on the shapley prevalue for fuzzy cooperative games(last version)

7/25/2019 On the Shapley Prevalue for Fuzzy Cooperative Games(Last Version)

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ON THE SHAPLEY PREVALUE FOR FUZZY

YEREMIA MAROUTIAN

[email protected]

Abstract. In this paper we deal with the problem of extending the concept of Shapley value

on fuzzy cooperative games. Interest in that type of research connected with that fact that it

allows to generalize the Shapley value for classical cooperative games. At the same time we

formalize Shapley axioms for fuzzy cooperative games. By using one of the known approaches

for definition of classical Shapley value [3], we define the Shapley prevalue for fuzzy games, as

well research its properties.

Keywords:Fuzzy cooperative games; fuzzy coalition; fuzzy Shapley prevalue..

JEL Classification C71.

1. Introduction.

Let N= {1, 2 n} be the set of all players. an n-dimensionalvector for each A cooperative fuzzy game with theplayers set N is a pair , where [ ], is the set of fuzzy coalitions and is thecharacteristic function of that game which maps a real number to each fuzzy coalition.

1) Results that included in this paper have been part of authors doctoral dissertation. A draft of

it published in the magazine of Lithuanian Academy of Sciences Math Methods in Social

Sciences 1987, vol. 20 pp. 47-61(in Russian).


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Cooperative fuzzy games reflect type of situations in which for players allowed to take part in

coalitions with participation levels that can vary from non-cooperation to full cooperation.

Fuzzy coalitions describe levels of participation at which players involved in cooperation. The

reward for players in this type of games defines depending on their level of cooperation.

An important topic of research for fuzzy cooperative theory is the extension of existing in

classical theory decision concepts on fuzzy games. It is known, that not every classical concept

has its natural counterpart for fuzzy games. At the same time some results in classical theory

allow to be transformed on fuzzy case, with of course significant differences.

2. Some preliminary facts and definitions.

For classical cooperative games has been proved [3] that the Shapley vector is a point of the

following set of preimputations

= ,on what gets its minimum the functional

(x, v) = || || || .That property of the Shapley vector allows using it as an alternative definition. Defined that

way Shapley vector in its turn can quite naturally be extended on fuzzy cooperative games.

Let be the set of fuzzy coalitions, i.e [ ]. It is known, that each coalition in itsclassical sense is a peak of the cube T, or more precisely, that peak for what

1 if S0 if SFor a concentrated on peaks of the cube [ ]and accepting values || || || measure we can rewrite the expression for (x, v) in a simpler form:

=


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On the cube T instead of one can take an arbitrary measure and for agame as a point on what the newly obtained functional will reach its minimum.We will choose a measure such that the function has been possessing with propertiesdefined by axioms, similar to Shapley axioms for classical cooperative games.

Below we formulate analogs of Shapley axioms for fuzzy cooperative games.

A1 (Symmetry) Let isan injection from the set onsuch, that for each coalition ;

Where

is a map, for what

=

.

Then, for all A.2 (Pareto optimality)

N = Some unusual form of this axiom compared with its classical analogue caused by the fact

that we dont exclude the possibility of .A.3 (Inefficiency of dummy) If in the game is a player such, that || =0),

for arbitrary , then . The expression ||=0) means, that the value ofthe coalitionin the game equals to the same value for the coalition with =0.

A.4 (Aggregation) If and are games with the same set of coalitions and , then

Besides A.3 we will also deal with its strengthened form. Will be a need the following

definition.

Definition1.We will say that the game (,derived from the game (,by includingin it a dummy player , if for every coalition and every [] takes place:


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A3. (he strengthened axiom of dummy) For arbitrary holds true

.

he A3 otherwise means that of including a dummy in the game gains of other players dont

get changed. It is obvious, that if A.2 remains true, then A.3 follows from A.3.

In general, the choice of measure is not unique. Partially, as such a measure can be also .The axiom A.1 holds true, if a measure has been defined the same way on all sides of the cube.

Let now consider the following sets. For arbitrary and denote by the follow

< 1,

L

= 1,

;

= 0,

N (

L)}.

|| | |, L .Sets actually are the faces of the cube [ ] We will deal with the case, when on

each face of the cube a measure defined by its density function . To satisfy axiom A.1will be accepted that are symmetric and none negative functions of their arguments.As a result, we will deal with the following functional,

= where || | |, L .

Our goal is finding conditions for functions so that minimizing the functionalfunction ) has satisfied to axioms A.3 and A.3. 1. Prevalue for fuzzy cooperative games.

Below we will find explicit view for the minimizing functional function At thesame time for functions participating in the expression of functional , we willformulate requirements at what for the function hold true the Shapley axioms.

The magnitudesdefined here will be in use further.


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+ ;

= = =

Let extend definitions of for and for by accepting that . Besides that, let also accept

Further we will need to deal with numbers:

Let denote:

= ,and

=

Preposition 1.If , then the functional Q(x, v) on the set= accepts its minimum by on the vector with the following components:


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{ + ( ) -

( ) ]

Proof.The vector on what functional reaches its minimum, according to the methodof Lagranges multipliers should satisfy to the following system of linear equations:

To write these equations in explicit form we need to find the partial differentials of

We have that

= ]We will also need the following notations:

+ +

We need to prove that

=

For that reason let rewrite magnitudes of

and

in different forms:

() ( ) ( ) ( ) ( )


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() ( ) () ( ) () ( ) == .

