A Convex Hull’ Characterization

(1)

Published online May 20, 2014 (http://www.sciencepublishinggroup.com/j/pamj) doi: 10.11648/j.pamj.20140302.13

A convex hull’ characterization

Franco Fineschi, Giovanni Quaranta

Department DISAG, University of Siena, Italy

Email address:

[email protected] (F. Fineschi), [email protected] (G. Quaranta)

To cite this article:

Franco Fineschi, Giovanni Quaranta. A Convex Hull’ Characterization. Pure and Applied Mathematics Journal. Vol. 3, No. 2, 2014, pp. 40-48. doi: 10.11648/j.pamj.20140302.13

Abstract: Conditions so that a vector belongs to a convex hull are obtained. Multilinear convex functions are considered. If

these maps are defined on a convex set, it is obtained the algebraic expression. As an application, infinite games, with linear convex payoff, are studied.

Keywords

:

Convex, Linear, Game

1. Introduction

Some properties of the class of the k-linear convex functions are considered. These maps seem to be the natural extention of k-linear functions if the domain is bounded. Given a linear convex mapping, defined on a body, it is possible to extend its definition if an argument is outside of the domain. This property enables us to find the analytical form of a k-linear convex functionφ:Xk →Y

,

if X is a convex set which contains the null vector. An arbitrary element of a convex set can be represented in three different ways using finite sequence of elements of the set; these representations are obtained by the following theorems: the Shapley-Folkman lemma, Carathéodory’s theorem and Fenchel-Bunt’ theorem.

For example by the last theorem, for any element sof a convex, connected set X, a sequence of

n

elements of X exists such that s is expressed by a convex combination of the sequence. In spite of this, it is well known that an arbitrary convex set has not a “convex basis”, that is, it is impossible to find an unique sequence, with finite number of elements of the set, which span, by linear convex combinations, any other. In Algebra it is proved that ℜn is free over the standard basis, this implies some properties, one of these is that an arbitrary sequence of a module A may be written as a linear combination of the vectors

) ( , ), (e₁ te_n

t … for an unique morphism :ℜn

→

. This concept of free set is now moved to convex sets by linear convex functions. One result obtained in this paper is that a convex combination for an element of a convex, connected set A is expressed by a unique convex combination of vectors φ(e1),…,φ(en) for a linear convex function

φ

, that

is, any element

a

of A determines a

φ

such that ∑ξ_iφ(e_i)=a.

Conditions are obtained in order that a vector belongs to a convex hull if it is spanned by a finite set. The conditions give new expression to the convex hull.

It is known, see [7] and [8], that two persons, infinite, symmetric games, with a compact, convex set as strategy space and a linear convex function as payoff, have solutions. In the last section we obtain the form of the payoff of these games.

2. Free Convex Sets

Let S be a set of vectors in ℜn, with the null vector S

∈

0 , then denote by X =coS the convex hull of S . Suppose k ≤n the maximal number of linearly independent vectors in X and x1,…,xk be a set of

linearly independent vectors of X.

A k-linear convex mapping : , for , is defined by

∑

= =

r

i

k i i

k r

i i i k

i a a b a a b a a

a

1 1 1

1

1, , , , )= ( , , , , )= ( , , , , )

( … … φ … λ … λφ … …

φ

where 0 and∑ 1, ℜn.

Example 2.1Consider the function

b y a x xy

y x

f( , )=2 0≤ ≤ , 0≤ ≤

Let , then f(x₁,y)=2x₁y and

y x y x

(2)

) , (x y

f is not a bilinear function. Whereas, for 0<λ<1,

y x x

f(λ1+(1−λ) 2, )=2(λ 1+(1−λ) 2)

y x y

x₁ 2(1 ) ₂

2

= λ + −λ

) , ( ) (1 ) , (

=λf x₁ y + −λ f x₂ y

that is, f(x,y) is a convex linear function of each variable separately.

The following is a known basic statement

Proposition 2.1 Let

φ

:Xk →ℜm be a k-linear convex

mapping and X a convex subset of ℜn . The image

) (Xk

φ

is convex in ℜm.

Proof. The image under

φ

of every convex

combination of elements of

X

is a convex combination in

ℜ

m.

A point of a convex hull has three known representations by sequences of elements of the same convex set.

a)The first representation is founded on the Shapley-Folkman lemma, see [1], [2] and [10].Consider a finite family , 1, … , " of subset of

ℜ

n, the Minkowsky sum # ∑ # is defined by element-wise addiction of vectors

# $# % # : # &

The Minkowsky sum has an immediate property: for any

nonempty finite family of sets of ℜn'( ∑ ∑ '( .

So if # ∑ '( and # # % # with # '( .

The expression of sby # % # depends on the same point s. For the representation # # % # , in the

space

ℜ

n, holds

Lemma (Shapley-Folkman) If the number k of is greater than the dimension n of the space, the point s has the expression

# ) #

* *+

) #, +- *,*

where # '( and #_, _,.

