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Fuzzy games: A description of the concept
Fuzzy Sets and Systems 1 (1978) 181-192. © North-Holland Publishing Company
FUZZY GAMES: A DESCRIPTION THE CONCEPT
OF
Dan BUTNARIU Str. Tepes Voda Nr. 2, lagsy 6600, Romania
Received January 1977 Revised November i977
O. Introduction The present paper is the first in a series of three papers having the same aim: an heuristical description and a mathematical approach to fuzzy games. The other two papers in this series will be entitled "Two-Person Fuzzy Games" and "N-Person Fuzzy Games" respectively. The first section of the present paper is dedicated to a general presentation of the notions, notations and results on fuzzy sets, which we are going to use later. The notion of"fuzzy set" (into a universe) which is introduced here is analogous, but not the same, with the notion offuzzy set introduced by Zadeh [23]. A great number of results, wellknown in the Universal Theory of Sets (Cohn I10]) can be extended to the theory of fuzzy sets in a universe. The second section of the present paper contains an heuristical discussion of fuzzy games and their relationship to the classical concept of a game, which is the object of the yon Neumann and Morgenstern theory [19]. We hope that the reader familiar with the content of this section will easily understand the intuitive meaning of the mathematical notions and results which we shall explain in the following two papers of the announced series.
1. Fuzzy sets into a universe
1.1. Universes and unambiguous sets The fuzzy set theory explained in the papers of Zadeh 1"23],Goguen 1-14l, Kaufmann [16], has as basic objects the "fuzzy subsets of a usual set". Let a usual set U be given. If U is a set such as the set of all the natural numbers, we observe that the Cartesian product of two fuzzy subsets of it is not generally a fuzzy subset of U. Also a great number of other properties, which appear in the usual set-theory, cannot be extended to fuzzy subsets of a usual set without reference to fuzzy subsets of a usual set other than U. Now we are going to present a notion of a "fuzzy set" which will eliminate a great number of these difficulties. We hope that the reader is familiar with the content of the first and second chapters of !81
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[10], i.e. with the axiomatic form of the theory of sets and classes explained here. The basic term of this theory is that of"class". A class A is called a set if and only if there exists another class B so that A is an element of B, i.e., formally, if and only if A 6 B. A function between two classes A and B is a sub-class of the cartesian product A x B, subclass having the usual functional property. A set U is called Universe if and only if the following axioms hold:
(UI) x 6 U {U2) x ~ U
implies
x _ U.
implies ~ ( x ) e U . (U3) x 6 U implies {x,y}~U, for any y~U. (U4)JeU, Fj~uo¢J) implies ~ Fj~U. where ~_, ~(x), [,.)j~j denote the usual inclusion of sets, the Boolean set of x and the union of the J-indexed family, respectively. We denote {x, "y}~ by (x, y). The empty set 0 and the family of all finite sets are universes. According to the axioms (U 3), for any two sets X, Y ~ U, the Cartesian product X x Y (i.e. the set of all pairs (a, b ) with a ~X and b ~ Y) is an element of U. Now we shall consider U as a universe with 06 U. An unambiguous set X is a set X such that X EU. According to (U1), for each unambiguous set X, we have X _ U. We can prove:
Proposition 1.1. 7he set N of al! natural numbers, the set Q of all rational numbers and the set R of all real numbers are unambiguous sets. The reader must note that by the previous remarks we have that for any n ~ N, the set R n and the set ~ ( R n) are unambiguous sets.
1.2. The fundamental CLOSG Here we must remember the notion of complete-latticial-ordered-semigroup (CLOSG) introduced by Goguen [ 14]. A CLOSG L is an ordered semigroup with nullelement denoted 0 and with a unit-element denoted 1, so that the order of the semigroup L gives rise to a complete latticial structure over L and 0,1 are the first and last elements relative to this order over L. Now we suppose that the semigroupal operation of L is infinite distributive relative to the union (denoted " V " ) and the intersection (denoted "/~") of the lattice L. A CLOSG L is called distributive if and only if the lattice L is infinite distributive (see Birchoff [1]). A CLOSG L is called with complementarity if and only if there exists a unary operation " - " over L so that, for any unambiguous set J and for any x, x ~ L(j e J) we have:
Vxj=Ax~ j~J
and
~=:x
and
0=1.
(1.1)
jeJ
Proposition 1.2. The real interval [0,1-] is a distributive CLOSG with complementarity, where the order is that of real numbers, and the semigroupal operation is the usual multiplication of real numbers. The complementarity is defined by: ~ = 1 - x for any x in I'0,1].
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Let LI and L2 be two CLOSGs (respectively CLOSGs distributive, CLOSGs with complementarity). We say that they are isomorphic if and only if there exists a one-toone and onto function from LI to L2 so that it is simultaneously an isomorphism of complete lattices and an isomorphism of semigroups (preserving the distributivity and preserving the complementarity respectively). Now we consider a CLOSG L with complementarity and we accept that the following axioms hold: (L I) L is a CLOSG with complementarity; L is isomorphic to [0, 1]. (L2) The subjacent set of L is unambiguous. (L3) L has a topological structure so that the previous isomorphism which is supposed to exist in (L1) is a homeomorphism (here [0, 1] is a topological space with the induced topology from R). The elements of L will be named grads. The grad which corresponds by the isomorphism to a number x from [0, 1-1 will be denoted by x. According to the axioms (L) we can operate with grads in the same manner as with the numbers from [0, 1]. The CLOSG L will be called.fundamental CLOSG.
1.3. Fuzzy sets in the universe U A fuzzy set (in the universe U) is a function from U to the fundamental CLOSG L. We observe that the universe U is a set in the usual sense. Hence the usual properties of the fuzzy sets (seen as L-fuzzy sets [14])must hold. We shall denote by Uj~ j Aj and ~ j ~z Aj the union and the intersection of a given family {Aj }i ~J of fuzzy sets, defined by"
for any x in U. If A is a fuzzy set we shall denote by ,4 the fuzzy set defined by:
A(x)=A(x) for any x in U. A will be called the contrary of the fuzzy set A. The reader must note that the union and the intersection of fuzzy sets are infinitely commutative, infinitely distributive one about the other, idempotent and absorbant operations. Concerning the contrarity the reader must note that De Morgan's laws also hold, i.e. ),=A
and
UAj=~AJ j~d
and
j~d
('/AJ=I,3AJ jeJ
and
0=I,
j~d
where {Aj}jej is a family of fuzzy sets, 0 is the null fuzzy set, I is the one fuzzy set, A is a fuzzy set. It is clear, according to (L1), that the proof of these properties is the same as that given for the homonimous properties in 1-14]. If A is an unambiguous set, we can assimilate it with the fuzzy set defined by
~a(x)=
10 if x ~ A , if x~A.
(1.2)
x~U.
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In the following we shall use A(x) instead of ~a(x), hence there is no distinction between the unambigous set A and the fuzzy set defined by it. It is easy to see that (according to (L2), (U2), (U3)), any fuzzy set is an element of U. Hence a natural question is: "What does "A(A)" mean if A is a fuzzy set?". If A is an unambiguous set the axioms of the classical set theory tell us that " A ( A ) = 0 " because "A ¢ A" (see (1.2)). Now we suppose that all the fuzzy sets considered in the present explanation have the same property, i.e. A(A)=O. Let A and B be two fuzzy sets. We shall say that A is included in B, and we shall denote this by A~_B if and only if A ( x ) < B ( x ) for any x in U. Two fuzzy sets A and B will be called equal if and only if A (x) = Blx) for any x in U. The inclusion is a transitive, non-symmetric a.,d antireflexive relation between fuzzy sets. Let us denote the class of all fuzzy set included by A by L(A). According to a previous remark we have L ( A ) c U. However L(A) can be seen as a fuzzy set defined by"
L(A)(x)=
1 if x6L(A), 0 if xeL(A),
x~U.
