The Nucleolus For Fuzzy Cooperative Games. YEREMIA MAROUTIAN1 Technical Carrier Institute 320, w.31Str. New York, New York 10001 USA ymaro221917@live.tcicollege.edu Abstract. In this paper we deal with the problem of extending the concept of nucleolus on fuzzy cooperative games. Described a newly invented definition of nucleolus for fuzzy cooperative games. We prove that the newly defined nucleolus on the set of classical cooperative games coincides with the previously existing one. It is proved that for separate classes of fuzzy games the nucleolus exists and unique. The process of finding the nucleolus illustrated on an example of a concrete type of fuzzy games. Keywords: Fuzzy cooperative games; fuzzy coalition; fuzzy nucleolus. JEL Classification C71. 1. Introduction. Let N= {1, 2… n} be the set of all players. A fuzzy coalition is an n-dimensional vector π = (π1 , π2 , … , ππ ) with 0 ≤ ππ ≤ 1 for each π ∈ π. A cooperative fuzzy game with the players set N is a pair (π, π£) where π ⊂ [0, 1]π is the set of all fuzzy coalitions and π£ is the characteristic function of that game which maps a real number to each fuzzy coalition. Cooperative fuzzy games reflect situations where for players it is allowed to take part in coalition with different participation level varying from non-cooperation to full cooperation. The obtained reward in this type of games defines depending on the level of cooperation. The participation levels at which each player is involved in cooperation gets described by fuzzy coalition. An explanation for the use of fuzzy coalitions made by Aubin (1981) when he first introduced in game theory the fuzzy cooperative games is the following: every player can choose his level of participation in a coalition and not only whether to participate or not. A justification for the mentioned type of approach may be considered for example reluctance of individual player to invest all of available resources in enterprise where that coalition involved. For fuzzy cooperative theory it is an important topic of research the extension of existing in classical theory decision concepts on fuzzy games. It is known, 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 work we aimed to establish an important in classical theory optimality principle i. e. nucleolus on fuzzy games. 1 PO Box 1958 Canal Str. Sta. New York, NY 10013. 2. Basic definitions and results. Together with the fuzzy theory of nucleolus we are also going to deal with classical theory of the same concept. For that reason we need to reproduce here some preliminary facts that concern to 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 [1] has defined the nucleolus as an imputation what is the best in the sense of some βΊαΉ½ relation. Let G=<N, αΉ½> is a classical cooperative game and Y (αΉ½) = {xΟ΅ RN / xi ≥ αΉ½({i}), ∑(i=1,n) xi =αΉ½ (N)} is the set of all imputations. The nucleolus of the game G consists of the following imputations: ν(Y) = { x ∈ Y/ x βΊαΉ½ y for every y ∈ Y}. The relation βΊαΉ½ defined as below. Definition.2.1 For a set Y ⊂ Rn and a characteristic function v, the set ν(Y) ⊂ R|N| is the nucleolus of Y if vectors from ν(Y) are minimal in the sense of relation βΊv : ν(Y) = { x ∈ Y/ x βΊαΉ½ y for every y∈ Y}. Theorem (D. Schmeidler, 1969). For every nonempty, convex and compact set the nucleolus exists and consists of only one vector. Theorem (A. Sobolev, 1976) Let for a game G=<N, αΉ½> the set of payoff vectors defined as below X (αΉ½) = {xΟ΅ RN / ∑ (i=1,n) xi = αΉ½ (N)}. Then the game G has a nonempty nucleolus ν(X) = { x ∈ X/ x βΊαΉ½ y for every y∈ X }, which contains only one vector. For outcomes from X (αΉ½) the condition of individual rationality has been violated. That is the reason because of what the set of payoff vectors X (αΉ½) is different of the set of imputations Y (αΉ½), i.e. it is not compact. Despite of that the statement about existence and uniqueness of nucleolus continues to remain true. Fuzzy cooperative games possess infinite number of coalitions. By that reason it’s impossible to extend this concept on the set of fuzzy cooperative games by using the approach that based on the idea of lexicographic order. From there arrives a need for a new definition of nucleolus on fuzzy games. To be valid the needed definition should be equivalent to 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, 1]n is the set of all fuzzy coalitions and v: T ο R 1 is the characteristic function of that game. Below we will prove that the nucleolus we