aa r X i v : . [ m a t h . GN ] J u l EQUILIBRIUM UNDER UNCERTAINTY WITH FUZZY PAYOFF
TARAS RADUL
Institute of Mathematics, Casimirus the Great University of Bydgoszcz, Poland;Department of Mechanics and Mathematics, Ivan Franko National University ofLviv, Universytettska st., 1. 79000 Lviv, Ukraine.e-mail: [email protected]
Key words and phrases:
Non-additive measures, equilibrium under uncer-tainty, possibility capacity, necessity capacity, fuzzy integral, t-norm[MSC 2020]28E10,91A10,52A01,54H25
Abstract.
This paper studies n-player games where players beliefs abouttheir opponents behaviour are capacities (fuzzy measures, non-additive prob-abilities). The concept of an equilibrium under uncertainty was introducedby J.Dow and S.Werlang (1994) for two players and was extended to n-playergames by J.Eichberger and D.Kelsey (2000). Expected utility (payoff function)was expressed by Choquet integral. The concept of an equilibrium under un-certainty with expected utility expressed by Sugeno integral were consideredby T.Radul (2018). We consider in this paper an equilibrium with expectedutility expressed by fuzzy integral generated by a continuous t-norm which isa natural generalization of Sugeno integral. Introduction
The classical Nash equilibrium theory is based on fixed point theory and wasdeveloped in frames of linear convexity. The mixed strategies of a player are prob-ability (additive) measures on a set of pure strategies. But an interest to Nashequilibria in more general frames is rapidly growing in last decades. For instance,Aliprantis, Florenzano and Tourky [1] work in ordered topological vector spaces,Luo [19] in topological semilattices, Vives [30] in complete lattices. Briec and Hor-vath [3] proved existence of Nash equilibrium point for B -convexity and MaxPlusconvexity.We can use additive measures only when we know precisely probabilities of allevents considered in a game. However, it is not a case in many modern economicmodels. The decision theory under uncertainty considers a model when proba-bilities of states are either not known or imprecisely specified. Gilboa [14] andSchmeidler [27] axiomatized expectations expressed by Choquet integrals attachedto non-additive measures called capacities (fuzzy measures), as a formal approachto decision-making under uncertainty. Dow and Werlang [6] used this approach fortwo players game where belief of each player about a choice of the strategy by theother player is a capacity. They introduced some equilibrium notion for such gamesand proved its existence. This result was extended onto games with arbitrary finitenumber of players in [11]. Another interesting approach to the games in convexcapacities with pay-off functions expressed by Choquet integrals can be find in [20].An alternative to so-called Choquet expected utility model is the qualitativedecision theory. The corresponding expected utility is expressed by Sugeno integral.This approach was widely studied in the last decade ([8],[9],[5],[26]). Sugeno integralchooses a median value of utilities which is qualitative counterpart of the averagingoperation by Choquet integral. The equilibrium notion from [6] and [11] for a game with expected payoff functiondefined by Sugeno integral was considered in [24]. The sets of pure strategies arearbitrarily compacta. Let us remark that in [6] and [11] attention was restrictedto convex capacities which play an important role in Choquet expected utilitytheory. There are two important classes of capacities in the qualitative decisiontheory, namely possibility and necessity capacities which describe optimistic andpessimistic criteria [8]. The existence of equilibrium expressed by possibility (ornecessity) capacities is proved in [24]. Since the spaces of possibility and necessitycapacities have no natural linear convex structure, some non-linear convexity wasused.Let us remark that the equilibrium notion from [6] supposed that players are al-lowed to form non-additive beliefs about opponent’s decision and answer with purestrategies. Another approach was considered [18] and [15] for games with Choquetpayoff where players are allowed to form non-additive beliefs about opponent’s de-cision but also to play their mixed non-additive strategies expressed by capacities.The same approach for games with Sugeno payoff was considered in [23]. Gameswith strategies expressed by possibility capacities were recently considered by Hosniand Marchioni [16]. They considered payoff functions represented by Choquet in-tegral and Sugeno integral. Games with expected payoff functions represented byfuzzy integrals generated by the maximum operation and some continuous triangu-lar norm (a partial case is the Sugeno integral which is generated by the maximumand the minimum operations) were considered in [25].We consider the equilibrium notion from [6] with beliefs expressed by possibility(or necessity) capacities and payoff functions represented by fuzzy integrals in thispaper. We prove existence of such equilibrium.2.