According to the present and to the preceding remarks we can reformulate the propositions given as axioms (U) using fuzzy sets instead of sets, L instead of ~ , and the union of fuzzy sets instead of the union of sets. The propositions obtained in this way are true.
1.4. Products of fuzzy sets: Fuzzy relations Let A and B denote ~ ,o fuzzy sets. Their product is the fuzzy set A x B defined by:
(Ax B)(z)={o4(x)"B(y)
if z = (x, y) for some x, y e U, if the contrary holds.
(1.3)
Here "." denotes the semigroupal operation of L. It is clear that, if A and B is unvoid, then A x B is unvoid because, if x, y e U exists so that Aix)~O and B(x)~O, then z = ix, y) e U and A ix )" B(y) = (A x B )(z ) ~ O. We must remark that, generally, if U is not a universe, then A x B has not this property. In fact, if U is the set of all natural numbers, and A, B are L-fuzzy subsets of it, and A, B are different from 0, A x B = 0 because for any x, y~. N the pair (x, y)¢ N. Now, let us return to our universe U. Let A and B be two fuzzy sets. A fuzzy relation between A and B is an element of L(A x B). It is clear that any fuzzy relation is a fuzzy set. Let R, S be two fuzzy relations with R ~ L(A x B) and S ~ L(B x C). The composition of R with S is the fuzzy set given by:
(So R)iz)=
t
V (R(x,t)'R(t,y)) 0
if z=(x,y) for some x,y~U,
teU
if the contrary holds.
(1.4)
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185
Proposition 1.3. S o R is a fuzzy relation between A and C. The composition (1.4)for fuzzy relations is an associative and not a generally commutative operation. Relative to this operation the relation 0 is a null element and the diagonal relation is a unit. Notations. IfA is a fuzzy set, we shall denote by R A the fuzzy relation contained in L(A × I) defined by: RA(Z)={o(X ) if Z=(X,y)for some x , y ~ U , if the contrary holds.
The fuzzy relation S A E L ( I x A) is defined by:
,
(1.5)
.
SA(Z)={o(y ) if z = ( x , y ) f o r some x, y e U , if the contrary holds.
(1.6)
If A, B, C, D are fuzzy sets and A _~C, B ~ D and R ~ L(C x D), let us denote by R[A] and R-~[B] the fuzzy relations SA ~R and R o Sa respectively. We have that R[A] ~_D and R- ~[B] ___C.
Theorem 1.4. l f X , Y are fuzzy sets, S, R, Ri~ L(X x Y), A, B, Ai~ L(X )(i ~ J with J an unambiguous set), then we have: (i) A c B implies REAl c RIB]; (ii) N,,,REA,], (iii) REU, ,A,] = (iv) R ~_S implies R[A] ~_S[A]; (v) = (vi) (vii) O[A] =O: (viii) if T ~ L ( Y xZ), then (T,:
R)[A]=T[R[A]].
2. Classical games and fuzzy games In this section we want to give an intuitive description of the concept of a "fuzzy game". Necessarily this concept must be seen as a special case of a more general concept which unifies the essential properties of a large class of phenomena named usual "games". 2.1. The general concept of a game Neumann and Morgenstern's "Theory of games" (1944) is a mathematical approach to the exchange of values (money) in a given social and political medium. A basic postulate of that theory is" The exchange of the values and the social and political medium are in the same relation as the mechanical movements and their physical medium.
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Accepting this last idea, we extend the area of the concept of a game by saying that a game consists of a set of exchanges of information between two or more partners (players). These exchanges have specific rules which are determined by the objective possibilities of transmitting and receiving information, i.e. by the external medium of the exchanges. By information we understand any object which can be transmitted, identified and received such as money, words, scientific ideas etc. A game can be defined by the following data: (a) the number of partners; (b) each partner must have his individual domain of choices or, more particularly, the number of his "pure strategies" seen as essential and irreducible directions of transmitting and receiving information in the game; (c) for any player must be given a method for deciding if a given exchange of iifformation is preferable to another; this must consist of an individual criterion of choosing the best of two different possible alternatives from the point of view of the individual player. Now, according to this description we must say: (1) From our point of view what becomes of yon Neumann and Morgenstern's concept of a game? (2) What is a fuzzy game?
2.2. The classical model of a two-person game According to von Neumann and Morgenstern's theory, a two-person game consists of the following data: (a') two players denoted 1 and 2 respectively; (b') for any player k, a set ~'.k= ~O'(k)l,''',°k"r(k)~sis given where ~k is called the set of pure strategies of k; (c') for any pair (tr~~, tri2 t2~) from ~1 x ~'.2, there is a unique real number u ~.,~1~ tr!~t called the gain of k. 12 ! In such a game, a possible individual choice of the player k consists of a mixed strategy of k, where by mixed strategy we understand an nk-vector p = (p~,..., P,k) with non-negative components and with ~'~ ~p~= 1. The number Pi is a measure of the importance of the pure strategy tr~k~for the behaviour of k, when the mixed strategy p is chosen in a specified play. The set Sk of all the mixed strategies of k is the domain of the individual choices of k in the given game. An exchange of information in the given two-person game consists of the following facts: player 1 chooses the mixed strategy ~ e $1 and player 2 chooses the mixed strategy ¢~ e $2- is the given information from each of the two players; player I receives his gain 4h(¢ 1, 4 2) and player 2 receives his gain 4~2(¢1, ¢2), where: nl
n2
aj
(2.1)
i=I j=l
with ~k=(¢k,...,~kk) , k = 1, 2 - t h i s is the received information of the two players. It is clear that any possible exchange of information in such a game can be completely described by a pair (~1, ~2)e $1 x $2. Let two possible exchanges of information (¢1, ¢2) and (~1, ~2) be given. We say that (~, ¢2) is preferable to (~, ~2) from the point of view
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187
of the player k, and we denote (¢~, ~2):~(~t, ¢2) if and only if {2.2)
Hence a classical two-person game is also a game according to the general definition given in 2.1. However, an important critical remark must be made on the classical two-person game (i.e. the two-person game in the sense of yon Neumann and Morgenstern): all the elements ~k ~ Sk are equally possible choices o f the player k. There exist many social, political or economic phenomena which are games in our enlarged sense, presented in Section 2.1 and which can be studied satisfactorily with this restriction, but not every game (in the extended sense) has this property. In the following paragraph we shall illustrate this idea. 2.3. A model o f a two-person.h~zzy game
Let us consider an exchange of information between two persons (called players~ named 1 and 2, so that the player k s {1, 2} is an investor who knows the set ~k = {,,% .... : '.~} (nk e N), where tr~k~ signifies the name of an objective (social, political economic...) such that an investment of k is made in order that ,,i _~k~ can be realized legally. Let us consider that any investment consists of a sum ~:f money. If the player/, ,k with the view of O'~g'(i-- 1, . . , nk) we obtain an Ilkdecides to make the investment ~,~ .k vector wk={w~ .... ,w,~) called the strategic composition o.[ k. The global financial possibilities of any player k are known, i.e. an unambiguous set Yk c_ R"' is known where w k ~ Yk if and only if wk is a realizable and legal, strategic composition of k. In fact Ykis the domain of individual choices of the player k. Let us consider that player k ~ {1, 2} supposes that player pc { 1,2j' - { k] will choose the strategic composition wr; hence player k can estimate the gain Fklw t, w z ~ which hc will obtain if he chooses the strategic composition wk. If lhe players are making their choices acting only under the rule of conduct ot'the s~!fest maximal gain, then we obtain a model of the classical concept of a two-person game. The reality shows that in a "game" where the choices are investments, the principle of the safest maximal gain is not the only principle which determines an individual choice of a strategic composition. Another rule of conduct of the behaviour of human beings in making choices is that of the best alternati~,e. What does this mean? In choosing a strategic composition w k from Yk, the player k takes into account the aesthetic, the moral or the psychological (...) aspects of the individual possibility w k. From his point of view not all the individual alternatives are equally possible t (or equally feasible). The grade of membership of an individual alternative w k to the set of feasible alternatives of k is dependent on the behaviour of the partner p~{1, 2}-{k}. E.g. if partner p behaves immorally from the point of view of k, then the rules of the game can force player k to behave in the same manner. Let us suppose that player 2 chooses w 2 in a given play. Then the set of alternatives of player 1 can be seen as a fuzzy set E~.,,: ~_ Y~, where, for any w ~~ 1:1, E~.,,.-'(w ~) is the E.f,. a pacifist k, considers from his subjective point of view, that an investment ~t:'with a little w~ is better than a ~ with a great investment ~ if a] hj means"war industry" but the latter can be more profitable for him than the first choice.
