The Pre-nucleolus for Fuzzy Cooperative Games

Transcription

1 Journal of Game Theory 207, 6(2): 43-5 DOI: 05923/jjgt The Pre-nucleolus for Fuzzy Cooperative Games Yeremia Maroutian 39, E98 Str Broolyn, NY 236 Abstract In this paper invented by D Schmeidler (969) for characteristic function games concept of nucleolus has been extended on fuzzy cooperative games The fuzzy pre-nucleolus defined by a new way On the set of classical cooperative games proved its coincidence with the already existed one For a class of fuzzy games the pre-nucleolus exists and unique The process of finding of pre-nucleolus illustrated on an example of a fuzzy game Keywords Fuzzy cooperative games, Fuzzy coalition, Fuzzy pre-nucleolus Introduction Let N {, 2 n} be the set of all players A fuzzy coalition is an n-dimensional vector τ τ, τ 2,, τ n with 0 τ i for each i N A cooperative fuzzy game with the players set N is a pair (T, v), where T 0, n is the set of all fuzzy coalitions and v is the characteristic function of that game which maps a real number to each fuzzy coalition Cooperative fuzzy games reflect situations in which for players allowed to tae part in a coalition with participation levels that may vary from non-cooperation to full cooperation The obtained reward in this type of games defines depending on the level of cooperation Participation levels at which players involved in cooperation gets described by fuzzy coalitions Aubin (98) has been explaining use of fuzzy coalitions by following way when he first introduced them in game theory Every player can choose his level of participation in a coalition instead of whether to participate in it or not As an example in favor of that approach can be considered case, when individual players reluctant to invest all of the available resources in enterprise where that coalition involved For fuzzy cooperative theory extension of classical decision concepts on fuzzy games is an important topic It is nown, that not every concept of classical theory has its natural counterpart for fuzzy games At the same time some results in classical cooperative games allow to be transformed on fuzzy games with of course significant differences In this wor we aimed to establish an important in classical theory optimality principle, ie nucleolus on fuzzy games 2 Basic Definitions and Results Together with fuzzy theory of nucleolus we are also going to deal with the classical version of the same concept For that reason we need to reproduce here some preliminary facts from the classical theory of nucleolus At the end of this paragraph we will bring the definition of nucleolus for fuzzy cooperative games For classical cooperative games D Schmeidler [] has defined nucleolus as an imputation what is the best in the sense of a preference relation ṽ To define the nucleolus we need the following notations For the game v and the imputation x R n, denote x S ί S x i Let G<N, ṽ> is a classical cooperative game and is the set of all imputations for the game G Y (ṽ) {xϵ R N / x i ṽ( i ), (i,n) x i ṽ (N)} * Corresponding author: ymaro2297@livetcicollegeedu (Yeremia Maroutian) Published online at Copyright 207 Scientific & Academic Publishing All Rights Reserved ) Results that included in this paper have been part of author s doctoral dissertation written in early 990 s There is available a copyright certificate from the US Copyright Office

