Lexicographic Strategic Games' Nonstandard Analysis

Published Online June 2013 in MECS DOI: 10.5815/ijisa.2013.07.01

Let discuss m -dimensional lexicographic noncooperative game [1-7] ,

standard mexed extension by ^ ^{( Г )} . In such kind of games the main problem is that in every Г game there doesn’t exist the equilibrium situation or mayby this set

^ ( Г )   0

will be empty -       =

.

In the following article for ^Г game we take a player’s new type of concept of a nonstandard mixed

Г=< NЛх.К n ,{ H i L N >^

( Г ⁰, Г ¹

p m —1 ^

strategy, that is distributions of Г

a m -dimensional probability

Г ⁰

game’s on

m — 1

,...,

By using such distributions payoffs’

criterion. functions

where every ' ^{G N}   t¹’^-", ^n} player’s payoff vector-

H i ⁰,

...,

,       H,      , .   H, = ( H”

function ⁱ we note by ^{i       i}

,...,

H im - ¹) , its

H m — 1

ⁱ sum forms decreasing sequence. For this

reason, methodology of nonstandard analisis is capable [8]. Namely analisis of hyperreal infinitesimal numbers.

For conducting such analysis it is necessary to define vectors lexicographic product operation by ⊗ notation.

Let note that in the conditions of b) case the requirement of difference from first components’ xero

Let we have

m -dimensional vectors

(a0, is essential in

...,

_am - 1₎

vector.

a = ( a ⁰

_am - 1₎

and

b = ( b ⁰

_bm - 1₎

we note

Let say that ( Ω , A ) is measurable space, where

a⊗b=(a0b0,a0b1+a1b0, a0b2 +a1b1 +a2b0, a0bm-1 +...+ am-1b0)

1               2

0  0        k 1-k        k 2-k            km

( a b , ∑ a b , ∑ a b ,  , ∑ a b )

k=0           k=0

Ω ⊂ R^m .

µ : A → R

Let discuss the transformation + m , that fulfills the following conditions: if

A i ∈ A i = 1,2,...,

,

U 4e A i=1

, then ^{i = 1}

^A i _∩ ^A j ≠ ∅

^ ⁽ U A i ) = Z ^ ⁽ A )

i = 1

.

i ≠ j

,                         ,

It is obvious, that m -dimensional a ⊗ b vector’s components are the following form of one variable m -degree

a(t) =a0 +a1t+...+amtm and

b(t)=b0+b1t+...+bmtm coefficients of polynomials multiplication, only in difference with, that in the process of their multiplication m -degree polynomial can be conservated.

In the following lemma some essential properties of vectors lexicographic product is established.

Lemma 1.1 Lexicographic product’s ⊗ operation defined by (2) has got the following properties:

• 1)    Commutativity - a ⊗ b = b ⊗ a ;

( a ⊗ b ) ⊗ c = a ⊗ ( b ⊗ c )

• 2)    Associativity -                                  ;

• 3)    Distributivity of summation –

( a + b ) ⊗ c = a ⊗ c + b ⊗ c

;

• 4)    “Monotonicity in two cases:

• a) Weak monotonicity. If

α = ( α ⁰,..., α ^{m - 1}) ∈ R^m      ( a ⁰,..., a^{m - 1}) ^L

,,         ⁺ and    ,,        ≼

^{( b 0,..., b m - 1) ( aL} ≼ ^{b )} , then ^{a ⊗ α L} ≼ ^{b ⊗ α} ;

As a role of e = (1,ε,ε2,...,εm-1)

unit we take

, where ε

infinitesimal number, and in addition components’ is equal to number

1 + ε + ε ² + ... + ε ^{m - 1}

and its standard part is equal to 1.

In the second section of

Γ = ( Γ ⁰, Γ ¹

,...,

m - 1

game

a vector

b) Strict monotonicity.

α = ( α ⁰,..., α ^{m - 1}) ∈ R^m

Let                 ,,           ⁺         and

( a 0 ,..., a m - 1 ) > L (0,...,0)   _Tf ( a ⁰,..., a m ^{- 1}) L ^

.

⁽ b 0 ,..., b m ^{- 1)} and a ⁰ ^, then a ® ^а L ^ b ® ^a .

is the

-

hyperreal sum of e

ε ^m

1 - ε

,

the article in

i ∈ N player’s

Χ nonstandard mixed strategy i is given by the form of p ×m           p a i      matrix, where i is an amount of ∈

Χ player’s pure strategies. The set of     i strategies is

noted by

Χ= ( Χ

ⁱ . The set of situations in Γ game is Σ ,

1,...