In other words # ∑ _{* *+}'( ∑_{+- *,*} _,.

b)By the Carathéodory’s theorem, any s∈coS⊂ℜn may be represented as a convex combination of n+1 elements of S .

c)In particular, if

S

is connected, the Fenchel-Bunt’ theorem, see [3] and [8], proves that any s∈coS can be expressed as a convex combination of

n

elements of

S

. Thesethree representations of an element do not establish the existence of a “convex basis” for coS, because the

n

elements may depend on the particular s to be computed, see [6]. Nevertheless, the next definition asserts the condition in order that a convex set is free. For a definition of a free module on a subset, see [4].

In the following we present a different representation for

elements of convex set based on the definition of free convex set.

Definition 2.1 Let K be a subset of a convex set X in

n

ℜ and let j:K→X be the insertion of K in X.

Denote by A a subset of

ℜ

m, then X is free over K if, for every function f :K→A, an unique linear convex

mapping

φ

:X →A exists such that

φ

j= f .

Example 2.2. Let . $ , & / ℜ2be a subset and '(. its convex hull. We want to prove that '(. is free over ., i.e., for every : . an unique linear convex function

: '(. exists such that 0 1 .

Unicity. The condition 0 1 means that

and . Then for every : . , we have

to prove the unicity of : '(. such that

and . For any " '(. and for

every linear convex function holds "

so is

univocally defined.

Existence. The function , defined by

, is linear convex, in fact for every

2 2 '(.

2 2 2 2 2 2

Explicitly the expression of the unique linear convex is given by

"

' , ' , '3 ' , ' , '3 =

4'' ''

'3 '3 5 6 7 896 7

Where it is supposed ' , ' , '3 ,

' , ' , '3 and , : ℜ3 ,for any " '(. with fixed a and b. Then,

φ

is the matrix 89.

The underlying property of the example above will be proved in a more general setting.

Some properties of the mapping

φ

:Xk →A follow. Since

X

is convex, ∀a_i∈X,0<α<1 ,

X a

a_i+ −α α _i∈

α (1 )0= . In particular

α

a_i∈X . Moreover

) , , 0 ) (1 , , ( = ) , , , ,

(a1…αai … ak Φ a1…αai+ −α … ak

Φ

(1)

supposed a_i=a_jand Φ skewsymmetric it follows

(2)

comparing 1 and 2, it follows Φ skewsymmetric and

) , , , , , , ( = ) , , , , , ,

(a₁… a_i … a_j … a_k a₁ …αa_i … a_j … a_k

αΦ Φ

) , , , , , ,

(a₁…αa_j …a_j … a_k Φ

− (3)

) , , 0 , , ( ) (1 ) , , , , (

=αΦa₁… a_i … a_k + −α Φ a₁… … a_k

) , , , , 0 , , ( ) (1 = ) , , , , , ,

(a1… ai…aj…ak − Φa1… …aj …ak

(3)

Let

β

∈ℜ+ and m∈N−{0} such that0 ₌< 1, X

a_i∈

∀ , by the (3) it holds

) , , , , , , ( = ) , , , , , ,

( ₁ _i _j _k ₁ a_i a_j a_k

m a m a a a

a … … … φ … β … …

βφ

) , , , , , ,

( 1 aj aj ak

m a

m

φ

…

β

… …

− (4)

Proposition 2.2 Let

λ

∈ℜ+(

λ

∈ℜ−) and X

x∈

λ , > |1|, then αx∈X(−αx∈X), for 0<α <1.

Proof. If 0<β<1 satisfies

α

=

β

⋅

λ

, then

0 ) (1 ) (

=β λ β

αx x + − , so αx∈X . If −

α

=

β

⋅

λ

then

0 ) (1 ) (

=β λ β

α + −

− x x and −αx∈X .

The next proposition extends the (1) and defines a linear convex mapping if its argument is outside the body

Proposition 2.3 Let x1,…,xi,…,xk be vectors in X

and δ∈ℜ, then a linear convex mapping

φ

:Xk→A

satisfies

) , , 0 , , ( ) (1

) , , , , ( ) , , , , (

1

1 1

k

k i k

i

x x

x x x x

x x

… …

… … …

…

φ δ

δφ δ

φ

− +

=

(5)

Proof. Let 0<

λ

,

µ

,

ν

<1 such that @ _AB C DA_A E and

λ

|

δ

|<1. It is always possible to determinate such numbers

λ

,

µ

,

ν

. Indeed, if _F <1, the relation @ _AB C DA_A E becames

δ

=

αµ

−(

α

−1)

ν

, then, for example supposeE , for

α

>2|

δ

|+1 , it holds

1 < 2

1 2 = < 0

α α δ

µ + − .

Then, by the Proposition 2.2, the vectors

µ

x

_i

,

ν

x

_i are in

X

and the mapping

φ

is defined on them.

