4_Ilie Mitran. 100KB Jun 04 2011 12:08:15 AM

ON THE MAXIMUM PROBABILITY CRITERIA

CONCERNING THE SEQUENCIAL DECISION - MAKING

PROBLEM

by

Ilie Mitran

Abstract. This paper presents, originally,some results about an important criterion used in the theory of the decisions. Here are treated the partial and the total co-operative cases and, finally, will be presented an application for a market competitional problem who will finish with a ruining problem.

INTRODUCTION

A problem of sequencial decision is described by the ensemble ([1],[3]):

   



 _{∈ ∈ ∈ ∈ ∈}

= X X X− M u i M D i M n N x X f n N

S _i i_{x n n}

n, , , , ,

, , , , , , ₀

Where the significance of the elements is as follows:

1) The set

X

represents the space of the positions and is a topological linear space (real) and BX is the

σ

−algebra generated generated by the topology of the

space

X

.

To be measurable space

(

X,BX

)

,the set µ

( )

BX of all measures of probability

defined in B_X is associated. The measure of probability P_X ∈µ

( )

B_X is associated which each state x∈X .

2)

−

X

X₀, represents the set of initial states and final states respectively and they

are supposed to be compact sets in X

3) The set M =

{

1,2,...,m

}

represents the set of deciders taking part in the decision-making process and i X →R

−

:

µ represents the utility function of the deciders i∈M (

u

i is supposed to be continuous)

The set

   



 _{∈ ≥}

= − − −

−

i i

i x X u x a

X : ( ) is called the target set of the decider M

i∈ .

4) The evolution of the decision-making process is described with the help of the recurrence relations:

(

x d

)

x X n N f

xn+1= n n, n , 0∈ 0,∀ ∈

where

( )

∏

( )

=

=

∈ n

i

n i n

n D x D x

d

1

and

( )

n i

x

D represents the set of the decision that can be made in the state xn∈X by the decider i M D

( )

x

i

(

∈ is supposed to be a topological linear space x∈X,i∈M).

The application f fn :XxDX → X,n∈N are called transition functions and

they are supposed to be continuous and bounded (

U

( )

X x

X D x

D

∈

= ).

If x_n ∈X− , then f_n

(

x_n,d_n

)

=x_n,∀d_n∈D

( )

x_n .. When there is no risk of confusion Di

( )

x_n is written as D_ni and D

( )

x_n as D_n.

The notion of inferior semi-continuity(i.s.c.), higher semi-continuity(h.s.c.) and continuity in the Haussdorf sense, both for the univocal and multivocal application will be the basic elements in the proves of some theorems.

2. THE EXISTENCE OF THE GUARANTEED OPTIMAL STRATEGIES

We shall put ourself in the position of decider in the case of the problem of sequencial decision described in introduction. Two situations will be analyzed:

a) the partial co-operative case; b) the total co-operative case;

The purpose of this paragraph is to specify the margins of the interval within which the maximum profit of deciders1 lies, as well as the strategies(simple or mixed) throught which these margins are reached.

If the decision-making process has evolved to the state xn∈X \X− , the

adoption by decider 1 of the criterion of maximum probability implies the adoption of the problem([3]):

( )

   

   

≥    



 _≥

     ∈

∈ − − − −

        −

p a x u X x P

R p

P i i

d d x f d

n

n n n n n

: :

sup

: ~

∏

₌

∈









=

m j j n n n n

n

d

d

D

x

D

d

2 1 ~ 1 1

,

As decider 1 will decide first and deciders j∈M \

{}

1 adopt,simultaneously, the following notations will be considerated:

∏

₌ =

= m

j j n

n D D D

D 2 2 1 1 ,

The following functionals are introduced:

(

1 2

)

( , , ) 1 1

(

1 2

)

1 2 2

1 , , : , ,

:

2

1 x X u x a d d DxD

P d d F R xD D

Fn n f_n x_nd d ∈

      _≥      ∈ = → − − −

(

)

( )

(

1 2

)

1 2 * * , , 2 1 2 1 , , ) ( : ) ( : , , : 2 1 xD D d d a x u X x P a x u X x P d d g R xD D g M

i i M

i i

x

M

i i M

i i d d x f n n n n n ∈       ≥ ∈ − −       ≥ ∈ = →

∑

∑

∑

∑

∈ ∈ − − − ∈ ∈ − − −

and the multivocal application:

( ) ( )

,

{

,

(

,

)

0

}

:D₁ →P D₂ B d₁ = d₂∈D₂ g d₁ d₂ ≥

Bn n n

For greater convenience we shall write

F

,

g

,

B

instead

of

(

)(

)

2 1 , , ,

, ,

, n n ₁ D ₂ D

n g B D d D d

F are assumed to be compact spaces.