D. Butnariu
188
grade of membership of w I in the set of feasible alternatives of player 1, when player 2 chooses w2. Similarly we can introduce the fuzzy set E2. w, of the feasible alternatives of player 2, when player 1 is considered to choose'the strategic composition w 1. Now, let us remark that there exists a unique fuzzy relation El (respective E2) between Y2 and Y~ (respectively between Y1 and ~ ) such that E I [{ w2}] = E1.,2 for any w2 contained in Y2 (respectively E2[{w~}]=E2.,,,, for any w ~ contained in I"1). In a large class of games the moral, aesthetic and, philosophical (...) principles of the players are known a priori. This means that the fuzzy relations Ek are prescribed for such a game. Let F be a game where ~1, ~2, 1"1,Y2and El, E2 are known. What is a play in such a game ? To say that we must observe that the possibilities of any player in a given state of the game are strongly restricted by the behaviour of the other player, i.e. to choose a wk from Y~, the player k~ {1,2} mus! know if this alternative is or is not possible in the future conjuncture of the game. Precisely, in order to know his own possibilities player k must estimate the future behaviour of the player p6{1, 2} - i k j, the estimation of the future behaviour of p is realizable based on the actual knowledge about how p will behave, k's actual knowledge about how p will behave consists of a set of more or less plausible data 2 about the fact that a specified wPe Yp will be the strategic composition which p will choose in the next step of the game. According to these data vlaver k can define a fuzzy set Ap~L(Yp), so that Ap(wp) is the grade of plausibility (from the point of view of k) of the assertion" ~v,
o,
"w p will be the next choice of the player p in the game F". If the player k considers the assertion as definite and indubitable, then Ap(w p) is exactly I. If k considers that the assertion is impossible, then A,,(w p) is exactly 0. In the other cases Ap(wp) is a grade between 0 and 1. If the fuzzy set Ap is determined by k, he must estimate its own set of possibilities in choosing strategic compositions. We suppose that Ek(Wp, wk)'Ap(w p) is the grade of surety that wk is to be chosen as a response of the choice wp of p. (If wp is considered by k to be the next choice of p with the certainty Ap(wP).). We postulate that the measure of the possibility of the strategic composition wk is given by:
V w
p
eY
[Ek(Wp, wk)'Ap(wP)] =Ek[Ap](wk),
(2.3)
p
i.e. the grade of membership of wk to the, set of the possible choices of k (when the player k estimates the future behaviour of p by Ap) is given by Ek[Ap](wP). 3 The $ fuzzy set Ek[Ap] will be called the set of possibilities of k under the estimation Ap of the future behaviour of his partner p. Now we can give the answer to the ~luestion concerning what a play in F is. A play in F is a pair (w ~, A2; A1, w2) where A k ( k = l , 2 ) a r e fuzzy subsets of Yk) and 2These data can be obtained by a study of the former behaviour of the partner in the game or by a predictive estimation of his future behaviour or by other methods. The rules of the game must specify how these data can be obtained in a regular way. 3The estimation Ap is not necessarily correct. In the mathematical approach of the model, the case of incorrect estimation {based on false data) will be treated.
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Et,[Ak](wP)4:O(k, p= 1,2, K :/:p): here Ak is the estimation of the player p about the future behaviour of k and w~ is a strategic composition with a non-zero grade of membership to the set of possibilities of k under the estimation Ap of the future behaviour of p. A game with two participants, where for any strategic alternative of one of the participants, there exists only a fuzzy set of strategic alternatives of the other which are possible fr,am his point of view, will be called a two-person juzzy game.
2.4. The concept of a (non-fuzzy) N-person game Now we wish to explain how N-person games (N > 3) can be viewed as exchanges of information in the sense of 2.1. It is important to observe that a game with N ( > 3) players is essentially different to a two-person game (see [19]). We consider a classical N-person game being given if the following data are known: (a") the set J = {1, 2,..., N} which elements are called players; (b") for any i~d, the set ~ = {~r~°, ..., cr,,,-"~,of pure strategies of the player i; (c"} a set .Jg ~.0/.(j) of regular coalitions in the game; for any K ~ . ~ the set ~k =~-~i~K~ is ~he set of pure strategies of the coalition K. Let F be a cl:,ssical N-person game. We denote by: 12.4) a~F. K
for any K. SK is called domain of the individual alternatives of the coalition K. As usual we consider {i} ~ X', and we denote Si instead of S.~. For any K ~ ~ and i eK, and s ~ SK a unique number v~(K,s) is known; that is the gain of the player i obtained by his participation in the coalition K, ilK chose the alternative s frc,m :ih- in a given play of F. We define the functions wi : >U x@K ~ , S~--,R (i ~ J) given by the following formula:
wi(K,s)={oi(K's~
if i e K a n d s ~ S if the contrary holds.
They are called gain-function of the game F. define a game in the sense of We are going to prove that F and the family {~i~i~J, ' Section 2.1. For this purpose we note that for an N-person game a play consists, for any player, in two important actions: bargaining the position of the player relative to the regular coalitions of the game and choosing individual alternatives. The first action is considered to be executed by any individual player before any decision about choosing individual alternatives is made. In this way the players of the given game define a partition of the set d. This partition of J will be called a coalitionai structure of the discussed play; its elements r,'ust be contained in ~ and they are modeling the class of simultaneous alliances bargained at the moment of the given play. The second action in the work of the coalitions which are elements of the coalitional structure of the play. It consists of choosing alternatives so that any regular coalition of the coalitional structure chooses an individual alternative from its own set of alternatives. In a given play. an individiaal player i~J cannot generally choose individual alternatives for himself (excepting the special case where ~t'~j is an element of the coalitional structure of the play).
