1. Introduction

The optimality principle for TU games, which in game theory known as nucleolus has been introduced by D. Schmeidler (1969). Since then it has attracted many researchers working in field of cooperative games. E. Kohlberg (1971) has described new properties of the concept, by Megiddo N. (1974) proved its nonmonotonicity, L. Zhou (1991) had a paper about weak coalitional monotonicity (WCM) of nucleolus. Sobolev A. (1976), Potters J. (1991), Sniders C. (1995), Voorneveld M, Nouveland A. (1998), Orshan and Sudholter P. (2003), have works that characterize the decision rule through various sets of properties.

After Aubin J.-P. (1981) introduced the fuzzy concept in game theory extension of existing in the classical theory decision rules to fuzzy cooperative games has become matter of principal importance. Core has been generalized by Aubin (1981) and Shapley value by Tsurumi et. al. (2001). Maroutian Y. (2017) has extended to fuzzy games classical nucleolus and in Maroutian Y. (2019) in a setting that varies of what described in Tsurumi et. al. (2001) generalized for the fuzzy case Shapley value.

Section 2 devoted to preliminaries that are known from the classical theory and the background material. The latter mostly refers to inductively defined sets [, ]. That inductive process after finite number of steps provides with prenucleolus. There are direct existence proofs of prenucleolus in Section 3. For three different classes of games: strongly bounded games, games with lattice polytopic sets of preimputations and for belonging to same class games but possess with certain continuous selection functions we give direct existence proofs for the prenucleolus. In section 4 we extend to fuzzy games result of L. Zhou on weak coalitional monotonicity of prenucleolus (WCM).

Section 5 characterizes prenucleolus through four properties: Nonemptiness, WCM, Consistency and Converse Consistency.

2. Preliminaries and Background

Nucleolus of a classical cooperative game is an imputation that is the best in sense of some preference relation Let for game

is the set of all imputations for game G.

By magnitude denotes excess of coalition S from vector x. Vector of excesses defined with components

Clear that components of ordered decreasingly. On set defined quasi-order following way. For where is for lexicographic order. That means exists a number such that

and

Definition. For and characteristic function the set is nucleolus for Y if vectors from are minimal in sense of relation i.e.

Theorem (Schmeidler (1969)) For every nonempty, convex and compact set the nucleolus exists and consists of only one vector.

Theorem (Sobolev (1975)). Let for a game payoff vectors are preimputations from set

Then game G has a nonempty prenucleolus, which consists of only one vector.

Due to violation of individual rationality for outcomes from that set no more is compact and hence it is different of the set . Despite that, the statement on existence and uniqueness of prenucleolus keeps remaining true.

For a fuzzy game where is the set of fuzzy coalitions let is the characteristic function of that game. Set of preimputations is the set of vectors that satisfy to condition of efficiency:

We inductively define sets by accepting that

For we define sets the following way:

and sets :

where and is the distance between point and set :

, where

When k increases sets do not decrease:

If for some it turns out that , then that in its turn entails stabilization of corresponding set , or otherwise, with increasing of k does not decrease any more. That kind of stabilization of sets after finite number of minimization steps can happen in case of games with no piece-wice affine characteristic functions.

Vectors from mentioned kind of sets we call prenucleolies in different of unique vectors that are prenucleoluses.

3. Direct Existence Proofs

3.1. The Strongly Bounded Games

Definition 3.1.1 A game is strongly bounded if there is a number M>0 such that for every vector and coalition inner product It is clear that for a strongly bounded game set of division vectors is compact and convex.

Definition 3.1.2 is a piece-wice linear characteristic function game if there is a collection of simplexes , such that for each i, j if as well for , where and is a linear function.

The following is a direct proof of existence of fuzzy prenucleolus for strongly bounded piece-wice linear characteristic function games.

Theorem 3.1.1. A strongly bounded piece-wice linear characteristic function game possesses with a unique prenucleolus.

Proof. Let the following equation holds true on some simplex :

Below we deal with minimization problem (MP):

where

and

This problem has a solution because set X is a polytope. Solution of that minimization problem is the set:

is a polytope because It also is convex and bounded. From there sets , where are compact and convex. For and inner product is a linear function of . Hence, instead of we can deal with inner product , where is some vector. Expression reaches at some coalition for every k. It is well known that reaches only on extX.

From there, as far as in the problem for is the set of points of minimum for expression hence Because set of preimputations X is a polytope, so for a bounded game it also bounded, closed and convex. So is set ext X of X. The solution of MP can be represented by a mapping From follows that

and for each is a nonempty and convex set. So, according to Kakutani’s theorem mapping possesses with a fixed point, i.e. with an such that and is a subset of set From the written right now it follows that for

To conclude the proof remains to consider the case, when and

Started from some number m and as well for arbitrary m. Holds true the inclusion: Denote: and where is an arbitrary number.

Accept we have that and Due to inclusion above H. From there, The latter one completes the proof that H* is the prenucleolus of game

3.2. Games with Lattice Polytopic Sets of Preimputations

In case of strongly bounded games set of preimputations to deal with obtains as a result of excluding from the initial set X big chunks of preimputations. This is a restrictive measure, which can cause fair objections. Dealing with games that have lattice polytopic sets of preimputations is a way for evading of blame.