The following hipothesis are made:

1) the forming of a coalition in the sense of maximum probability is allowed; 2) if the first decider has adopted the strategy d₁∈D₁, the other decider will adopt only strategies from B

( )

d₁ .

Remark 2.1 Hypothesis 2 is based on the following argument:if the choice of the pair of the strategies

(

d1,d2

)

∈D1xB

( )

d1 increases in the state

x

n+1 the value:

( ) { } { }         ≥ ∈

∑

∑

∈ ∈ − − − 1 \ 1 \ ,

,1 2 : ( )

M j j M j j d d x

f x X u x a

P

n n

then this choice will suit deciders j∈M \

{}

1 ; if in the state

x

_n+1the value:

( ) { } { }         ≥ ∈

∑

∑

∈ ∈ − − − 1 \ 1 \ ,

, _{1 2} : ( )

M j j M j j d d x

f x X u x a

P

n n

doesn’t increase (as against the value:

{ } { }         ≥ ∈

∑

∑

∈ ∈ − − − 1

\ \1

) ( :

M

j j M

j j

x x X u x a

P

n ), but

(

d₁,d₂

)

≥0

{}

1 \

M

j∈ will be favored again. In his turn, decider 1 will be favored as he has the possibility of improving his control over deciders j∈M \

{}

1 .

Having introduced these notations, we can formulate problem (P1n) in the

following way:

( )

1 :

n

P determine

( )

1 2

* 2 * 1 *

,d DxD d

d = ∈ which verifies the equality:

( )

_{( )}

(

1 2

)

, * 2 *

1, sup , 2 1 2 1

d d F d

d F

xD D d

d ∈

= .

The solving of the problem

( )

P₁n represents however the ideal case for decider 1 as in concrete situations it hardly ever happens for all the deciders of the set M to have the same target set (in other words, all the deciders of the set M have the same target).

As for any d2∈D2 −

the following inequalities occur:

( )

(

1, 2

)

sup 1, 2 sup sup_{( )}

(

1, 2

)

inf sup

1 2 1 1 1

1 1

2 1 1

d d F d

d F d

d F

d B d D d D

d d

B d D

d ∈ ∈

− ∈

∈

∈ ≤

   

≤ .

The following functionals will be introduced naturally

( )

( )

(

1 2

)

1

1 1

1: , sup , 1 2

d d F d

f R D f

d B d∈

= →

( )

1 _{( )}

(

1 2

)

2 1

2: , inf , 1 2

d d F d

f R D f

d B d∈

=

→ .

Theorem 2.1. The following results occur:

1) If

F

is h.s.c.,

g

is continuous( as an univocal application in the topology generated by

2 1xD

D

d

), B

( )

d₁ is closed in the metric space

( )

D2 ,d , d1 D1

P  ∀ ∈

  



 −

(

−

d

is the Haussdorf metric drawn with the help of the 2

D

d

metric) and

B

is closed( as a multivocal application), then there is

d

₁*

∈

D

₁ so the following equality takes place:

( )

_{( )}

(

1 2

)

* 1

1 max max , 1 2 1 1

d d F d

f

d B d D

d∈ ∈

= ;

2) If

F

is h.s.c.( as an univocal application in the topology generated by

2 1xD

D

d

),

B

is continuous( as a multivocal application in the topology generated by the Hauswsdorf metric d drawn with the help of

2

D

d

metric), then there is

d

₁**

∈

D

₁ so the following equality occurs:

( )

_{( )}

(

1 2

)

* * 1

2 max inf , 1 2 1 1

d d F d

f

d B d D

d∈ ∈

Proof

1)We first prove that if g is continuous, then

B

is h.s.c.(as a multivocal application). Let us consider:

( )

( ) ( )

2

* 2 ` 1 2

1 * 1 1 1

, ,

,d d d Bd d d

D d

n n n n

n n n

n ⊂ → ∈ → .

