>> Yuval Peres: Good afternoon. So most of you have probably wondered what you actually
need to do in order to get a Nobel Prize. And so you certainly hear one possible route. But
unfortunately, that one’s been already taken. And more seriously, stable matching is a favorite
in many places, and in particular, here at MSR. Many of you probably know the beautiful
pictures that Ander[phonetic] has drawn that relate to stable matching, but now we'll hear the
real story of how it started in many of the connections. So we are very happy to have Professor
Neyman from the Hebrew University to tell us about it.
>> Abraham Neyman: Thank you. So first, an apology. None of the work that I'll be talking
here is my own work. But the Hebrew University we have a [inaudible] that whenever
somebody get the Nobel Prize in economics and is working in game theory, one of us, the game
theory presents it to the general audience of the mathematicians, and you [inaudible] recorded
me to do the same out here. So I'll present this work that’s the 2012 Nobel Prize in economics.
And let me start with the press release because the press release almost says it all. For those
who don’t know anything, at least you could read this press release and know something about
it.
So stable locations from theory to practice. And indeed this Prize was from theory, a small
corner in game theory, its theory and its practice. And they say this year's Prize concerns a
central economic problem, how to match different agents as well as possible. For example,
students have to be matched with schools and donors of human organs with patients in need of
a transplant. How can such matching be a completed as efficiently as possible? What methods
are beneficial to what group? The Prize rewards two scholars who have answered these
questions on a journey from abstract theory on stable allocation to practical design of market
institutions.
And it goes on. It credits first, Lloyd Shapley, then Alvin Roth. Lloyd Shapely used competitive
game theory to study and compare different matching methods. The key issue is to ensure that
the matching is stable in the sense the two agents can be found who would prefer each other
over the current counterparts. Shapely and his colleagues devised specific methods, in
particular, the so-called Gale-Shapely algorithm that always ensures stable matching. These
methods also limits agents’ motives for manipulating the matching process. Shapely and Gale
were able to show how the specific design of a method may systematically benefit one or the
other sides of the market. Essentially, everything that is written here will be proved and
emphasized in this talk. And let me just say, I'll continue with a personal list on Alvin Roth, but
just a few words about Lloyd Shapely. In the game theory community we were expecting Lloyd
Shapely to be the first one to receive a Nobel Prize in economics for his work of game theory
and not necessarily for this work. He has several other important books. Each one, it lists in my
view, is at least as important, if not more than this one.
And let's go to the second part of the reward for Alvin Roth that recognized the Shapely
theoretical results could clarify the functioning of important markets in practice. In a series of
empirical studies, Roth and his colleagues demonstrated that stability is the key, understanding
the success of particular market institutions. Roth was later able to substantiate this conclusion
and systematic lab laboratory experiment. He also helped to redesign existing institution for
matching new doctors with hospitals, students with schools, and organ donors with patients.
These reforms are all based on the Gale-Shapely algorithm, along with modifications that take
into account specific circumstances and ethical restrictions, such as the preclusion of side
payments. So also I wrote about Al Roth. He was really a driving force for like 30 years. He was
running with this topic and trained to pursue every word that he go in school admissions,
hospitals, in the matching of, in terms, into hospitals he was really going on with the hammer
and trying to apply and modify, wherever needed, these type of initial algorithms into the
practical world.
>>: [inaudible].
>> Abraham Neyman: Hmm?
>>: [inaudible].
>> Abraham Neyman: Why? Yes.
>>: [inaudible].
>> Abraham Neyman: You're right. But there are some, I'll touch on it later because there are
many extra knowledges, and all this work doesn't [inaudible] and one is to modify things and,
I'll tell you later. At the very end there will be nice story when I come to show that everything is
part of game theory; a nice story of your whole department and my Institute in Jerusalem when
there was an issue of recording and some comment of [inaudible] that show that basically you
can mix everything, all the decisions altogether if you accept some principles of [inaudible].
And even though these two researchers wrote independently of one another, their
combination of Shapely basic theory and also empirical investigation experiments and practical
design have generated a flourishing field of research and improved the performance of many
market. This year's Prize is awarded for an outstanding example of economic engineering. So
that's really a type of, we call it game theory in general, but it's economic engineering, that you
take something from theory and you really engineer it to the real world whenever there is a
need for a small adaptation, you do the small adaptation.