(

1 )

)

,

)

,

(

=

)

,

(

x

₁

…

µ

x

_i

…

x

_k

φ

x

₁

…

λδ

λ

ν

x

_i

…

x

_k

φ

+

−

) , , , , ( ) (1 ) , , , , (

=

λφ

x₁…

δ

x_i … x_k + −

λ

φ

x₁…

ν

x_i … x_k that is

)

,

(

)

(1

)

,

(

=

)

,

(

1

1 1

k i

k i k

i

x

…

ν

φ

λ

µ

φ

δ

λφ

−

by the (1)

) , , 0 ) (1 , , ( 1 = ) , , , ,

(x1… xi… xk φ x1… µxi µ … xk

λ δ

φ + −

) , , 0 ) (1 , , ( ) (1

1 xi xk

x …ν ν …

φ λ

λ ₊ ₋

− −

) , , 0 , , ( ) (1 ) , , , , (

= x1… xi… xk φ x1… … xk

λµ φ

λ

µ ₊ −

) , , , , ( ) (1

1 xi xk

x … …

φ λλν

− −

) , , 0 , , ( ) )(1 (1

1 xk

x … … φ

λ ν

λ −

− −

) , , , , ( ) ) (1 (

= φ x1…xi…xk λλν

λ µ₋ −

) , , 0 , , ( ) ) )(1 (1 ) (1

( φ x1… … xk

λ ν

λ λµ

− − − − +

) , , 0 , , ( ) (1 ) , , , , (

=δφ x1… xi… xk + −δ φ x1… … xk

Theorem 2.1 Let

φ

:Xk →Y be a k-linear convex

function and X a convex set with 0∈X , then

) , , (a₁… a_k

φ

may be expressed by a linear combination of

) , , (

1 jk

j x

x …

φ , where x X

i

j ∈ span the vectors ai∈X .

Proof. The nonnull vectors a₁,…,a_k in X may be written as

G % G G , … , G

i i

b

a

k

||

=

) 1

1 (

= 1 ₁

k i ki

i i

i x

a k x

a k a k

|| || ||

|| ||

||

γ

+⋯+

γ

j i ji k

j

i

x

a

k

a

k

||

1 γ

=

1 =

∑

{0} 1

=

1 =

− ∈

∑

b x k N

k a

k ji j

k

j i|| ||

where −1≤ = ≤1

|| || i

ji ij

a

b

γ

, i,j=1,…,k.

By k applications of the Proposition 2.3 to the arguments of

φ

, it follows

) ,

, (

= ) , ,

(a₁… a_k

φ

k||a₁||b₁… k||a_k||b_k

φ

) , , , 0 ( ) (1

) , , , (

=k||a₁||

φ

b₁ a₂… a_k + −k||a₁||

φ

a₂ … a_k

) , , , , (

=k2||a₁|| ||a₂||φ b₁ b₂ a₃ … a_k

) , , , 0 , ( )

(1 ₂ ₁ ₃

1 k a b a ak

a

k|| || − || || φ …

+

) , , , , 0 ( ) (1

|| ₁ ₂ ₃

2 k a b a ak

a

k|| − || || φ …

+

) , , , 0 , 0 ( ) )(1

(1−k||a₁|| −k||a₂|| φ a₃…a_k +

⋯ ⋯ ⋯ ⋯

=

) , , ( =kk||a₁||⋯||a_k||φ b₁… b_k

⋅ −

+

∑

− 1 −1 +1 (1 )

1

1 =

|| || || || || || || || ||

|| i i k i k

k

i

a k a a

a a

k ⋯ ⋯

⋯ …

… +

(4)

⋅ −

−

+

_∑

− ₋ ₍₁ ₎ ₍₁ ₎

1 1

< < 1

|| || ||

|| || || || ||

r i i

r k j j r k

r i i

a k a

k a

a

k ⋯ ⋯

⋯

⋯ …

…

… +

⋅ ( , , =0, , =0, , )

1

1 bi bi_r bk

b

φ

) 0 , , 0 , 0 ( ) (1

) )(1

(1 || ₁|| || ₂|| ⋯ || || φ …

k

a k a

k a

k − −

−

+ (6)

where , … , is the complement of 1 , … , 1 _D with respect to the set 1, … , " and the sum in (6) is over all the subsets , … ,

Now use the convex linearity on all the vectors b_i

) 1 , , 1 ( = ) , , (

1 = 1 1 =

1 jk j

k

j j j k

j

k b x

k x

b k b

b … φ

∑

…

∑

φ

) , , , ( 1

= 1 2

1 =

k i i k

i

b b x b k

…

φ

∑

⋯ ⋯ ⋯ ⋯

=

) , , ( 1

=

1 1 1 1

k j k k j j j k j j

k b x b x

k

…

∑

φ

∑

where, by the (1)

) ,

, ( = ) ,

, (

1 1 1 1

1

1 j jkk jk j j jkk jk

j x b x b x b x

b … φ …

φ

) , , , 0 ( ) (1

2 2 2 1

1 j j jkk jk

j b x b x

b φ …

− +

⋯ ⋯ ⋯ ⋯

=

) , , ( , , =

1 1

1 jkk j jk

j b x x

b … φ …

⋅ − +

∑

₁1, , ₋₁−1, ₊₁+1, , (1 )

1

= i

j k k j i i j i i j j k j

j i j

b b b

b

b … …

⋯ …

… +

⋅ ( , , =0, , )

1 ji jk

j x x

x

φ

⋅ − − +

−

∑

(1 ) (1 )

1 1

< < 1

r i i r k s s r i i

b b b

b ⋯ ⋯

⋯

⋯ …

…

… +

⋅ ( , , =0, , =0, , )

1

1 i ir jk

j x x x

x

φ

) 0 , , 0 , 0 ( ) (1 ) (1

1

…

⋯ φ

k j

j b

b −

−

+ (7)

where i₁,…,i_r is the complement of

r k

s

s₁,…, ₋ with

respect to the set 1,…,k and the sum in (7) is over all the subsets i₁,…,i_r.

