1. Introduction

In classical theory refer to excess based solutions the nucleolus and several other types of nucleolies. Still in nineteen forties game theorists in RAND recommended lexicographic minimization of objective functions. The recommendation published in (Brown G, 1950) and in (Dresher M., 1961).

In (Schmeidler D. 1969) introduced the nucleolus of TU Games. After Schmeidler D., E. Kohlberg in (1971) has described new properties of nucleolus for TU Games. In N. Megiddo (1974) has proved nonmonotonicity of nucleolus. L. Zhu, (1991) has proved the weak coalitional monotonicity of the decision rule. Zhu’s result has been extended to fuzzy games in Y. Maroutian (2024).

Section 2 describes the background and current model. Devoted to prenucleolies Section 3. As superdifferential represented sets X^k of prenucleolies. Interpreted geometrically the prenucleolus’s behavior while it is in the process of construction and described structure of sets X^k of prenucleolies.

We prove the following properties for games to prenucleolies mappings ν(.), the superlinearity in section 4.1 and Section 4.2 devoted to proving of nonmonotonicity by Megiddo for prenucleolies.

In section 5, we have proved that the prenucleolus remains invariant under operation of monotonic covering.

2. The Background and Current Model

2.1. Some Preliminaries on Prenucleolies

In theory, use of the term prenucleoli has a common meaning and varies depending on type of games to what it applies. Refer to nucleolies in case of coalitional TU games the nucleolus (Schmeidler [1969], Owen [1977]) and prenucleolus (Sobolev [1975]). It has more general meaning in (Mashler et al [1992]), where nucleolies are decision rules that obtain as a result of lexicographic minimization. In (Mashler, Peleg [1977]) defined generalized nucleoli for set valued dynamic systems. Potters and Tij’s [1992] introduced matrix nucleoli and established that it is an analogue of Kohlberg’s characterization of nucleolus for TU games in terms of balanced sets (E. Kohlberg [1971]).

In fuzzy theory process of prenucleolus’s finding when characteristic function of a game is not of affine type may end up with a solution, which does not possess with a unique value. This kind of decision rule we call prenucleoli of a fuzzy TU game following to classical use of that term.

2.2. Definitions and a Decision Rule

We denote by FG and CFG the sets of all fuzzy and fuzzy concave characteristic function games respectively. FC is the sets of all fuzzy coalitions:

For a fuzzy game (T, ν), where is the game’s set of fuzzy coalitions, assume

is it’s characteristic function. Set of preimputations is a set of vectors that satisfy to condition of efficiency:

For k=0, 1, … ,p inductively defined sets X^k, where it is assumed that and The X^k+1 are following sets:

For k = 1, …, p sets T^k are as below:

is excess of coalition from division vector and

With we denote distance between and set T^k

If started from some number k₀, for k > k₀, T^k+1 = T^k, then that in its turn entails the stabilization of corresponding set X^k or otherwise with increase of k sets X^k do not decrease anymore. That kind of stabilization of sets X^k after finite number of minimization steps may happen in case of games with no piece-wice affine characteristic functions.

We call vectors from sets X^k mentioned above prenucleolies in different of unique vectors that are prenucleoluses.

3. The Prenucleolies

3.1. The Prenucleolies as Superdifferentials and Their Structure

Definitions ([13]). A vector is a supergradient of concave function at a point

The set of supergradients of f at is the superdifferential of f at . We accept farther v∈CFG.

Below we prove that for a piece-vice affine characteristic function v if v CFG, at each one of minimization steps set of solutions where . The proof based on the fact that when k increases sets decrease monotonic, i.e. , if k>m.

Let us now consider MP discussed at some step m for finding of prenucleolies, i.e.

If at the same time consider MP, which is at step k, for k>m, then because sets {X^𝑙} decrease with 𝑙 increasing, so we will have that there is x∈X^m such that it belongs also to X^k and that way satisfies to MP of step k:

As far as k>m and in each one of the following to m steps of minimization magnitudes of excesses for coalitions decrease compared with the ones that have been at step m, hence

from where

The last inequality means that each is a supergradient of characteristic function from game (T, ν) at or . The opposite inclusion, i.e. one can obtain by taking a vector . The latter one, i.e that really there is , and hence, that one can claim based on concavity of . We will come to already obtained inclusion’s opposite one by reversing the proof above. That results in the following:

Proposition 3.1.1. For a piece-wice affine concave characteristic function , where , is superdifferential of at .