190
D. Butnariu
An exchange of information in such a game can be seen as a pair (R, {sK }R) where R is a relation of equivalence over J such that its classes of equivalence are contained in Jg" and {sr }Rdenote a set of alternatives with K being elements of the canonical partition of R. It is clear that any play in F can be presented as an exchange of information where the equivalence R is given by the coalitional structure of the play and the set of alternatives is defined by the intrinsic choice of the play. This means that in a play of r the mformauon exchanged between the players concerns the alliances and the payments. An exchange of information (R, {sK}a) is preferable (from the point of view of the player i e J) to another exchange of information (R', {s'}g,) if and only if we have: wi(K, st()> ~,~(K , ,,s'~.),
(2.5)
for any K and K' contained in the canonical partition of R and R' respectively. This concept of an N-person game is satisfactory for d mathematical study of a large class of real phenomena. However, an extrinsic criticism can be made. It is based on the following two observations: (1) if a coalitional structure R is bargained in a classical game, then the player i transfers all his rights for individual choices to the coalition K of the coalitional structure, coalition which contains i as a member; this coalition K becomes a lawful decident deciding in the respective play instead of all its members; (2) if the coalitional structure R of a play is bargained, a player i e J is contained only in one coalition K of R. These two conditions are specified for an N-person game in the classical sense (see also [ 19, 20]), but they are not generally verified by all phenomena which are games in the sense of the general concept described in Section 2.1. E.g., in political games, it is usual that a country which is a partner in a given game cannot generally transfer all its decisional rights to a coalition (seen as a representative decident of its members) but a country can be simultaneously a member in many coalitions. "~ 2.5. N-person f u z z y games
In the present section we wish to explain intuitively a model of a game in which the two extrinsic limits of the classical concept of an N-person game which we have shown in the previous section are eliminated. We consider the exchanges of information between N-persons who can emit their information individually or together. If two or more partners emit their information in a correlated manner, we shall say that they form a coalition in the given game. Each coalition is defined by a statute which defines the rights and the duties of a member of this coalition. The statute can be viewed as a conjunction S of logical propositions having as basic variables the names of the players. Let d = {1,...,N} be the set of the players (where N is a natural number greater than 3). To become a member of a given coalition, a player i must define his attitude concerning the statute S of the coalition. If the player i accepts completely the statute S, he becomes a rightful member of the coalition; then we say that the grade of membership of i to the given coalition is 1. If the 4This manner of behaviour is illustrated by the position of the United Kingdom which refuses to transfer its economic decisional rights to the West-European Parliament, but it is a member of EEC and of the Commonwealth in spite of their contradictory requests in many questions.
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player i refuses to accept the statute S, then the grade ofmen~, ership of i to the coalition with statute S will be considered as 0. In a "classic" N-person game as in those described by yon Neumann and Morgenstern, only these two alternatives are possible. We postulate that a partial acceptance of the statute S is also possible; the grade of membership in the coalition of a player i who accepts partially (i.e. in a conditional manner) the statute S of the coalition will be a number from the fundamental C L O S G but different from 0 and from 1. Hence, if the statute S is given, then there exists a fuzzy set A ~ L(J) so that A (i) is the grade of acceptance of i for the statute S. We call A (i) the grade of membership of i to the coalition with the statute S. Mathematically, we say that the fuzzy set A is exactly the coalition with the statute S. Now we remember that in a classical N-person game a coalitional structure is a relation of equivalence whose associated partition is a set of simultaneous coalitions of the given game. If R is such a coalitional structure, we have iRj if and only if the players i and j are members of the same coalition of the game. In fact, a coalitional structure is modeling the bilateral relations between the N players in a given state of the game. It is clear that the notion of coalitional structure is an essential one for any mathematical approach of the concept of an N-person game. What is a coalitional structure of a game with fuzzy coalitions ? A coa|itional structure in a game with fuzzy coalitions can be presented as a fuzzy binary relation R ~ L ( J A J), where Ri is a possible coalition of the game and R(i, j) is the measure (i.e. the grade) of the intensity of the collaboration of the players i and j in making decisions in the game. From the above presentation of a coalitional structure in games with fuzzy coalitions it is clear that it is natural to consider any coalitional structure as a symmetrical relation, i.e. R must have R (i, j) = R (j, i ) for any i, j from J. That is the expression of the fact that any collaboration between two players must be equally reciprocal, i.e. any two players i and j must collaborate one with the other in the same measure, independent of whether i is seen as a collaborator ofj or./is seen as a collaborator of i. It is also natural to consider that the bilateral relation between two players i and k has an influence over the behaviour of the other players in the given game. In many real games the grade of direct bilateral collaboration of two players i and j is greater than the grade of bilateral collaboration of i and j through the agency of any other player k. Mathematically this can be expressed as:
R ( i , j ) > R(i, k ) . R ( k , j ) for any k e J, or equivalently,
R(i,j)>= v R ( i , k ) . R ( k , j ) . k~J
We postulate that this is true for any i and j from J, i.e. we have R : R ~_R, which means that R is a fuzzy transitive relation. An N-person game with regular fuzzy coalitions and with regular fuzzy coalitional structures will be called an N-person fuzzy game. The information in N-person fuzzy games is exchanged by bargaining the coalitional structure of the game and by making
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D. Butnariu
payments at the end of each play. The value paid by a player is dependent on the bargained coalitional structure of the game and the coalitional structure is bargained such that the players" attitude concerning a regular coalition of the game is dependent on the amount which can be obtained by each of them if they become members of the discussed coalition. The future mathematical approach of the problem of N-person fuzzy games must describe the dynamism of the coalitional structures and payments. That is the aim of the last paper of the announced series.
Rd'erences* [1] G. Birchoff, Lattice Theory (Amer. Math. Soc., Colloq. Publ., Providence, RI, 1968). [2.] N. Bourbaki, Th6orie des Ensembles (Hermann, Paris, 1970). [3] D. Butnariu, Fuzzy games and their minimax theorem (in Romanian) St. Cert. Math. 28 (2) (1976) 142160.
[4] D. Butnariu, Equ::ibrium points in fuzzy games (in Romanian) St. Cerc. Math. (to appear). [5] D. Butnariu, (E, L)-Fuzzy topological spaces, Ana|s Sci. Univ. "AI. I. Cuza"-lasi 23 Seria I, Fasc. 1 (1977).
[6] D. Butnariu, L-Fuzzy topologies, Bull. Math. Soc. Sci. Math. R.S.R. 19 [3) (1975) 227-236. [7] D. Butnariu, Three-person fuzzy games (in Romanian) St. Cerc. Math. (2) (1977) 1-10. [8] C.C. Chang, Infinite-valued logic as basis for set theory, Proc. 2nd Int. Congress of Logic, Methodology and Philosophy of Science (North-Holland, Amsterdam, 1964).
[9] G. Ciucu, M. Iosifescu and R. Teodorescu, Theory of Games (in Romanian) (Ed. Tehnica, Bucuresd, [10] [11] [12] [13] [14]
[15] [16] [17] [18] [19]
[20] [21]
1965). P.M. Cohn, Universal Algebra (Harper Row, New York, 1965). A. Frankel and Y. Bar-HiUel, Foundation of Set Theory (North-Holland, Amsterdam, 1958). R. Feron, Ensembles al6atoires flous, C.R. Acad. Sci. Paris, 22 (4) (1976) 903-906. R. Feron, Economie d'6xchange al6atoire floue, C.R. Acad. Sci. Paris, 282 (9) 1379-1382. J.A. Goguen, L-fuzzy sets, J. Math. Anal. Appi. 18 (1967) 145-174. V. Isrr.tescu, Introduction to fixed points theory, Ed. Acad. R.S.R. (1973). A. Kaufmann, Introduction fi ia Th6orie des Ensembles Flous, Vol. 1 IMasson and Co., Paris, 1973). Gr. C. Moisil, Ensembles flous et logiques/t plusieurs valeurs, Centre de Recherches Math6matiques, Universit~ de Montr6al, CRM-286 (Mai 1973). C.V. Negoita and D. Ralescu, Applications of Fuzzy Sets to System Analysis (Birkhauser Verlag, Basel, 1975). J. yon Neumann and O. Morgenstern, Theory of Games and Economic Behaviour (Princeton University Press, 1970). G. Owen, Game Theory (Saunders Co., Philadelphia, 1968). A. Rose and J.B. Rosser, Fragments of many-'valued statement calculi, Trans. Amer. Math. S0c. (1958) 1-53.
[22] E.S. Santos, Maximin, minimax and composite sequential machines, J. Math. Anal. Appl. 24 (1968) 246-259.
[23] L.A. Zadeh, Fuzzy sets, Information and Control 8 (1965) 338-352. [24] L.A. Zadeh, A system theoretic view ofbehaviour modifications, in: Harvey and Wechler (eds.) Beyond the Punitive Society (Wildwood-House, London, 1973). *These references are used in all the series of papers.
