n w by ~w, for any/xl,/X2 ~ and >- m, > w c~M,w:
/Xl ~ ' / X 2 iff /xl(m) >- ,. /x2(m),
tz, a ' / z 2 iff ~zl(w ) >-w/~2(w), t ~ t
/Xl ~ m / x 2 , /X2 ~ m / x l i f f /xl(m)=/x2(m), I
/X,~ ' /X2, /X2~../Xl iff /X,(w)=/X2(w ).
An important observation here is that the so- cial choice problem that results has a restricted domain, consisted of only profiles that are in- duced by some ~M,w. Both Arrow's and Plott's negative results consider unrestricted domains and may not strictly apply here. Indeed, we prove that within the domain so restricted by the mar- riage model, weak Condorcet winners always exist.
Definition 4. Binary majority tournament T is a binary relation such that for each pair of alterna- tives a, b c A , aTb if and only if more voters strictly prefer a to b. A weak Condorcet winner is an alternative a that never loses in binary major- ity tournaments, i.e. ] b c A such that bTa.
Let 7//- be the set of all weak Condorcet win- ners. The following theorems establish the gen- eral existence of weak Condorcet winners for the social choice problem induced by a marriage model.
Theorem 2 (Roth [4]). In a marriage problem in which the sets of men and woman are of unequal sizes, there is at least one stable matching in which all the members of the smaller set are matched. Furthermore, all members of the smaller set are matched.
Theorem 3. The set of all stable matchings ~ is a subset o f the set of all weak Condorcet winners ~,¢r, in all matching problems.
Proof. Without loss of generality, let n > r, i.e. there are more men than women. By Theorem 2, all women will be matched in any stable matching /xs c.JC's. Consider/xs and any other matching /x. Partition men and women respectively according to their preferences between/xs and/X as
M = X U X ' U X " U UU U ' U I
and
W = Y u Y' w Y",
where: (i) X and Y (respectively X ' and Y') are the
subsets of men and women matched in both /X~ and /X, and strictly prefer their /xs-partners to
352 W.-Y. Ng et al. / European Journal of Operational Research 88 (1996) 248-357
/x-partners (respectively /x-partners to /xs- partners);
(ii) X " and Y" are the subsets of men and women matched to the same partner in ei ther/xs or /x;
(iii) U are men matched in/X~ but not in/X; (iv) U' are men matched in /X but not in /X~; (v) I are men unmatched in both /X~ and /X.
Note that [U[ = [U'I and members of X", Y" and I are indifferent between/X~ and/X. The net number of voters that prefer/xs to/X is therefore
8 : = ( I X I + I Y I ) - ( I X ' I + I Y ' I ) .
Since/x~ is stable, the/x~-partners of men in X ' cannot be from Y', nor from U, and therefore must be from Y. Consequently, I YI>~I X'I. Simi- larly, I X I >t I Y'I. Thus 15 >i 0 always. A stable matching /X~ is therefore a weak Condorcet win- ner who never loses any binary tournament to any other matching, i.e. I ¢ a 7f. []
Theorem 4. A weak Condorcet winner always ex- ists in a social choice problem induced by a mar- riage model.
Proof. This is a direct consequence of the general existence of stable matchings [8] and Theorem 3. [3
Note however that a weak Condorcet winner can be an unstable match. This springs from the fact that 6 may not increase when a stable match is destabilized by swapping two pairs so that while one original pair becomes a blocking pair in the new match, the other pair is broken for union with their more preferred partners.
5. Arbitration
Now we consider the communication between candidates and the central matchmaking author- ity.
Consider a simple example of matching a set of two actors A = {al, a 2} to a set of two mates T = {t 1, t2}, whose mutual preferences are
P ( t l ) = [a l , a2], P ( a l ) = [t2, t l ] ,
P ( t 2 ) = [a2, a l ] , P ( a 2 ) = I l l , t2].
The only two possible matchings are
/Xl = [ (ml , a l ) , (m2, a2)] ,
/x 2 = [ (ml , a2), (m2, a l ) ] ,
Note that both matchings are stable. Matching /Xl may be called the actor-optimal stable match- ing which matches actors to their first choices, and subsequently actors their least preferred choices. Matching/x2 is the actor-optimal match- ing that matches actors to their first choices, and subsequently mates to their least preferred choices.