Remark. Set of concave functions intersects with the set of piece-wice affine characteristic functions. While proving proposition 3.1.1 as by product we have obtained no emptiness of sets X^k. Theory of superdifferentials is a convenient tool for assessing sets X^k and researching through them prenucleolies of concave characteristic function games.

Below we prove another presentation for sets X^k.

Proposition 3.1.2. for a concave ν and .

Proof. We need to show only that .

First lets prove that . Because ν∈CFG then by τ concave is also Hence, and there is for . Then for a coalition

Because so for there is such that and then for ε small enough we will have that from where, as far as can be arbitrarily small, hence can be so as well which means that

i.e. , and from there

The other inclusion one can obtain by repeating the same arguments in reverse order.

The proved right now together with Proposition 3.1.1. concludes the proof of our statement.

Proposition 3.1.3. Let be a fuzzy game, where ν is a piece-wice affine, concave characteristic function such that and is a linear function. Then for exists satisfying to where ext is the set of extreme points of .

Proof. Let v is a concave piece-wice affine characteristic function, it’s superdiffrential at some point τ∈T and is an arbitrary vector.

Below we will prove that for games possessing with either prenucleolus or prenucleoli and for vector holds true equality in case if .

We assume that ν∈CFG, and as they are above. Then for arbitrary From there Farther, let denote by .

If we will have that for every We can obtain the same inequality as well based on for . By Proposition 3.1.2. for the games that we are dealing with

when From there statement of Proposition 3.1.3. takes place if for exists T^k such that For x, y∈X^k if then which means that const for all

. The same remains true as well for Hence, for every

To conclude the proof let assume that for linear function holds true only when and is an arbitrary vector. By the other side because is a linear function, hence exists such that From there we are obtaining that

which is a contradiction. That proves our statement, i.e. exists such that and from there in its turn also

Statement 3.1.4. If is a prenucleolus of game and k is a number such that then

Proof. It is immediate and follows from definitions of sets X^k and extX^k.

Statement 3.1.5. is a convex set if there is number k such that .

Proof. Follows from Proposition 3.1.2. and convexity of sets X^k.

Statement 3.1.6. For a game with linear characteristic function ν if

and then

Proof. According to Proposition 3.1.3. for where From there we have that if then

The following refers to structure of sets X^k for linear characteristic function games.

Proposition ([13]). If is compact convex set then C=conνh(extC). We note by conνhA the conνh of set A.

This proposition can be applied to sets where ν is a strongly bounded game.

Proposition 3.1.7. For set where

Proof. By Proposition 3.1.3., for some where From there follows the first equation, which is above, i.e. . Farther, as far as by definition , hence . This is what has been required.

Remark. Based on Proposition ([13]) in formulation of Proposition 3.1.7 for strongly bounded games instead of inclusion sign takes place equality.

3.2. A Geometric Description on the Prenucleolus’s Construction

The way sets X^k behave during the process of prenucleolus’s constructing allows to describe it geometrically. That description based on theory of superdifferentials and refers to .

The position of a set X^k defines by position of its supporting hyperplane. Point through what passes supporting hyperplane belongs to of map and supports with each step of minimization. The position of supporting hyperplane changes relative to position of similar set that has been at previous step. On set position’s changes and equivalently changes in directions of supergradients containing in during constructing of prenucleolus one can get an idea by comparing expressions of supergradients at two consecutive steps of minimization. At the same time takes place change in position of the supporting hyperplane at each step of the prenucleolus’s construction. From there, one can say that while constructing a prenucleolus the supporting hyperplane and together with it the entire set X^k oscillate around a certain position. Sets X^k keep oscillating with each step until they reach their final position at the last step. That is when defines position of the prenucleolus. The latter one of course refers to games that possess with unique prenucleolus.