FUZZY GAMES: A DESCRIPTION THE CONCEPT
OF
Dan BUTNARIU Str. Tepes Voda Nr. 2, lagsy 6600, Romania
Received January 1977 Revised November i977
O. Introduction The present paper is the first in a series of three papers having the same aim: an heuristical description and a mathematical approach to fuzzy games. The other two papers in this series will be entitled "Two-Person Fuzzy Games" and "N-Person Fuzzy Games" respectively. The first section of the present paper is dedicated to a general presentation of the notions, notations and results on fuzzy sets, which we are going to use later. The notion of"fuzzy set" (into a universe) which is introduced here is analogous, but not the same, with the notion offuzzy set introduced by Zadeh [23]. A great number of results, wellknown in the Universal Theory of Sets (Cohn I10]) can be extended to the theory of fuzzy sets in a universe. The second section of the present paper contains an heuristical discussion of fuzzy games and their relationship to the classical concept of a game, which is the object of the yon Neumann and Morgenstern theory [19]. We hope that the reader familiar with the content of this section will easily understand the intuitive meaning of the mathematical notions and results which we shall explain in the following two papers of the announced series.
1. Fuzzy sets into a universe
1.1. Universes and unambiguous sets The fuzzy set theory explained in the papers of Zadeh 1"23],Goguen 1-14l, Kaufmann [16], has as basic objects the "fuzzy subsets of a usual set". Let a usual set U be given. If U is a set such as the set of all the natural numbers, we observe that the Cartesian product of two fuzzy subsets of it is not generally a fuzzy subset of U. Also a great number of other properties, which appear in the usual set-theory, cannot be extended to fuzzy subsets of a usual set without reference to fuzzy subsets of a usual set other than U. Now we are going to present a notion of a "fuzzy set" which will eliminate a great number of these difficulties. We hope that the reader is familiar with the content of the first and second chapters of !81
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D. Butnariu
[10], i.e. with the axiomatic form of the theory of sets and classes explained here. The basic term of this theory is that of"class". A class A is called a set if and only if there exists another class B so that A is an element of B, i.e., formally, if and only if A 6 B. A function between two classes A and B is a sub-class of the cartesian product A x B, subclass having the usual functional property. A set U is called Universe if and only if the following axioms hold:
(UI) x 6 U {U2) x ~ U
implies
x _ U.
implies ~ ( x ) e U . (U3) x 6 U implies {x,y}~U, for any y~U. (U4)JeU, Fj~uo¢J) implies ~ Fj~U. where ~_, ~(x), [,.)j~j denote the usual inclusion of sets, the Boolean set of x and the union of the J-indexed family, respectively. We denote {x, "y}~ by (x, y). The empty set 0 and the family of all finite sets are universes. According to the axioms (U 3), for any two sets X, Y ~ U, the Cartesian product X x Y (i.e. the set of all pairs (a, b ) with a ~X and b ~ Y) is an element of U. Now we shall consider U as a universe with 06 U. An unambiguous set X is a set X such that X EU. According to (U1), for each unambiguous set X, we have X _ U. We can prove:
Proposition 1.1. 7he set N of al! natural numbers, the set Q of all rational numbers and the set R of all real numbers are unambiguous sets. The reader must note that by the previous remarks we have that for any n ~ N, the set R n and the set ~ ( R n) are unambiguous sets.
1.2. The fundamental CLOSG Here we must remember the notion of complete-latticial-ordered-semigroup (CLOSG) introduced by Goguen [ 14]. A CLOSG L is an ordered semigroup with nullelement denoted 0 and with a unit-element denoted 1, so that the order of the semigroup L gives rise to a complete latticial structure over L and 0,1 are the first and last elements relative to this order over L. Now we suppose that the semigroupal operation of L is infinite distributive relative to the union (denoted " V " ) and the intersection (denoted "/~") of the lattice L. A CLOSG L is called distributive if and only if the lattice L is infinite distributive (see Birchoff [1]). A CLOSG L is called with complementarity if and only if there exists a unary operation " - " over L so that, for any unambiguous set J and for any x, x ~ L(j e J) we have:
Vxj=Ax~ j~J
and
~=:x
and
0=1.
(1.1)
jeJ
Proposition 1.2. The real interval [0,1-] is a distributive CLOSG with complementarity, where the order is that of real numbers, and the semigroupal operation is the usual multiplication of real numbers. The complementarity is defined by: ~ = 1 - x for any x in I'0,1].
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Let LI and L2 be two CLOSGs (respectively CLOSGs distributive, CLOSGs with complementarity). We say that they are isomorphic if and only if there exists a one-toone and onto function from LI to L2 so that it is simultaneously an isomorphism of complete lattices and an isomorphism of semigroups (preserving the distributivity and preserving the complementarity respectively). Now we consider a CLOSG L with complementarity and we accept that the following axioms hold: (L I) L is a CLOSG with complementarity; L is isomorphic to [0, 1]. (L2) The subjacent set of L is unambiguous. (L3) L has a topological structure so that the previous isomorphism which is supposed to exist in (L1) is a homeomorphism (here [0, 1] is a topological space with the induced topology from R). The elements of L will be named grads. The grad which corresponds by the isomorphism to a number x from [0, 1-1 will be denoted by x. According to the axioms (L) we can operate with grads in the same manner as with the numbers from [0, 1]. The CLOSG L will be called.fundamental CLOSG.
1.3. Fuzzy sets in the universe U A fuzzy set (in the universe U) is a function from U to the fundamental CLOSG L. We observe that the universe U is a set in the usual sense. Hence the usual properties of the fuzzy sets (seen as L-fuzzy sets [14])must hold. We shall denote by Uj~ j Aj and ~ j ~z Aj the union and the intersection of a given family {Aj }i ~J of fuzzy sets, defined by"
for any x in U. If A is a fuzzy set we shall denote by ,4 the fuzzy set defined by:
A(x)=A(x) for any x in U. A will be called the contrary of the fuzzy set A. The reader must note that the union and the intersection of fuzzy sets are infinitely commutative, infinitely distributive one about the other, idempotent and absorbant operations. Concerning the contrarity the reader must note that De Morgan's laws also hold, i.e. ),=A
and
UAj=~AJ j~d
and
j~d
('/AJ=I,3AJ jeJ
and
0=I,
j~d
where {Aj}jej is a family of fuzzy sets, 0 is the null fuzzy set, I is the one fuzzy set, A is a fuzzy set. It is clear, according to (L1), that the proof of these properties is the same as that given for the homonimous properties in 1-14]. If A is an unambiguous set, we can assimilate it with the fuzzy set defined by
~a(x)=
10 if x ~ A , if x~A.
(1.2)
x~U.
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D. Butnariu
In the following we shall use A(x) instead of ~a(x), hence there is no distinction between the unambigous set A and the fuzzy set defined by it. It is easy to see that (according to (L2), (U2), (U3)), any fuzzy set is an element of U. Hence a natural question is: "What does "A(A)" mean if A is a fuzzy set?". If A is an unambiguous set the axioms of the classical set theory tell us that " A ( A ) = 0 " because "A ¢ A" (see (1.2)). Now we suppose that all the fuzzy sets considered in the present explanation have the same property, i.e. A(A)=O. Let A and B be two fuzzy sets. We shall say that A is included in B, and we shall denote this by A~_B if and only if A ( x ) < B ( x ) for any x in U. Two fuzzy sets A and B will be called equal if and only if A (x) = Blx) for any x in U. The inclusion is a transitive, non-symmetric a.,d antireflexive relation between fuzzy sets. Let us denote the class of all fuzzy set included by A by L(A). According to a previous remark we have L ( A ) c U. However L(A) can be seen as a fuzzy set defined by"
L(A)(x)=
1 if x6L(A), 0 if xeL(A),
x~U.
