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 𝜏 =
(𝜏1 , 𝜏2 , … , 𝜏𝑛 ) with 0 ≤ 𝜏𝑖 ≤ 1 for each 𝑖 ∈ 𝑁. A cooperative fuzzy game with the players set N is a pair (𝑇, 𝑣)
where 𝑇 ⊂ [0, 1]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 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 ≺ṽ relation.
Let G=<N, ṽ> is a classical cooperative game and
Y (ṽ) = {xϵ RN / xi ≥ ṽ({i}), ∑(i=1,n) xi =ṽ (N)}
is the set of all imputations. The nucleolus of the game G consists of the following imputations:
ν(Y) = { x ∈ Y/ x ≺ṽ y for every y ∈ Y}.
The relation ≺ṽ 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 ≺v :
ν(Y) = { x ∈ Y/ x ≺ṽ 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, ṽ> the set of payoff vectors defined as below
X (ṽ) = {xϵ RN / ∑
(i=1,n)
xi = ṽ (N)}.
Then the game G has a nonempty nucleolus
ν(X) = { x ∈ X/ x ≺ṽ y for every y∈ X },
which contains only one vector.
For outcomes from X (ṽ) the condition of individual rationality has been violated. That is the reason
because of what the set of payoff vectors X (ṽ) is different of the set of imputations Y (ṽ), 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 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:
𝑋 (𝑣) = {𝑥𝜖 𝑅𝑁 / ∑
𝑖=1,𝑛
𝑥𝑖 = 𝑣 (1)}
We will inductively define sets X 𝑘 , 𝑇𝑘 by accepting that
𝑋 0 = 𝑋, 𝑇0 = ∅ .
(3.1)
For 𝑘 = 0, 1 … 𝑝 we will define sets 𝑋 𝑘+1 the following way,
𝑋 𝑘+1 = 𝑎𝑟𝑔𝑚𝑖𝑛 𝑥∈𝑋𝑘 𝑠𝑢𝑝 𝜏∉𝑇𝑘 [(e (𝜏, 𝑥) − 𝑒0 )/𝜌(𝜏, 𝑇𝑘 )]
(3.2)
and sets 𝑇𝑘 for 𝑘 = 1,2 … 𝑝
𝑇𝑘 = {𝜏𝜖𝑇 /x𝜏= y𝜏, for every 𝑥, 𝑦 ∈ 𝑋 𝑘 }
(3.3)
where e(𝜏, 𝑥) = 𝑣(𝜏) − 𝑥𝜏 , 𝑒 0 = 𝑚𝑖𝑛 𝑥 𝑚𝑎𝑥 𝜏 𝑒(𝜏, 𝑥)
and 𝜌(𝜏, 𝑇𝑘 ) is the distance between the point 𝜏 and set 𝑇𝑘 :
𝜌(𝜏, 𝑇𝑘 ) = 𝑖𝑛𝑓 𝜏′ 𝜖𝑇𝑘 𝜌(𝜏, 𝜏′).
ρ(x, y) = 𝑚𝑎𝑥 𝑖 |x i − yi ||.
Sets { 𝑇𝑘 } increase. When k increases, 𝑇𝑘 does not decrease: 𝑇𝑘+1 ⊇ 𝑇𝑘 . If for some 𝑘 0 it is turning out that
𝑇 𝑘0+1 = 𝑇𝑘0 , then that entails the stabilization of the corresponding set 𝑋 𝑘0 , or otherwise by increasing
𝑘, 𝑋 𝑘 will not decrease any more. The set 𝑋 𝑘0 obtained that way we will call the nucleolus for fuzzy game
(𝑇, 𝑣).
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 𝐺 =< 𝑁, 𝑣 > means a classical cooperative game, where 𝑁 = {1, 2 … 𝑛} is the set of all players
and
𝑣: 2 𝑁  𝑅 1
is a characteristic function that satisfies to the condition 𝑣(∅) = 0.
First we should pay attention that in case of classical cooperative games relations (3.1) - (3.3) accept the
following view:
𝑋 0 = 𝑋, 𝑇0 = ∅.