But let us go to the starting point. The starting point was a paper by David Gale and Lloyd
Shapely in 62. And let me explain, first, the problem by an example. I would assume, let me
maybe take here a vote. Who knows this paper, the Shapely, who doesn't know? A few
doesn't know. Okay, so I'll go quickly on the model from A-Z, and stop me if you think I'm
rushing too fast or speed me up if you think I'm going too slow at the first part of the talk.
So there are three men, alpha, beta, gamma. And three women, A, B, and C, and matching, in
this particular set up, the one to one would be the right concept in later applications is the I
want to 1 to 1 function from the set of men to the set of women. But we are interest, so if you
look on matching just there, it’s all [inaudible]; you could list them. And each man has a
preference over the women, and each woman has a preference or there's an example of a
preference, just a notation, how I write the preference; and each woman is a preference over
the men. So here's an example of a woman A, woman B, and woman C of their preferences
over the men's alpha, beta, and gamma. In a statement matching, our matching’s were there is
no unmatched couple preferring each other to their mates. Very simple. Is that clear what's
[inaudible] matching? Because if we don't understand that we will be lost in the [inaudible].
And later, I'll may be like try [inaudible]. Even if it's not understood, I'll explain it later, it is
exactly what we call the CORE of a comparative game.
So it's an instance of a model where the CORE just reduces to this simple definition of what is
called [inaudible]. So let us continue the example that we just mentioned. If you would have
looked of the preference and you remembered them, I don't, here is the preference. So men
alpha prefers, his first linking is A, then B, then C. Beta first is B, then C, then A. Gamma first
choice is C, second A, and third is B. And similarly, the second column presents the preference
of the woman over the man. And let us look on the red example. This is not stable because
gamma and A are not matched. But gamma prefers A on B on this next one is going to compare
to three, and A prefers gamma on alpha who is, they are a match, and therefore this is not a
stable matching. And if you look on the following matching, this is a stable one. If you want, I'll
go over the letters, but I think it's not needed.
And let’s look on another example. Here we have four men and four women. And the
question is, is there stable matching? So most of you know the solution finally, but [inaudible]
it was a question. Is there a stable matching? And you could ask the same question, is there
stable matching for three, for four, for five, for any number of men and women? You could ask
this question. I must confess that I was feeling very bad as an undergraduate when I was
unable to solve this question when it was posed to me by Peroles[phonetic], especially after
seeing the solution. I said, well, come on. Maybe I should leave mathematics because I
couldn't solve that. And then when I met Gabe, in 78 I think, I told him the story. He said, oh,
don’t be so frustrated. I was struggling with this problem for many years and asking many
mathematicians; none could have solved it. I sent the letter to Lloyd Shapely, and in return,
mailed the next day I go down.
So once you see the solution, once you know about it, you say, how could you miss it? Yes? I
think even if you will tell somebody that the reason I'll go with them [inaudible], but usually
mathematicians ask is the one and you don't necessarily think of building an algorithm. So the
red matching is not stable. Here are the older pairs. The blue ones that could, are unmatched
and would prefer each other to their own mate. Let's look on just one such example. Let's say
that C and beta, they are not matched, but C prefers beta because rank three over four and
beta prefers C over is matched, which is four, which is B. And this is the unique, in this case
there is a unique stable matching which is done here in the blue. And, just to tell you the story
that it's not [inaudible] so trivial, the problem, and after you see the proof you will think it is
trivial, is that [inaudible] when used tell somebody okay, yes, there is a stable matching, then
you generalize a little the problem and you see that there need not be a stable matching in very
small volume. It's very important two-sided market, what we call.