Because g is continuous, we obtain that g

(

d_*1,d_*2

)

= lim

(

1n, n2

)

n g d d ., As

(

,

)

0

2 1

≥

n

n

d

d

g

,

N n∈

∀ , it means that g

( )

d1n,dn2 ≥0, therefore

( )

1 2

* B d

d ∈ and consequentely

B

is h.s.c..

If d₁

( )

D₁ d_*1

n

→

∈ , from the fact that F is h.s.c. and B

( )

d₁ is closed ∀n∈N, it follows that there is

d

_n2

∈

B

( )

d

_n1 so that the following conditions occur:

( )

( )

( )

( )

( ) (

1 2

)

2 1 2

1 1

1 sup , max1 , , 2

1 2

n n n

d B d n

d B d

n F d d F d d F d d

d f

n n

= =

=

∈

∈ .

From the fact that

B

is h.s.c. it follows that there is d_*2∈B

( )

d_*1 so that _*2 lim _n2

n d

d = .

As

F

is h.s.c. we shall have:

( )

(

) ( )

_{( )}

( ) ( )

1

* 1 2 1 * 2

* 1 * 2

1 1

1 lim , , max ,

lim

1 2

d f d d F d

d F d d F d

f

d B d n

n n n

n = ≤ ≤ ∈ =

− −

From limf₁

( ) ( )

d1_n f₁ d_*1

n ≤ it follows that f1 is h.s.c.; as D1 is compact it means that

there is d_*1∈D₁ so that the below equality is verified:

( )

_{( )}

(

1 2

)

1

* max max , 1 2 ` 1 1

d d F d

f

d B d D

d∈ ∈

= .

2).In order to demonstrate the existence of

d

₁** it is sufficient to prove that f₂

is h.s.c.;let us consider any d₁0∈D₁. Also, let us consider any sufficiently small 0

>

ε

. As

( )

( )

(

2

)

0 1 0

1

2 inf 0 , 1 2

d d F d

f

d B d∈

=

there is d₂0∈B

( )

d₁0 so that F

(

d₁0,d₂0

) ( )

≤ f₂ d₁0 +ε.

F

beeing h.s.c. for the closen

ε

there will be

δ

so that:

(

)

≥

(

)

−ε ∀

(

)

∈

(

(

)

(

0

)

)

<ε

2 0 1 2 1 2

1 2 1 2

1 0

2 0

1,d F d ,d ₂, d ,d D xD ,d 1 2 d ,d , d ,d

d

F DxD .

As

B

is continuous, there is γ >0 so that d

(

B

( ) ( )

d₁ ,B d₂

)

≤δ . Let us consider

( )

( )

{

_{1 1}: ₁, ₁0 minδ,γ

}

1 0

1 = d ∈D d d d ≤

V_d D .

For any 0

1 1 Vd

d ∈ , there is a d₂∈B

( )

d₁0 with dD₂

(

d2,d20

)

≤δ . Henceforth, for any 0

1 1

V

_d

( )

(

,

)

(

,

)

₂

( )

₂

2 1 2 2 1

0 2 0 1 0

1

2 d +ε ≥F d d ≥F d d −ε ≥ f d −ε

f

and so

( )

( )

1 1 1

2 0 1

2 d f d , d Vd

f ≥ −ε ∀ ∈ . This means that f₂ is h.s.c. and so there is

1 * * 1 D

d ∈ so that the following equalities occur:

( )

_{( )}

(

1 2

)

* * 1

2

max

inf

,

1 2 1 1

d

d

F

d

f

d B d D

d∈ ∈

=

.

Remark 2.2 Theorem 2.1 specifies the existence of the strategies where the boarders of the interval in which decider 1 will obtain his maximum profit can be reached( and which is the maximum of the probability of realization of the target set in the state

1 +

n

x

).