Games with non-additive beliefs
By compactum we mean a compact Hausdorff space. In what follows, all spacesare assumed to be compacta except for R and maps are assumed to be continuous.Let A be a subset of X . By Cl A we denote the closure of A in X . By F ( X ) wedenote the family of all closed subsets of X .We need the definition of capacity on a compactum X . We follow a terminologyof [21]. A function ν : F ( X ) → [0 ,
1] is called an upper-semicontinuous capacity on X if the three following properties hold for each closed subsets F and G of X :1. ν ( X ) = 1, ν ( ∅ ) = 0,2. if F ⊂ G , then ν ( F ) ≤ ν ( G ),3. if ν ( F ) < a , then there exists an open set O ⊃ F such that ν ( B ) < a for eachcompactum B ⊂ O .If F is a one-point set we use a simpler notation ν ( a ) instead ν ( { a } ). A capacity ν is extended in [21] to all open subsets U ⊂ X by the formula ν ( U ) = sup { ν ( K ) | K is a closed subset of X such that K ⊂ U } .It was proved in [21] that the space M X of all upper-semicontinuous capacitieson a compactum X is a compactum as well, if a topology on M X is defined bya subbase that consists of all sets of the form O − ( F, a ) = { c ∈ M X | c ( F ) < a } ,where F is a closed subset of X , a ∈ [0 , O + ( U, a ) = { c ∈ M X | c ( U ) > a } ,where U is an open subset of X , a ∈ [0 , M X simply capacities.A capacity ν ∈ M X for a compactum X is called a necessity (possibility) capac-ity if for each family { A t } t ∈ T of closed subsets of X (such that S t ∈ T A t is a closedsubset of X ) we have ν ( T t ∈ T A t ) = inf t ∈ T ν ( A t ) ( ν ( S t ∈ T A t ) = sup t ∈ T ν ( A t )). QUILIBRIUM UNDER UNCERTAINTY WITH FUZZY PAYOFF 3 (See [31] for more details.) We denote by M ∩ X ( M ∪ X ) a subspace of M X con-sisting of all necessity (possibility) capacities. Since X is compact and ν is upper-semicontinuous, ν ∈ M ∩ X iff ν satisfy the simpler requirement that ν ( A ∩ B ) =min { ν ( A ) , ν ( B ) } .If ν is a capacity on a compactum X , then the function κX ( ν ), that is definedon the family F ( X ) by the formula κX ( ν )( F ) = 1 − ν ( X \ F ), is a capacity aswell. It is called the dual capacity (or conjugate capacity ) to ν . The mapping κX : M X → M X is a homeomorphism and an involution [21]. Moreover, ν is anecessity capacity if and only if κX ( ν ) is a possibility capacity. This implies inparticular that ν ∈ M ∪ X iff ν satisfy the simpler requirement that ν ( A ∪ B ) =max { ν ( A ) , ν ( B ) } . It is easy to check that M ∩ X and M ∪ X are closed subsets of M X .For each capacity ν we consider an upper semicontinuous function [ ν ] : X → [0 , x ∈ X to ν ( x ). Observe that for a possibility capacity ν ∈ M ∪ X and a closed set F ⊂ X we have ν ( F ) = max { ν ( x ) | x ∈ F } , and ν is completelydetermined by its values on singletons. It means that ν is completely determinedby the function [ ν ]. Conversely, each upper semicontinuous function f : X → I with max f = 1 determines a possibility capacity ( f ) ∈ M ∪ X by the formula( f )( F ) = max { f ( x ) | x ∈ F } , for a closed subset F of X .For a continuous map of compacta f : X → Y we define the map M f : M X → M Y by the formula
M f ( ν )( A ) = ν ( f − ( A )) where ν ∈ M X and A ∈ F ( Y ). Themap M f is continuous. In fact, this extension of the construction M defines thecapacity functor in the category of compacta and continuous maps. The categoricaltechnics are very useful for investigation of capacities on compacta (see [21] for moredetails). We try to avoid the formalism of category theory in this paper, but wefollow the main ideas of such approach.3. Tensor products of capacities
For a continuous map of compacta f : X → Y we define the map M f : M X → M Y by the formula
M f ( ν )( A ) = ν ( f − ( A )) where ν ∈ M X and A ∈ F ( Y ). Themap M f is continuous. In fact, this extension of the construction M defines thecapacity functor in the category of compacta and continuous maps. The categoricaltechnics are very useful for investigation of capacities on compacta (see [21] for moredetails). We try to avoid the formalism of category theory in this paper, but wefollow the main ideas of such approach.The tensor product operation of probability measures is well known and veryuseful partially for investigation of the spaces of probability measures on compacta(see for example Chapter 8 from [12]). General categorical