Now let recall again that the present description is about the class of games that possess with superdifferentials, i.e. games with concave characteristic functions. That class intersects with piece-wice affine games. From there though not every concave characteristic function game has a prenucleolus and hence, it is not always that the described oscillations of sets X_k will come to a final state and stop, however described oscillations will take place for all those games that possess with prenucleolies.

4. Properties

4.1. The Superlinearity of Mapping vu (.)

This part together with property of superlinearity of mapping as a function of ν contains as well a result on nonmonotonicity of prenucleolies by Megiddo.

This property of mapping insighted from Schmeidler (1969) where it relates to nucleolus of classical characteristic function. D Schmeidler defined it in terms of lexicongraphic minimization and hence, for proving the superlinearity the methods that used there are different of ours.

Proposition 4.1.1. The set of CFG of all fuzzy concave characteristic function games is a convex and closed cone. For every is a superlinear mapping.

Proof. That CFG is a convex and closed cone proves directly. That is the reason for omitting it here. Farther, we are going to deal with sets as superdifferentials We have proved in Proposition 3.1.2 that for

Suppose and are games, where By definition of superdifferentials for

and similarly, for

Also,

That means for takes place:

or otherwise, Hence,

To prove the superlinearity of for our needs we will use relation for vectors from set for if for all

We require that for vectors where and Otherwise, from there it would follow that violates condition of The latter one means that for arbitrary holds true inequality z≥x+y, which in its turn means that . From there we have really possesses with property of superlinearlity.

4.2. The Nonmonotonicity of Fuzzy Prenucleolies by Megiddo

On this property there is the work of N. Megiddo (1974) for the nucleolus of classical characteristic function games. We extend his result to fuzzy prenucleolies.

Let and are two fuzzy TU games and are sets of prenucleolies for games and respectively. We denote by B the following collection of coalitions:

For all i.e.

Definiton 4.2.1. We will say that prenucleolies for games and are monotonic if for and arbitrary where We call it monotonicity by Megiddo.

Proposition 4.2.1. For fuzzy games , when , its prenucleolies are not monotonic by Megiddo.

Proof. First, it is clear that prenucleolies and are solutions of known MP’s for games (T, ν) and (T, u) at some steps k and k′ when each one of these sets get stabilized. We have proved in Proposition 3.1.2 that where and Let consider superdifferential for coalitions and , where and as well for vectors

Because hence

Now we are going to choose coalitions τ and τ′ the following way:

And

In case of chosen that way coalitions and the inequality above holds true only for which means that or From there, is not monotonic. The proposition proved.

Remark. The nonmonotonicity of fuzzy prenucleoluses, which corresponds to the case when , follows directly from proposition we have proved.

5. On the Prenucleolus of Monotonic Covers

Definition 5.1. For a piece-wice affine characteristic function fuzzy game its monotonic cover is a game that for a coalition defines as:

The preference relation between coalitions we will consider in this section in sense of vector ordering

Theorem 5.1 For a fuzzy piece-wice affine characteristic function game its prenucleolus invariant under operation of monotonic covering.

Proof. Let is a fuzzy game that mentioned in theorem’s formulation. Below is a minimization problem (MP) for :

(1)

We have proved in Y. Maroutian (2017) that solution for LPP (1′) below is also solution for MP(1).

(1')

here is set of peaks of simplexes that cover the set of T.

It has been proved that a game possesses with prenucleolus when minimization problems below, where have solutions:

We assume that set is the solution of MP above. There will be need for dealing parallelly with similar constructs that refer to either one of the games: or . Deserves for separate discussion the case when points of maximum of function for are falling out of sets . If for then because from it follows that also hence the left hand side of the inequality, which contains the utility function in MP for becomes negative due to equality to zero of its right hand side. From there to find points of maximum of considering only set does not suffice. Hence, instead one should consider the entire set . Let as above is the solution of MP (1) for game . We show first that exists solution of MP for such that

For where, . Every set is compact. In its turn, the characteristic function ν(τ) is continuous. From there on each one of sets , there is

such that =

Due to the finite number of points of maximum exists such that

Based on that

From there for and as well for coalitions and such that holds true

For be the following set:

for pairs of coalitions such that

The set because and there are coalitions such that refers to the set of all preimputations of . From there, the MP:

possesses with solution

Farther we describe sets and for arbitrary k through three statements.