According to the present and to the preceding remarks we can reformulate the propositions given as axioms (U) using fuzzy sets instead of sets, L instead of ~ , and the union of fuzzy sets instead of the union of sets. The propositions obtained in this way are true.
1.4. Products of fuzzy sets: Fuzzy relations Let A and B denote ~ ,o fuzzy sets. Their product is the fuzzy set A x B defined by:
(Ax B)(z)={o4(x)"B(y)
if z = (x, y) for some x, y e U, if the contrary holds.
(1.3)
Here "." denotes the semigroupal operation of L. It is clear that, if A and B is unvoid, then A x B is unvoid because, if x, y e U exists so that Aix)~O and B(x)~O, then z = ix, y) e U and A ix )" B(y) = (A x B )(z ) ~ O. We must remark that, generally, if U is not a universe, then A x B has not this property. In fact, if U is the set of all natural numbers, and A, B are L-fuzzy subsets of it, and A, B are different from 0, A x B = 0 because for any x, y~. N the pair (x, y)¢ N. Now, let us return to our universe U. Let A and B be two fuzzy sets. A fuzzy relation between A and B is an element of L(A x B). It is clear that any fuzzy relation is a fuzzy set. Let R, S be two fuzzy relations with R ~ L(A x B) and S ~ L(B x C). The composition of R with S is the fuzzy set given by:
(So R)iz)=
t
V (R(x,t)'R(t,y)) 0
if z=(x,y) for some x,y~U,
teU
if the contrary holds.
(1.4)
Fuzz), games
185
Proposition 1.3. S o R is a fuzzy relation between A and C. The composition (1.4)for fuzzy relations is an associative and not a generally commutative operation. Relative to this operation the relation 0 is a null element and the diagonal relation is a unit. Notations. IfA is a fuzzy set, we shall denote by R A the fuzzy relation contained in L(A × I) defined by: RA(Z)={o(X ) if Z=(X,y)for some x , y ~ U , if the contrary holds.
The fuzzy relation S A E L ( I x A) is defined by:
,
(1.5)
.
SA(Z)={o(y ) if z = ( x , y ) f o r some x, y e U , if the contrary holds.
(1.6)
If A, B, C, D are fuzzy sets and A _~C, B ~ D and R ~ L(C x D), let us denote by R[A] and R-~[B] the fuzzy relations SA ~R and R o Sa respectively. We have that R[A] ~_D and R- ~[B] ___C.
Theorem 1.4. l f X , Y are fuzzy sets, S, R, Ri~ L(X x Y), A, B, Ai~ L(X )(i ~ J with J an unambiguous set), then we have: (i) A c B implies REAl c RIB]; (ii) N,,,REA,], (iii) REU, ,A,] = (iv) R ~_S implies R[A] ~_S[A]; (v) = (vi) (vii) O[A] =O: (viii) if T ~ L ( Y xZ), then (T,:
R)[A]=T[R[A]].
2. Classical games and fuzzy games In this section we want to give an intuitive description of the concept of a "fuzzy game". Necessarily this concept must be seen as a special case of a more general concept which unifies the essential properties of a large class of phenomena named usual "games". 2.1. The general concept of a game Neumann and Morgenstern's "Theory of games" (1944) is a mathematical approach to the exchange of values (money) in a given social and political medium. A basic postulate of that theory is" The exchange of the values and the social and political medium are in the same relation as the mechanical movements and their physical medium.
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D. Butnariu
Accepting this last idea, we extend the area of the concept of a game by saying that a game consists of a set of exchanges of information between two or more partners (players). These exchanges have specific rules which are determined by the objective possibilities of transmitting and receiving information, i.e. by the external medium of the exchanges. By information we understand any object which can be transmitted, identified and received such as money, words, scientific ideas etc. A game can be defined by the following data: (a) the number of partners; (b) each partner must have his individual domain of choices or, more particularly, the number of his "pure strategies" seen as essential and irreducible directions of transmitting and receiving information in the game; (c) for any player must be given a method for deciding if a given exchange of iifformation is preferable to another; this must consist of an individual criterion of choosing the best of two different possible alternatives from the point of view of the individual player. Now, according to this description we must say: (1) From our point of view what becomes of yon Neumann and Morgenstern's concept of a game? (2) What is a fuzzy game?
2.2. The classical model of a two-person game According to von Neumann and Morgenstern's theory, a two-person game consists of the following data: (a') two players denoted 1 and 2 respectively; (b') for any player k, a set ~'.k= ~O'(k)l,''',°k"r(k)~sis given where ~k is called the set of pure strategies of k; (c') for any pair (tr~~, tri2 t2~) from ~1 x ~'.2, there is a unique real number u ~.,~1~ tr!~t called the gain of k. 12 ! In such a game, a possible individual choice of the player k consists of a mixed strategy of k, where by mixed strategy we understand an nk-vector p = (p~,..., P,k) with non-negative components and with ~'~ ~p~= 1. The number Pi is a measure of the importance of the pure strategy tr~k~for the behaviour of k, when the mixed strategy p is chosen in a specified play. The set Sk of all the mixed strategies of k is the domain of the individual choices of k in the given game. An exchange of information in the given two-person game consists of the following facts: player 1 chooses the mixed strategy ~ e $1 and player 2 chooses the mixed strategy ¢~ e $2- is the given information from each of the two players; player I receives his gain 4h(¢ 1, 4 2) and player 2 receives his gain 4~2(¢1, ¢2), where: nl
n2
aj
(2.1)
i=I j=l
with ~k=(¢k,...,~kk) , k = 1, 2 - t h i s is the received information of the two players. It is clear that any possible exchange of information in such a game can be completely described by a pair (~1, ~2)e $1 x $2. Let two possible exchanges of information (¢1, ¢2) and (~1, ~2) be given. We say that (~, ¢2) is preferable to (~, ~2) from the point of view
Fuzzy games
187
of the player k, and we denote (¢~, ~2):~(~t, ¢2) if and only if {2.2)
Hence a classical two-person game is also a game according to the general definition given in 2.1. However, an important critical remark must be made on the classical two-person game (i.e. the two-person game in the sense of yon Neumann and Morgenstern): all the elements ~k ~ Sk are equally possible choices o f the player k. There exist many social, political or economic phenomena which are games in our enlarged sense, presented in Section 2.1 and which can be studied satisfactorily with this restriction, but not every game (in the extended sense) has this property. In the following paragraph we shall illustrate this idea. 2.3. A model o f a two-person.h~zzy game
Let us consider an exchange of information between two persons (called players~ named 1 and 2, so that the player k s {1, 2} is an investor who knows the set ~k = {,,% .... : '.~} (nk e N), where tr~k~ signifies the name of an objective (social, political economic...) such that an investment of k is made in order that ,,i _~k~ can be realized legally. Let us consider that any investment consists of a sum ~:f money. If the player/, ,k with the view of O'~g'(i-- 1, . . , nk) we obtain an Ilkdecides to make the investment ~,~ .k vector wk={w~ .... ,w,~) called the strategic composition o.[ k. The global financial possibilities of any player k are known, i.e. an unambiguous set Yk c_ R"' is known where w k ~ Yk if and only if wk is a realizable and legal, strategic composition of k. In fact Ykis the domain of individual choices of the player k. Let us consider that player k ~ {1, 2} supposes that player pc { 1,2j' - { k] will choose the strategic composition wr; hence player k can estimate the gain Fklw t, w z ~ which hc will obtain if he chooses the strategic composition wk. If lhe players are making their choices acting only under the rule of conduct ot'the s~!fest maximal gain, then we obtain a model of the classical concept of a two-person game. The reality shows that in a "game" where the choices are investments, the principle of the safest maximal gain is not the only principle which determines an individual choice of a strategic composition. Another rule of conduct of the behaviour of human beings in making choices is that of the best alternati~,e. What does this mean? In choosing a strategic composition w k from Yk, the player k takes into account the aesthetic, the moral or the psychological (...) aspects of the individual possibility w k. From his point of view not all the individual alternatives are equally possible t (or equally feasible). The grade of membership of an individual alternative w k to the set of feasible alternatives of k is dependent on the behaviour of the partner p~{1, 2}-{k}. E.g. if partner p behaves immorally from the point of view of k, then the rules of the game can force player k to behave in the same manner. Let us suppose that player 2 chooses w 2 in a given play. Then the set of alternatives of player 1 can be seen as a fuzzy set E~.,,: ~_ Y~, where, for any w ~~ 1:1, E~.,,.-'(w ~) is the E.f,. a pacifist k, considers from his subjective point of view, that an investment ~t:'with a little w~ is better than a ~ with a great investment ~ if a] hj means"war industry" but the latter can be more profitable for him than the first choice.
