Scaling Behavior in the Stable Marriage Problem

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Marie-José Oméro, Michael Dzierzawa, Matteo Marsili, Yi-Cheng Zhang

To cite this version:

Marie-José Oméro, Michael Dzierzawa, Matteo Marsili, Yi-Cheng Zhang. Scaling Behavior in the Stable Marriage Problem. Journal de Physique I, EDP Sciences, 1997, 7 (12), pp.1723-1732.

�10.1051/jp1:1997166�. �jpa-00247483�

J.

Phys.

I IFance 7

(1997)

1723-1732 DECEMBER1997, PAGE 1723

Scaling Behavior in the Stable Marriage Problem

Marie-Jos4 Omdro,

Michael Dzierzawa

(*),

Matteo Marsili

(**)

and

Yi~cheng Zhang

Institut de

Physique Th60rique,

Universit6 de

Fribourg,

1700 Fribourg, Switzerland

(Received

15

July199( accepted

18

August 1997)

PACS.05.20.~y

Statistical mechanics PACS.01.75.+m Science and society

PACS.02.50.Le Decision

theory

and game

theory

Abstract. We

study

the optimization of the stable marriage

problem.

All individuals at-

tempt to optimize their _own satisfaction, subject to

mutually conflicting

constraints. We find that the stable solutions _are

generally

not the

globally

best solution, but

reasonably

close to it. All the stable solutions form _a

special

sub~set of the meta~stable states,

obeying interesting scaling

laws. Both numerical and

analytical

tools _are used to derive _our results.

1. Introduction

Optimization problems

have become _an

interesting

_area ^of research in statistical

physics. They usually require

to find the minimum

(or minima)

of _a

global quantity

_e.g.

Hamiltonian). Spin- glasses

_are

just

_one of the _current well-known

examples.

^In the context of social science _or economy, decision-makers _are individuals _or

companies. They

have their _own rather selfish

goals

to

optimize,

which _are often

conflicting.

This

situation,

which is

typical

in game

theory,

cannot be described

by

_a

global

Hamiltonian

II, _2].

One of its

simplest

realizations is in the

classical stable

marriage problem [3-5]. Despite

its

simple

^definition the solutions have a very rich structure.

The stable

marriage optimization problem

does not

require

the

"energy"

to be the smallest

possible,

but that the

resulting

state be stable

against

the

egoistic attempts

of individuals to lower their _own

"energy".

This _new

concept

of

equilibrium

is

typical

in game

theory

where _one deals with _a number N of distinct

agents,

each of whom is

trying

to maximize his

utility

at the _same time. It is not true in

general

that the state with the

largest

total

utility

will be _an

equilibrium

state. Indeed it is

possible

that in such _a state _an

agent

would benefit from

making

an action which increases his

utility

at the expense of others. This leads to the

concept

^of Nash

equilibrium _[6],

which is _a state characterized

by

^the

stability

with

respect

to the action of any

agent.

^{In other}

words,

_a state is stable if _any

change

in _an

agent's strategy

^is unfavorable for

himself.

The

marriage problem

describes _a

system

where two sets of N persons have to be matched

pairwise.

We shall _assume that these sets _are

composed

of _men and _women who _are to be

(*)

Present address: Institut for

Physik,

Universitbt

Augsburg,

86135

Augsburg, Germany

(*^* e~mail: [email protected]

@

Les

#ditions

_de

Physique

1997

married.

Clearly

the

marriage problem

is

applicable

in many different contexts where two distinct sets have to be matched with the best satisfaction. For

instance,

_{one can} consider N

applicants facing

^N'

jobs.

Each

applicant

has _a

preference

list of

jobs

and each

job~owner

ranks the

applicants

in order of

preference.

For the convenience of

presentation

we

exclusively

use the

paradigm

of

marriage

between _men and _women.

Suppose

the _men and the _women in the two sets know each other well. Based _on his

knowledge

each _man establishes _a wish-list of his desired _women, in the

descending priority order,

I._{e, on} the top ^{of his list is his dream}

girl;

the bottom is _a mate whom he has to marry

only

in the worst _case when all the other _women

reject

him. The _women do

exactly

the _same to the _men. Note that in the convention of this

model, everybody

must marry.

