for compact convex sets - American Mathematical Society

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 49, Number 2, June 1975

ON A GAMETHEORETIC NOTION OF COMPLEXITY

FOR COMPACTCONVEX SETS EHUD KALAI AND MEIR SMORODINSKY ABSTRACT.

The

duced by Billera

sets

in R"

of complexity

1. Introduction. model of market valued

notion

zz. Also

Let

defined

An allocation

u.,

x>>0

for every

(2) V¡=lx\= We define

the function

occur

z?z-dimensional

= (uy(x\,

x\,

of zz is a compact

if for some

u.(xy,

the bundle

portion

...,x)),

of R" (see

convex

set

of n traders

on Im, where Received

amount

paper

C C R",

Key words market

*£

[l]).

lm.

by

•",*;)).

A subset

we say that V is

("players")

available

by Billera the set

VCR"

is

ix)\.

p < q if p¿ < 3 there

• • • , zz be zz concave

1 for i= 1,2,-..,

u((x[))

(For

definitions

«,,

on the

convex

that

for zz = 3 the maximal

(of the commodities)

zzz; / = 1, 2, • ■• , n, satisfying (1)

for compact It is shown

The following

games.

functions

of complexity

and Bixby is considered.

m < nin - l).

They

13, 1973 and, in revised

(1970). convex

that for any compact

for some iu,,I u~, • • •, u)n ¿ defined

the coz7z-

form, May 13, 1974

Primary 90A15, 52A20; Secondary sets,

concave

utility

functions,

com-

games. Copyright © 1975. American Mathematical Society

416

License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

A GAME THEORETIC NOTION OF COMPLEXITY plexity

of V (or C) to be the minimal

plexity

of n (com(zz))

ttz for which

to be the maximal

complexity

this

417

is true,

and the com-

of an attainable

set in

R". It follows is easy

from the work of Billera

to see that

an attainable

set in R3 with complexity

conjectured

that

n and,

that for every

therefore,

paper

2, i.e.

com (zz) = n - 1 for all

In §2 we prove plexity

and Bixby that

com (2) = 1, and in their

com (tí) < 72(72- l).

they gave

an example

It of

2 < com (3) < 6.

One of them

is an attainable

set of com-

72.

n, n > 3, there

com (72) > 72 for n > 3.

3 we show that com(3) = 3. We are grateful

to Nimrod

Megiddo

who made

valuable

comments

about

the problem.

2. Complexity in R". Lemma following

(a)

1. Suppose

V* is an attainable

set

in R",

n > 3, satisfying

the

properties:

the

72+1

points

p* =(a,

a, •••,

a),

P* = (0, 1, 0, -..,

p* = (1, 0, • • •, 0),

0), ....

P* = i0, ■•■, 0, 1)

belong to V* with 0 < a< 1; (b) for each p = iy y, ■■• , y ) £ V*. if for some j, 1 < ;' < 72,y. > a, then y, < a for every k 4 jThen com V* > n. n — Proof.

Assume

that generate

u {x',

V*.

that Let

u = iuy,

■ • • , u ) are the utility

(x) = (x;.) be the allocation

• • ■, x' ) = a for /: = 1, 2, ■• ■, 72. Consider

er in this X. » 1.

allocation, If / 4 0,

We claim

that

(x",I

■ ■• , x"). zzz

assume there

without

Let loss

is an index

functions

for which

of generality 1< i

i.e.

the bundle of the 72th play-

/ be the set of indices J

j,,

on lm

zz(x) = p*,

that

< k, such

i for which

] = \k + 1, • ■ • , m\. that

x\y = 0 for

every /', /' 4 1» n. If not, put 8. = 2. ,} nxi > 0 for 1 < i < k and 8 = min1 3, the allocation

and

we find

1 is arbitrary,

z'j > k, which

2nd and

that there

j, j 4 2,n,

1 to conclude

= 0 for every

x?, = 1 and

x\. = 0 for every

n > 3, this is a contradiction, Remark.

that

to conclude

Observe

from which

m = n - 1 so that

the choice

argument

inductively,

k such

72 to player

> 0 and x\

the above

m > k > n - 1.

that

by a. on B . So

it must be a global

the requirement

x\2 = 0 tot every

are 72- 2 commodities Suppose

and by concavity

72th players

1 and

that

zz is bounded

> 0, for if xjy = 0 then

Continuing

and all between

(b) that

of the claim.

repeat

that

z, 4 i2-

in B

contradicts

x¿.

Next,

1 < i- < k, such

implies

from property

for the point

1 that in the case

p* is the unit matrix

where

up to a renaming

of the commodities. Theorem

1.

The attainable

vex hull of the 72+1

set

V

Pl = iVi,V2,■•■,lA), P2*= (o, 1,0, ...,o), has complexity Proof. to show that

in R",

n > 3, generated

by the con-

points

p* = ii,o,o,.-.,o), ••-,

p* = io,---,o,D

n.

To show that the complexity V

satisfies

condition

of V

is at least

(b) of Lemma

1.