X 𝑘+1 = argmin 𝑥𝜖𝑋 𝑘 max
(3.4)
𝑆∉𝑇𝑘
e(S, x)
𝑇𝑘 = { 𝑆 ∈ 2𝑁 / ∑(𝑖∈𝑆) 𝑥𝑖 = ∑(𝑖∈𝑆) 𝑦𝑖 𝑓𝑜𝑟 𝑒𝑣𝑒𝑟𝑦 𝑥 , 𝑦 ∈ 𝑋𝑘 }
(3.5)
(3.6)
Because the set 2 𝑁 is finite, so after finite number of steps the process of construction of sets X 𝑘 , T 𝑘 will get
abrupt. The last set X 𝑘 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 𝑥, 𝑦 𝜖 𝑋, 𝑥 ≠ 𝑦 and
𝑇 = {𝑆 / 𝑥(𝑆) = 𝑦(𝑆)} ≠ 2 𝑁 ,
if 𝑚𝑎𝑥 𝑆∉𝑇 𝑒(𝑆, 𝑥) < 𝑚𝑎𝑥 𝑆∉𝑇 𝑒(𝑆, 𝑦), 𝑡ℎ𝑒𝑛 𝑥 ≺𝑣 𝑦.
Let numbers ∝1 ≥∝2 ≥ ⋯ ≥∝𝑝 are all of the different values that accept components of the vector
𝜃(𝜈(𝑋), 𝑣).
Below we will consider 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−1 , if 𝑚 ≤ 𝑙; and 𝑒(𝑆, 𝑥) ≤ ∝𝑙 for 𝑆 ∉ 𝐵 𝑙 }.
Lemma 3.2. If X 𝑘 = 𝑌𝑙 𝑡ℎ𝑒𝑛 𝑇𝑘 ⊇ 𝐵𝑙 .
Lemma 3.3. For all 𝑘 = 1, … , 𝑞 exist numbers 𝑙 𝑘 𝑎𝑛𝑑 𝑠𝑒𝑡𝑠 𝑋 𝑘 such that
1 = 𝑙1 < 𝑙2 < ⋯ < 𝑙𝑘 , and 𝑋 𝑘 = 𝑌 𝑙𝑘
(3. 7)
Theorem. 3.1There is a number q such that X 𝑞= 𝜈(𝑋).
4.1 Fuzzy games with finite sets of coalitions.
4.1. 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 nucleolus.
Lemma 4.1. Let X is a convex politope and χ is the solution for the next linear programming problem:
𝑚𝑖𝑛 𝜀
x𝜏 𝑗 + 𝜀 𝑐𝑗 + 𝑒0 ≥ 𝑎𝑗 𝑤ℎ𝑒𝑟𝑒 𝜏 𝑗 𝜖 𝑇 ′
𝑥𝜖 𝑋.
Then exists a vector τj0 ϵ T ′ such that for every x, y ∈ χ
𝑥𝜏 𝑗0 = 𝑦𝜏 𝑗0 .
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 {∑𝑗} what covers 𝑇: 𝑇 =∪ ∑𝑗 , ∑𝑘 ∩ ∑𝑙 = ∅ 𝑖𝑓 𝑘 ≠ 𝑙, and 𝑓𝑜𝑟 𝜏𝜖 ∑𝑗 , v
(𝜏) = u 𝑗 (𝜏)-∝𝑗 , 𝑤ℎ𝑒𝑟𝑒 u 𝑗 (𝜏) is a linear function and ∝𝑗 ≥ 0. Then the game (𝑇, 𝑣) 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, 𝑣 > with the following characteristic function 𝑣(𝜏):
𝜏
𝜏2 1−𝜏1
𝜏0
𝜏0 1−𝜏0
𝑣(𝜏) = min { 1 ,
,
,
1−𝜏2
1−𝜏0
1
}, for 𝜏 = (𝜏1 , 𝜏2 ) ∈ [0,1]2 and 𝜏0 < .
2
It is clear that for this game 𝑣(1) =1 and 𝑋 = {𝑥𝜖 𝑅2 / 𝑥 1 = −𝑥 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