As far as we redefined above the magnitudes , so,while figuring out the difference we can formally extend summing by the indexfrom 0 toand by the index from to . As a result we will have the required equalityinstantly.

Further, by using the notations that have been defined above, we can rewrite

in

an equivalent form:

Summing equations (2) byand taking in account the (3), we will obtain, that

[ ( ]That expression for together with the system (2) (3) gives as, that

[ - (- ]= [ [ ].Let submit inthis formula instead of(its expression and unite integrals with coinciding

domains of integration. Then for we will have the following expression: [ [


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]=[

+

( ) ]}.As a result we have obtained the required formula (2.1).

Remark.1.In general, solution for the system is unique for arbitrary (andnot only for not negative) functions

, that participate in the expression of

Solution for system (2) - (3), obtained that way still can be accepted as some analogue for the

Shapley function, despite of that it already will not minimize the functional Remark.2.If given for an arbitrary measure ,then the formula for will

obtain a simple expression. So let

= By using the following notations,


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functions . provided by the equation , and the linearity takes place due to .Here, what requires most of the work is the checkup of axiom

Below we will find such conditions for functions at what hold true the axiomWhile doing it we should mention, that these conditionswill be some relations betweenfunctions that take place almost anywhere. So, for brevity, while writing thoseequalities we will not mention each time that they may be violated on some null measure sets.

Proposition 2. Let For the functionto satisfyaxiomit is necessary andsufficient, that take place the following equations:

+

for pairs that , besides pairs ). At thesame time will also take place following equations: = ) ( ) (

Proof. Conditions based on the axiom. While doing that we will

prove the necessity of our statement. Due to reversibility of necessary for thatargumentsthat

way we will also prove the sufficiency of this proposition.

We will accept, that dummy is the player , i.e. ||=0), for all .he integral equations below take place due to dummies property:

[ ] By using equalities from the formula for the component we will

obtain the following expression:


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{[ ) ( )]- [ )( ] +

] }. based on the axiom. At the same time, as well as , andis for arbitrary coalition , so the expressions in square brackets should be equal to As a

result we will obtain the equations and To conclude it is remaining to show, that the equations can be obtained from to

what will be devoted rest of the proof.

Let consider the following magnitudes:

[ ] [ ] (2.10)

That statement we will prove trough the four lemmas below.

Lemma1.Takes place the following equation:

(2.11)Proof. Let and . We will consider the integral below:


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Let transform the sub integral expression by using the symmetry of functions andnotations invented in the beginning:

[ ].If to apply the condition (5) to the integral we will obtain that:

[ () () What has been required to prove.

Lemma 2.For every , takes place the following presentation: = (12)

Proof. This statement we will prove through the method of mathematical induction.

1. Let prove first, that the statement correct for . Really, as far as

, so

= 2. Accept that the expression (12) is correct for arbitrary . We will use the formula of

the Lemma1, as well as equalities: .Below we have done some quite natural transformations: =


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The lemma has been proved.

Lemma3. = ) ( ).Proof. For

according to presentation from the Lemma2,

= + .By using formulas and , In the obtained expression let submit values for the following magnitudes:

= ) ) ,The final expression will be:

=

) (

).

Lemma4. = .Proof. Below we are integratingthe equality by the domain and using the notations

above:

+

+ +


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= Multiplying the equalities above by

and summing them by all pairs

, for what takes place, we will obtain:= [ + ] As far as so we will have, that=

[

] [ ] [ ] [

] Expressions in the square brackets are equal to each other:

=0.As a result .Now recall that takes place for pairs where that ,

besides pairs

).

If to separate from the sum of written above expression members, that obtain for pairs then the remaining will be equal to 0, because of The separatedmembers, that correspond to pairs will give us the following expression:


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By submitting values for magnitudes that take part in it, we will obtain what has been

required:

=

.

The last two lemmas conclude the proof of preposition 2.

1. Properties of the function Let the prevalue for the game That means, it is the point, on what functional

with functions

reaches its minimum, and the functions

satisfy to

specified for Shapley axioms conditions.In this paragraph we will show, that for a class of fuzzy games obtained from the classical

cooperative games through the Owens multilinear extension, the prevalue coincides withShapley prevalue for classical game. Further, we will obtain necessary and sufficient conditions

at what the function satisfies to the strengthened dummies axiom. We will also prove apreposition allowing constructing functions

(

that satisfy to axiom

for fewer players.

4.1. The fuzzy games we are going to deal with get defined through some dimensionaldistribution functions, ... , , that are continuous by each one of their arguments andsatisfy to the following condition:

, ... , +, ... , =, ..., where

, ... ,

is some

dimensional distribution function of

variables , ... , and possesses with same kind of properties as functions, Preposition 3. Let is a fuzzy game obtained from a classical game by the

following way:


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where

= { and is a function satisfing to the condition . If is the Shapley prevalue forgame the prevalue for fuzzy game then .