Each

person's

satisfaction

depends

_on the rank of the partner ^he

/she gets

to marry. Thus the rank _can be _{seen as a} cost function. If the

top

^{choice is}

attained,

the cost is the

least,

the bottom choice has the

highest

cost. Two _{men may}

happen

to put ^the same woman as their top

choice,

or two _women may

happen

to

prefer

the _{same man.} There _are

necessarily conflicting

wishes which

give

a

special complexity

to the

problem.

We shall deal here with the _case in which each

person's

wish~list is

randomly

and

independently

established.

2. The Model

In order _to set up the notation let _us look at the

following example

of three _men and three

women. The

preference

lists for all persons are shown below.

1: 1 2 3 1: 31 2

2: 1 3 2 2: 1 2 3

3: 2 3 1 3: 3 2 1

Men's

preference

lists Women's

preference

lists

The detailed cost for each

perion

can be set

observing that,

_e.g, if _man I marries _woman 2 his cost is 2 since she is in the second

position

of his

preference

list

and,

vice _versa, the cost

for _woman 2 is I. If _man 2 marries _woman

I,

his cost is I and if _man 3 marries _woman 3 his

cost is 2. The costs of _women I and 3 in these _{cases are} 3 and

I, respectively.

It is convenient to introduce _a

representation

of the lists in terms of

rankings:

We define the matrices F and H for _women and _men

respectively,

such that

f(w, m)

denotes the

position

of

a man m in the list of _{a woman w.}

Equivalently h(m, w) yields

the rank of _{a woman w} in m's

list. Rank

plays

here _a similar role to energy in statistical

mechanics,

_{so we} shall

frequently

refer to

f(w, m)

_or

h(m, w)

as

energies.

A realization of the

preference

lists is also called an

instance. A

matching

is _a set of N

pairs

fi4

=

((m~,w~),

I

= I,.

,N),

and there _are Nl

possible matchings

in _an instance of size N.

The

problem

is to find _a stable

matching

fi4

=

( (m~,

w~

))

such that _one cannot find _{a man m~}

and _{a woman} wj ^who are not married

(I # j)

but would both

prefer

to marry each other rather

than

staying

^with their

respective partners

_w~ and mj. ^Such a

couple

is called _a

blocking

_pair.

A

blocking pair (m~, wj)

is then such that

f(wj, m~)

<

f(wj,mj)

and

h(m~, wj)

_<

h(m~, _w~).

If _no such

pair exists,

the

matching

is called stable.

One _can calculate the _{energy per} person, for _women and _men in _a

given matching

_as

N°12 SCALING BEHAVIOUR IN THE STABLE MARRIAGE PROBLEM 1725

and the energy

£(fi4)

=

£F(fi4)

+

£H(fi4)

_per

couple.

Here and in the

following

the

subscripts

F and H stand for _women and men,

respectively.

3. The

Gale-Shapley Algorithm

A lot of work has been

spent

in

developing

fast

computer algorithms

to find all stable

matchings

for _a

given

instance of size N

[4, 5j.

These

algorithms

_are based _on the classical

Gale-Shapley (GS) algorithm

_[3] which

assigns

the role of _{proposers to} the elements of

one

set,

^the men say,

and of

judgers

to the elements of the other.

The man-oriented GS

algorithm

starts from _{a man m}

making

_a

proposal

to the first _woman

w on his list. If she

accepts they get married,

if she refuses _m goes on

proposing

to the next

woman on his list, _w accepts a

proposal

when either she is not

engaged

_or she is

engaged

with _{a man} m' _worse than the _one

proposing (m).

In the latter case, m' will have to go on

proposing

to the _woman

following

w on his list. When all _men have _run

through

their lists

proposing

until all _{women are}

married,

the

algorithm stops

and the

matching

thus reached is

a stable

matching.

As _an illustration let _us consider the

example

of the

preceding paragraph.

The man~oriented GS

algorithm

_{goes as} follows: _man 1 proposes ^to woman 1 who accepts and

they

form the

pair (I, I).

Then 2 proposes ^to

I,

^{but she refuses. So} man 2 proposes ^to woman 3 and

they get

married.

Finally

man 3

happily

marries _woman 2. This results in the

matching

MH

"

ii i,1),

_12,

3),

_13,

2)1'

The GS

algorithm

_can be _run

reversing

^{the roles}

(woman~oriented)

_to

yield

the _woman-

optimal

stable

matching.