So φ(a1,…,ak) is obtained by a linear expression in

) , , (

1 jk j x x …

φ , where some

i j

x

can be the null vector

0

.

In the n-dimensional vector space ℜn, denote by

S

_n the convex hull of the vectors {e₁,…,e_n} of the standard basis. S_n is a compact, connected, convex set and its elements may be expressed by convex combinations of the

unit vectors {e₁,…,e_n}.

Theorem 2.2 The set Sn is free over the standard

basis {e₁,…,e_n} of ℜn.

Proof. Let K ={e₁,…,e_n} , the condition

φ

j= f implies

φ

(e_i)= f(e_i)=a_i and the function

φ

is defined by

A a a

e

e + + _n)=

∑

_i _i _i∈ℜ+,

∑

_i =1, _i∈

(ξ₁ ₁ ξ ξ ξ ξ

φ ⋯

Uniqueness. If

φ

is a linear convex function with

i i a

e)= (

φ

, then, for any k∈S_n

1 = =

) ( = ) (

= )

(k φξ1e1 ξnen ξiφ ei ξiai ξi

φ +⋯+

∑

so

φ

is unique.

Existence. φ(k)=

∑

ξ_ia_i defines a function

φ

:S_n→A. Let us prove that

φ

is a linear convex function. ∀k∈S_n let k=ξ₁e₁+⋯+ξ_ne_n,

∑

ξ_i=1, then

) ( )

( = =

) (

= )

(k φξ₁e₁ ξ_ne_n ξ_ia_i ξ₁φ e₁ ξ_nφ e_n

φ +⋯+

∑

+⋯+

By the Fenchel-Bunt’ theorem, any element

a

of a compact, connected, convex set A is expressed as a convex combination of the sequence a₁,…,a_n of vectors of A, that is a=

ξ

₁a₁+⋯+

ξ

_na_n. By the theorem 2.2 exists an unique linear convex function

φ

such that

) ( = = =

)

(ξ₁e₁ ξ_ne_n ξ_ia_i a ξ_iφ e_i

φ +⋯+

∑

so, any element a∈A may be expressed as a convex combination of the vectors

φ

(e_i),…,

φ

(e_n). In other words, any a∈A determines a linear convex function

φ

such that

∑

ξ_iφ(e_i)=a.

Example 2.3 Let A be a convex, connected set in ℜ2. If a=ξ₁a₁+ξ₂a₂,ξ_i ≥0,

∑

ξ_i =1,a_i =(a_i₁,a_i₂) is an element of A, then, by the theorem 2.2, it follows

) ( ) ( = =

) (

= ₁e₁ ₂e₂ ₁a₁ ₂a₂ ₁ e₁ ₂ e₂

a

φ

ξ

+

ξ

+

ξ

φ

+

ξ

φ

where φ:S2→ A is linear convex. This implies

2 2 1

1)= , ( )=

(e a φ e a

φ _{, and so}

2 22

12 21 11

) ( =

)

( x x S

a a

x _ ∈

  

 

φ

3. A Convex Hull’ Characterization

(5)

} ) , , ( = , { = } , ,

{ 1

1 =

1 i i m m

m

i

m a

a a

co …

∑

α α α …α ∈Σ

where

Σ

_m is the unit simplex in

ℜ

m . The next proposition lays down a condition so that a vector belongs to a convex hull.

Theorem 3.1 Let co{a₁,…,a_m} be a convex hull where )

, , (

= a₁ a_m

A … is a matrix with |A|≠0 , then

} , ,

{a₁ a_m

co

b∈ … iff

(i). =0

1 1

1

1 11

1

mm m

m

a a

b

a a

b

… … … … …

… …

(ii). The determinants

| | |, , , , , | , |, , , , , | |, , , ,

|ba₂… a_m a₁b a₃… a_m … a₁ a₂… a_m₋₁b A

have the same sign.

Proof. Suppose b∈co{a₁,…,a_m}, then _ia_i b

m i=1

α

=

∑

for

α

=(

α

1,…,

α

m)∈Σm, that is, Aα=b. Then

| |

| , , , , , , |

= 1 1 1

A

a a

b a

a _j _j _m

j

…

… ₋ ₊

α

is the solution of the system Aα =b . Impose 1

= 1 = j

m j a

∑

, we obtain

| |=| , , , , | | , , , , | | , , ,

|ba₂…a_m + a₁ba₃…a_m +…+ a₁a₂…a_m₋₁b A and so the (i).

Conversely, if (i) and (ii) hold, moving backward, we get

α

such that Aα=b, then b∈co{a₁,…,a_m}.