2 44 Yeremia Maroutian: The Pre-nucleolus for Fuzzy Cooperative Games The difference e S, x v S x S is the excess of coalition S regarding to x Defined that way excess can be interpreted as a measure for complain of coalition S from imputation x Let consider the vector of excesses θ x, v (θ x, v,, θ 2 n x, v ) with components: θ m x, v max { U 2 N, U m} min S U v S x S From the definition of θ m x, v it is clear, that the components of θ x, v ordered decreasingly For the game v on the set R n defined a quasi-order the following way Let x, yεr N x (v) y if θ x, v L θ y, v, where θ x, v L θ y, v is for the lexicographical order It means, exist a number m such that θ x, v θ y, v for,, m and θ m x, v < θ m x, v Definition 2 For Y R n and characteristic function v, the set ν(y) R N minimal in the sense of relation v : ie ν Y { x Y/ x ṽ y for every y Y} is nucleolus for Y if vectors from ν Y are Theorem (D Schmeidler, 969) For every nonempty, convex and compact set the nucleolus exists and consists of only one vector Theorem (A Sobolev, 976) Let for a game G<N, ṽ> as a set of payoff vectors defined the set of pre-imputaions: Then the game G has a nonempty pre-nucleolus X (ṽ) {xϵ R N / x i (i,n) ṽ (N)} ν X { x X(ṽ) / x ṽ y for every y X }, that contains only one vector For outcomes from X (ṽ) the condition of individual rationality has been violated For that reason the set of payoff vectors X (ṽ) is not compact and hence, it is different of the set Y (ṽ) of imputations Despite of that the statement about existence and uniqueness for pre-nucleolus continues to remain true Fuzzy cooperative games possess infinite number of coalitions That fact does not allow using the approach based on idea of lexicographic order to extend this concept on fuzzy games From there arrives a need for a new definition of pre-nucleolus on fuzzy games To be valid the needed definition should coincide with the existing one for classical games and at the same time to allow extending that concept on fuzzy cooperative games Let (T, v) is an arbitrary fuzzy game, where T 0, n is the set of all fuzzy coalitions and v: T R is the characteristic function of that game Below we will prove that the newly defined pre-nucleolus coincides with already existing one We will consider the set of only collectively rational payoff vectors, ie pre-imputations: X (v) {xε R N / Inductively defined sets X, T by accepting that For 0,,, p we will define sets X + the following way and sets T for,2,, p x i i,n v ()} X 0 X, T 0 (3) X + argmin x X sup τ T [(e (τ, x) e 0 )/ρ(τ, T )] (32) T {τεt /xτ yτ, for every x, y X } (33) where e(τ, x) v τ xτ, e 0 min x max τ e(τ, x) and ρ(τ, T ) is the distance between the point τ and set T : ρ(τ, T ) inf τ εt ρ(τ, τ ) ρ(x, y) max i x i y i For sets { T } true the following: when increases, T does not decrease: T + T If for some 0 it is turning out that T 0 + T 0, then that entails the stabilization of corresponding set X 0 or otherwise, by increasing, X does not decrease any more The set X 0 obtained that way we will call the prenucleolus for fuzzy game (T, v) 3 About the Pre-Nucleolus for Classical Cooperative Games In this paragraph first will be described the new definition of pre-nucleolus for the set of classical cooperative games For that type of games below has been proved that pre-nucleolus defined both of the ways coincide

3 Journal of Game Theory 207, 6(2): Let the pair G < N, v > means a classical cooperative game, where N {, 2,, n} is the set of all players and v: 2 N R a characteristic function satisfying to the condition v( ) 0 First, let pay attention that in case of classical cooperative games relations (3) - (33) accept the following view: X 0 X, T 0 (34) X + argmin xεx max S T e(s, x) (35) T { S 2 N / i S x i i S y i for every x, y X } (36) N Construction of sets X, T after finite number of steps will be abrupt because finite is the set 2 The last set X will contain a unique vector, coinciding with the pre-nucleolus in sense of its initial definition Taes place the following lemma: Lemma 3 Let x, y ε X, x y and N T {S / x(s) y(s)} 2, if max S T e(s, x) < max S T e(s, y), then x v y Proof: For a given vector x X