D. Butnariu
188
grade of membership of w I in the set of feasible alternatives of player 1, when player 2 chooses w2. Similarly we can introduce the fuzzy set E2. w, of the feasible alternatives of player 2, when player 1 is considered to choose'the strategic composition w 1. Now, let us remark that there exists a unique fuzzy relation El (respective E2) between Y2 and Y~ (respectively between Y1 and ~ ) such that E I [{ w2}] = E1.,2 for any w2 contained in Y2 (respectively E2[{w~}]=E2.,,,, for any w ~ contained in I"1). In a large class of games the moral, aesthetic and, philosophical (...) principles of the players are known a priori. This means that the fuzzy relations Ek are prescribed for such a game. Let F be a game where ~1, ~2, 1"1,Y2and El, E2 are known. What is a play in such a game ? To say that we must observe that the possibilities of any player in a given state of the game are strongly restricted by the behaviour of the other player, i.e. to choose a wk from Y~, the player k~ {1,2} mus! know if this alternative is or is not possible in the future conjuncture of the game. Precisely, in order to know his own possibilities player k must estimate the future behaviour of the player p6{1, 2} - i k j, the estimation of the future behaviour of p is realizable based on the actual knowledge about how p will behave, k's actual knowledge about how p will behave consists of a set of more or less plausible data 2 about the fact that a specified wPe Yp will be the strategic composition which p will choose in the next step of the game. According to these data vlaver k can define a fuzzy set Ap~L(Yp), so that Ap(wp) is the grade of plausibility (from the point of view of k) of the assertion" ~v,
o,
"w p will be the next choice of the player p in the game F". If the player k considers the assertion as definite and indubitable, then Ap(w p) is exactly I. If k considers that the assertion is impossible, then A,,(w p) is exactly 0. In the other cases Ap(wp) is a grade between 0 and 1. If the fuzzy set Ap is determined by k, he must estimate its own set of possibilities in choosing strategic compositions. We suppose that Ek(Wp, wk)'Ap(w p) is the grade of surety that wk is to be chosen as a response of the choice wp of p. (If wp is considered by k to be the next choice of p with the certainty Ap(wP).). We postulate that the measure of the possibility of the strategic composition wk is given by:
V w
p
eY
[Ek(Wp, wk)'Ap(wP)] =Ek[Ap](wk),
(2.3)
p
i.e. the grade of membership of wk to the, set of the possible choices of k (when the player k estimates the future behaviour of p by Ap) is given by Ek[Ap](wP). 3 The $ fuzzy set Ek[Ap] will be called the set of possibilities of k under the estimation Ap of the future behaviour of his partner p. Now we can give the answer to the ~luestion concerning what a play in F is. A play in F is a pair (w ~, A2; A1, w2) where A k ( k = l , 2 ) a r e fuzzy subsets of Yk) and 2These data can be obtained by a study of the former behaviour of the partner in the game or by a predictive estimation of his future behaviour or by other methods. The rules of the game must specify how these data can be obtained in a regular way. 3The estimation Ap is not necessarily correct. In the mathematical approach of the model, the case of incorrect estimation {based on false data) will be treated.
Fuzzy games
189
Et,[Ak](wP)4:O(k, p= 1,2, K :/:p): here Ak is the estimation of the player p about the future behaviour of k and w~ is a strategic composition with a non-zero grade of membership to the set of possibilities of k under the estimation Ap of the future behaviour of p. A game with two participants, where for any strategic alternative of one of the participants, there exists only a fuzzy set of strategic alternatives of the other which are possible fr,am his point of view, will be called a two-person juzzy game.
2.4. The concept of a (non-fuzzy) N-person game Now we wish to explain how N-person games (N > 3) can be viewed as exchanges of information in the sense of 2.1. It is important to observe that a game with N ( > 3) players is essentially different to a two-person game (see [19]). We consider a classical N-person game being given if the following data are known: (a") the set J = {1, 2,..., N} which elements are called players; (b") for any i~d, the set ~ = {~r~°, ..., cr,,,-"~,of pure strategies of the player i; (c"} a set .Jg ~.0/.(j) of regular coalitions in the game; for any K ~ . ~ the set ~k =~-~i~K~ is ~he set of pure strategies of the coalition K. Let F be a cl:,ssical N-person game. We denote by: 12.4) a~F. K
for any K. SK is called domain of the individual alternatives of the coalition K. As usual we consider {i} ~ X', and we denote Si instead of S.~. For any K ~ ~ and i eK, and s ~ SK a unique number v~(K,s) is known; that is the gain of the player i obtained by his participation in the coalition K, ilK chose the alternative s frc,m :ih- in a given play of F. We define the functions wi : >U x@K ~ , S~--,R (i ~ J) given by the following formula:
wi(K,s)={oi(K's~
if i e K a n d s ~ S if the contrary holds.
They are called gain-function of the game F. define a game in the sense of We are going to prove that F and the family {~i~i~J, ' Section 2.1. For this purpose we note that for an N-person game a play consists, for any player, in two important actions: bargaining the position of the player relative to the regular coalitions of the game and choosing individual alternatives. The first action is considered to be executed by any individual player before any decision about choosing individual alternatives is made. In this way the players of the given game define a partition of the set d. This partition of J will be called a coalitionai structure of the discussed play; its elements r,'ust be contained in ~ and they are modeling the class of simultaneous alliances bargained at the moment of the given play. The second action in the work of the coalitions which are elements of the coalitional structure of the play. It consists of choosing alternatives so that any regular coalition of the coalitional structure chooses an individual alternative from its own set of alternatives. In a given play. an individiaal player i~J cannot generally choose individual alternatives for himself (excepting the special case where ~t'~j is an element of the coalitional structure of the play).