Definitions. For vectors x preferable of y in sense of preference or if there is a set such that and for some but .

All of the followed operations of ordering defined in sense of preference.

For denotes least of upper bounds (biggest of lower bounds) of A in X. With respect to preference X is a partially ordered set (POS).

A POS X is a join (meet) lattice if for every

At this point we require that for our polytope X it’s set of peaks has been a join lattice in sense of preference

Theorem 3.2.1. A piece-wice linear characteristic function game with a lattice polytope set of preimputations possesses with a unique prenucleolus.

Proof. Let for a linear game on is the set of solutions of MP that corresponds to , i.e.

and is a mapping.

In difference of strongly bounded games sets here not bounded and hence they are only closed and convex. Mapping for games from this class can as well be represented as a mapping We have proved in the previous theorem that which means that is a set of extreme points itself. Farther, we will deal with the least upper bound of set in sense of preference . We need to show that exists For the latter recall first what we have written related to mapping i. e

Then, which is l.u.b. of set is sense of -preference also is l.u.b. for some set . Because for each there is only one upper bound that belongs to , hence due to closedness of and linearity of excess as a function of .

Again for the reason that every is a linear function, there is a such that So for some numbers where in representations of above linear functions for each one of is a l.u.b. for set . The same are as well fixed points for mappings . Because so from there Prenucleolus is the only vector that is biggest in sense of preference, or which is the same as it is vector that minimizes expression At the same time for some or . To resume, vector , which is the prenucleolus of is the least of upper bounds for lattice polytope X and hence minimizes the expression with above.

3.3. Mapping φ Possesses with Continuous Selection Functions

Below we prove existence of prenucleolus in case when mapping possesses with continuous selection functions. We assume as well that there are both kinds of them there: monotone increasing and decreasing.

Definition 3.3.1 A mapping decreases (increases) if for and , from follows that

Definition 3.3.2. A selection function for a multivalued mapping is a function

such that for every

For the needs of this part we assume that the set of preimputations X endowed with sup norm topology. It is not worth a lot that in the definition of a game inherent a topological assumption regarding the payoff functions. We just need it to ensure the convergence of sequences and continuity of selection functions for .

The existence proof of prenucleolus becomes pretty direct for mapping that possesses continuous, monotonic selection functions.

Theorem 3.3.1. Let is a piece-wice linear characteristic function game, with a lattice polytope set of preimputations X, and maps to solutions of it’s MP’s. Then possesses with a unique prenucleolus if there are monotonic increasing and decreasing continuous selection functions for .

Proof. First we prove existence of a fixed point for selection function f and then show not emptiness of set of fixed points: After what we show that the obtained this way fixed point, which contained in is greatest in sense of

As we have mentioned that earlier, where is the set of solutions of MP that corresponds to the game with simplex of fuzzy coalitions . Farther for and

We separate two cases.

Case 1. Let and . As least upper bound of

The monotone increase of argument x we consider in sense of preference .

Let for selection function f assume that it is continuous, monotonic increases and is a sequence of f -iterates from That means and . From there, is a monotone increasing in sense of preference sequence. Since X is a lattice politope, so there is an upper bound of X, which bounds and from where converges to a point Because so due to continuity of f converge both of the sequences and , which means that is a fixed point for f, i.e. By the other side, because hence That in its turn means, i.e. the same ē as well is a fixed point for and set is not empty.

Let now show that is the biggest fixed point in sense of preference . Suppose that and It is clear that in the sense of preference.

Assume that f′ is a monotone decreasing selection function for . Sequence of f′ - iterates of monotone decreasing. From there, for some number and for some other number which again holds true due to continuity of f ’. By induction we obtain that Hence, in this case prenucleolus is fixed point

Case 2. and In this case is the biggest point in that can minimize the maximal excess. From there it is the prenucleolus of

An example of a monotonic selection function.

Example 3.3.1 Let is a monotone decreasing mapping and for be the following set: such that and

when

For

Defined this way function g(.) is a decreasing selection function for .

Remark 3.3.1. In the example above if to replace the condition of monotonic decreasingness for mapping by the same kind of increasingness then function g(.) will become monotone increasing.

4. The Weak Monotonicity Property of Fuzzy Prenucleolus

Let and (T, w) are two fuzzy games and is a set of fuzzy coalitions such that

and for

Denote by and the sets of prenucleolies for games and respectively.

Definition. The fuzzy prenucleoli is weakly coalitionally monotonic if for and .

The theorem below extends L. Zhu’s (1991) property of weak coalitional monotonicity for the classical prenucleolus on prenucleolies for fuzzy cooperative games.

Theorem 4.1 For any fuzzy game its prenucleolies when they exist possess the property of weak coalitional monotonicity.

Proof. Let construct for game be such that it results to prenucleoli . Then there are sets such that .