Next step is the extension of the Theorem 3.1 to a convex hull given by the vectors a₁,…,a_n, in ℜm, with

m

n> and r(A)=r. For this purpose it is necessary to introduce some generalizations of elementary definitions in linear algebra. For more details see [5].

The multindex

I

_rn of length

r

is defined by

}

<

1 :

)

,

{(

=

i

1

i

1

i

n

I

r r

n

r

…

≤

…

≤

besides we define, for a fixed natural number

k

,

} 1

, < < = < < 1 : ) , , {( = )

( ₁ ₁

n k where

n i k i i i i i

I_rn _k _p _r _p _r

≤ ≤

≤

≤ … …

… …

Let F be a field with

char

(

F

)

≠

2

, then, for an

arbitrary matrix, consider the linear form in the parameters

F

∈

α β

λ

given by

Definition 3.1 Let  ∗

         

× _→

∆ : ( r )

n r m n m

r F F be a map

defined by

. , ,

| | =

,

n r m r

rA A ∈I ∈I

∆

∑

λ

α

β

β α β β α

If m=r=r(A), then |Aα_β |, in ∆_mA, are the Plücker

coordinates of the subspace spanned by the rows of A.

Example 3.1 For the matrix

   

 

23 22 21

13 12 11

=

a a a

A , with

2 = ) (A r

23 22 21

13 12 11 2 =

a a a

a a a A

∆

12 23 23 22

13 12 12 13 23 21

13 11 12 12 22 21

12 11

=

λ

a a

a a a

a a a a

a a a

+ +

The definition of cofactor

α

_ij can be extended to an arbitrary matrix.

Definition 3.2 Let A∈Fm×n be a matrix with r

A

r( )= , the cofactor

α

_ijr of the entry

a

_ij of

A

, for

n

m

<

, is defined by

α

λ α

α β

∈

≠ +

− ∆

i if only

0 and 1 = a n, , 1, j 1, j , 1, = for x 0 = : =

0 1

0 =

ij 1

1 11

… …

… … …

… …

… … …

ix r

i

mn m

n

r ij

a A

a a

| )

( |

= 1

1 1

1

' j e row th i r i i i

r j j j r i i i

r j j j r j j j r i i i

A

−

∑

……

… … …

… … … … … … …

λ

where in the last determinant the i-th row is ,0)

,1, (0,

= … …

' j

e , i.e. the j-th unit vector. The sum is over

all i₁,…,i,…,i_r∈(I_rm)_i and

j n r r I

j j

j₁,…, ,…, ∈( ) .

Similarly, for

m

>

n

,

j mn m

in i

n

r ij

a a

… …

… … … … …

… …

… … … … …

… …

0 1 0

=

1 1

1 11

α

(6)

) 0 1 1 0 1 0 ( ) 0 1

1 0 1 0 ( = 0 1 0 0

0 1 0 0 = 0 1 0 0 =

23 34 34 33 23 23 33 32 23 13 33 31 12 34 14 13

12 23 13 12 12 13 13 11 23

34 33 32 31

12 14 13 12 11 2

34 33 32 31

14 13 12 11 23

λ λ

α

a a a a a a a

a

a a a a a a a a

a a a a

r

+ +

As in the example, it will be useful to write the numbers of the rows (columns) involved in the expansion of the cofactor like apexes (indexes).

Let C=(α_ijr) the r-cofactor matrix of A . The transpose of C, i.e. the r-adjoints of A will be denoted by Aad.

Theorem 3.2 Any consistent linear system Ax=b,b≠0, where A∈Fm×n, r(A)=r, has as set of solutions

n j

A b b

x

r m r mj r

j

j= =1, ,

1 1

… ⋯

∆ +

+ α

α

for any

λ

αβ∈F such that r A r

ad₎_≥

( .

Proof.(∆_rA)−1Aad are the (1)-inverses of

A

, then the solutions of Ax=b are x=(∆_rA)−1Aadb.

The Theorem 3.2 is an evident Cramer’s rule generalization.

All that is necessary to the next proposition is now ready.

Theorem 3.3 Let co{a₁,…,a_n} be a convex hull where )

, , ( = a₁ a_n

A … is a matrix with _r₍_A₎₌_m, a i n

i≠0 =1,…, ,

then b∈co{a₁,…,a_n} iff

(i)

γ

₁₁m+b₁

γ

₂₁m+⋯+b_m

γ

_mm₊_1,1=0

where the cofactors

γ

_ijm are obtained by the matrix

    

 

    

 

mm m

m

a a

b

a a

b C

… … … … …

… …

1

1 11

1

1 1

1 =

(ii) The sums αm_jb +⋯+α_mjmb_m

1

1 , with

n

j=1,…, , have the same sign of ∆_mA.