Theorem 2.2 If

B

is h.s.c. and closed(as a multivocal application),

F

is h.s.c.( as an univocal application) and B

( )

d is compact for every d∈d₁, then there is d*∈D1 so

that the following equality occurs:

( )

* _{( )}

(

1 2

)

1 max max , 1 2 1 1

d d F d

f

d B d D

d∈ ∈

=

Proof In order to prove the existence of

d

* having the required property it is sufficient to show that

f

is h.s.c. Let us consider

( )

*1

1 1 1

lim

, d d

D

d n

n n

n ⊂ = . From the

fact that

F

is h.s.c. and B

( )

D is compact, it follows that there is dn0∈B

( )

dn1 so that:

( )

1

( )

(

1 2

)

( )

(

1 2

) (

1 0

)

1 sup , ₂max₁ , ,

1

2 _Bd n n d BD n n n n

d

n F d d F d d F d d

d f

N n n

n

= =

=

∈

∈ (1)

From the fact that

B

is h.s.c. (as multivocal application) we obtain that there is

( )

1 2

* Bdn

d ∈ so that _*2 lim _n0

n d

d = . Because B is also closed( as a multivocal application), we get that

( )

*1

2 * Bd

d ∈ .

F

is h.s.c. so:

(

) (

)

( )

( ) ( )

1

* 1 2 1 * 2

* 1 * 0

1

, max ,

, lim

1 * 2

d f d d F d

d F d d F

d B d n

n

n ≤ ≤ ∈ =

−

(2)

which means that f₁ is h.s.c. and consequently there is d_*∈D₁ so that the below equalities occur:

( )

* _{( )}

(

1 2

)

1 max max , 1 2 1 1

d d F d

f

d B d D

d∈ ∈

= (3)

Corollary 1 If

F

is continuous,

B

is continuous and B

( )

D is compact,

1

D d∈

∀ , then there are d₁*,d*₂* so that for every d−₂∈B

( )

d₁ we have:

( )

_{( )}

(

)

_{( )}

(

)

( )

*

1 1 2 1 2

1 2

1 *

* 1

2 max min , max , max max , 1 2 1 1 1

1 1

2 1 1

d f d d F d

d F d

d F d

f

d B d D d D

d d

B d D

d ≤ =

    ≤

= _{∈ ∈ ∈} − _{∈ ∈}

There are the following two cases:

a) The partial co-operative case. It corresponds with the situation when

( )

* 1 1 1

f

d

p

n

>

. In this case the decider 1 must give up the coalition idea (because in the next stage

x

_n+1 is led through an inferior gain to the gain

p

₁n according to the

x

_n stage).

Remark 2.4 The term ”partial co-operative” comes from the fact that the deciders of the set M \

{}

1 can form a coalition in this case( forming the total coalition or distinct coalitions), even though the decider 1 might not belong to any coalition.

The decider 1 has the following alternatives:

i) The deciders of the set M \

{}

1 adopt a prudent strategic behavior( whichmeans that they adopt maxmin or minmax strategies). In this case, according to ”the equalization criterion” [4], the decider 1 owns a strategy

~ 1

n

d

which, if he adopts it, he will obtain p₁n+1> p₁n( that means that the maximum probability criterion is equivalent with the equalization criterion).

ii) The decider 1 has no information regarding to the strategic behaviorof the other deciders. In this case, if the following conditions are fulfilled:

a) the target sets Xj j∈M

−

, , realize un unfolding of X− ; b) the strategy sets D1nare compact sets in R n N

k ∀ ∈

, ;

c) application

F

is continuous in both arguments and convex in the second, there is a finite subset D1_n ⊂D₁

−

so that a necessary condition for solving the problem

( )

P₁n is the solving of the following problem:

(

n n

)

n n

d n n d n D d n

I H

P _{1 , 1}

~

1 :max_∈ + + +

  

where

∑

₌

+ =−

n

i

d j n d

j n d

nn P n P n

H

1

, ,

represents the undeterminancy( Shannon entropy) provided the choice of the strategy:

( )

∏

₌ − − −

− −

=    



 _{∈ ≥}

=

∈ n

j

d j n j j

d x f j n n

n

n D D x D P_{n n n} x X u x a P n

d

2

, ,

1 _{, : ( )}

∑

₌ + +

+ =

n

j

j n d

j n d

j n d

n n

P P P

I

n n

n

1

, 1 ,

1 1

, ln

represents the mean information profit( in the Renyi sense) obtained through passing on form the state

x

_n to the state

x

_n+1 as a result of the adoption of the strategy

n

d

.