Χ )∈Σ n       is a situation in nonstandard

mixed strategies, and Γ^Σ is a mixed extension of Γ

Γ game. The equilibrium situation is defined in Σ game

and their set is noted by G ( Γ^Σ )

.

Γ

In the third section a mixed extension ^Σ of a matrix game and a saddle point in nonstandard mixed strategies are discussed. The property of transformation on nonstandard mixed strategies (lemma 3.1) and necessary and sufficient conditions of a saddle point in Γ

^Σ game are proved.

In the fourth section the examples of a lexicographic matrix games are discussed and it is shown that if in a game does not exist a saddle point in standard mixed strategies, then there can’t be a saddle point in nonstandard mixed strategies either (example 4.1). If in a lexicographic matrix game does not exist a saddle point in standard mixed strategies, there can be a saddle point in nonstandard mixed strategies (example 4.2).

II. Lexicographic    Noncooperative    Game’ s

Nonstandard Mixed Extension

Let in

Г       x — ( x, ,

^S game      ^{x 1}

...

, x ) e x .

ⁿ        is a situation

Let define in lexicographic

Г = ( Г 0 , г ¹,.

j-ч m — 1

in pure situation

t t .       . X — (Xt, strategies, and           1

...

, X n) eS is a is a

game nonstandard mixed strategies and then make ^Г

game’s mixed extension. For this, in advance, note m -

dimensional probability distribution on ^{X i} set in ^Г game for i ^{e N} player, that has ^Pl amount of pure strategies in the following form:

X_t ( X 1 ) = ( X °( X 1 ),..., X m - 1 ( X 1 )) e R m

,

x = ⁽ x¹,..., x p i )

,

In this connection

p

£ X i ( x ) = (1, s , s ²

l = 1

s m - 1 )           „

1,..., n

where ^s is a hyperreal infinitesimal number.

Let note m -dimensional probability distribution

X i ( x i )

X on i set of strategies for

form of following matrix:

^{1 e N} player in a

situation

in nonstandard mixed strategies. Then in ^{X i e N} player’s payoff is

H i ( X ) — £ H ( x ) 0X ( x )

^{x e} X

£ ... £ H i ( x „..., x n ) 0 X , ( x , ) 0 ... X n ( x, )

^x 1 ^e X 1    x n ^e X n

f ( X-(x ¹),

X i =    ..........

I ( X -( xn

x m ^- 1 ( x 1 )) "

x m ^- 1 ( x p )) J

( X i ⁰,..., X m - 1 )

,

pi                             pi

£ X"( x) — 1 £ Xk (x-) — s l—1                           l—1

,

k — 1,

m — 1

For example let write (3) in case of two players i.e.

. N — {1,2} T .   (X., X?)  4 v f t when             . In the     1    2 situation for first player we have

H (Xj, X2) — X 0 H 0XT —

£ £ X 1 ( x j0X 2 ( x 2 )

^e X 1 x 2 e X 2 .

According to the definition of multiplication of ⁰ operation and its 2) property for x ⁽ x ¹, x ² ) ^{e X} we write:

X 1 ( x 1 ) 0 H 1 ( x ) — ( X 10 ( x 1 ) H 10 ( x ), ...

, X 1 ⁰ ( x 1 ) H m - ¹ ( x ) + ... + X 1 ^m - ¹ ( x 1 ) H 1 ⁰ ( x ))

,

X 1 ( x 1 ) 0 H 1 ( x ) 0X T ( x 2 ) —

( X 1 ⁰( x 1 ) H 1 ⁰( x ) X 2 ⁰( x 2 ),...,

X 10 ( x 1 ) H 10 ( x ) X m - 1 ( x 2 ) +

X 10 ( x 1 ) H m - 1 ( x ) X 20 ( x 2 ))

X

This      ⁱ matrix is called

nonstandard mixed strategy. It is

/ ^e N player’s

. X obvious, that ⁱ

strategy gives probability distributions on ^Г game’s г 0     г m — 1              .

,...,        criterion for ^{i e} N player.