Note the negative symmetry in this particular preference profile - each candidates' most pre- ferred partner prefers himself least. For instance, actor a 1 prefers mate tl most, but t 1 prefers a 1 least. There is clearly no unbiased deterministic choice. In this case we consider the only defensi- ble fair approach to be a random choice between
{/xl,/x2}- In the general case, equity from randomization
is defensible only when the choice set being ran- domized is reasonable in some sense. The set of all stable matchings ~ is a possibility, especially when stability is consistent with Cordorcet opti- mality in the collective choice framework as shown above. However, the set ~ is potentially large and Theorem 1 shows that full knowledge of the preference profile is needed for its generation. This poses processing overhead to the matchmak- ing authority in terms of both elicitation and computation.
We have designed an interactive approach called arbitrated matching to reduce the process- ing overhead as much as possible. A central matchmaking arbitrator interacts with all candi- dates, constructs the choice set and matches ac- to r -mate pairs in an incremental manner. Arbi- trated matching is composed of two stages as described below.
5.1. Stage I: Top-down preference elicitation
Stage I aims at eliciting just sufficient prefer- ences so that ~ can be enumerated. A top-down elicitation protocol (Fig. 1) elicits the contents of the preference lists on a need-to basis, starting
W.-Y. Ng et al. / European Journal o f Operational Research 88 (1996) 248-357 353
with the most preferred choices of each candi- date. In round i, i = 1 ,2 , . . . , the arbitrator com- piles and sends each unmatched actor a' an up- dated agenda, of a set of unmatched mates Uff ), and each unmatched mate t' an agenda of a set of unmatched actors /_It ,"). Agenda updates are always deletions and never additions, i.e
Ua ~i) C Ua ~i- l) and UtS i) c UtS i- 1).
Every unmatched candidate p ' then reveals he r /h i s most preferred partner from the agenda given her /h im. Let A~ ) c A and T. ~i) c T be the sets of actors and mates who remain unmatched in round i respectively, and f ( i )(p)be unmatched candidate p"s most preferred partner in he r /h i s agenda U (i).
All candidates are unmatched in the first round and all agendas handed out are entire sets, namely,
A~ ) :=A, T~ l) := T,
U~9 ) := T Va' ~ A
and
Ut !l) : = h Vt' ~ T
Also, the arbitrator maintains for each un- matched candidate p a partial preference list L(i)(p) ~ {f(1)(p), f (Z)(p) , . . . , f(i)(p)}, for which the ranking is completely known. The algorithm shown in Table 1 is then executed for round i.
By deleting (a",t'), we mean deleting a" from all t"s future partial preference lists and agendas, and t' from those of a"'s. By construction, a pair (a",t') is deleted for its destabilizing effect on any matching.
)Elicit n i (p) choices from each candidate p
l Identify unstable pairs and reduce Up
Feedback Up to [ each candidate p
N Complete?
Stage H
Fig. 1. Top-down preference elicitation.
Table 1
Va' ~ A~j), Vt' ~ T~ ° set a' and t' Free (i.e. N O T Engaged) while some unmatched actor a' is Free do begin
if L(i)(a ') is empty set a' Idle
else begin t' := most preferred mate in L(i)(a ') if(a' ~ L(i)(t'))
set a' Idle else begin
if 3a", (a", t') Engaged Disengage (a", t') and set both Free set the pair (a', t ') Engaged
foreach successor a" o f a' in L(i)(t ') Delete the pair (a ' , t ')
end end
end
Stage I terminates after the round in which preference of all updated agenda items are known, i.e. round N where
T(N+ 1) II(N+I) cL~ N + l ) - v VP ~A~u N+l) n - , "
This algorithm is an adapted version of the ex- tended Gale-Shapley (EGS) matching algorithm [1], which acts upon a fully known preference profile. The adapted version essentially executes (EGS) 'as far as possible'. In each iteration, a free actor a' makes proposal to the first free mate on his list. With only partial information, problem arises if either 1) a"s preference list is empty or 2) the most preferred mate m' on a"s list has not reveal her preference regarding m'. In the modified algorithm, whenever 1) or 2) hap- pens, the proposer is set to idle, and the algo- rithm continues with other proposals. The algo- rithm ends whenever there is no free mate. So, by executing 'as far as possible', we mean processing as many proposals as possible. In the mean time, the algorithm identifies and deletes unstable pairs.
The final partial preference list U~ N+l) com- prises of all stable partners of candidate p and is called the GS-list of p [1]. In any stable matching, the partner of p is selected among he r /h i s GS- list. The set of all stable matchings may then be
354 W.-Y. Ng et al. / European Journal of Operational Research 88 (1996) 248-357
enumerated by enumerating and screening for stability all possible matchings generated from the GS-lists. Some efficient algorithms are found in [1].