Remark 2.5 Let us consider

d

_n* the solution of

(

)

~ 1

n

P

. Not always

P

_ndn

p

n 1 1 ,

*

>

. In [3] is proved that if the inequality:

( )

n n

n d

n n d

n I p p p

H n n

1 1

1 1 ,

1 ln ln1 *

*

− − −

≥ + + +

holds, then

P

nd

p

n

n 1 1 ,

*

>

( which means that the reached probability in the state

x

n+1 of

the target set

− 1

X

is greater than the reached probability of the target set

− 1

X

by the decider 1).

b). The total co-operative case. It corresponds with the situation when

<

n

p1

( )

* 1 1 d

f .Therefore, forming a coalition and adopting the strategy d₁*, at the state

1 +

n

x the decider 1 increases his own reached probability of the fixed target set

− 1

X (that is Pnd,1n p1n *

≥ ).

It appears the natural problem of finding what are the conditions for the decider 1 to obtain lim ₁n =1.

n p That is, if the decider 1 forms a coalition with the

other deciders and it is formed the total coalition, which are the conditions for this coalition to reach the target set at the end of the decisional process. In order to do this, we first introduce some important notations and notions.

The sequence of multivocal applications

( )

B

_{n n} is defined through reccurence. The sequence is defined as follows:

( ) ( )

1 0

{

1 1 0

(

0 0

)

0 0

}

0

1:X P X ,B x x X :x f x ,d ,d D

B → = ∈ = ∈

( ) ( )

0

{

1

(

1 1

)

1 1 1 1

( )

0

}

0 , : , , ,

:X P X B x x X x f x d d D x B x

Bn → n = n∈ n = n− n− n− n− ∈ n− n− ∈ n− ,

1

>

n .

The set Bn

( )

x₀ represents the set of the states which can be reached in n

(

)

( )

( )

   



 _{∈ ≥}

=

→ x∈B x − − m − m n

n

a x u X x P

x x F R X X x X F

n : ( )

, , \

:

0 0

0 0

( )

x _{( )}F

( )

x x n N R

R X

R n

x B x n

n

n

∈ =

→

∈sup , ,

,

: 0 0 0

0

(we wrote

∑

∑

∈

∈ =

=

M i

i M

M i

i

M u a a

u , ). We denote:

( )

U

( )

n

k k n

x B x

T

1 0

0 ,

=

=

( )

U

∞

( )

= ∀ ∈

=

1

0 0 0

0 ,

n n

X x x T x

T . The sets T

( ) ( )

x0 ,T x0

n

represent the sets of trajectories of n duration that start from

x

0 and the sets of trajectories that start

from

x

0, respectively.

The transition functions

f

nare supposed to be Lipschitzian with the same

Lipschitz constant

M

0

,

∀

n

∈

N

.We further on attempt to prove that the problem of sequencial decision under consideration there are convergent trajectories and optimum trajectories.

Theorem 2.3 We have the next results:

1) If for every n∈N,Fn is h.s.c., then there are n

( )

n n

x B

x *,0

*,

0 ∈ so that:

(

x x

)

_{( )}F

( )

x x

F n

x B x X x n n n

n

, max

max

, 0

* *, 0

0 0 0∈ ∈

= .

2) If Fn is h.s.c. and x n =x0,∀n∈N

*,

0 (i.e. all of the optimum trajectories

start from the same end), then all of the optimum trajectories converge towards

−

M

X .

3) Let us take

( )

xnm _n,m ⊂T

( )

x₀ ,x₀m =x₀,x₀ fixed. There is always

( ) ( )

nm m n k n m

n x

x k

,

, ⊂ and

( )

xn n ∈T

( )

x0 so that lim − =0.

X n m n

k x x

k

Remark 2.6 1) From the teorem 2.3 it results that for any n∈N there is

( )

n n n n

x B x X

x *,0

* 0 *,

0 ∈ , ∈ so that whatever the trajectories from

( )

n n

x

T *,0 of ends *

*, 0 , n

n

x

x , maximum profit is guaranteed in the end x*n.