For every x ^{1 e X 1} and x ^{2 e X 2} by summing given expressions we get:

Xj 0 H 0 X T — ( X 00 H 0 X ₂⁰,

001  100  010

.^k J .^ H J .X. 2 +    -^X J .^ H J .^k -^ ^H .^x J .^ H J .^x 2 ,

V" 0 Lj0 "Vm — 1  1     1   "V^ Tjm - 1 vO^

.^x ^ .^H ^ .^x ^ I ... I  -^X J -^H J    -^x <^ )

...,

Let note in ^Г game ^{i e N} player’s nonstandard

S, mixed set of strategies by i and a set of situations by s—nsi ieN . Let define г game’s nonstandard mixed extension by

Analogically is for a second player too

H2 (Xt, X2) — Xt 0 H2 0XT

.

.

Let note that m -vector’s components are players

г ⁰ г ¹

payoffs certain sums in scalar ,

,...

, г ^m - ¹

games.

r_s —< N ,{ S i } i e N ,{ H i } i e N >

.

X ^a 1 H ^a 0 X ^a 2

In these sums each summand has a form ^{1 i 2}

( i — 1,2). _{T t}         _  H “ ⁰( X a ¹ ,X a ²)   .

Let change it by ^{i     1     2} and say

a + a + a — p

^{0     1     2}       . After this it is obvious that m -

vectors components are such summands

Hα0(Xα1,Xα2)          p i     1     2   , for which numbers are constant and       accordingly       are       equal       to

p = 0, p = 1,

...,

p = m - 1 . Thus, in lexicographic

Γ bimatrix ^Σ

game

i ∈ {1,2}

player’s payoff in

( Χ 1 , Χ 2 )

situation is equal to

H i ( Χ 1 , Χ 2 ) = ( H i ⁰( X 1 ⁰, X 2 ⁰),

∑ H i ^{α 0}( X 1 ^{α 1}, X 2 ^{α 2}),..., ∑ H i ^{α 0}( X 1 ^{α 1}, X 2 ^{α 2})).

p = 1                             p = m - 1

Let note by G ( Γ^Σ ) a set of equilibrium situations in Γ

^Σ game.

With the preceding definitions and notations that are all about lexicographic noncooperative Γ game’s Γ nonstandard mixed extension of Σ , it is clear to see the difficulty of the given apparatus. It seems that some classical results about scalar noncooperative games Γ transfer to Σ games, but some do not. Some important Γ circumstances take place for Σ games. We discuss some such kind of results for lexicographic matrix games [9].

If we note for any n

H ^{α 0} ( X ^{α 1} ,.

in ^{i       1}

^X n ^{α n )}

p=α +α +...+α , expression        0     1         n then analogically to (4) equality (3) has the following form:

H i ( Χ ) = H i ( Χ 1 ,..., Χ n ) =

( H i ⁰( X 1 ⁰,..., X n ⁰),

∑ α ₀    α ₁       α_n

Hi (X1 ,..., Xn ),..., p=1

∑Hiα0(X1α1,...,Xnαn)), p=m-1                     i ∈ N

.

Let take a notation

Χ || i Χ i = ( Χ 1 ,..., Χ i - 1 , Χ i , Χ i + 1 ,..., Χ n )

.

• III.    Lexicographic Matrix Game’s Nonstandard Mixed Extension

Let discuss lexicographic p × q matrix game

m - 1

Γ = ( Γ ⁰, Γ ¹,.

by matrix of payoff

H = {(ai0j, ai1j,..., aimj-1)} i=1,..., p j=1,...,q

;                                             .

In this game 1 and 2 players’s commom (standard) mixed strategies are:

X = ( x ₁,..., x_p )   ∀ x ≥ 0

,

Y = ( y 1 ,..., y q )   ∀ y j ≥ 0

x + ... + x = 1 ;

y ₁ + ... + y_q = 1

then analogically to (4) we write:

H i ( Χ || i Χ i ) =

( H i ⁰( X 1 ⁰,..., Χ i ⁰,..., X n ⁰),

∑α0    α1      αi      αn i    (    1 ,..., i ,..., n ),..., p=1

0 ∑ H i ^{α 0}( X 1 ^{α 1},..., Χ ^α iⁱ ,..., X ^α nⁿ )) p = m - 1

Definition 2.1 We say that a situation in nonstandard *

mixed strategies Χ ∈ Σ is an equilibrium situation in

Γ noncooperative Σ game, if

*

*

H i ( Χ ) _≽ L H i ( Χ || i Χ i )_, ∀ i ∈ N _, ∀ Χ i ∈ Σ i . ₍₆₎

In Γ matrix game let note a set of equilibrium situation by σ(Γ) . As we have already stated above, this set could be empty -

σ ( Γ ) ∅

.