Example. The above algorithm may be general- ized by asking candidate p to reveal her /his top k (p ) > 1 choices, with which iterations may be ended earlier. This is done for the following example with eight actors and eight mates.
Round 1: U a = M,Va ~ A;
U m =A, Vm ~ M .
The arbitrator requests 8 choices from each mate and 3 choices from each actor. The collected preference information is shown in Tables 2 and 3.
After the second round, each actor has already completely ranked all his stable partners. So, the preference information collected is sufficient to generate all stable matchings as listed below:
12,1 = { ( m l , a s ) , ( m 2 , a3) , (m3 , a s ) , (m4 , a6) ,
(ms, a7), (m6, a l ) , (m7, a2), (ms, a4)},
/z 2 = { (ml , a8), (m2, a3), (m3, as), (m4, a6),
(m5, a7) , (m6 , a l ) , (m7 , a2 ) , ( m s , a4)},
A final match may be randomized directly from the set of all stable matchings .Ze. However, we are going to show that it is possible to shrink the randomized set to the advantage of all candi- dates when risk preference of the candidates is elicited and considered. Such advantage is a re- sult of reduced uncertainty when a smaller set is randomized while changes to candidates' ex- pected utilities are kept strictly non-negative.
Let .~¢ be a set of matchings, and .Z~a~ be a lottery by which a matching is selected with equal
W.-Y. Ng et al. / European Journal of Operational Research 88 (1996) 248-357 355
power set Expected Utility of d ~ ~ X ~ o f s
"--Q )
Expected Utility of d Fig. 2. Status quo in each bargaining stage.
chance from ~ . Let u a : M ~ a ~ be the utility of actor a towards the mates, and u m :A ~q~ be that of mate m towards the actors. The expected utility of - ~ for candidate p is therefore
E Ep[ . .~ , ] = u . ~ ' I~1
Let ~'a, ~'2 be two sets of matchings, and Sajl, Sa~2 be the lotteries of randomly selecting from "~1 and "-'~2 respectively. The preference of candidate p between -~s~l and -~s~2 is then
S a ~ p.Z~a~: iff g~[.~j~] >_Ep[-~,2].
Define the lowest certainty superior (LCS) of a lottery .~m for actor a as the mate LCS=(.~j) M whose utility value is smallest yet larger than E~[.~n]. That is,
L C S a ( - ~ ) := arg min Ua(m ) • m ~M,m ~ . . . ~
LCS for a mate is similarly defined.
Let Nj (a ,m) be the proportion of matchings in ~ ' which match actor a to mate m. A bargain- ing procedure is carried out in rounds as follows. The initial bargaining set is initialized to the set of all stable matchings, i.e. ~'1 '=-~trr In round i, the arbitrator compiles and returns Nj~(a,m) to actor a and mate m, Va ~ A , Vm ~ M , and re- quests each candidate to give her /h is LCS. The arbitrator then computes a new bargaining set
~,~i+1 := {/-/, E~-~ill-t(a) ~a LCSa(--~.~i) and
/z(m) ~ m LCSm('~J~),
Va ~ A , m ~ M}.
The bargaining set ~'i+ 1 is so constructed to retain only matchings in ~'i that guarantee each candidate partners no less preferable than their LCS's. The bargaining terminates in round n when .~'. + 1 is empty.
Theorem 5. If ~'i+ 1 ~ 0 , "~,+1 ~ p S a ~ Vp ~ A UM.
Proof. For candidate p ~ A u M,
Ep[ S'%~+ 1 ] - Ep[2'~,]
E~ l/" ~ i + 1 Up(1.1,) --~ [ ~i+11 - U p ( t f S p ( - ~ q ~ i ) )
E l J" ~ ' ~ i + 1 [ Up( ~/, ) -- Up( LCSp(2s~ ~))] I i+ll
> 0 .
Therefore, . ~ ,+ . ~ p - ~ c []
Expected Utilility of d I i certainty
X !i X X ~ ~ e q u i v a l e n c e X
x x x-.-.x.--x- . . . .
X ! X :v X x X ........... -X ..........................
~ s t a t u s X quo X X X
x ..x x X :X )
Expected Utilility of dz
Fig. 3. Certainty equivalence in round n.
356 W.-Y. N g et al. / European Journal o f Operational Research 88 (1996) 2 4 8 - 3 5 7
6. Discussion
Each round i can be viewed as a standard bargaining problem over the set of lotteries {_ow~ I~ ' ~ 2 ~i} with status quo at . ~ , .Z, ea~ 1 =.g¢~ in particular, set by the arbitrator. Suppose the bargaining stage terminates at round n. Note that
.ZP~ ~ p . ~ , _ ~ p . . . ~ p . 5 ~ Vp ~ A U M .