defined coincides with the already existing one. Similar to classical games here too we will consider the set of all only collectively rational payoff vectors: π (π£) = {π₯π π π / ∑ π=1,π π₯π = π£ (1)} We will inductively define sets X π , ππ by accepting that π 0 = π, π0 = ∅ . (3.1) For π = 0, 1 … π we will define sets π π+1 the following way, π π+1 = ππππππ π₯∈ππ π π’π π∉ππ [(e (π, π₯) − π0 )/π(π, ππ )] (3.2) and sets ππ for π = 1,2 … π ππ = {πππ /xπ= yπ, for every π₯, π¦ ∈ π π } (3.3) where e(π, π₯) = π£(π) − π₯π , π 0 = πππ π₯ πππ₯ π π(π, π₯) and π(π, ππ ) is the distance between the point π and set ππ : π(π, ππ ) = πππ π′ πππ π(π, π′). ρ(x, y) = πππ₯ π |x i − yi ||. Sets { ππ } increase. When k increases, ππ does not decrease: ππ+1 ⊇ ππ . If for some π 0 it is turning out that π π0+1 = ππ0 , then that entails the stabilization of the corresponding set π π0 , or otherwise by increasing π, π π will not decrease any more. The set π π0 obtained that way we will call the nucleolus for fuzzy game (π, π£). 3. About the nucleolus for classical cooperative games In this paragraph we will first describe the new definition of nucleolus for classical cooperative games. For that set of games will be proved that nucleolus defined by both of the ways coincide. Let the pair πΊ =< π, π£ > means a classical cooperative game, where π = {1, 2 … π} is the set of all players and π£: 2 π ο π 1 is a characteristic function that satisfies to the condition π£(∅) = 0. First we should pay attention that in case of classical cooperative games relations (3.1) - (3.3) accept the following view: π 0 = π, π0 = ∅. X π+1 = argmin π₯ππ π max (3.4) π∉ππ e(S, x) ππ = { π ∈ 2π / ∑(π∈π) π₯π = ∑(π∈π) π¦π πππ ππ£πππ¦ π₯ , π¦ ∈ ππ } (3.5) (3.6) Because the set 2 π is finite, so after finite number of steps the process of construction of sets X π , T π will get abrupt. The last set X π will contain a unique vector what will coincide with nucleolus in the sense of its initial definition. Takes place the following lemma: LEMMA 3.1. Let π₯, π¦ π π, π₯ ≠ π¦ and π = {π / π₯(π) = π¦(π)} ≠ 2 π , if πππ₯ π∉π π(π, π₯) < πππ₯ π∉π π(π, π¦), π‘βππ π₯ βΊπ£ π¦. Let numbers ∝1 ≥∝2 ≥ β― ≥∝π are all of the different values that accept components of the vector π(π(π), π£). Below we will consider sets B π and Y π defined following way: B π = {π / π(π, π(π)) ≥ ∝π } π π = {π₯ ∈ π/ π(π, π₯) =∝π for π ∈ π΅ π\ π΅ π−1 , if π ≤ π; and π(π, π₯) ≤ ∝π for π ∉ π΅ π }. Lemma 3.2. If X π = ππ π‘βππ ππ ⊇ π΅π . Lemma 3.3. For all π = 1, … , π exist numbers π π πππ π ππ‘π π π such that 1 = π1 < π2 < β― < ππ , and π π = π ππ (3. 7) Theorem. 3.1There is a number q such that X π= π(π). 4.1 Fuzzy games with finite sets of coalitions. 4.1. Let (π’, π£) is a fuzzy cooperative game, where π’ ⊂ π is some finite set of fuzzy coalitions. Below we will prove that in presence of some conditions this type of games possess a unique nucleolus. Lemma 4.1. Let X is a convex politope and χ is the solution for the next linear programming problem: πππ π xπ π + π ππ + π0 ≥ ππ π€βπππ π π π π ′ π₯π π. Then exists a vector τj0 Ο΅ T ′ such that for every x, y ∈ χ π₯π π0 = π¦π π0 . 4.2. Fuzzy games with piece-wise affine characteristic functions In this paragraph we will prove a theorem about existence and uniqueness of nucleolus for fuzzy cooperative games with piece-wise affine characteristic functions. Theorem 4.1. Let (T, v) is a fuzzy cooperative game with piece-wise affine characteristic function v. That means, exists a collection of simplexes {∑π} what covers π: π =∪ ∑π , ∑π ∩ ∑π = ∅ ππ π ≠ π, and πππ ππ ∑π , v (π) = u π (π)-∝π , π€βπππ u π (π) is a linear function and ∝π ≥ 0. Then the game (π, π£) has a nucleolus that consists of a unique point. Below is an example of a game the nucleolus for what has been calculated. Considered game G = < [0, 1] 2, π£ > with the following characteristic function π£(π): π π2 1−π1 π0 π0 1−π0 π£(π) = min { 1 , , , 1−π2 1−π0 1 }, for π = (π1 , π2 ) ∈ [0,1]2 and π0 < . 2 It is clear that for this game π£(1) =1 and π = {π₯π π 2 / π₯ 1 = −π₯ 2 }. The game with characteristic function possesses following nucleolus : ν(X) = (0, 0). REFERENCES [1] SCHMEIDLER D. 1969 The nucleolus of a characteristic function game. –SIAM J. OF MATH. 1, vol. 17, pp. 1163-1170. [2] SOBOLEV A. 1976.Characterization of the optimality principles in cooperative games by functional equations. – Math Methods in Social Sciences. - Vilnius, pp. 94-151. (In Russian) [3] AUBIN JP (1981). Cooperative fuzzy games. Math Oper.es 6: 1-13