In _our

example,

this leads to

fi4F

~

Ill,1), (2, 2), (3, 3)).

The

energies

for _men and _women in these

matchings

are

(£H, £F)

_#

(4/3, 7/3)

and

(6/3,

5

/3) respectively

for

fi4H

and

fi4F.

As _seen in this

example

who proposes is

always

better off than who

judges.

It _can be shown [4] that the man~oriented GS

algorithm yields

the

man~optimal

stable

matching

in the _sense that _{no man can} have _a better

partner

in any other stable

matching.

It is rather

surprising

that

though

_men

get rejections nearly

all the time and

just

one

positive

answer,

they

_are far better off than _women. On the other hand _women who take the

pleasure by saying

_no to almost all the suitors

except

one who is best among her

suitors,

^will end up ⁱⁿ

a

marriage

that is the worst among all

possible

stable _ones. The lesson is that the _person who takes initiative is rewarded.

In order to

quantify

_more

precisely

this statement, it is

enough

to observe that I) once

a woman is first

engaged,

she will remain

engaged (eventually

with different

men) _forever,

and that

it)

the total _energy for _men in the

man~optimal

GS

equals

^the total number of _pro~

posals

_men need to make to marry all _women.

Proposals by

men define _an intrinsic time in the

algorithm. Imagine

that at time tk

(I.e.

after tk

Proposals)

the kth _woman

gets

en~

gaged.

In view of the randomness of the

preference lists,

the

probability

that the next pro~

posal

is addressed to one of the N k free _women is I k

IN.

On average, ^men will need

N/k

=

(tk+i

_tk

proposals

to engage ^one more woman. Since the total energy for _men is

N£H

# tN, we find

~

£H(fi4H)

_#

~

G~

log

^N + C

11)

~_~

k

where C

= 0.5772. is Euler's constant.

Taking

into account that _men do not propose twice

to the _{same woman}

yields

_a correction of

O((logN)~ IN)

to

equation (I).

On the average

each _woman receives

log

N + C

proposals

^of men who _are

randomly

distributed between I

and N _on her

preference

list.

Keeping only

the best

proposal

the _women arrive at _an energy

of the order of

£F(fi4H)

_t

N/ logN

which is much

larger

than the

corresponding

result

(I)

for the _men.

4. Stable

Matchings

in the

Large

N Limit

Coming

back to our N

= 3

example,

it is also

interesting

to note that _none of the two stable states we found is the _one with the smallest energy, the

"ground

state". Indeed the state

fi4o

#

Ill, 2), (2,1), (3, 3))

has

(£H. £F)

#

(5/3, 5/3),

and _a total energy

£(fi4o)

#

10/3

which is lower than those of the other two states. This state is however unstable. Indeed there is _one

blocking pair ii, I).

This

simple example already

shows _some

interesting

features of the stable

marriage problem:

I)

minimum energy I.e. maximum

global

satisfaction does not

imply stability land

vice

versa)

and

2)

there _can be _more than _one stable

matching (the

GS

algorithm guarantees

that there is

always

at least _one stable

matching).

In order to decrease the total energy of _a

given

stable

matching,

it would be necessary that

some individuals _pay the

price

of

accepting

a worse

partner

^than the _one with whom

they

_are

actually

married. But in absence of _a

supervising body (like government

or

parents) they

do not consider to

help

others

by switching.

The inherent selfishness and

conflicting optimizations

in the stable

marriage problem

lead to stable solutions that _are not

globally optimal.

It is known that the _average number of stable _states in _an instance of size N is

proportional

to N

log

N [5].

Starting

from the

man-optimal Gale~shapley

solution all other stable

matchings

can be obtained

by performing cyclic exchange

_processes

1/l1, bJl);

,

(/lr, bJr)

~

(/ll, _bJ2);

,

(/~r, bJl) (2)

within _a

properly

chosen group of

pairs

of _men

~1~ and _{women ~~.} These

exchange

_processes

are called rotations. The number _r of

pairs

involved in _a rotation _can be

regarded

as the

"distance" between the two stable

matchings

connected

by

this rotation. Efficient

algorithms

exist for

finding

all rotations and therefore all stable

matchings

in _a

given

instance

[5].