So let's look quickly on what's called the roommate problem. And in the roommate problem,
we wish to make two two-person themes from a group of A, B, C, and D. You could think of
domes. You have four individuals; they have to be matched in two domes. And each one has a
preference of his roommate in the dome. So it's one side. There is no distinction here between
employees and employers, schools and students, women and men. So one group, and the
preference of each member for the partitions, so each one, basically if you adjust four, the
preference is exactly on the partition because if you partition, if you know who is your
roommate, you know everything. In the other cases of the matching, if there are externalities,
it's a different story. So the preference here is identical with your own preference over whom
you are being matched. So preference A, A prefers B over C, over D. B, C over A over D. C over
A, and D is independent. S these, the last one in each one of these preference match. There is
no stable matching here because in any stable matching, D will be matched with one of A, B,
and C. But each one of them is the top ranked of one of the others, A, B, C. So he will depart
with the guy, well he's the top mark, and so in the woman problem there is no stable matching.
So existence results, Gale and Shapely, they proved that there always exist the stable matching.
And as I said, it's obtained by any iterative procedure, a very simple one, a two line description
of the algorithm goes in stages. In stage K, each man proposes, you’ve all told me that in
Microsoft I shouldn't use men and women.
>>: Why don’t you switch to the multi-[inaudible]?
>> Abraham Neyman: Because this was the Nobel Prize of all this here, I started my
undergraduate game theory CORES with this problem. The first lecture. And I was studying the
solution, and then in the exam, we had just finished checking the, there was, the algorithm was
described in so, such nice words, one would use, he was more attractive, and one was using, I
love you more, he loves you more; each one used his own connotation for the story of the
matching. So, stage K, each man proposes to his favorite woman among those who have not
rejected him in the past. So obviously, in the first date, means that each one, nobody was
rejected, so each one applies to the one ranked top. Each woman who receives more than one
proposal, rejects all but the favorite from among those who have proposed them.
Okay? Now we have to, that's an algorithm. So we have stages. Well it stops. The procedure
stops on the first stage with no rejections. So if there is a given date with no more rejections, it
is being stopped. Obviously, this procedure stops in at most, n square stages, at a stable
matching and you see that it's a stable matching. It is one, let me see if I[inaudible]. So let me
just point to two simple arguments in the matching. If you look on the proposed woman for
each man, these are decreasing in order. None rejected present men that is proposing to him
and is an increasing function. They never go down. They always go up. And therefore, if there
would be a man that is not matched with the woman that prefer each other, it means that the
man was visiting this woman on a previous day. But if he was rejected, it means the one that is
left there is more preferred by [inaudible] over that.
So let's [inaudible] the procedure, so here is that paper. This example where we said there is a
unique one. So the first day, if you look on this algorithm, both alpha and beta are going to
their first most preferred alpha and beta. Gamma goes to B, and delta goes to gamma. The
next day, so who is rejected here? Lady A looks on alpha and beta. She prefers alpha, she
rejects B, so the red one is the rejected one. So beta goes to his next preference, which is B,
and so on, and basically that's the longest procedure that could be, this example is the longest
one. And it's not exactly n square; you could save a little if you wanted to really think how
many steps it takes. But you get the longest one and you could iterate that to [inaudible]
longest one and. Okay.
Now let's look on variations of the story that I was just telling you. First of all, the number of
men can differ from the number of women. Nothing will change. Also, the way that I define
the stopping of the algorithm is the same. Each woman or college, W, to be politically correct
here we have to switch to colleges, can desire to be matched with K, W men students. Okay?
And the third application, which I haven't seen it matched being discussed in the literature, but
it's an HOL[phonetic] extension, which is true. Each club C can have K, C members, and each
person P can be a member of n, P, different clubs. Okay? And each one of these models, if the
preference is indeed without externality, each individual has preference over each one of the
other clubs or without, in clubs usually it's important, really for the application, who are the
other members of the club. Yes, but if, and then it wouldn’t work for this type of stability. But
in the case that there are no externality, each one is ranking the clubs in his order, and who is
the first, second and third and so on. And each club is ranking each member independently of
the others, then in this case, exactly you could define what is a stable matching in a nonobvious
way. And you will get that there is a stable matching.
Now let's raise a few issues arising in this set up. One is optimality. Maybe there are several
stable matchings. We just proved that their exist ones, they were matching. Then
bachelorhood, if there are more men than women than some would be, would remain
bachelors. And can we say something about the size of bachelors in this such a society and
stable matchings? Then there's the issue of the incentive compatibility. What is incentive
compatibility in general? You think okay, we found an algorithm. So maybe we could ask the
matchmaker. We will submit to him and our preferences and he will perform his algorithm and
generate the match. And the question it is incentive compatible if there is an incentive for each
member to give his [inaudible] friends and not to bluff by inserting another preference. So
incentive compatibility means that given that the others are promoting the truth, then you
would like to report your truth as well. And non-strict preference, it’s trivial, an extension in
the story want to sort of include it but it doesn't cause any difficulty.