190
D. Butnariu
An exchange of information in such a game can be seen as a pair (R, {sK }R) where R is a relation of equivalence over J such that its classes of equivalence are contained in Jg" and {sr }Rdenote a set of alternatives with K being elements of the canonical partition of R. It is clear that any play in F can be presented as an exchange of information where the equivalence R is given by the coalitional structure of the play and the set of alternatives is defined by the intrinsic choice of the play. This means that in a play of r the mformauon exchanged between the players concerns the alliances and the payments. An exchange of information (R, {sK}a) is preferable (from the point of view of the player i e J) to another exchange of information (R', {s'}g,) if and only if we have: wi(K, st()> ~,~(K , ,,s'~.),
(2.5)
for any K and K' contained in the canonical partition of R and R' respectively. This concept of an N-person game is satisfactory for d mathematical study of a large class of real phenomena. However, an extrinsic criticism can be made. It is based on the following two observations: (1) if a coalitional structure R is bargained in a classical game, then the player i transfers all his rights for individual choices to the coalition K of the coalitional structure, coalition which contains i as a member; this coalition K becomes a lawful decident deciding in the respective play instead of all its members; (2) if the coalitional structure R of a play is bargained, a player i e J is contained only in one coalition K of R. These two conditions are specified for an N-person game in the classical sense (see also [ 19, 20]), but they are not generally verified by all phenomena which are games in the sense of the general concept described in Section 2.1. E.g., in political games, it is usual that a country which is a partner in a given game cannot generally transfer all its decisional rights to a coalition (seen as a representative decident of its members) but a country can be simultaneously a member in many coalitions. "~ 2.5. N-person f u z z y games
In the present section we wish to explain intuitively a model of a game in which the two extrinsic limits of the classical concept of an N-person game which we have shown in the previous section are eliminated. We consider the exchanges of information between N-persons who can emit their information individually or together. If two or more partners emit their information in a correlated manner, we shall say that they form a coalition in the given game. Each coalition is defined by a statute which defines the rights and the duties of a member of this coalition. The statute can be viewed as a conjunction S of logical propositions having as basic variables the names of the players. Let d = {1,...,N} be the set of the players (where N is a natural number greater than 3). To become a member of a given coalition, a player i must define his attitude concerning the statute S of the coalition. If the player i accepts completely the statute S, he becomes a rightful member of the coalition; then we say that the grade of membership of i to the given coalition is 1. If the 4This manner of behaviour is illustrated by the position of the United Kingdom which refuses to transfer its economic decisional rights to the West-European Parliament, but it is a member of EEC and of the Commonwealth in spite of their contradictory requests in many questions.
Fuzzy games
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player i refuses to accept the statute S, then the grade ofmen~, ership of i to the coalition with statute S will be considered as 0. In a "classic" N-person game as in those described by yon Neumann and Morgenstern, only these two alternatives are possible. We postulate that a partial acceptance of the statute S is also possible; the grade of membership in the coalition of a player i who accepts partially (i.e. in a conditional manner) the statute S of the coalition will be a number from the fundamental C L O S G but different from 0 and from 1. Hence, if the statute S is given, then there exists a fuzzy set A ~ L(J) so that A (i) is the grade of acceptance of i for the statute S. We call A (i) the grade of membership of i to the coalition with the statute S. Mathematically, we say that the fuzzy set A is exactly the coalition with the statute S. Now we remember that in a classical N-person game a coalitional structure is a relation of equivalence whose associated partition is a set of simultaneous coalitions of the given game. If R is such a coalitional structure, we have iRj if and only if the players i and j are members of the same coalition of the game. In fact, a coalitional structure is modeling the bilateral relations between the N players in a given state of the game. It is clear that the notion of coalitional structure is an essential one for any mathematical approach of the concept of an N-person game. What is a coalitional structure of a game with fuzzy coalitions ? A coa|itional structure in a game with fuzzy coalitions can be presented as a fuzzy binary relation R ~ L ( J A J), where Ri is a possible coalition of the game and R(i, j) is the measure (i.e. the grade) of the intensity of the collaboration of the players i and j in making decisions in the game. From the above presentation of a coalitional structure in games with fuzzy coalitions it is clear that it is natural to consider any coalitional structure as a symmetrical relation, i.e. R must have R (i, j) = R (j, i ) for any i, j from J. That is the expression of the fact that any collaboration between two players must be equally reciprocal, i.e. any two players i and j must collaborate one with the other in the same measure, independent of whether i is seen as a collaborator ofj or./is seen as a collaborator of i. It is also natural to consider that the bilateral relation between two players i and k has an influence over the behaviour of the other players in the given game. In many real games the grade of direct bilateral collaboration of two players i and j is greater than the grade of bilateral collaboration of i and j through the agency of any other player k. Mathematically this can be expressed as:
R ( i , j ) > R(i, k ) . R ( k , j ) for any k e J, or equivalently,
R(i,j)>= v R ( i , k ) . R ( k , j ) . k~J
We postulate that this is true for any i and j from J, i.e. we have R : R ~_R, which means that R is a fuzzy transitive relation. An N-person game with regular fuzzy coalitions and with regular fuzzy coalitional structures will be called an N-person fuzzy game. The information in N-person fuzzy games is exchanged by bargaining the coalitional structure of the game and by making
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payments at the end of each play. The value paid by a player is dependent on the bargained coalitional structure of the game and the coalitional structure is bargained such that the players" attitude concerning a regular coalition of the game is dependent on the amount which can be obtained by each of them if they become members of the discussed coalition. The future mathematical approach of the problem of N-person fuzzy games must describe the dynamism of the coalitional structures and payments. That is the aim of the last paper of the announced series.
Rd'erences* [1] G. Birchoff, Lattice Theory (Amer. Math. Soc., Colloq. Publ., Providence, RI, 1968). [2.] N. Bourbaki, Th6orie des Ensembles (Hermann, Paris, 1970). [3] D. Butnariu, Fuzzy games and their minimax theorem (in Romanian) St. Cert. Math. 28 (2) (1976) 142160.
[4] D. Butnariu, Equ::ibrium points in fuzzy games (in Romanian) St. Cerc. Math. (to appear). [5] D. Butnariu, (E, L)-Fuzzy topological spaces, Ana|s Sci. Univ. "AI. I. Cuza"-lasi 23 Seria I, Fasc. 1 (1977).
[6] D. Butnariu, L-Fuzzy topologies, Bull. Math. Soc. Sci. Math. R.S.R. 19 [3) (1975) 227-236. [7] D. Butnariu, Three-person fuzzy games (in Romanian) St. Cerc. Math. (2) (1977) 1-10. [8] C.C. Chang, Infinite-valued logic as basis for set theory, Proc. 2nd Int. Congress of Logic, Methodology and Philosophy of Science (North-Holland, Amsterdam, 1964).
[9] G. Ciucu, M. Iosifescu and R. Teodorescu, Theory of Games (in Romanian) (Ed. Tehnica, Bucuresd, [10] [11] [12] [13] [14]
[15] [16] [17] [18] [19]
[20] [21]
1965). P.M. Cohn, Universal Algebra (Harper Row, New York, 1965). A. Frankel and Y. Bar-HiUel, Foundation of Set Theory (North-Holland, Amsterdam, 1958). R. Feron, Ensembles al6atoires flous, C.R. Acad. Sci. Paris, 22 (4) (1976) 903-906. R. Feron, Economie d'6xchange al6atoire floue, C.R. Acad. Sci. Paris, 282 (9) 1379-1382. J.A. Goguen, L-fuzzy sets, J. Math. Anal. Appi. 18 (1967) 145-174. V. Isrr.tescu, Introduction to fixed points theory, Ed. Acad. R.S.R. (1973). A. Kaufmann, Introduction fi ia Th6orie des Ensembles Flous, Vol. 1 IMasson and Co., Paris, 1973). Gr. C. Moisil, Ensembles flous et logiques/t plusieurs valeurs, Centre de Recherches Math6matiques, Universit~ de Montr6al, CRM-286 (Mai 1973). C.V. Negoita and D. Ralescu, Applications of Fuzzy Sets to System Analysis (Birkhauser Verlag, Basel, 1975). J. yon Neumann and O. Morgenstern, Theory of Games and Economic Behaviour (Princeton University Press, 1970). G. Owen, Game Theory (Saunders Co., Philadelphia, 1968). A. Rose and J.B. Rosser, Fragments of many-'valued statement calculi, Trans. Amer. Math. S0c. (1958) 1-53.
[22] E.S. Santos, Maximin, minimax and composite sequential machines, J. Math. Anal. Appl. 24 (1968) 246-259.
[23] L.A. Zadeh, Fuzzy sets, Information and Control 8 (1965) 338-352. [24] L.A. Zadeh, A system theoretic view ofbehaviour modifications, in: Harvey and Wechler (eds.) Beyond the Punitive Society (Wildwood-House, London, 1973). *These references are used in all the series of papers.