Proof. Let is the solution for system of game Byusing the explicit expression for and formula we can prove that allowssome linear representation that depends on (.

For ( hold true due to properties ofSo to prove ourstatement, we need to show, that (takes place the dummies axiom. For that reasonwe should prove, that if is a dummy in the game then is so also in Let for all , Then= ) [ +( ) ( ) [

By the other side: (|| ) [ ] = ( ) So from there

(|| ), and hence,

As a result we have obtained that the prevalue () satisfies all of the axioms of Shapley, whichmeans, that Remark.In a special case, when players participate in coalition independently of each other,

i.e., when


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( )then is a multilinear extension of Owen. It is easy to check, that then for functions takes place the condition .Below we will derive relations that concern to the strengthened axiom of dummy.

Preposition 4.Let function in an person game satisfies to thestrengthened dummies axiom if and only if, when for all pairs that , besides pairs and takes place the following equation:

[ ] and the equation for the players game:

() Proof. Necessity.The equation follows from the Proposition2, because, if the axiomA2.

holds true, thenA3follows from A3.

Similar to the formula

let rewrite the expression for

by separating the player

, while we will sum by the side of the cube and replacing by ,based on the equation { [ + ( ) - ( )

( )( ) ( ) ( )( ) ( )( ) ]


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The formula also gives as an expression for . According to the strengthened axiomof dummy, the gain for the same player in both of the games (,and (,should be the same. For that reason the obtained for and expressions should beequal. Let write that equation:

{ + ( ) - ( ) ] { [ ( ) ( )( )

(

)(

)

( ) ( ) ( )( ) ]By equalizing coefficients at for similar coalitions in both of the sides of equation we will

obtain relations connecting functions and . The form of these coefficientswill depend on to which one of the sides belongs the coalition More precisely, to which oneof the sets or belongs the player Besides that have special view coefficients atand As a result it will turn out, that the equality (i.e. the axiomis equivalent to the following five relations.

The equality of coefficients at for , where (besides the side )gives as the relation: ( ) ( ) ( ) ( )( )

for all pairs that , besides the pair .For the sides

, where K

besides the side

, we will obtain the following

relation: [( ) ( ) ( )( ) ] which takes place for all pairs that , besides the pair .


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For the sides , where we will obtain the following relation: ( ) ( )

(

)

which is correct for all pairs that .Finally, by equalizing the coefficients at we will obtain

[ = . (19). [ = . (20).

By adding to each other equalities we will obtain that takes place for pairsthat .By adding to each other equalities and dividing the sum by we will obtain thattakes place for pairs that .Subtracting of each other equalities and dividing the result by we willget that takes place for pairs that besides the pair So, we have proved that the equality

takes place for all pairs

that

besides the pairs and The Proposition has been proved fully.

Below we will prove one more proposition allowing to construct functions ( for satisfing equations (5) if they take place for .Proposition5. Let for some and for functions defined by following way

(

=

[

] (21)

take place equations Then =c and ( satisfy to the equation (5) too.Proof. =c can be proved by using the formula we knew earlier, but

this time for i.e. = ,


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as well the equations . To check that functions ( satisfy it is enough by using theformula (21) to figure out written for the all magnitudes that take part in and afterthat sum them by taking in account that for the game take place

3. Below constructed examples of none negative functions that satisfy to dummies axiom

for some fixed value of Example1.The simplest is the following set of functions :

= { In this case the gain of a player depends only on the value of the game on peaks of the

cube , which are the coalitions in the classical sense. The functional in its turn coincides withthe functional (x, v), defined for classical cooperative games.

Example2. Let define for some functions the following way: ,

and recurrently, by using the following formula, define : ( ) },where, if one can choose

arbitrary, but big enough to get

none negative.

For the follows immediately from the formula that recurrently defines functions .For we have:

( )

( )

We will apply further theProposition, by accepting that = c. Let check, that for ,

can be defined by the formula :


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= So, we have constructed example of functions that satisfy to the axiomfor the numberof players that does not succeed to some However, unfortunately this method does not

allow to construct set of functions satisfyingfor arbitrary We should note that in this example are essential only those coalitions that can be described by

peaks and connecting pairs of adjacent peaks of the cube Example3.For to axiom the following defined on square functions:

. . ) ).Remark 4. The constructed in examples above satisfy to the following condition:

,which can be considered as another property for the distribution .

As a conclusion author would appreciate any information on copyright breach regarding results that

included in this article.

REFERENCES

[1] AUBIN JP (1981). Cooperative fuzzy games. Math Oper.res 6: 1-13

[2] MAROUTIAN Y. (1987). An analogue of Shapley function for fuzzy cooperative games. -

Math Methods in Social Sciences.vol. 20, pp.47-61, Vilnius. (In Russian)

[3] SOBOLEV A. (1978). Nonlinear analogues of Shapley function.Modern Directions in Game

Theory - Vilnius, pp.119-126. (In Russian)


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on the shapley prevalue for fuzzy cooperative games(last version)

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