So let's address first of the issue of optimality. In college, and it's feasible for a student S if S is
matched with C in some stable matching, there could be many, we say there is none; we
haven't yet said anything about uniqueness. Okay? Even though in the work of, for different
reasons of Yuval and Ander[phonetic], there is a stable matching for, a unique stable matching
their for a good reason; but there could be more than one and you say that a pair is feasible to
each other if there's some stable matching that they are matched. Theorem, I think it's already
again in Shapely’s 62 paper. A college that rejects a student is not for the, feasible for him. So
we are looking on the student, the algorithm is called the deferred acceptance algorithm. So
we look here on the proposals by the students to the colleges, and if there is a college that to
reject some student, it's not feasible for him it means that there's no other stable matching in
which they will be matched. Who knows the proof of that?
No. Okay. [inaudible] So I’ll give the proof. It's again, very simple. But again, it’s one of these
examples that some people save oh, I proved it, or I know how to prove it, and they have to
reconstruct and they have difficulties with the reconstruction. So it's very short proof.
Otherwise, in the student proposing the deferred acceptance algorithms, there is a first stage
where a college C reject this. So let S one up to S, K, C, be the students proposing on that day
to Cand not rejected yet in that stage. If C is feasible for S, there is a state of matching such
that C is matched with S and one of this S, J is matched with C, 1. But if this is a stable
matching, it means it's impossible that, we know that C prefers S, J to S. But by stability, it must
be that C, 1 is preferred by this S, J over C. Otherwise, C will go with S, J and will object to this
stability. But this implies that S, J was rejected by C, 1 in an earlier stage in [inaudible].
Okay. Let's work on some corollaries. Some of them are amazing if you stated [inaudible] and
you asked somebody to proof, you will find difficulties. Even once you see these few lines of
proofs, you get the proof of it. If you look on the third accepting algorithm in the matching
which is, in which every member X of the proposing side, so there are two sides, the one that
proposes and the one that rejects. So is matched with its smallest preferred feasible partner.
So you look on all possible stable matchings, yes? And this algorithm gives the most preferred
outcome to the proposing side of the market. And this is because of the previous theorem
because if not, there is somebody more preferred that will be matched in some feasible, in
some stable matching. But the previous theorem says that some of these rejected, a student is
rejected from a college, there is no stable matching by which they are being matched.
Bachelorhood. The set of bachelors is the same in all stable matchings. If you look on more
men than woman, again, this is an exercise to a smart mathematician that is an familiar with
the topic, even though [inaudible] everything Roth’s.
Now let me say, I say that there need not be a unique, but you could look on the reverse
algorithm, say a student proposing to the colleges. We could have looked on the opposite
example by which colleges are proposing to the students. So maybe it'll end up with the same
stable matching. So what has been just written on the blackboard? On the slides. Basically
shows that if the two side deferred accepted algorithms result in the same stable matching,
there is a single stable matching. But typically, there need not be a same stable matching. A
unique stable matching. How much time?
>>: [inaudible]. When I first heard this first corollary, I found it kind of shocking because it
naively seems that it is perhaps the long way around because naively it seems that the people
who are doing the rejecting, the women in the first case, have all the power. But it's>> Abraham Neyman: Exactly. How mucyh time do I have in the talk? Okay. And is the pace
okay? For you, a little slow, I know. So incentive compatibility. I already defined what it is
when I raised the issue. And there is a theorem. Two places where it's still being mentioned in
Dubins and Freedman, 81, and Roth, 82. I think I knew about it in the 60’s as a problem that
[inaudible] assigned in the class. But I [inaudible] confess. So this came up in various set up.