2)The conditions in which the existence of the optimum trajectories has been demonstrated, in the case of the finite horizon and infinite horizon (theorem 2.3 and 2.4), are very hard. If these conditions are loosen , it is only the existence of convergent trajectories that can be demonstrated ( without securing their conditions of optimality) on the basis of the following theorem:

1) there is a trajectory which converges in the target set

−

M

X

;

2) there is a sequence of trajectories which converge at this trajectory.

3. APPLICATION IN A RUINING PROBLEM

Below it will be given an market competitional problem which leads at the end to a ruin problem

Let us consider a sequencial decision problem in which the deciders are formed a coalition in two coalition C1,C2 having the final state sets

− −

2 1,X

X which

form a partition for X− :X− = X−₁∪X−₂,X−₁∩X−₂ =Φ

We also consider the model of the following market phenomen:in their struggle for supremacy in taking hold of a certain commodity market, the deciders from C₁, intending to eliminate the deciders from C₂which control the market, want in a first stage to take hold of at least one strategic point of the existing k in this market. Having one penetrated the commodity market, the deciders in C1 will try the

complete elimination of the deciders in C2 by ruining them.

We interpret the decision process of first stage as a game made up of k-simultaneous periods. We assume that in this stage the capitals of the two coalitions from

A

and

B

, each banking unit of a decider from C₂can ruin mjmonetary units of

the

A

capital in the game j, j=1,...,k.

Let D1j,D2j be the set of the strategies of C1and C2 respectively, in the

game

j

,

j

=

1

,...,

k

for any

(

)

C

k

(

)

C

j

k

j j k

j k

D d

d d d D d

d d d

1 1

2 2

2 2 1 2 2 1 1

2 1 1 1

1 , ,..., , , ,...,

= = ∈ =

∈

= . We shall have

∑

₌ =

∑

₌ =

=

≥ k

j

k

j j j

j j

B d A d k j

d d

1 1

2 1

2

1, 0, 1,..., , , .

We introduce the utility function

C

k

j

j j

R xD D u

1

2

1 ,

:

= →

(

)

∑

(

)

= −

= k

j

j j j k

k

d d m d

d d d d d u

1

1 2 2

2 2 1 2 1 2 1 1

Let us calculate the guaranteed optimum strategy for C₂(maxmin strategy) as well as the maxmin value of the noncooperative game between C1 and C2. We will use the

result ([3]):

(

)

∑

(

)

∑

_{= }= ₌ − = −



 ₋

= k

i

i i d d i i i d

d k

i

i i i d

d md d md d md A

V

1

2 1

2 1

1 2 1

1 2 1

2 1

2

min max min

max 0

, min

min max

By introducing the partial utility function : 1 ,

~

R D

u_i i → u_i

( )

d2i =m_id2i −A

~

we shall have

( )

0

( )

0

~ 1 ~

u A

ui =− = and hence it results (from equalization principle)

that among the optimum strategy will be strategies of the form

(

2,0,...,0

)

j

d so that

( )

j j d

u

V 2

~

1 = (where

j

is determinated from the condition: m d A

(

md A

)

i i k i j

j − = −

≤

≤ 2

1

2 min .

It will results directly that the guaranteed optimum (simple) strategy for C₂ will be:

∑

₌

= k

i j

j j

m m

B d

1 2

1 ,

the maximum value beeing:

   

 

   

 

− =

∑

₌ 1 ,0

min

1

1 k

i i

A m

B

V .

For the determining of the guaranteed optimum for C₁(minmax strategy) as well as og the minmax value we shall first observe that the uefficiency function is concave in

(

k

)

d d d

d₂ = ₂1, ₂2,..., ₂ and so the V₁ maxmin value for C₂ will be equal to the value V of the game ([3],[4]):V =V₁. The minmax value is:

(

)

{

min ,0

}

min

1

2 m B A

V

i

k

i −

= _≤≤ .