according to definition we have that Let denote by Similarly, Further, for the components of Θ x, v, let Θ x, v max S (v S x S ) argmax Sϵ2 N e S, x Θ x, v 2 max v S x S S Σ 2 argmax S Σ e S, x Θ x, v max (v S x S ) S argmax S e S, x Let S be an arbitrary coalition If T, then Θ x, v v(s ) x(s ) v(s ) y(s ) max S (v S x(s) ) Θ y, v If it taes place the strong inequality then the lemma s statement proved We will accept now, that for all S taes place only equality Let also assume, that for some < n T, T and for every S l l where l, Let consider excess where S If S T, then θ l x, v v S l x S l v S l y S l θ l y, v θ x, v v S x S, θ x, v v S x S max s T (v(s) x(s))<max s T (v(s) y(s))<max S e(s, y) θ y, v Ie x v y Let now T too and S is some coalition: θ x, v v S x S v S y S max S v S y S θ y, v Again, if it taes place the strong inequality, then x v y If for all tae place only equalities then x y, which contradicts the condition of lemma Let numbers 2 p are all of different values accepted by the components of vector θ ν X, v Below we will deal with the sets B l and Y l defined following way: B l {S / e(s, ν(x)) l } Y l {x X/ e S, x m for S B m\ B m, if m l; and e S, x l for S B l } Lemma 32 If X Y l then T B l Proof Let X Y l and S B l be an arbitrary coalition It is clear, that for some m l, S B m\ B m According to definition of Y l for every x ε X Y l, e S, x α m But then Sε T which means that T B l

4 46 Yeremia Maroutian: The Pre-nucleolus for Fuzzy Cooperative Games Lemma 33 For all,, q exist numbers l and sets X such that l < l 2 < < l, and X Y l (37) Proof For the relation (37) follows from definitions of sets X and Y Accept it already has been proved that for some and l X Y l According to lemma 2 supposed to tae place inclusion T B l If T 2 N, because B p 2 N, then exists l + > l such that B l T and B l + T Necessary to prove that it will entail the coinsidence of sets X + and Y l+ : X + Y l + Let now for some l > l T B l and T B l + Then will exist set S 0 such that S 0 B l + and S 0 T Subsequently, S 0 B l Which means, that S 0 B l +\ B l Because ν X Y l X, so according to lemma 3, By the other side, as far as S ε 0 B l + \ B l then From there it follows that Further, because T Or otherwise, Which means, that min x X max S T e(s, x) max S T e(s, ν X ) e(s 0, ν X ) e(s 0, ν X ) l + min x X max S T e(s, x) l + max S T e(s, ν X ) max S Bl e(s, ν X ) l +, B l and l + is the first value of e(s, ν X ), which is smaller than l As a result, X min max S T e S, x l + x X + { x X / max e S, x l S T +} X + Y l + Let now prove the opposite inclusion, ie if x X +, then x Y l + too As far as T every S B m \ B m, where m l e(s, x) e(s, ν X ) m Besides that, for every S B l, e(s, x) l + It is remaining to proof that for arbitrary S B l + \ B l e(s, x) l + Accept that for some coalition S 0 B l + \B l e(s 0, x) < l + Let consider the vector z εx + ε ν X B l and ν X X, so for For coalitions S B l e S, z e S, ν X, for S B l +, e(s, z) e(s, ν X ), and S 0 B l + e (S 0, z) < e (S 0, ν X ) From there it follows that for a number ε > 0 small enough, and every S B l +, S 2 B l +, e S, z > e S 2, z Which means that the constructed above vector z is more preferable than ν X : z v ν X That contradicts to the fact that ν(x) is pre-nucleolus for the game < N, v > Hence, for every S B l + \ B l supposed to hold true the equality e(s, x) l + But then the received equality would mean, that x Y l +, and Y l + X + The last inclusion concludes the proof of our lemma Theorem 3 There is a number q such that X q ν X Proof Because ν X Y l for every l, so according to lemma 33, we will have, that also ν X X If X contains more than one point then it is obvious that T 2 N Then based on lemma 33 and lemma 32, B l 2 N and according to lemma we will be able to construct the next set X + The constructed that way last set X + will consist of only the nucleolus ν X 4 Fuzzy Games with Finite Sets of Coalitions 4 Let (T, v) is a fuzzy cooperative game, where T T is some finite set of fuzzy coalitions Below we will prove that in presence of some conditions this type of games possess a unique prenucleolus Lemma 4 Let X is a convex polytope and χ is the solution for the next linear programming problem:

5 Journal of Game Theory 207, 6(2): xτ j + ε c j + e 0 a j where τ j ε T xε X Then exists a vector τ j 0 ε T such that for every x, y χ Proof Let for every τ j ε T exists a vector x j ε χ such that e 0 + x j τ j + ε 0 c j > a j, where ε 0 is the solution of mentioned above linear programming problem Consider now the vector Because of convexity of the set χ, x ε χ and x m j m x j e 0 + xτ j + ε 0 c j > a, j for every τ j, which contradicts to the condition that (ε 0, χ) is the optimal solution for our minimization problem So, exists a vector τ j 0 ε T such that for every x χ e 0 + xτ j + ε 0 c j a j From there the assertion of lemma 34 follows Theorem 4 Let (T, v) is a fuzzy game, where T is a finite set of fuzzy coalitions that also contains coalitions τ i (0,, τ i,,0) for arbitrary iε N Then the game (T, v) possesses a unique pre-nucleolus Proof We need to prove that after finite number of steps the process of construction of sets X, T will be abrupt and the last set X will consists of a unique point The set X is solution for the following minimization problem: j v (τ ) xτ j ε for every τ j ε T (38) xε X In problem (38) the number ε bounded below Really, if τ i (0,, τ i,,0), then ε v(τ i ) x i for iε N Summing all these inequalities by i N we will obtain that from where nε iεn ( v(τ i ) x i ) ε n ( iεnj v τ i v()), what has been required to prove When ε accepts its minimal value we obtain the solution of our problem: The corresponding set T is: Further we need to find the X {xε X / v τ i xτ i ε, for all τ i ε T } T { τεt / x τ x 2 τ for every x, y ϵx } argminx ε X max τ T [ (v τ x τ e 0 ) / ρ(τ, T )] That is the same as solving the following minimization problem: v(τ) τx e 0 ερ (τ, T ) for τ T x X The solution X 2 for this problem is a convex politope and the set T 2 strictly contains the set T : T 2 T The same will tae place on the following steps too As far as the set T is finite, so construction of sets X, T will be abrupt after finite number of steps Let now T p T It is remaining to prove that X p If x, y ε X p then from T p T will follow that τ l x τ l y for arbitrary τ l ε T, from where x y That concludes the proof of our theorem 42 Fuzzy Games with Piece-Wise Affine Characteristic Functions Below proved a theorem about existence and uniqueness of pre-nucleolus for fuzzy games with piece-wise affine characteristic functions

6 48 Yeremia Maroutian: The Pre-nucleolus for Fuzzy Cooperative Games Theorem 42 Let (T, v) is a fuzzy cooperative game with piece-wise affine characteristic function v That means, exists a collection of simplexes { j j l j } what covers T: T, if l, and for τε, v ( τ) u j(τ) - j, where u j (τ) is a linear function and j 0 Then the game (T, v) has a pre-nucleolus that consists of a unique point Proof According to definitions of sets X, T X m argmin x Xm sup τ Tm [(e (τ, x) e 0 )/ρ(τ, T m )] T m {τεt /xτ yτ, for every x, y X m } The set X is the solution for following minimization problem: Let consider the following linear programming problem: v τ xτ ε (39) x X v τ j xτ j ε x X, where the {τ j } is the set of all peas of simplexes { } Accept that the pair (ε 0, X ) is the solution for that problem, where X is a convex politope It is clear that e 0 ε 0 We will prove that ε 0 also is solution for the problem (39) For that reason we will need to show that the inequality