Statement 5.1. For a vector takes place also if and only if when for pairs of coalitions σ and τ₀ such that and holds true

Proof. Necessity. Let That means for solution of MP on game (T, ν), which is (ε₀, X^k(ν)) holds true the following inequality:

(S1) where

To state that the same vector belongs as well to means requiring that a similar inequality took place for MP, which this time refers to game and has solution

(S2) where

By the other side

where is a coalition for what

To obtain that vector in the chain above first we should provide the condition (S1), i.e. requiring what is below:

when

Of what we have had now first follows that really and second that it takes place, when

Sufficiency. Let We need to prove that if then it also belongs to

From we have that for arbitrary

where

From it follows that and hence,

Statement 5.2. For when exists a pair such that

Proof. If for then there is a coalition such that

then because for arbitrary so from there, by the property of triangle for metrics.

That is what required. As a result we obtain that

for arbitrary pair of coalitions and such that and . Hence,

Remark 5.1. Sets because where is the set of all preimputations of game .

Statement 5.3. For arbitrary

Proof. Let is the solution of k’th MP for game . That means we can state the following: if , then for arbitrary is the minimal value of in the inequality:

From there, because so and , for arbitrary and . At the same time also minimizes the expression because . That means for .

Farther, for arbitrary where Because the same hold true for hence .

Let’s now move ahead and prove the opposite inclusion, i.e. .

Assume that . Then for and holds true the following inequality:

There is such that and

In case, when the last inequality in the above chain remains true, because then and from where it’s left side is negative. As far as so for . From there, for .

By definition of metrics we have that if then

So the inequality entails that

As a conclusion we will have that

or just , where .

The latter one means (ε₀, x), the pair, which minimizes (**) inequality, does the same also with the inequality (*). Something that refers this time to MP for game .

So, . From there, . The proved right now inclusion with the opposite one, which we have proved above provides as with one required by statement 5.3.

Remark 5.2. Based on equality from statement 5.3. we can claim the following properties for sets :

a.) Sets are convex

b.) Sets are monotonic increasing

c.) With increase of k increases also

Remark 5.3. Sets are convex.

The proof of this statement is immediate. To prove increasingness of we don’t use the mentioned property of sets Now we are ready to conclude the theorem’s proof. As we mentioned that above the equality provides growth of with each pair of new sets and of game that are constructed. From there, after finite number of steps due to convexity of sets last set will coincide with T and the corresponding to it set will consist of only one vector, which is prenucleolus of game . This results to conclusion of theorem’s proof with following statement.

Statement 5.4. If is a piece-wice affine fuzzy characteristic function game and is it’s set of preimputations then the prenucleolus of monotonic cover of game conincides with prenucleolus of , which is .

ACKNOWLEDGEMENTS

The author is very thankful to an anonymous referee, an Associate Editor and the editor of the Journal of Game Theory for their helpful comments.

Glossary and List of Notations

– fuzzy game with set of coalitions and characteristic function . Briefly, we say also game or game .

– set of preimutations of game , which is a set of vectors that satisfy to condition of efficiency:

– inductively defined sets where . Each one of sets corresponds to k-th step of construction of prenucleolus:

– set of coalitions corresponding to k’ th step of constructing of prenucleolies:

excess of coalition from division vector x, where

is inner product of vectors x and .

.

– distance between coalition and set .

, where

prenucleolus – unique vector to which results process of constructing of sets after finite number of steps. Possess with prenucleolus piece- vice affine games.

Prenucleoles- set of vectors that obtain as a result of stabilization of sets started from some number , i.e. for

MP – a minimization problem that discussed at some step m for finding of prenucleolies.

FG, CFG – sets of fuzzy and fuzzy concave characteristic function games.

FC – set of all fuzzy coalitions: partially

– the prenucleolus of characteristic function game (T, v).

and – superdifferentials respectively of game at point and excess .