Existence of equilibrium point

From section 4.1, we define the payoff function 

 

x and the function 

 

x ,

by the equation 4.1-2, 

 

x,y is continuous in x and yand is concave in y

for every fixedx.We now prove the existence theorem for the concave n- person game.

Definition 4.2-1: A single-valued mapping f :X Y sends a point x of X to a point f

 

x of Y. But on some occasions, we need to consider a mapping f that lets correspond to each point x of X a subset f

 

x of Y. Such a mapping is termed a set-valued mapping or a point-to-set mapping.

Definition 4.2-2: An -net of a metric space X is a finite subset



^{ai |i1,2,...,s}



of X such that the family of  -neighborhoods



^N

 

^{ai, |i1,2,...,s}



is a covering of X . Here, N

 

a, 



x|dis

 

x,a 



denotes an -neighborhoods inX .

Figure 4.2-2 The -neighborhoods

Definition 4.2-3: A set X is compact if any sequence of its points contains a sub-sequence that converges to a point in X .

35

Corollary 4.2-4: If a metric space X is compact in the sense of definition 4.2-3, it has an -net for any  0.

Proof. Suppose that X had no  -net. Take any one point a¹.

Since there is no -net, the -neighborhoods ^N

 

^a1, cannot coverX . So that some a²X does not belong to ^N

 

^{a1, .}

Again, for the same reason, there is some a³ belonging to none of ^N

 

^{a1, and}

 

^a2,

N .

Continuing this procedure, we obtain a sequence

 

^av such that

 

^,

1

1 v

v

i

v N a

a





  .

By construction, the sequence has the property ^dis



^au,av



^{ for uv. Such a}

sequence has no convergent sub-sequence, contradicting the compactness of X , Q.E.D.

Definition 4.2-5 (convexity): A vector y in Rⁿ is said to be a convex combination if y can be written as





 ^s

i i ix y

1





 s 

i i 1

 1, _i 0 , for i1,...,k (4.2-1)

Definition 4.2-6: If X are convex subsets in _i Rⁿ



i 1,2,...,s



, their linear combination



 n

i i iX

1

 is also convex.

36

Definition 4.2-7: (upper semi-continuous)

(a) Point : A numerical function defined on X is said to be upper semi-continuous at x , if, to each ₀  0, there corresponding exist a neighborhood N_

 

x₀ such that

     



   

N x₀ f x f x₀

x (4.2-2)

Figure 4.2-3 An upper-semi continuous function

(b) Mapping: Let  be a mapping of a X Y. Let x be a point of ₀ X . We say

 is upper semi-continuous at x if for each open set G containing ₀ x_o

there exists a neighborhood ^N_

 

^x₀ such that

 

^{x x G}

N

x _{ 0}  

Corollary 4.2-8: (Brouwer, 1909,1910). Let X be a nonempty compact convex set in Rn, and f :X X be a continuous mapping that carries a point x of X _to some point f

 

x of X . Then f has a fixed point xˆ so that xˆ f

 

xˆ

37

Figure 4.2-4 One dimensional case(fixed point)

Theorem 4.2-9: (Kakutani, 1941). Let X be a nonempty compact convex set in Rⁿ, and f :X 2^X be a set-valued mapping which satisfies

(a) For each xXthe image set ^f

 

^x is a nonempty convex subset of X ;and (b) ^f is a closed mapping

Then ^f has a fixed point.

Figure 4.2-5 Fixed point for set-valued function

Proof. Since X is compact, recall the Corollary 4.2-4 on the existence of  -net , for every  0, it has an -net



 



 a i s

N  ⁱ | 1,..., . Next choose an arbitrary point b^iof ^f

 

^ai . Then, we define the continuous functions _i^

 

x on X by

 

^max



^{i ,0}



i^ x  x a^

   



^i1,...,s_



(4.2-2)

38

According to Definition 4.2-2, Since N an _  -net, for each x we have a i

x ^

   for some i, so that we have _i^ 0 for this i. With these function we can obtain the weight functions

   

Using these weight function, we define a single-valued continuous mapping

    

Because of the convexity of X which define in Definition 4.2-5. Then we obtained a single-valued continuous mapping f^ :X X for every  0. By the Brouwer fixed-point theorem (Theorem 4.2-6), there is a fixed point x^,

 

^

Without loss of generality, assume that we have chosen a sequence

 

_v of positive numbers fulfilling following constraints

39

a. lim 0



 v

v 

b. x ^v x

vlim  ˆ







c. ^{xv  fv}

 

^xv (4.2-6)

We tend to show that xˆ is a desired fixed point of f . Consider the set

 

_

 f x U

O  ˆ 







  u u 

U | for a  0

If xˆO_ for any  0, we have dis



xˆ,f

 

xˆ



0, which entails xˆf

 

xˆ

because f ˆ is closed in

 

x X . The subsequent discussion will clarify that xˆO_ for any  0.