v (τ) xτ ε 0 holds true for all τ T, when x X Let τ εt is an arbitrary coalition and is a simplex with peas τ j, τ j 2 τ j n, which contains τ Then τ λ,n τ j, where λ 0 and,n λ Because v (τ) is an affine function on v (τ) x τ λ v(τ j ) x,n so we will have that λ τ j λ,n,n ( v(τ j ) x τ j ) ε 0 From there, ε 0 really is a solution for the problem (39) So, we will have that X is the following set: X argmin xε X sup τεt {v (τ) xτ} According to definition of sets T m, for a fixed τ T m and every x X m, xτ c τ, what means that the product xτ is constant for every x X m Let now τ, τ 2 ε T m are such coalitions that xτ c, and xτ 2 c 2 From there it will follow that if for some numbers λ and λ 2 λ τ + λ 2 τ 2 T, then for every x X m x ( λ τ + λ 2 τ 2 ) c λ + c 2 λ 2 The latter one means that set T m is the intersection of the set of all coalitions T with some hyperplane and subsequently is a convex set, because of convexity of T Next we will rewrite the definition of X m in a different form: X m argmax x Xm inf τ Tm [(e 0 e(τ, x))/ρ(τ, T m )] The set X m defined that way is solution for the following maximization problem: max ε e 0 v(τ) + xτ ε ρ(τ, T m ) for every τ T m (30) x X m As it was in the beginning of the proof besides this problem also let consider the corresponding linear programming problem for peas of simplexes { } that does not belong to T m- : max ε e 0 v (τ j ) + xτ j ε ρ(τ j, T m ) for every τ j ε (39') \ T m (30 ) x X m The problem (30 ) has a solution because it is a linear programming problem and X m is a convex polytope Let

7 Journal of Game Theory 207, 6(2): denote that solution by (ε, X m ) and prove that inequalities (30 ) remain true for all xε X m and τ j ε \ T m For τ j ε \ T m the inequality (30 ) follows from definition of sets X m For τ j ε { } T m (30 ) is true because for that ind of τ j the right side of (5 ) is equal to 0 and the left side is not negative as far as X m X Let now x X and τ T m Accept that is a simplex for what τ ε and τ j,, τ j n are peas for that simplex According to the Karatheodory s theorem: e0 v τ + xτ e 0 λ v(τ j,n ) + x λ τ j,n,n λ ( e v τ j + xτ j 0 ) ε (,n λ ρ( τ j, T m )) ε ρ( τ, T m )) The last inequality in the chain above taes place because of convexity of metric ρ x, T by the variable x As a result, has been proved that the solution (ε, X m ) for the problem (30 ) also is solution for (30) From there according to lemma 4 exists j 0 such that τ j 0 T m and for arbitrary x X m taes place equality in (30 ) Then because T m T m and τ j 0 T m so τ j 0 ε T m As a result to that the dimension of T m will increase by at least one From there because as its proved above the sets T m are convex, so after finite number of steps T m will coincide with T and the corresponding set X m will contain only one point 43 An Example for Calculation of Pre-Nucleolus The paragraph below devoted to finding of the pre-nucleolus for fuzzy game from a parameterized class Let considered 2 a game G < [0, ], v > with the following characteristic function v(τ): v(τ) min { τ, τ 2, τ, τ 2 }, for τ τ τ, τ 2 0, 2 and < 0 2 It is clear that for this game v() and X {xε R 2 / x x 2 } Solving of the problem will start from dividing the square T [0, ] 2 on eight triangle subsets and figureing out values of v (τ) on each one of them Let denote these subsets by Ω i (i 8) and start to describe them () Ω { τ ε T / τ τ 2 } for τ ε Ω, v(τ) τ (2) Ω 2 { τ ε T / τ 2 τ } for τ ε Ω 2, v(τ) τ 2 (3) Ω 3 { τ ε T τ + τ 2 2, τ 2 } Based on inequalities that define Ω 3 it is obtained that τ 2 τ and τ τ 2 from there v (τ) τ (4) Ω 4 { τ ε T τ + τ 2 2, τ } For τ ε Ω 4 are true the following inequalities: τ 2 τ, τ 2 τ From what it follows that v (τ) τ 2 (5) Ω 5 { τ ε T / τ + τ 2 2, τ ( ) ( τ ) } The definition of Ω 5 implies that for τ ε Ω 5 hold true the inequalities: So, for τ ε Ω 5 : v (τ) τ τ 2 τ τ τ 2 ; τ τ 2 ; τ (6) Ω 6 { τ ε T / τ + τ 2 2, τ 2 ( ) ( τ 2 ) } Analogically to τ ε Ω 5 in this case too Further, because τ 2 τ τ 2 τ τ 2 τ ; τ 2 (7) Ω 7 { τ ε T / τ τ 2, τ ( ) ( τ 2 ) } From inequalities that define Ω 7 follows that for τ ε Ω 7 τ τ τ 2 (8) Ω 8 { τ ε T / τ 2 τ, τ 2 ( ) ( τ ) } On Ω 8 hold true the following inequalities: τ τ 2 τ τ, so v (τ) τ 2 τ 2 ; τ 2 τ i e v (τ) τ 2 ; τ 2 τ So, v (τ) τ