It’s not too difficult, the proposal. There is no incentive for what college [inaudible] to
misrepresent its preferences over order when the college deferred acceptance algorithm is
used. So the proposing aside, for the proposing sides, there is no incentive ever to change its
preference. It is not true for the other side of the market. For the other side of the market, you
could easily cook examples where there is an incentive, given some preferences of the others,
to misrepresent its preference. Roth, in 84, mentioned that there is no stable mechanism for
which stating that true preference is a dominance strategy for every agent. So you may think
maybe there are other algorithms, other of the mechanisms that people will have to plug in
their preferences, maybe we could cook something more sophisticated that will disable an
incentive for any side of the market to give, to misrepresent its preferences. But there is none
like that. Less well known, I don't know how known that this is here to the audience, Yuval, did
you know that?
>>: No.
>> Abraham Neyman: Okay. So at least we could>>: [inaudible].
>> Abraham Neyman: And Roth and Peranson, Roth was basically doing, not only taking the
exact theory and trying to prove particle designs of some markets, but also when there are
flows, like the issues of the incentive compatibility, which theoretically you could say, okay, you
cannot avoid them, are they really essential in large markets? Some of the markets are really
large. So in the paper by Roth and Peranson, they looked on computer simulations with
randomly generated data, as well as on the data from the National Resident Matching Program.
I think one of the first examples of these markets, do you know the story about the National
Resident Matching Program?
>>: [inaudible].
>> Abraham Neyman: In accepting interns into hospitals, there is an old story of how they
really did this matching. And it started, basically the functioning of this market shows that
when there are, this issue of stability causes a lot of instability in these markets. So it started by
universities starting to offer earlier and earlier to make offers to the students up to which a
point, I don't remember the year, that they were offered the position to the students in the
second year of their study. But obviously, we did not know enough information; there is a lot
of information, and they required students gave a positive answer at a very early stage. So the
student wouldn't know yet exactly their preferences, and eventually it was very noisy market
for many years and many switches and they didn't know what to do, until essentially, they
found the Gale-Shapely algorithm in about 1950s. In the practical way they did something very
similar.
So they looked on the data of this program that they really give the preferences and so on, and
also some [inaudible], and it is suggested, and I haven't read this paper, so I don't know what
does it mean, it is suggested what is really quantified there, that in large markets very few
agents could benefit by manipulating their preferences. So maybe there is the issue of
incentive compatibility might be very residual in very large markets. And then there is, the
previous things are not difficult, then there is a deeper theorem. Not deep, but more involved
theorem of Dubins and Freedman from 81, that says that not only it is impossible for one’
member of the proposing side of the market to manipulate this preference, everyone, if
everybody else is recording its correct preference, each one would have to record his own
preference, but they also show it for a group of proposals. So there is no correlation of
proposals that they could get together in the room and say, okay, we know what are the
preferences of the others, let us report different preferences ourselves. Let us report different
preferences of ourselves, and once we reported, maybe each one of us will come better off
from the algorithm that will match us. And they proved that this is impossible. And I think that
what happens is, a small story about that, how those things are usually you think you have it,
but then it's, you don't get it exactly, and you have to, I think in 79, Dubin's told me that they
solved it. Okay? But it took them about two years to reconstruct the proof. They couldn't find,
they were sitting, they were wanted to write in 79, they couldn't write the proof. And then
eventually, they played with it and. Who knows this theorem? Also new. Okay, okay sorry. I
was afraid that>>: I heard this from you.
>> Abraham Neyman: So just an earful for those who want some people enjoy to do the proofs
themselves and necessarily to read the pros. And many of these proofs would basically go by
some clever type of induction on variations of the Gale-Shapely mechanism. If you look on the
Gale-Shapley mechanism, usually, okay there was a story to show this path to this guy about
what is going on. But you could do the mechanism in a different way. You don't have to, all the
men proposing on the first day. You could take them out of order, for instance. Or some order,
yes? A first man comes into the room, proposes to the, to his most preferred woman. Then
the second can deduct, and does the woman proposes. If he proposes to the same woman who
already has somebody and this lady reject him, he goes back to the room. And this order you
could play with it, and it’s the same outcome as the men proposing algorithm. Okay? So
playing with this type of algorithm and doing some clever induction, you get this.