Depending on if coalitions belong to set or not one can describe the containing them sets from these constructs as well their mutual locations in and If

and for a vector sets, for example and contain x then they coincide, because the vectors containing in both of the sets minimize equal excesses.

Let now for a coalition discuss first it’s mutual positions in constructs and . It is clear that there is a number j such that . Because so the position of coalition in construct compared with the same in obtains by moving it up through the column of to a certain set , where

Set that corresponds to consists of vectors minimizing the excess of coalition in game . Therefore if and are prenucleolies of games and respectively and then for such that

From there it follows that

Remark 4.1 The proved property remains true as well in the case when a game possesses with prenucleolus.

Remark 4.2 When set contains as well classical coalitions then inequality obtained

above this time refers also to coalitions For we receive Zhou’s classical property of weak coalitional monotonicity i.e.,

when and instead are prenucleoluses and v,w are piece-wice affine games.

5. A Characterization of Fuzzy Prenucleolies

5.1. Some Preliminaries

For we denote by the set of all fuzzy coalitions that belong to cube .

To a corresponds fuzzy coalition , which is the vector if and Fuzzy coalition corresponds to the empty player coalition. We denote by the class of all fuzzy games that possess with prenucleolies.

For a game a payoff vector is a function . x is efficient if

The set of preimputations of is:

Restriction of payoff vector x to S is vector and for denotes inner product of vectors

5.2. The Properties

Below are the properties by what we are going to characterize fuzzy prenucleolies.

P1. Non-emptiness (NEmpt.)

P2. Weak coalitional monotonicity (WCM)

P3. Consistency (Cons.)

P4. Converse Consistency (ConCons)

A solution on is a mapping which associates with each game a set

that satisfies to properties P1-P4. The domain for solution is the set of fuzzy games .

As that proved in Maroutian Y. (2017) the minimization process by solving of linear programming problems (LPP) after finite number of steps provides with a single solution, which is the prenucleolus for piece-wice affine characteristic function games.

For not piece-wice affine characteristic function games started from some step k solution of the corresponding MP may not decrease any more, i.e. may be for following problem. Vectors that belong to stabilized this way sets we call prenucleoli.

We will use Y - A Hwang’s (2007) extension of classical reduced game to games from that has been introduced by Davis and Mashler (1965).

Definition. Let and Reduced game with respect to S and x is , where

Below are definitions of properties P2-P4.

• Consistency (cons.) If and then ,

• Converse consistency (CCons.) If and then

• Weak coalitional monotonicity (WCM).

Let and is a set of fuzzy coalitions defined in Part 4. If and

are prenucleolies of and respectively, vectors and then fuzzy prenucleolies and are weakly coalitionally monotonic if when .

As far as this property of fuzzy prenucleolies we have proved in the theorem of Part 4, so we will just refer to WCM without proving it again here.

5.3. The Results

Lemma 5.3.1 Let , and For arbitrary is the coalition of maximal excess then holds true following inequality

Proof. Let and are the same as they are in lemma’s formulation. Then

We need to show that if is an arbitrary coalition then

Accept that for some in contrary takes place the opposite inequality. By replacing in the inequality above first by it’s expression and then doing the same with in case if takes place the opposite inequality then that would mean existence of a set such that and at the same time for a but The latter one then would contradict to our assumption about vector x, i.e. With this contradiction our lemma proves.

Lemma 5.3.2. The prenucleolies satisfy consistency.

Proof let . For and due to it’s definition . In reduced game is efficient because of and definition of game From there it remains to show that for all is an arbitrary vector then

The latter one follows from lemma 5.3.1 and the inequality in its turn means that is a prenucleoli for i.e.

Lemma 5.3.3. prenucleolies satisfy to converse consistency.

Proof. Let , with is prenucleoli of game and vector We assume that for all coalitions and for vector reduced game

i.e. is a prenucleoli of We need to prove that x is a prenucleoli for as well, i.e

Let be an arbitrary vector and for is the coalition of maximal excess in game

Then by Lemma 5.3.1

We need to show that for arbitrary

Otherwice, if there is such that

Then it would mean that for some coalition where if and hence for that coalition S in reduced game would not be it’s prenucleoli. The latter one violates precondition on converse consistency of That means the inequality (5.3.1) is correct and x is a prenucleoli of game

Farther we will prove the uniqueness. It based on Elevator Lemma introduced by Thomson (2005), The variant of this lemma we use says that if a solution б is consistent and on the subdomain of all proper games of it is contained in a solution which is conversely consistent then the inclusion always holds.

Theorem 5.1. A solution satisfies NEmpt., WCM, Cons. and CCons. if and only if for all

Proof. That satisfies to NEmpt. follows from belongs to domain , where every game possesses with a set of prenucleolies. For each one of the rest of properties a corresponding statement we have proved.

Assume that the solution also in its turn satisfies to WCM, Cons, and CCons. Let . To prove uniqueness we will use the method of induction on number . For . Suppose that if and

The case Let Based on Con. of for all with

By CCon. of the prenucleoli From there, The inclusion one can show by starting from and come to by applying similar proof. Hence,