Γ = ( Γ ⁰ Γ ¹ ... Γ ^{m - 1})

Now discuss matrix         ,  ,,       game in nonstandard mixed extension. Let note i≡x = 1,..., p j≡ y =1,..., q i           ;        j            and write H vector in the following for

H = { H ( x i , y j )} = ({ H ⁰( x i , y j )},..., { H^{m - 1}( x i , y j )})

Analogically to a lexicographic noncooperative game, in the given matrix game players’ nonstandard mixed strategies for 1 and 2 players accordingly have the following form:

X =

p

( X 0 ( x 1 ),..., X m ^{- 1}( X 1 ))

( X ⁰( X p ),..., X m - * ( X p ))

₌( X ⁰,

...,

X m ^{- 1}) ^{, p}

X X ⁰( X i ) = 1 X X k (x, ) = г

l = 1

l = 1

,

, k = 1,..., m -1

.

Y =

( ( Y ⁰( y 1 ),..., Y m ^{- 1}( y j) )

( Y ⁰

y m - 1 )

, l = a + a + a where       1     2     3

.

From the (6) definition of noncooperative game’s equilibrium situation we get the following definition.

Г

Definition 3.1 In ^z matrix game with matrix of

**

H payoff we call ^{( X} , ^Y ) situation the equilibrium (sadle point), if for any ^X and ^Y nonstandard mixed strategies take place

**

*

Г® H ®Y L < X® H ®Y

or (7) accordingly

* ⁰

( X ⁰ H ⁰ Y

,...,

( ( Y ⁰( y q ),..., Y m ^{- 1}( y q )) J

qq

X y ⁰( y , ) = 1 X y‘ ( y , ) = ^

j = 1                          , = 1

,, k = 1,...,m -1

* ⁰         * ⁰

( X H ⁰ Y

X X ^a 1 H ^{a 2} l = m - 1

* ^a 1

,...,

*

( X ⁰ H ⁰ Y ⁰

,...,

*

L

* ^a3

^Y) L

≼

* «3

XX Ha2 Y )

l=m-1

* ^a1

XX H ^{a 2}Y ^a3) l = m-1

L

≼

,

.

Let note ^Гmatrix game’s mixed extension in nonstandard mixed strategies as well as in the case of

Г noncooperative games by z .

If the (8) conditions fulfill **

(X, Y) e G(Г, )

we

write

.

Let define in     , y ^j

situation the first player’s

Г

It is clear, that in noncooperative ^z

games the

payoff analogically to the operations held in (4) it is

equal:

property of transformation on nonstandard mixed

Г strategies is fulfilled. Let prove it for matrix ^zgame.

H (X, y,) = X® H., =

(X⁰H.0,XX^1-kH.^k,..., X X^m-1-kH.^k) k=0                  k=0

j = 1,...,q

.

Lemma 3.2

(Y mixed Y = player’s and (a are fulfilled

0,

,.

Г

Let in ^zgame are any nonstandard

...,

...,

Y m-1 )

am^-1)

strategies of the second

is a vector and inequalities

We get the second player’s payoff in the same way

(X, Y) ч v i      situation

Hi.®Y L < (a⁰,

...,

m-1

a ) i = 1,...,p

,

.

00    k

H(x,,Y) = Hi.®Y = ⁽i ,^Xi y 1-k

Then

X = ( X⁰,

for     any     nonstandard     mixed

...,

inequality is fulfilled

X m-11

strategy of the first player the

,...,

m-1

jjk ym-1-k k=0

) i = 1

,

,...

^,p .   (7)

X® H ®Y L < (a⁰,

...,

am^-1)

.

The first player’s payoff in analogically to (5) is equal

(X, Y)  . v situation

Proof. According to (7) we write

(H".Y⁰, X Hi.

k=0

v 1- k

_Y ,...,

H (X, Y) = X® H ®Y =

(X⁰H⁰Y 0, X X ^a1H ^{a 2}Y ^a3 l=1

,...,

X X ^a1H ^a2 Y ^a3) l = m-1

m-1

У H^k .Y^m-1-k )

i0

k=0               ^L< ^(a ,

...,

m-1

a ) i = 1, ^,

...

,p

.

,...

Each inequality for every i¹

, p we multiply

..  ,        X( x,) = (X 0( x,),..., Xm^-1( x,))

accordingly on       ^{i            i              i}

.