Proof. Suppose b∈co{a₁,…,a_n}, then _a _b

i i n i=1α =

∑

for α=(α1,…,α_n)_∈_Σ_n, that is, Aα =b. Then, by Theorem

3.2

A b b

m m m mj m

j

j _∆

+

+ α

α

α ⋯

1 1 =

is the solution of the system Aα=b . Impose 1

= 1 = j

n j a

∑

using linearity by rows, it follows

1 1

=

1 1,1

1 11

1

2 21

1

… …

… … …

…

⋯

… … … …

… …

n m m

n

m

mn m

n m

a a

b

a a

b A

− −

+ + ∆

and so the (i). By α_j≥0, j=1,…,n, we obtain the (ii). If

α

_r is null in the solution of the system Aα=b, then b∈co(a₁,…,a_r₋₁,a_r₊₁,…,a_n). As a special case, if only α_i,α_j are nonnull in the solution of Aα=b and

j i a

a, are extreme points, then

b

is an extreme point of

the convex hull.

Example 3.3 A vector

b

is in ∈co{a₁,a₂,a₃}, with 0

,

2 _≠

ℜ

∈ i

i a

a , r(A)=2, if it satisfies the following

conditions

1,2,3 = 0 ) (

1 2

2 2 2 1 1 2

j b

b

A j+ j ≥

∆ α α

0 = 1 1 1 1

23 22 21 2

13 12 11 1

a a a b

Given any matrix

A

, a matrix

A

(2) such that (2)

(2) (2)

=A

AA

A is called a (2)-inverse of

A

. Let

n m

C

A∈ × and r(A)=r. Then (∆_rA)−1Aad is the set of (2)-inverses of

A

iff r(Aad)≤r, see [9].

A (2)-inverse

A

_RN(2) with prescribed range

R

and null space

N

is the matrix

(

∆

_r

A

)

−1

A

ad such that

(i). (∆_rA)−1AadAx=x x∈R

(ii). (∆_rA)−1Aady=0 y∈N

By the next proposition, a matrix

A

c such that

α

= A

Ac with

α

∈Σ_n is obtained.

Theorem 3.4 Let A∈Cm×n and r(A)=m. The matrix

c

A

is given by

0. , )

) ( ) ( ( ) (

= A ++Q I − A A + Q∈C × Q≠

A m mn

c _α _α _α _α

where (Aα)+ denotes the Moore-Penrose inverse of

α

A .

Proof. It is known that the matrix equation XB=C

(7)

equation, B= Aα and C=α, since (A

α

)+A

α

=1 the equation has the general solution

0. , ) ) ( ( ) (

= A ++Q I −A A + Q∈C × Q≠

Ac α α _m α α m n

Theorem 3.5 Let co(a₁,…,a_n) be a convex hull. The following four statements are equivalent

(a). b∈co(a₁,…,a_n),

(b). the linear system Aα=b has the solution

n Σ ∈

α

,

(c). co(A,b)=co(A),

(d). Acb=

α

.

Proof. It is immediate that (a) and (b) are equivalent. Let us show that (b) and (c) are equivalent. Suppose

) ( = ) ,

(Ab co A

co

b∈ , then b=

α

1a1+,⋯,+

α

nan , so

b

Aα= with

α

∈Σ_n. Conversely, suppose Aα=b with

n Σ ∈

α

, that is b=

α

₁a₁+,⋯,+

α

_na_n and b∈co(A), but it also b=

α

₁a₁+⋯+

α

_na_n+0b where

1

,0)

(

α

∈Σ_n₊ , then )

, (Ab co

b∈ . Moreover ∀k∈co(A,b) is

b a

a

k=

δ

₁ ₁+⋯+

δ

_n _n+

δ

_n₊₁

) (

=

δ

₁a₁+⋯+

δ

_na_n+

δ

_n₊₁

α

₁a₁+⋯+

α

_na_n

n n n n

n )a ( )a

(

=

δ

₁+

δ

₊₁

α

₁ ₁+⋯+

δ

+

δ

₊₁

α

since ) ( ) ( = )

(δ1+δn+1α1 +⋯+δn+δn+1αn δ1+⋯+δn +δn+1α1+⋯+αn

1 1

=δ +⋯+δ_n+δ_n₊

1 =

and (

δ

_i+

δ

_n₊₁+

α

_i), i=1,…,n , are nonnegative, then

)

(

A

co

k

∈

.

(b)

→

(d) . Let A

α

=b,

α

∈Σ_n, then there exists

c

A

such that AcA

α

=

α

, so Acb=

α

. Conversely, the relation Acb=

α

is satisfied by b=Aα, then there exists

) , , (

=A coa₁ a_n

b

α

∈ … .