First note that O is an open set containing _ f ˆ . This can be seen by noting that

 

x

 



^

 x U

O  

This union taken over all xf

 

xˆ , and also the openness of U .By the definition _ 4.2-6, we know the convexity of O_

f is upper semi-continuous. Since O is an open set containing _ ^{f ˆ}

 

^x , there is an

-neighborhood ^V ^



^{x| xxˆ ,xX}



of xˆ such that ^f

 

^V_ ^O_. By ()(),we have

2

_V  and x^V V_2 for large v.

40

For these large v, w_i^v

 

x^v 0 implies

2





^vi x ^v  _v 

a , so that

x x x a x

a^vi  ˆ  ^vi  ^v  ^v  ˆ

<  



 2

2 .

In summary for large v, we have a^vi V_ for i with wi^v

 

x^{v 0}, which entails

 



^{ }

 

 f a f V O

b ^vi  ^vi   . In view of

  

 _i^{v v vi}

v w x b

x^{   }

x^v turns out to be a convex linear combination for only b^vi lying in O for large _

v. The convexity of O therefore implies _ x^vO_ for large v. Letting ^v tend to

infinity in view of

 

 , we have in the limit xˆ O 2.

The replacement of  _by 2 in the resulting relation is due to the possibility that  xˆ , the limit of

 

^xv , may lie on the boundary of U . 

However, xˆ O ^2 for any  0 is equivalent to xˆO^ for any  0_{, whence}

 

x f

xˆ ˆ in the light of the preliminary discussion above. This completes the proof, Q.E.D.

The following part we use Rosen(1965) method to proof the existence and uniqueness of the Equilibrium point.

Theorem 4.2-10: An equilibrium point exists for every concave n-person game.

41

Proof:

Consider the point-to-set mapping xx, given by

   



^{y x y x z}



x | , maxz R  ,

  (4.2-1)

It follows from the continuity of 

 

x,z and the concavity in z of 

 

x,z for fixed x that  is an upper semi-continuous mapping that maps each point of the convex , compact set R into a closed convex subset of R. Then by the Theorem 4.2-9, there exists a point x^*R such that x^*x^*, or



^{x x}

  

^{x z}

R

z ,

max

, ^{* *}

*  

  (4.2-2)

The fixed point x^* is an equilibrium point satisfying equation (4.1-1), which we rewrite below..

 

^

   

1^{* *}



^



*

* 1

* max _i ,..., _i,..., _n | ,..., _i,..., _n

y x y x x y x

x

i



 i 1,...,n

If we suppose that x^* were not be the equilibrium point. Then, say for il, there would be a point x_l  x_l such that ^x 



^x₁^*,...,^xl,...,^xn^*



^Rand ₁

 

x _l

 

x^* . Then we have ^

 

^{x*,x }



^x*,x*



, which contradicts (4.2-2).

Then the proof is completed.

Recall the network game problem, if we assumed the set

 



^{| 0}





 x h x

42

Then the Kuhn-Tucker conditions equivalent to (4.1-1)  with  given by (4.2-3) can now be stated as follows

 

^{x0 0}

We shall also use the following relation by the Theorem 4.1-4, which holds as a result of the concavity of h_j

 

x . For every x⁰,x¹R we have

Now consider the network game, if we consider the weight value of the all player, to find the weighted nonnegative sum of the functions _i

 

^x , then we have

43

Definition 4.2-12: The function 

 

x,r will be called diagonally strictly concave for xR and fixed r0 if for every x⁰,x¹R we have



^{x1 x0}

   

^{'g x0,r  x0 x1}

  

^{'g x1,r 0}

We can also represent as Theorem 4.1-5

   



^{x1 x0,g x0,r g x1,r}



>0 (4.2-11)

Theorem 4.2-13: There exists a normalized equilibrium point to a concave n-person game for every specified r 0.

Using the fixed point theorem as in Theorem 4.2-9 (Kakutani fixed point theorem), there exists a point x^* such that

44



^{*, *,}



^max



^



^{*, ,}



^|

^{ }

^0



 x x r x y r h y

y (4.2-13)

Then by the necessity of the Kuhn-Tucker conditions, ^h

 

^{x* 0, and u* 0, such}

that ^u*'h

 

^{x* 0 and}

   

⁰

1

*

*

*   





 k

j

j i j i

i i x u h x

r 



i1,...,n



(4.2-14)

Let

i j

ij r

u u

*

*  , which the same with equation (4.2-7) , are sufficient to insure that x^* satisfies (4.1-1); x^* is therefore a normalized equilibrium point for the specified value of r r.

The proof is completed.

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