^In this _way it has been found that stable states _are

organized

in _very

peculiar graph~theoretical

structures

[5j.

We shall first

analyze

the statistics of stable states and then return to the

concept

^{of rotations.}

It is

possible

to derive _a

general

relation between the

energies

of _men and _women in the stable states. Our aim is to evaluate the energy of _men,

given

^{that of} women in _a stable state.

We _can then _assume that _a stable

matching,

in which _{woman w} has energy £w, exists. In

order _to find the stable

matching

_we consider the

"hypothetical"

situation in

which,

for _some

reason, the _women know that

they

_can reach _a stable

matching

where _{woman w} has energy

£w.

Knowing this,

the best

strategy

^of women becomes that of

refusing

all

propositions

from

men

ranking higher

than _£w. The best

strategy

for _{men, on} the other

hand,

remains that of the GS

algorithm.

The

dynamics

of this modified GS

algorithm

will

clearly

reach the stable

state foreseen

by

_women: Each _{man m} makes his

proposals sequentially

until he hits _{a woman}

w such that

f(w, m)

< £w. In order to

compute

the average man energy, consider _{a man m}

and,

in order to

simplify

the notations, let his wish list be

h(m, w)

= w, for _w

= I..

,

N

(~)

His

proposal

to the wth _woman will fall

randomly

within I and N in the

rankings

of _{woman w}

and, according

to the above

strategy,

it will be

accepted

with _a

probability

_£w

IN. Otherwise,

if it is

refused,

_{man m} will consider the _w + lst _woman. The

probability

that this _man will

(~) ^This can

always

be achieved with _a permutation of _women indices.

Clearly him'. w) #

_w for m'

~

_m, in

general.

N°12 SCALING BEHAVIOUR IN THE STABLE MARRIAGE PROBLEM 1727

get

^his

qth

choice is then

q-1

~ ~

PH(q)

"

fl _(1~ ~) _{# (3)}

w=1

We _{can now} take the average on realizations of the above

equation.

Under the

assumption

that _{£w are}

independent

random variables with average £F, we find

PH(q)

₌

(1- ))~

^~

).

₁₄₎

From this _{we can}

compute

the average energy for the mth _{man £H}

"

£ qPH(q).

This leads to the relation

£H eF #

N.

(5)

In other

words,

in stable

matchings, for

random _instances

of

the _marriage

problem,

the energies

for

men and _{women are}

inversely proportional.

Of course,

reversing

the roles of _men and _{women, one} finds the _same conclusion

equation (5)

and _a distribution

PF(q)

_t

exp(-q/£H)/£H

of _women

energies

which

depends

on £H.

It is _now

possible, returning

to the

assumption

of

independence

of the

£w's,

to show that

only

a weak correlation exists _so that

equations (4, 5)

_are exact in the limit N _{~ cc.} In order to do this _{we can run} the above

argument

in its women~oriented version

(with

_women

proposing)

to derive the

joint probability

distribution of the

energies

of two women. With

independent

men

energies,

it is clear that unless two women propose ^to the _{same man,} there will be _no correlation between their

energies.

The

probability

that both propose to the _{same man} is of the order of

£)/N~,

which

implies

_a weak correlation of the form

&Ej&Ez =

)

Ez

)~

16)

among

energies.

^This

clearly

holds both for _women and for _men under the

assumption

of

the

independence

of _{men or women}

energies, respectively.

It _can be

easily

_seen that _women

(men) energies

are also

weakly correlated,

as in

equation (6),

if _men

(women) energies

are

weakly

correlated. We therefore conclude

selLconsistently

that

energies

are

weakly

correlated in stable

matchings.

In _presence of _a weak correlation of the form

(6),

the _same

procedure,

from

equation (3)

to

equation is)

leads to £H£F #

Nil

₊

4c/N).

Therefore

equation is)

is exact in the limit N _{~ cc.} We found

numerically

that the _{constant c} is

generally negative (c

cf -0.3 in _man

optimal states).

This correlation is similar to the _one

occurring

_among N variables whose _sum is constrained.

Our numerical results indicate that the relation

is)

is

already

satisfied

approximatively

for

a rather small number of

pairs. Figure

I shows

log

_{£H as a} function of

log

_£F for all sta~

ble

matchings

found in systems of size N

=

50,100,200,500,1000.