8 50 Yeremia Maroutian: The Pre-nucleolus for Fuzzy Cooperative Games To find sets X and T enough to calculate the following magnitude: min x max τ e(τ, x), where e(τ, x) v τ + x(τ 2 τ ) Let now to calculate the magnitude of max T e(τ, x) by the scheme below: max T e(τ, x) max ( i 8) e(τ, x) Further by turn will be figured out magnitudes of the following inner maximums: max Ω e(τ, x) max Ω { τ max Ω2 e(τ, x) max Ω2 { τ 2 max Ω3 e(τ, x) max Ω3 { τ max Ω4 e(τ, x) max Ω4 { τ 2 max Ω5 e(τ, x) max Ω { τ max Ω6 e(τ, x) max Ω6 { τ 2 max Ω7 e(τ, x) max Ω7 { τ 2 max Ω8 e(τ, x) max Ω8 { τ + x(τ 2 τ ) x if x > 0 if x 0 + x(τ 2 τ ) x if x < 0 if x 0 + x(τ 2 τ ) x if x > 0 if x 0 + x(τ 2 τ ) x if x < 0 if x 0 + x(τ 2 τ ) x if x > 0 if x 0 + x(τ 2 τ ) x if x < 0 if x 0 + x(τ 2 τ ) x if x > 0 if x 0 + x(τ 2 τ ) if x 0 x if x < 0 Now, when the values for max τ e (τ, x) by subsets Ω i, already have been found can be calculated value for the preliminary expression: min x max τ e(τ, x) min {min x 0 max τ e(τ, x), min x 0 max τ e(τ, x) min { min x 0 max (x,, x), min x 0 max (, - x, -x) min X max{x, Further, because for x with x > min X > max { x,} >, so from there it is clear that argmin X max τ e (τ, x) [,-] That value together with definition of the set T gives that T { } Let denote e 0 min x max τ e (τ, x) To find sets X 2, T 2 should be calculated the magnitude of max τ τ0 F (τ, x), where F (τ, x) v τ x τ 2 τ e 0 max τ, τ 2 } The magnitude of max (τ τ0 ) F (τ, x) also will be calculated by subsets Ω, i the same way as it has been done with the max τ e (τ, x) max Ω \{ }F(τ, x) max (Ω \{ }) if x 0 x if x < 0 max (Ω2 \{ })F(τ, x) max (Ω2 \{ }) x if x < 0 + x if x 0 max (Ω3 \{ })F(τ, x) max (Ω3 \{ }) τ τ0 x τ 2 τ τ max (Ω \{ }){ τ0 + x(τ 2 τ ) τ τ2 τ0 x τ 2 τ τ 2 max (Ω2 \{ }){ τ0 + x(τ 2 τ ) τ 2 τ τ0 x τ 2 τ τ max (Ω3 \{ }){ τ0 + x(τ 2 τ ) τ, if x 0 x, if x < 0

9 Journal of Game Theory 207, 6(2): max (Ω4 \{ })F(τ, x) max (Ω4 \{ }) + 2x if x 0 + x if x < 0 max (Ω5 \{ })F(τ, x) max (Ω5 \{ }) x+ if x < 0 impossible, if x 0 max (Ω6 \{ })F(τ, x) max (Ω6 \{ }) x if x > 0 impopssible if x 0 max (Ω7 \{ })F(τ, x) max (Ω7 \{ }) x if x 0 if x < 0 max (Ω8 \{ })F(τ, x) max (Ω8 \{ }) x if x 0 if x < 0 τ2 τ0 x τ 2 τ τ 2 max (Ω4 \{ }){ τ0 + x(τ 2 τ ) τ 2 τ τ0 x τ 2 τ τ 2 max (Ω \{ }){ τ2 τ0 x τ 2 τ + ( τ0 x)(τ 2 τ ) τ0 τ 2 τ max (Ω6 \{ }){ τ0 + ( x) (τ 2 τ ) τ τ2 τ0 x τ 2 τ τ 2 max (Ω7 \{ }){ x (τ 2 τ ) τ τ τ0 x τ 2 τ τ max (Ω8 \{ }){ x (τ 2 τ ) τ At the end it is remaining to calculate one more magnitude, which will give us the set X 2 : min (xε, ) max τ τ F(τ, x) min { min 0 xε 0, max τ min { min xε 0, max[, x -, - x -, 2x -, - x+ x+, x, -, x F(τ, x), min xε,0 max τ F(τ, x)} ], min τ xε,0 max [-, - x, 2x, + x, 0 ] } min { min τ xε 0, max[ +2x, x ], min 0 τ xε [,0) max [- -2x, - x+ ]} 0 max {, - - So, as a result it obtaines that X 2 argmin x, max τ τ F(τ, x)0 0 From there for the pre-nucleolus v(x) of initial game it follows that ν X (0, 0) REFERENCES [] SCHMEIDLER D 969 The nucleolus of a characteristic function game SIAM J OF MATH, vol 7, pp [2] SOBOLEV A 976 Characterization of the optimality principles in cooperative games by functional equations Math Methods in Social Sciences - Vilnius, pp 94-5 (In Russian) [3] AUBIN JP, 98 Cooperative fuzzy games Math Operes 6: -3