Example 3.4 Consider the convex hull

0 , ),

, ,

(a₁ a₂ a₃ a_i∈ℜ2 a_i ≠

co and let

3 3 2 1 3 3 2 1 2 3 2 1

1 _, _, ₎

(

= ∈Σ

+ + + + +

+λ λ λ λλ λ λ λλ λ

λ λ

α , then

                                  + + − + − − − − + − + + − + + − + − + + − − − + − − + + + − + − + + − + + − − − − + − + + − + − + − + } ) 1 ( ) ( ) , ) (1 ) ( ) { } ) ( ) 1 ( ) , ) ( ) (1 ) { } ) ( ) ( ) (1 , ) ( ) ( ) (1 { 2 1 = 3 23 2 22 1 21 3 32 23 31 13 2 32 22 31 12 1 32 21 31 11 3 13 2 12 1 11 3 32 23 31 13 2 32 22 31 12 1 32 21 31 11 3 23 2 22 1 21 3 22 23 21 13 2 22 22 21 12 1 22 21 21 11 3 13 2 12 1 11 3 22 23 21 13 2 22 22 21 12 1 22 21 21 11 3 23 2 22 1 21 3 12 23 11 13 2 12 22 11 12 1 12 21 11 11 3 13 2 12 1 11 3 12 23 11 13 2 12 22 11 12 1 12 21 11 11 λ λ λ λ λ

λ λ λ λ

λ λ

λ λ λ λ

λ λ

λ λ λ λ

λ λ

λ λ λ λ

λ λ

λ λ λ λ

λ λ λ a a a q a q a q a q a q a q a a a a q a q a q a q a q a q a a a a q a q a q a q a q a q a a a a q a q a q a q a q a q a a a a q a q a q a q a q a q a a a a q a q a q a q a q a q a Ac

the condition Acb=

α

implies for b=(b1,b2)

) , ( = 3 2 1 3 23 2 22 1 21 3 2 1 3 13 2 12 1 11 λ λ

λ λ λ

λ λ

λ

λ λ λ

λ + + + + + + +

+a a a a a

a b so } : { = ) , ,

(a₁ a₂ a₃ b _i∈ℜ+

co λ

4. Symmetric Two Persons Games

Denote by Γ₂=(X,X,Φ) the triplet of a two persons

infinite, symmetric game in normal form, with

X

the full strategy space, see [7] and [8].

X

=

coS

is the convex hull of subset S of Euclidean n-dimensional linear space

n

ℜ , with the null vector 0∈S, and let Φ:X×X →ℜ be the bilinear, skewsymmetric, convex payoff function.

Explicitly, impose the following axioms 1) If x₁

λ

₁+⋯+x_n

λ

_n =x∈X , where x X

i∈ and

0

≥ i

λ

, =1

1 = i

n

i

λ

∑

, then

X y y x y x y

x _i _i

n i i i n i ∈ Φ Φ

Φ( , )= (

∑

, )=

∑

( , )

1 = 1 = λ λ

likewise, if y₁

δ

₁+⋯+y_n

δ

_n=y∈Y, where y X

i∈ and

0

≥ i

δ

, =1

1 = i

n

i

δ

∑

, then

X x y x y x y

x i i

n i i i n i ∈ Φ Φ

Φ( , )= ( ,

∑

)=

∑

( , ) 1 = 1 = δ δ

2) For every x,y∈X×X , it holds Φ(x,y)=−Φ(y,x)

The payoff function Φ:X×X →ℜ satisfies the following properties

a)∀x∈X, by the second axiom, Φ(x,x)=−Φ(x,x), that is Φ(x,x)=0.

b)∀x,y∈X,0<

α

<1, since X is convex, X

x

x+ −α α ∈

α (1 )0= . Moreover

) , 0 ( ) (1 ) , ( = ) , 0 ) (1 ( = ) ,

( x y Φ x+ − y Φ x y + − Φ y

Φα α α α α that is

) , 0 ( ) (1 ) , ( = ) ,

( x y Φ x y + − Φ y

(8)

by x=y, it follows ) , 0 ( ) (1 = ) ,

( y y − Φ y

Φα α (9)

comparing (8) and (9), it follows

) , ( ) , ( = ) ,

(x y

α

x y

α

y y

α

Φ Φ −Φ (10)

In the same way of theorem 2.1, the analytic form of the payoff Φ is now obtained.

For any two vectors a1,a2∈X let

i i n i ni i i i j i ji n j i

i x n a b

a a n x a a n a n x a a n a n

a || ||

|| || || || || || || || ||

|| 1 = (1 1 )=

= 1 1

1 =

+ +

∑

⋯

then, by the proposition 2.3, it follows

) , ( = ) ,

(a₁ a₂ Φ n||a₁||b₁ n||a₂||b₂ Φ ) 0 , ( ) (1 ) , (

=n2||a₁||||a₂||Φb₁b₂ +n||a₁|| −n||a₂||Φb₁

) 0 , 0 ( ) )(1 (1 ) , 0 ( )

(1 1 2 1 2

2 − Φ + − − Φ

+n||a || n||a || b n||a || n||a ||

using convex linearity

) , ( 1 ) , ( 1 ( = ) , ( ₂ 1 1 2 1 1 11 2 1 2 2

1 x b

a a n b x a a n a a n a

a n _n

|| || || || || || ||

|| Φ + + Φ

Φ ⋯ ) 0 , ( 1 ) 0 , ( 1 )( (1 1 1 1 1 11 2

1 n xn

a a n x a a n a n a n || || || || || || ||

|| − Φ + + Φ

+ ⋯ ) , 0 ( 1 ) , 0 ( 1 )( (1 2 2 1 2 12 1

2 n xn

a a n x a a n a n a n || || || || || || ||

|| − Φ + + Φ

+ ⋯ ) 0 , 0 ( ) )(1

(1− ₁ − ₂ Φ

+ n||a || n||a ||

) , ( ) ( = 2 2 1 1 1 = 1 = 2 1 j j i i n j n i x a a x a a a a || || || || || || ||

||

∑

Φ

) 0 , ( ) (1 1 1 1 = 2 1 i i n i x a a a n a || || || || ||

|| − Φ

+

∑

) , 0 ( ) (1 2 2 1 = 1 2 j j n j x a a a n a || || || || ||

|| − Φ

+

∑

) 0 , 0 ( ) )(1

(1− ₁ − ₂ Φ

+ n||a || n||a ||

by the (1)