The

asymptotic

result

log

_eH +

log

_eF

#

log

N is indicated

by

lines. The

points

on the left and _on the

right

of each set of data

correspond

to the _man and the _woman

optimal

GS

solutions, respectively.

Figure

2 shows that the distribution of individual

energies

in _a stable

matching

_{agrees very}

well with the

predicted exponential

behavior

equation (4).

Although,

as shown

by

the GS

algorithm,

stable

matchings

can be _very

asymmetric regarding

the energy of _men and _women, there is nevertheless _a minimum _energy of order

tJ(log N)

that

cannot be reduced further without

losing stability.

Stable solutions where either _{men or women}

possess an average energy of

Oil

_are not

possible.

,

O Stable Matchings

' ,

N=I COO

', ^N=SOD

', ^N=200

, N=TOO

... ', N=50

' ' O Ground Stale

'l

°~

m

°

o

' O

~

~b'o_g§°

',,@

°

'i~~Q

".,

^"

'o '

"., _'

'o, '

", '

I ' '

2 3 4 5

log(

_E~

Fig. 1.

Average

_energy of _men

(eH)

and _women

(£F)

for all stable

matchings

in systems of size N = 50,100, 200, 500,1000

(from

left to

right)

_{on a} double logarithmic scale. The

analytic

result of equation

(5)

is indicated

by

lines. The

(o)

point below each line

corresponds

to the Ground State energy.

5.

Dynamics

between Stable States

Rotations

play

_a central role in the

algorithm

which finds all stable _states. As for the

GS,

this

algorithm

has _a man-oriented version and its woman~oriented

counterpart.

^{As mentioned} a

rotation is _a

cyclic permutation

^of

partners

within _a subset of persons in _a stable

matching fi4,

which allows to reach _{a new} stable

matching

^fi4'. In the man-oriented

algorithm,

to which _we shall restrict

attention,

the execution of _a rotation raises the energy of _{any man}

involved,

and lowers the energy of the

corresponding

_woman. In this way,

starting

from the

man~optimal

state, ^the execution of

rotations,

in all

possible

orders

(~),

allows to reach

sequentially

all other stable states until the _woman

optimal

_one. This process,

therefore,

_runs

through

the set of stable states shown in

Figure

1 from top

(the man~optimal state)

to bottom

(the

_woman~

optimal state).

We _can understand this process with _a

generalization

of the GS

dynamics. Imagine

that in _a stable state fi4

=

((m~, w~)),

_a woman ~i, for _{some reason} divorces. This event leaves

an

unpaired

_{man ~li,} and the GS

dynamics

starts

again:

~li will _run

through

his list

making proposals

to the _women

following

_~i until he finds _{a new woman ~2} which

prefers

him to her

partner

_~1~. This will

put

man ~12 in the _same situation _{as man} ~li before. The process will

continue, involving

other _men

~1~ and _{women ~~,} until _{a man ~lr} will make _a

proposal

to _woman

~i Under the GS

dynamics,

this

proposal

will be

accepted,

because _~i is free. It

might happen

that the _new

partner

^of ~i is better than the _one she left

f(~i, tlr)

_<

f(~i, tli).

In this _case

(2)

^{A rotation, like the} one in equation

(2),

_can be executed _{on a} stable state only if it is

exposed,

i,e.

only

if each _{man p~} is

paired,

in that state, with the

woman uJ~

specified

in equation

(2),

for

I =1,.. ,r.

N°12 SCALING BEHAVIOUR IN THE STABLE MARRIAGE PROBLEM 1729

O°

O men

. women

Eq.j4) 10~

SZ

£ ^lO~

#

r O-

0 ^~

~i§~

lo~

_.

o

2 4 6 8

x=E/E~,

E/E~

Fig.

2. Distribution of individual energies of _men

(open circles)

and _women

(full circles)

for all stable

matchings

in

an instance of size N

= 1000

on a

logarithmic

scale. The energies of _men

(women)

are scaled by the their average values: _x

=

h(m~,w~)leH

for _men, and _{x =}

f(w~, m~)lef.