According to lemma 1.1 inequalities are remained and by the definition of the operation ® we get:

a -2

(X0(Xi)HiY0,2X“1 (Xi)H  Y“3,..., l=1

^a2

2 X “1(Xi) H  Y “3)^

l = m-1«<

(a⁰X⁰( Xi), 2a^kX 1^-k (Xi),..., 2 akXm—1—k (Xi))

k=0

i = 1,...,p

Let sum these inequalities and take into account that in the components of vectors every summands except one that consists of ⁸  =¹infinitesimal numbers.

With the help of their ignorance we get:

(X⁰H⁰Y⁰,2X^a' H“ Y^a3,..., 2X“1H“2 Y“3)

l =1                          l=m -1

≼

1       p                   m -1

(a 0,2 a (2 X1-* (Xi)).....2 a"<2 Xm^-'^-k (Xi)))

k=0      i=1                      k=0

( a0 X 0( X), 2 akX 1- k (Xi),..., 2 akXm-1-k ( X)) =               k=0

(a⁰,2 a8^k,..., 2 a^k^"^-1^-k (Xi))

=      k=0

₌ (a⁰,..., am^-1)

Hence (9) is proved.

Analogically will be the transformation on nonstandard mixed strategies in other eniqualities.

Necessity. In (8) conditions by following consequences let suppose that

' (1.E,..., г"^-1)'

(0^,.,0)

...

I (0,0,...,)) J f (0,0,...,0) )

m-1\

⁽¹, £,..., 8    )

( (0,0,...,)) J f (0,0,...,0) " <⁰,⁰-^0,

\^(I, ⁸,..., ⁸     )J

Then, if we neglect summands, that take place

8,8 ^

m-1

, the left side of (8) has a form

(H".Y⁰,2 Hi.

k=0     Y1- k m-1

2 H^k Y^m-1-k ) k=0

**

L^X®H®Y

for any i '^,...^,p and thus it fulfills (10) left part.

Analogically, if we put in (8) the second player’s strategies by sequence

Y = ((1,8,..., 8m^-1) T ,(0,0,...,0) T ,...,(0,0,...,0) T ),

Y = ((0,0,...,0) T ,(1,8,..., 8) T ,...,(0,0,...,0) T )

**

Theorem 3.1

_K            (X, Y) e G (Г ) .

For this purpose v ,  7     v 2/ it

Y = ((0,0,...,0) T ,(0,0,...,0) T ,...,(1,8,..., 8) T )

is necessary and sufficient for every i   ',..., p and j = 1q        n                                  ,

,, the following inequalities should be fulfilled we get that it fulfills (10)’-s right part for every

^1,...’ q . The theorem is proved.

**

Hi. ®^YL_< X® H ®Y L_< ^X® ^H. j

Proof. Sufficiency. If we use the both sides of (10) lemma 3.1, we get (3).

_n и т . (X, Y) •     <           Г

Corollary. Let ^v ’  ⁷ is a saddle point in ^sgame

**

(X, Y) e G(Г) _T.                    .   ,

- v ’  7     v s J . Then according to (10) that for every i and j

*         *           *         *

**

(X,_Y) H0.Y⁰< X⁰H⁰Y⁰< X⁰H\

(у. ⁸) '

(0,1 - у )J

L< (ау, а8+Ру + ау +1 - а - у)

That means that

(X⁰, Y⁰)

situation in standard

mixed strategies is a saddle point in the first scalar

**

f Г0            (X0,Y0) e  а(Г0)  • matrix 1 game - v ,    7 e    v 7 in

Г = (Г⁰. Г¹Г m^-1)

lexicographic         ,   ,,       matrix game.

L

≼

/ ^(а. Р) (0,1 - а)

As in ^Гscalar game the first and the second players’ optimal strategies are accordingly

• IV.    Examples

On the basis of stated aparatus let conduct analysis of

. . Г = (Г⁰. Г¹)

a lexicographic matrix         ,    game, which is given by matrix of payoff. Firstly, discuss a matrix

Г = (Г0,Г1). ..  _ _ game         ,    , that was studied by P. Fishburn and showed that there is no saddle point in it -^(Г)= 0 [10].