) 0 , ( ) ( ) , ( = ) ,

( 1 2 2

1 = 1 = 2 1 1 = 1 = 2

1 i j i

n j n i j i j i n j n i x a a a x x a a a

a Φ + − Φ

Φ

∑

|| ||

) 0 , 0 ( ) )( ( ) , 0 ( )

( 1 1 2 2

1 = 1 = 2 1 2 1 = 1 = Φ − − + Φ −

+

∑

i j

n j n i i j i n j n i a a a a x a a

a || || || || || ||

) , 0 ( ) (1 ) 0 , ( )

(1 ₂ ₁

1 = 2 1 1 = i i n i i i n i x a n a x a n

a − Φ + − Φ

+

∑

|| ||

∑

|| || ) 0 , 0 ( ) )( (1 ) 0 , 0 ( ) )(

(1 1 2 2

1 = 1 1 2 1 = Φ − − + Φ − −

+

∑

i

n i i n i a a a n a a a

n|| || || || || || || ||

) 0 , 0 ( ) )(1

(1− ₁ − ₂ Φ

+ n||a || n||a ||

By the symmetry of the game, Φ(0,0)=0. The real value Φ(a₁,a₂) is so determined by Φ(xi,xj) and

) 0 , (x_i

Φ . Denoting by (x,y) a solution of the game, the choice of the Φ(x_i,x_j) and Φ(x_i,0) must satisfy the following inequalites     Φ ≥ Φ ≥ Φ Φ ≥ Φ ≥ Φ n , 1, = i ) , 0 ( ) , ( ) 0 , ( ) , ( ) 0 , ( ) 0 , ( … y y x x y x x

x i i

and     Φ ≥ Φ ≥ Φ Φ ≥ Φ ≥ Φ n , 1, = j i, ) , ( ) , ( ) , ( ) , ( ) , ( ) , ( … y x y x x x y x x x x x i j i j i j

Suppose

X

be a compact, convex set and define

X y y x y x X x x

maxy ={ : ∈ ⇒Φ( , )≥Φ( , )} ∈

* * and X x y x y x X y y

minx ={ : ∈ ⇒Φ( , )≤Φ( , )} ∈

* *

then, by the imposed conditions on Φ and X, maxy

and minx are not empty and compact, convex sets.

The map 2

) , ( }, {

:X×X→ minx×maxy ∀xy∈X

Φ is upper

semicontinuous and we can use the following proposition, see [8]

Theorem 4.1 (Kakutani’s fixed-point theorem) If }

{ ) , (

: x y → min_x×max_y

Φ is an upper semicontinuous

map on compact, convex X×X and {Φ(x,y)} is a convex set, then, at least an element (x,y)∈X×X exists such that (x,y)∈Φ(x,y).

By the theorem 4.1, an element (x,y) exists with

x

min

y∈ and x∈max_y . Since x∈max_y it holds

) , ( ) , (

, x y x y

x Φ ≤Φ

∀ and ∀y,Φ(x,y)≤Φ(x,y), then

) , ( ) , ( ) , (

,y X x y x y x y

x ∈ Φ ≤Φ ≤Φ

∀

that is, (x,y) is a solution of the game Γ₂ .

References

[1] M.Carter Foundations of mathematical economics, Cambridge MIT PRESS, 2001.

[2] J.W.S. Cassels Measures of the non-convexity of sets and the Shapley-Folkman-Starr theorem Mathematical Proceedings of the Cambridge Philosophical Society Vol. 78 pp. 433-436, 1975.

(9)

[4] C.Faith Algebra I Rings, Modules, and Categories Springer-Verlag, 1981.

[5] F.Fineschi, R.Giannetti Adjoints of a Matrix Journal of Interdisciplinary Mathematics. Vol.11, No. 1, pp.39-65. Taru Publications, 2008 .

[6] J.B.Hiriart-Urruty, C.Lemaréchal Fundamentals of Convex Analysis Springer-Verlag, 2001.

[7] T. Iimura, T. Watanabe Existence of a pure strategy equilibrium in finite symmetric games where payoff functions are integrally concave Discrete Applied Mathematics, Forthcoming 2012.

[8] S. Karlin Mathematical Methods and Theory in Games, Programming, and Economics - The Theory of Infinite Games Addison-Wesley, 1959.

[9] Y. Wey A characterization and representation of the generalized inverse A_RN(2) and its applicatations, Linear Algebra Appl. N. 280 pp 87-96, 1998.