The solid line is the

analytic

result of equation

(4).

the state thus reached will

again

^{be stable.} The

dynamical

_process described

above, exactly

represents

in this _case, the execution of the rotation

(2).

On the other

hand, if,

for _{woman uJi,} tlr is a worse partner than _~li, the state will be unstable.

Indeed

(~li, uJi)

constitute in this _{case a}

blocking pair:

^{both of them would indeed}

prefer

to

get together again

than to be married with _uJ2 and _~lr

respectively.

We _can, in this _case,

regard

the above _{process as a} "virtual" _process that does not lead to a new stable state and that leaves

the state

unchanged.

Note that the

probability

that the next

proposal

_uJi receives is better than the _one she holds in the stable state is

f(uJi,tli) IN

_ct

£F/N.

_£F

r~

4 _implies

_{that the}

probability

that any

woman

improves

her situation with _a divorce is very small

Oil Ilk).

With such _a small

probability

divorce is very

risky

^for _{any woman} under the

strategies

fixed

by

^{the GS}

algorithm (Note,

_on the other hand, that there will be

O(/k)

women uJi for which the above _process

leads to _{a new} stable

state).

This

dynamics

motivates _a

deeper study

of the statistics of rotation

leng~hs.

Indeed _{we can} understand the execution of _a rotation _as the _response of the

system

to the small

perturbation

which _causes the first divorce. While the response is

generally

linear in

equilibrium

statistical

systems,

we shall _see that _a small

perturbation

_{on a} stable

marriage

can cause a

large

_response,

I.e. _a

large change

in the

system.

This is reminiscent of the behavior of self

organized

critical

systems.

^The statistics of rotation

lengths

is also

interesting

to understand the

"geometrical"

nature of the

organization

^of stable states. Indeed the rotation

length

_{r measures} the "distance"

between two stable states, I.e. the number of

marriages

^{which differ in the} two

matchings.

o N=loo

V N=200

+ N=500

O N=IOOO

-exp(-x~)

_

O-1

U7 d Z

~

£

~ u~

"z

_o.Ol

+

o.ooi

_

0.5 1.0 1.5 2.0 2.5

r/

N°'~

Fig.

3. Scaled distribution of rotation

lengths

for systems of size N ₌ 100, 200, 500,1000. ^The

curve is _cc exp

(-x~ _/2).