Example 4.1 A lexicographic matrix game

Г = (Г⁰,Г¹).   .     .      . .    -

,    is given by matrix of payoff

Г ⁽1,^{0)  №0)}1

((0,0)  (0,1) J

*

X⁰= (а*,1 - а*) а e[0,1]

,

*

Y⁰ = (у’,1 - у*) у * = 0 _{л f}        ..

and                      ,         , therefore according to (12) the following inequalities must by fulfilled

(0,8)! (0.1) J

, ^(а‘.^в)

L_<(0,а‘8 +1 - а*) L _< 1(0,1 - а )

but that is impossible. Thus, in the given game there also does not exist a saddle point in nonstandard mixed strategies.

Example 4.2 Let there is a lexicographic matrix

Г = (Г⁰,Г¹)                .

game         ,     with a matrix of payoff

^(Г)

Here       =   .

Г ⁽0,D  (1.0) 1

I (2,0)  (0,1) J

. .   , -     . Г = (Г0, Г1)                ,   .   .

Let define for          ,      game nonstandard mixed strategies:

There is no saddle point in it - ^СТ(^Г) = ⁰

.

X Г  ^(а,^в  1

((1 - а, в - в) J

,

Y = ((у, 8) T ,(1 - у, в - 8) T )

Let check up in the given game if there is a saddle point in it or not in nonstandard mixed strategies. For this let define (11) strategies and check up the theorem 0

• 3.1.    As in ^Гgame players’ optimal strategies are

*

X⁰= (а',1 - а*) = (f.|)

In this case the vectors

H. 0Y

X0H.,   , j and

^{X 0}H 0 ^Yhave been already defined above as the following forms:

H. 0Y = (у, 8) H₂.0Y = (0,1 - у)

,                                                                    ,

X0 H_ч= (а, в) X0 H.₂= (0,1 - а)

,                                                                      ,

X 0 H 0 Y ₌(ау, а8+Ру + ау +1 - а - у)

According to the theorem 3.1 we write the following inequalities:

and

*

Y⁰= (7,1 - у") = (Ы)

, therefore the condition (10) has the following form:

(j,¹+^в -⁸ )!    Г2   4  2  .1

⁽*.| + 28")L ^^3,  9 ⁺3 ^в J L

f(b-2 + 2s’ - 2в)

_ 1   (2,1 + в)

*

These inequalities fulfill when

• ³. Thus in

Г = (Г⁰, Г1)

^* = о. в = 3, ;

game players’ optimal

nonstandard mixed strategies are accordingly to

*

х =

V ⁽з^,о7

*

Y = ((3,0) T ,(<,!) T)

and G (^Гх) * 0

**

H (X, Y) = (1,1)

.

The first player’s payoff is

Thus, offering construction considers necessities of vectorial criterion and is a generalization of a game’s mixed extension. It is obvious that if ^(Г) * ⁰, then G (F_z) * 0 .

• V. Conclusion

By nonstandard mixed extension of lexicographic strategic games a generalization standard mixed extension has been conducted by using of nonstandard analysis, namely by analysis hyperreal infinitesimal numbers. It has been shown that if in the given game there exists an equlibrium situation in nonstandard mixed strategies, then in the same game there exists also an equilibrium situation in nonstandard mixed strategies. Such kind of analysis helps us to solve the problem about the existence of an equilibrium situation in games with the help of simple formal procedures

• [5]    M. Salukvadze, G.N.Beltadze and F. Criado. Dyadic theoretical games models of decision – making for the lexicographic vector payoffs. International Journal of information Technology and Decision Making, Vol. 8, Issue 2, 2009, pp. 193-216.

• [6]    G. N. Beltadze. Lexicographic non-cooperative game’s mixed extension with criteria. International Journal of Systems and Software, ARPN Publishers, Vol 1, № 8, November 2011, pp. 247250.

• [7]    G.N. Beltadze. Lexicographic Multistage Games with Perfect Information. Informational and Communication technologies–Theory and Practice: Proceedings of the International Scientific Conference ICTMC-2010 Devoted to the 80^th Anniversary of I.V. Prangishvili. Nova Publishers, 664 pp. USA, 2012. pp. 275-281.

• [8]    M. Davis. Applied Nonstandard Analisis. Courant Institute of Mathematical Sciences, New York University, 1977.

• [9]    G. N. Beltadze. On the reduction of the solution of a lexicographic matrix game to the solution of its square subgame. Bulletin of the Academy of sciences of the Georgian SSR, 104, № 1 (1981), pp. 29-32 (in Russian)

• [10]    P. C. Fishburn. On the foundations of game theory: the case of non-Arximedean utilites. Inter. J.Game Theory, 1, № 2 (1972), pp. 65-71.

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