In other words, the status quo is being pushed towards the efficient frontier in the utility space (see Fig. 2).
However, the final lottery S '~, may not be non-dominated in the utility space. The termina- tion condition that ~ ' ,+ l is empty only means that no lotteries dominate the LCS point, i.e. giving every candidate a partner at least as good as he r /h i s LCS, but not that no lotteries may dominate .ows~ " (see Fig. 3).
The effectiveness of the bargaining is depen- dent on the risk attitudes of candidates. While two candidates ca, c 2 of the same side may have identical preference lists [ p l - < p z ~ . . " ~Ps], they may have different degrees of risk-averse- ness. The ease when c I is more risk-aversive than c 2 is illustrated by their different utility functions in Fig. 4.
In general, the less risk-averse the candidates are, the faster the bargaining will converge, but the larger the randomization set will be. The reason is as follows. Let Bi (c ) be the set of matchings in B i that give candidate c better
Utility Function
Cl .......... . . ' " " ../* • / .." ."
..w /" ,..*" .,.
II ... . ~ ' * ' " C 2 .. . . ,
#:i.....e...- -°''"
I I I I a •
Pl P2 I)3 P4 Ps Stable Matchings
Fig. 4. Utility functions of candidates with different attitudes.
partners than h i s /he r respective certainty equiva- lent of -~Bi" The less risk-averse c is, the more highly ranked will be h i s /he r certainty equivalent - subsequently, the smaller Bi(c ) will be. Obvi- ously,
n i + 1 = { O c ~ M c ~ A n i ( c ) }
will be more likely to be empty. In an extreme case, the candidates may be so risk-prone that their LCS's are all their top choices, in which case there is no ground for further bargaining. The arbitrator has to resort to the original status quo, namely, randomizing the entire J's.
7. Conclusions
We have shown that to generate a stable solu- tion for one-off bipartite matching requires elici- tation of ordinal preference of all candidates concerned. A progressive approach in which an arbitrator interacts with candidates on a need-to basis is more reasonable and reduces the elicita- tion effort involved. In particular, actor-mate pairs that prefer each other most are matched without eliciting their lower preferences which are irrele- vant.
Matching stability proves to be sufficient for weak Condorcet optimality in the social choice sense. We therefore suggest an arbitrator who declares a status quo action of randomizing the entire stable set. When the candidates are suffi- ciently risk-averse, randomizing among a well- chosen smaller stable subset proves to be a win- win move that results in strictly non-negative in- crease in their respective expected utilities of the randomization. We have designed an iterative bargaining procedure in which risk profile elicita- tion is simply in terms of certainty equivalent of a moving status quo. However, bargaining is only effective when the candidates are sufficiently risk-averse. In the extreme case, any reduction of the randomization set may not be possible and the arbitration would simply pick a stable match- ing by random. Characterization of the rationality in randomizing among the entire stable set re- mains an open question.
W.-Y. Ng et al. / European Journal of Operational Research 88 (1996) 248-357 357
References
[1] Gusfield, D., and Irving, R.W., The Stable Marriage Prob- lem: Structure and Algorithms, MIT Press, Cambridge, MA, 1989.
[2] Moulin, H., Axioms o f Cooperative Decision Making, Cor- nell University Press, Ithaca, NY, 1991.
[3] Roth, A.E., and Sotomayor, M.A.O., Two-Sided Matching - A Study in Game Theoretic Modeling and Analysis, Cornell University Press, Ithaca, NY, 1990.
[4] Mongell, S., and Roth, A.E., "Sorority rush as a two-sided matching mechanism", The American Economic Review 81/3 (1991) 441-464.
[5] Irving, R.W., Leather, P., and Gusfield, D., "An efficient algorithm for the optimal stable marriage", Journal of the ACM 34 (1987) 532-543.
[6] French, S., Decision Theory - An Introduction to the Mathematics of Rationality, Ellis Horwood, Chichester, 1986.
[7] Plott, C.R., "Axiomatic social choice theory: An overview and interpretation", American Journal of Political Science 20 (1976) 511-596.
[8] Gale, D., and Shapley, L., "College admissions and the stability of marriage", American Mathematical Monthly 69 (1962) 9-15.
[9] Arrow, K.J., Social Choice and Individual Values, 1st ed., Wiley, New York, 1951.