We found that the normalized distribution of the

length

_r of _a rotation satisfies

~~~' ~~

r0

N)

~

ro _N)

(7)

with the

typical

rotation size

scaling

with N _as

ro(N)

rw

/k.

The scaled distributions _are shown in

Figure

3. All data

collapse

on a

single

_curve which is

remarkably

well fit

by

_a Gaussian

p(x)

_i

f~ exp(-x~) (line).

This

scaling

behavior _can be

understood

considering

the above mentioned extension of the GS

algorithm.

Note indeed that the number of _men involved in the process is _r.

Using arguments

similar to the _ones

leading

to

equation (5),

_{one sees} that each _man

typically

needs an additional

N/£F Proposals.

So the total number of

proposals

received

by

_uJi is

r/£F.

The

length

_r of the rotation is obtained

imposing

that _uJi receives

rw 1

proposition.

This

implies

that _r will be

typically

of the order of _£F

r~

4.

It also

implies

that the men's energy difference between the two states is

d£H

_t

r/£F,

which is of order _one. This agrees, apart ^from

logarithmic corrections,

with the observation [5j ^that

there _{are rw} N

log

N stable

matchings.

6.

Globally Optimal

Solution

While there exist

powerful

numerical

algorithms

to obtain all stable

matchings

in _a

systematic

way it is _a much harder

problem

to find the

ground state,

I._e. the

matching

with minimal total energy. For _an

analytical approach

it is convenient to introduce the random variable

xlm,

_W)

=

( _lhlm,

W) +

flw, m)1

18)

N°12 SCALING BEHAVIOUR IN THE STABLE MARRIAGE PROBLEM 1731

which is the normalized energy associated with the formation of the

pair (m, w).

Since _we

are no

longer

interested in

stability

considerations it is not necessary ^to

distinguish

between the energy of _men and _women. In the

large

N limit

x(m, w)

can be treated as a continuous

random variable with distribution

p(x)

=

min(x,

2

x)

for 0 _{< x <} 2. The

problem

of

finding

the minimum of

N

£

(fi4

"

~j

X

(lUz,

Wz

(9)

1=1

over all

matchings

fi4 reduces to the

bipartite matching problem

which has been solved

by

MAzard and Parisi [7]

using

the

replica technique

for the disorder average.

Following

their

approach

_we obtain

£mjn =

1.6174. ₍₁₀₎

On the other

hand,

minimization of

£(fi4)

=

£F(fi4)

+

£H(fi4) subject

to the

stability condition,

I.e. to

equation (5), yields

£$(f~~

=

24. _(I I)

Thus

giving

up the constraint of

stability

allows for _a reduction of energy

by

19 percent.

7. Conclusions

We have

investigated

the classical stable

marriage problem

which

despite

its

simple

definition contains the full

complexity

of

highly

frustrated

systems

like

spin glasses.

In contrast to the

traditional

examples

of statistical mechanics where the

dynamics

of the

system

is

governed by

a

single global _quantity,

the

Hamiltonian,

the stable

marriage problem

fits _more

naturally

into the framework of game

theory

where the concept ^{of Nash}

equilibrium plays

the central role.

The

game~theoretical

definition of

stability

leads to the somewhat

paradox

result that

although

all individuals do whatever

they

_can in order to maximize their

personal

benefit the

resulting

stable states are not the

globally

best solution. In the

large

N

limit,

_we found stable states with total

energies ranging

from

£$](~~

=

N/ log

N to

£$])~~

=

2/k,

_whereas the

globally

best solution has _£m;n

=

1.617/k.

_Within

a stable

matching

the distribution of individual

energies,

say for the _men, is

decaying exponentially

where the

decay

constant is determined

by

the _mean energy of the women, and vice _versa. As _a consequence, the _mean

energies

of _women and _men

satisfy

the

simple

relation _{£H £F}

= N. We also studied the distribution of distances between stable

matchings

in _an instance of size N. This distribution turned out to be _a universal

function when the distances _are scaled with the

typical

rotation

length

_ro

r~

4. Introducing

a

simple dynamics,

^{this result} also

implies

that the

system

^is characterized

by

_a non-linear response to

perturbation

similar to the _one observed in self

organized

critical

systems.

There _are still many open

questions

to be

investigated

in this

problem.

It would be

interesting

to compare the distribution of individual

energies

ⁱⁿ the

ground

state, I.e. the

globally

best

solution,

with the _{one we} found in stable

matchings.

The

latter,

_as

shown,

_are well described

by independent

variables with _{a common} distribution. On the other hand _one

expects that,

in the

ground

state,

they

_are much _more

strongly (anti~)

correlated and that individual

energies

can fluctuate much _more

wildly (see

_e.g.

_[8]).

Another

interesting question

is how _many

blocking pairs

^there are in the

ground

state since this would be _{a measure} for its

degree

of

instability.

A very

rough argument suggests

that this

number is of order

£$;~ /4

_rw N.

We _are also

presently investigating

a

generalization

of the model where the

assumption

that the

preference

lists of different individuals _are uncorrelated is

replaced by

a more realistic

hypothesis.

Acknowledgments

This work _was

supported

in part

by

the Swiss National Foundation under grant 20~40672.94

/1.

References

iii Fudenberg

D. and Tirole J., Game

Theory (The

MIT

Press, 1991).

[2] Marsili M. and

Zhang Y.-C.,

Fluctuations around Nash

Equilibria

in Game

Theory,

to

appear in

Physica

A

(1997).

[3] ^{Gale D. and}

Shapley L.S.,

Am. Math.

Monthly

69

(1962)

9.

[4] Knuth D.

E., Mariages

Stables

(Les

Presses de l'UniversitA de Montr4al,

Montr4al, 1976).

[5j ^{Gusfield D. and}

Irving

R. W., The Stable

Marriage

Problem: Structure and

Algorithms (The

MIT

Press, Cambridge, Massachusetts; London, UK; 1989).

[6] Nash

J.,

Proc. Nat. Acad. Sci. 36

(1950)

48.

[7j ^{M4zard M. and Parisi}

G.,

J.

Phys.

Lett. France 46

(1985)

L~771.

[8j Marsili

M.,

J.

Phys.

A 29

(1996)

5405~5419.