The minmax (mixed) strategy will be:

k j

m m

d _k

i i

j j

,..., 1 , 1 1

1

1 = =

∑

₌

as for any

∏

=

∈ k

i i

D d

1 2

(

)

∑

∑

∑

∑

∑

=

= =

= =

=     

 

    

 

− =

    

 

    

 

− ≥

−

k

i

k

j j

k

i k

j j

i i i i

i k

j j

i

m B A

m m

d m A d

m A

m m

1

1 1

1 2 2

1

0 , 1 max

1 max

0 , max

1 1

1

1

0 , 1

min V

A m

B

k

j j

=     

 

    

 

− =

∑

₌

Remark 3.1 As a result of the concavity of the u functional in relation to

(

₂1, ₂2,..., ₂

)

,

2

k

d d d

d = the value of the game between the two coalitions will be equal to

2

V and consequently a decision-making behavior for C₂ which is based on keeping decisions does not favor this coalition. It is very important for C₂ to obtain additional informations on the strategic behavior of C₁.

Remark 3.2 The optimum solution ofC₁ consists in the concentration of the forces in a single game (in the

j

₀ game in which the condition

{ }

j

k j

j m

m

≤ ≤

=

1min

0 is realized), keeping the secret about the game in which it concentrates its forces. If C₂has no information on C₁, it has to distribute its forces uniformly.

After the first stage, the remaining capital reserves beeing A₁,B₁ ∈ N, the second stage, the ruining stage proper, takes place as a particular sequencial process:

{ }

a,b,a,b N,a b A1 B1;X0

{

A1,B1

}

;X

{

A1 B1,0

} {

0,A1 B1

}

X = ∈ + = + = − = + ∪ + .

If xn ∈X,xn =

{ }

a₁n,a₂n , we will have:

(

) (

1

)

2 1 1 2 1

1 , , +, + + = n n n n = n n

n f x d d a a

x

where:

(

n n

) (

{

n n

)

n n

} (

n n

)

n n

xD D d d a

a a a a

a₁+1, ₂+1 ∈ ₁ +1, ₂ −1, ₁ −1, ₂ +1,∀ ₁, ₂ ∈ _{1 2}

Remark 3.3 The sequencial process described before consists on a series of null sum games, the loss of the game in the state

x

_n by a coalition means its having to concede to the winning colaition a monetary unit aut of the available capital.

the state

x

n is

p

=

constant, n∈N. It results that

( )

n n

a₁ is a homogeneous Markov chain with the state 0,1,2,...,C =A₁+B₁, and with the passing matrix:

p q p q

p

M = −

                − − − − − − −

= , 1

1 0 0 ... 0 0 0 0 ... 0 0 0 0 0 0 ... 0 0 0 0 0 ... 0 0 1

The potential matrix

R

is given by:

1 0 0 ... 0 0 0 0 0 ... 0 0 0 0 0 0 ... 0 0 0 0 ... 0 0 −                                 − − − − − − − − = q p p q p I R

the elements r

( )

i, j of this matrix beeing:

( ) (

)

(

)

(

)

            =    > − ≤ − ≠         >             −             −     ≤         −             −             −     − = − − − 2 1 , , , 2 2 1 , , 1 , 1 1 1 1 2 1 , 1 1 1 p i j j C i i j i C j A p i j q p q p q p i j q p q p q p p j i r i j i C i i A j A

The mean duration

D

_mof the decision-making process will be:

( )

₂1

, 2 1 , 1 1 2 1 , 1 1 1 1 1 1 ≠           =               − −         −     − = =

∑

− = p p B A A q p q p q p p l A r D C B C c l m

The ruining probability of the C1 coalition is given by 1,

C r

   

   

 

= −

≠ −    

−    

=

2 1 ,

1

2 1 , 1 1

1 1

1

p C A

p

q p q p

P

C B

C r

and the runing probability of the C2 coalition will be: 1

2 ₁ C

r C

r P

P = −

REFERENCES

[1] Mitran,I. – Co-coperative and partial co-operative decisional models, monographs, AMSE, France, 1990;

[2] Mitran,I. – Optimalite minmax, monographs, AMSE, France, 1992.

[3] Mitran, I.- On the maximum probability, Proceedings of the International Conference on Approximation and Optimization, ICAOR, Cluj-Napoca, vol. II, pp169-178

Author.