Let

72, it is enough p be a point

in

the convex hull of \p^, ■■• , p*\; then p = £"=0 X¡p*, where X¡> 0 and 2^"=0^/

= 1. Denoting

p = iyy,

• ■• , yn),

we then have

for z 4 ]■>

y. + Jy.j = A„ < 1,' 0 + A. 1 + A. j — J1

which

proves

property

To show that

that generate

(b).

com Vn < n, we exhibit

72 utility

functions

V . Define


from

/" to R

A GAME THEORETIC NOTION OF COMPLEXITY

zzz.(x,,l

Denote

by V

• • • ,' x zz)'=:—-, \x. + I z

2L

the attainable

the allocation

For every

to player

; the total

show the converse and show that

less ality

assume

player

to player

amount

we start

of x under

p

different

combination

the changes

diagonal linear

player

/, x1. = ß . if i 4 ji x\ = a-

and

a>

is the image of the allocation

1 and

2 the same

consider

and

of the

entries

are zero except

the point

of players

2nd player

arrive

of such

2 iß2 + ßy,

p

obtained

1 and from p

2.

pisa

and

p

convex and making

where

We infer that

0-2 + ß y,

in the same

(or 1st in the case

allocations

ß..

that assigns

as in x, assigns

at allocations

for one row.

of the images

amounts

to player

p . Now starting

and so on, we finally

combination

is

which

the roles

in the bundles

the 3rd player,

• • ■ , y ) which

of gener-

Also

of p

3 V . To x = (xJj

loss

way as p , by interchanging linear

V

assigns

Without

from

ßy).

p = iyy,

;'th

that

allocation

• • • , p*\.

at

of the

Therefore

zz is a point

1 (ctj - ß Jt 0, • • • , 0) and assigns

ß2 + ßy , • • ■ , ß2+

amount

with an arbitrary

hull of ip*,

attained

at the allocation

of all the commodities.

in the convex

the point

/ the total

;', p* is attained

inclusion,

that for every

Consider to every

to each player

player

the image

than a point

for i = 1,

x.; II

IS/i« J

set of zz = (zzj, • • • , zzn). pg is

that assigns

commodity.

min

419

of p ) and

all the off

p is a convex

of the following

form:

xl = y and xí = 0 ií i 4 j fot every j'4 k; x* = 1 and xk = 1 - y for i 4 k (where

k is the row with the off diagonal

of such an allocation,

then

nonzero

p = iy y, ■• ■ , yn),

entries).

where

If p is the image

y. = Y2y ii j 4 k and

yk=V2 +ÍV2- My) = 1 - Viy- So p = yp* + (l - y)p*, and we conclude that p is a convex

3.

linear

Complexity

combination

in R3.

V in R5 can be attained

variable,

utility

of ip*,

In this by three

functions

■■• , p*).

section

we prove

continuous,

that any attainable

concave,

monotone

set

in each

on /^.

Theorem 2. com(3) = 3Proof.

In view of Theorem

Let C be a compact

convex

ity (see [l, p. 262]) that maxíyj: that

miniy,:

conditions iliary

(y,, for the

function

1 we have to show only that

set in R . Assume

com (3) < 3-

loss

of general-

iy y, y 2, y A) £ C tot some y2> y^\ = 1 and

y 2, y A £ C tot some 2nd and

without

3rd players.

y,,

yA = 0. Let

Also assume

V = C — R\.

on / ,


the same

Consider

the aux-

420

EHUD KALAI AND MEIR SMORODINSKY

gix y, x2) = max\S £ R: ix y + 8, 1 - X. +8, It is easy

to verify

Now we shall

functions

that

g is continuous

define

generate

utility

1 - x2) e V\.

and concave.

functions

on I* and prove

that these

utility

V. UyiXy,

X2,

Xj)

= Xy+

g^,

X^,

uAx - x,,13 xA, 2123 , x , xA - x. 1 + g(l ° zz3(xr Let

V be the attainable

sider any point

x2, x3) = min¡x2,

set of zz = iuy,

y = iy.,

x^.

zz,, zz,).

y 2, y A £ C. Choose

To see that

8Q such that y y + z5Q+ y, + (5n

= 1. It follows that 0 < y. + 8Q < I, i = 1, 2. Consider

.i .. . *

x3 = 0>

x, = y + (5„,

x, = 0,

x, = 1 - y,,

x^ = o,

*2 = y3'

*)->*•

' 2

0'

calculation that

that

Uy, u2, u,

loss

of generality

x3. If z5= g(xj, the

„i -

*2 = 2 -y3'

A straightforward

yj=Xj+z5,y2

..

the allocation

*1 = >1 + S0'

1

To prove

>r, 2

increasing

it can be assumed

1 - xp,

zz.'s for the given

that

'3

zz((xp) = y.

(x7.) = x be any allocation.

ate monotonically

= 1-Xj

3

of zz((x'.)) shows

V C V, let

that

V ^ V, con-

in each

It is easy

variable,

to see

so without

x* = 0, x2 = 0, x^ = 0, and

x^ =

then there is a point (y1? y2, y A £ V such that + S and

allocation

y, = x3.

But these

y¿'s

are the values

of

x.

REFERENCE 1.

Louis

J. Billera

and Robert

E. Bixby,

A characterization

of Pareto

surfaces,

Proc Amer. Math. Soc. 41 (1973), 261-267. DEPARTMENT OF STATISTICS, TEL AVIV UNIVERSITY, TEL AVIV, ISRAEL