Okay. Let me now go to few related models. There is the assignment game by Shapely and
Shubik. The assignment game is a game by which also there are economic surplus, like here in
matching for two sides of the market, like employers and employees, but in the previous
model, there were no side payments. There were no dowries. The preference will not on the
matching plus dowries or some side payments, but only on the actual preference you're making
from first to last. But in real, in some situations, like employer, employees, you could basically
balance the markets by adjusting the salaries. So they look on the model, which they call the
assignment game, which are two-sided markets with a surplus utility that is being generated,
but by which there is possibility of transferring utility from one to the other.
Another example is the one sided matching of the Shapely and Scarf. Again, this is a general
idea model than I will describe here. But I think the one that I'll describe here is the one that is
being, motivating the application to donations of organs in medicine. And this is, the simple
story is the following: you have K individuals. Each one is a house, but the preferences of the
houses is not necessarily that you like most your own house. So you may like more the house
of [inaudible], you may like more the house of [inaudible], and so on, and could be each one
has some preference over the houses of the others. And so the general concept that is
[inaudible] of stable matching and on the house, this particular game that I'm describing, which
is called the house swapping game, is that you look for a solution, which is called the point in
the CORE. The point in the CORE is an outcome such that no group of individuals can, by
themselves, improve upon this outcome. So for this particular of house, we look for relocation
of the houses such that there will be no possibility for it some proof individuals look on their
own houses, redistribute it differently than what was reallocated the houses, and each one be
better off. And the question was, does this game has a nonempty CORE? Is the [inaudible]?
And Scarf, Shapely and Scarf, maybe it should be Scarf and Shapely, yes? A, B, C you, sorry.
Scarf and Shapely developed a general theory of what's called the CORE of games within the
feasibilities and without side payments. So here you cannot buy the houses. It's only
redistributed in the solutions. And in particular, they proved that this particular house
swapping game is a nonempty CORE.
Obviously, the general machinery for that is not necessarily because there is a very trivial
algorithm for that which was proposed by David Gale, which is called the top trading cycle.
Familiar with that? No? So it's a very simple algorithm. So let's show. Each man points his end
on the house he most likes. Okay? So you have a graph between individuals on the houses.
This graph must have a cycle. So basically in the cycle, it means that if you trade these houses
among the members of these elements of the cycle, each one gets his most preferred house.
So take these houses and those individuals out and by induction theories, either continue the
algorithm or do the same.
There is another paper by Shapely and Shubik, which somehow is not mentioned in the, no, is
mentioned, is reference in the Nobel Prize citation, but is of less of degree which is called
market games. That's another class of games that they studied which is very analogous to
actual market performance, individuals of endowments, they have utility or they have
production functions they could trade between them, the endowments, and the question is
whether there is something in the CORE namely whether there is some allocation of the
resources by the individuals or of the profits of the firms such that no subgroup of individuals
could improve their outcome by just isolating them from society and trading between
themselves. And they give the conditions, relatively the conditions that guarantee that in this
type of model, the CORE is nonempty. And each time, I already mentioned a few times the
CORE, when I say what is the CORE? And the CORE is the set, you have a correlational game,
you have a group of individuals, each subset of individuals could force some set of outcomes.
And you will say that some solution that these proposed by an arbitrator of the society, if you
want, look at some society and proposes some outcome, that this outcome is in the CORE if
there is no subgroup of individuals that can, by themselves, depart the society and trade
between themselves and get or force an outcome that is better for each one of the members of
this group.
So each one of the previous cases is an example of economic models in which the CORE is
nonempty. And it is often argued that in these models where the CORE is empty, there are
always type of frictions and instability in the outcome, but unfortunately, we know that there is
a lot of instability. There are strikes, there are wars, etc., and the reason is that in most games
the CORE will be empty. And so here we proposed a few classes of examples which some
special structure for which the CORE was not empty. There was a solution which was stable.
Usually, there is no solution, which is in many examples now, no such solution, and if we want
to think out of the box, out of game theory, we have to think more boldly on comparative game
theory, and indeed there is one solution which I like most among the comparative game theory,
which is called the Shapely value, also by Shapely. And this basically will propose a kind of a
solution to any type of game in correlation [inaudible]. And obviously, as we have seen, there is
a lot in this story that was not only the comparative game, but also the kind of incentives of
individuals, of acting, participating in these type of markets. Yes.
>>: Just to clarify a question regarding market games. Are these non-comparative games?
>> Abraham Neyman: Comparative games.
>>: So all of these models are>> Abraham Neyman: All of these models are comparative games. There was a little mixture of
non-comparative game which was when we discussed the issue of incentive compatibility.
Because once you see that this is the solution, still, the issue is that when you apply
noncomparative, when you apply comparative games, there are some information that are
private to the individuals. Like the preferences. So often, we want to think that if we are
proposing something into such a solution, and individuals were left to be revealing their own
preferences, you want to make sure that there is no incentive for an individual to misrepresent
his preference. So there's a little mixture, a little interplay in this type of theory of noncomparative means. But most of it is comparative. There was another question here? Any
other questions?
>>: Okay. So maybe you can comment why you think that the residents market, how does this
Gale-Shapely algorithm, why do other markets like the [inaudible] market in universities doesn't
have it? I mean, you still have all these instabilities, offers being made earlier and earlier and so
on?
>> Abraham Neyman: I don't know, I think often when you make a postdoc as appointment
and you make more than one, right? And I'm not sure that your own preference over the
candidates are without externalities. So often, you would like to have a pair of postdocs that,
either because you want to work in two different areas and you want to have that diversity, and
sometimes because you want them to be in the same area, so that it could be more productive
together. So there are externalities. And let me point, thank you for the question, because
there is also similar issue with the interns going to hospitals. Because sometimes, basically,
when students are already graduating, often have a couple of marries. Married couples. And
the married couples have a preference, which is externality, because they want to be matched
to the same thing. Once you insert that, there is no solution that will be stable. Okay? If you
insert [inaudible]. But still, Roth was basically aware of this fact, aware of the theoretical
difficulties that this causes, but still, understanding these difficulties, was trying to help
matching the interns in our P market, how to overcome it with minimal friction. But there is
always friction. Yes?
>>: So is there an estimate of how fast a certain member for colleges reach their best
preference in the sense, is it true that the top ones reach their preferences faster than the
middle level ones? So the algorithm is running, right? And so the women are getting their
mates from the men, so we say something about when do the women get their mate? Do the
top-ranked women get them faster than the middle-ranked women? Yeah, so, I guess that is
the question. But I'm trying to think of it as a college, is a question of time, and then you have,
you probably answer, rejected or answer but a given time. So is there a time component, but
are there some groups which benefit more than other groups?
>> Abraham Neyman: The group that benefits is the proposing side of the market. They get
the best solution to themselves. Now, in terms of time, that you ask in times, in many of these
markets that got organized, like some schools in New York district area, you have the
applications, you have the preference, it’s essentially, you submit your preference, and the
program is deciding on the allocation.
>>: You, as in both sides.
>> Abraham Neyman: Hmm?
>>: When you say you, as both sides.
>> Abraham Neyman: Both sides submit their preferences. Yes?
>>: [inaudible] you decide that for the side that is doing the rejection [inaudible] can be
incentive for them to misrepresent their preferences. But that's willful information if they
know the whole [inaudible], so what about if they just have some partial information? Maybe
their preferences [inaudible] a little bit more. Is there some systematic way for them to
misrepresent their preferences so that they cannot be worse, so perhaps they do better
[inaudible]; it’s a vague question because I didn't say what vital information>> Abraham Neyman: So the results that say that there is, it's examples that they say that there
is, okay? I'm not familiar with any work, for somebody that will say their preferences are being
drawn from some distribution. And you don't know this distribution, to what extent this will be
phased out, these incentives.
>>: But it might, I mean it's not necessarily a distribution. It could just be, here’s a
deterministic algorithm if you know, for example, if each woman knows her own preferences,
and she also knows that the algorithm is being run exactly in stages, so in a position away, for
example, here's one person, is there some deterministic algorithm for the women to
sometimes make false rejections just given their own preferences and what has happened in
the past, which guarantees that they do no worse and gives them the possibility to do better.
[inaudible].
>> Abraham Neyman: Without knowing preferences of others.
>>: Right. But they still have the information of, you know, [inaudible]. I mean, that seems to
[inaudible] if you have a little [inaudible].
>>: We will take further questions off-line. Thank you.