definition of tensorproduct for any functor was given in [2]. Applying this definition to the capacityfunctor we obtain that a tensor product of capacities on compacta X and X is acontinuous map ⊗ : M X × M X → M ( X × X )such that for each i ∈ { , } we have M ( p i ) ◦ ⊗ = pr i where p i : X × X → X i ,pr i : M X × M X → M X i are the corresponding projections.A tensor product for capacities was introduced in [18]. This definition is based onthe capacity monad structure. An explicit formula for evaluating tensor productof capacities was given in [24] omitting the formalism of category theory. For µ ∈ M X , µ ∈ M X and B ∈ F ( X × X ) we put µ ⊗ µ ( B ) = max { t ∈ [0 , | µ ( { x ∈ X | µ ( p (( { x } × X ) ∩ B )) ≥ t } ) ≥ t } . The problem of multiplication of capacities was deeply considered in the possi-bility theory and its application to the game theory and the decision making theory
TARAS RADUL where the term joint possibility distribution is used. A standard choice of a jointpossibility distribution is based on the minimum operation (see for example [16]).For µ ∈ M ∪ X , µ ∈ M ∪ X and ( x, y ) ∈ X × X we put[ µ ⊗ µ ]( x, y ) = [ µ ]( x ) ∧ [ µ ]( y ) . (Let us remind that by [ ν ] we denote the density of a possibility capacity ν .) Thecoincidence of both definitions is proved in [25]. So, the difference is only in terms.It is also shown in [25] that the notion of tensor product coincides with the methodof aggregation of capacities considered in [7].A more general approach is also used where the minimum operation is changedby any t-norm (see for example [10]). We will use this definition in our paper butwe prefer the term tensor product.Remind that triangular norm ∗ is a binary operation on the closed unit interval[0 ,
1] which is associative, commutative, monotone and s ∗ s for each s ∈ [0 , ∗ and consider a tensor product ⊛ : M ∪ X × M ∪ X → M ∪ ( X × X ) generated by ∗ defined as follows. For possibility capacities µ ∈ M ∪ X , µ ∈ M ∪ X and ( x, y ) ∈ X × X we put[ µ ⊛ µ ]( x, y ) = [ µ ]( x ) ∗ [ µ ]( y ) . We also can generalize the above mentioned formula from [24]. For capacities µ ∈ M X , µ ∈ M X and B ∈ F ( X × X ) we put µ e ⊛ µ ( B ) = sup { t ∈ [0 , | µ ( { x ∈ X | µ ( p (( { x } × X ) ∩ B )) ≥ t } ) ∗ t } . It was shown in [25] that both definitions coincide in the class of possibility capac-ities.It was noticed in [18] that we can extend the definition of tensor product toany finite number of factors by induction. It is also true for the tensor productgenerated by a continuous norm.
Lemma 1.
The map e ⊛ : M ( X ) × · · ·× M ( X n ) → M ( X × · · ·× X n ) is continuous. Lemma 2.
Let X i be a compactum, A i ∈ F ( X i ) and µ i ∈ M X i such that µ i ( X i \ A i ) = 0 for each i ∈ { , . . . , n } . Then e ⊛ ni =1 µ i ( Q ni =1 X i \ Q ni =1 A i ) = 0 .Proof. Consider the case n = 2. Let B any compact subset of ( X × X ) \ ( A × A ).For any t > K t = { x ∈ X | µ ( p (( { x } × X ) ∩ B )) ≥ t } . If x ∈ A , then p (( { x } × X ) ∩ B ) ⊂ X \ A , hence x / ∈ K t . Thus K t ⊂ X \ A andwe obtain µ e ⊛ µ ( B ) = 0.The general case could be obtained by induction. (cid:3) Equilibrium under uncertainty with fuzzy payoff
Let us describe the fuzzy integral generated by a continuous t-norm ∗ withrespect to a capacity µ ∈ M X . Such integrals are called t-normed integrals andwere studied in [32], [33], [28] and [25]. Denote ϕ t = ϕ − ([ t, + ∞ )) for each ϕ ∈ C ( X, [0 , t ∈ [0 , ∗ and a function f ∈ C ( X, [0 , Z ∨∗ X f dµ = max { µ ( f t ) ∗ t | t ∈ [0 , } . Let us remark that existence of maximum in the previous definition follows fromthe semicontinuity of the capacity µ . If we consider a partial case when t-norm isthe minimum operation, we obtain the Sugeno integral.Now, we are going to introduce notion of equilibrium under uncertainty for gameswhere belief of each player about a choice of the strategy by the other player is a QUILIBRIUM UNDER UNCERTAINTY WITH FUZZY PAYOFF 5 capacity. We follow definitions and denotation from [11] with the only differencethat we use the t-normed integral for expected payoff instead the Choquet integral.Our approach is a generalization of [24] where expected payoff was expressed bythe Sugeno integral.We consider a n -players game p : X = Q ni =1 X i → [0 , n with compact Haus-dorff spaces of strategies X i . We assume that the function p is continuous. Thecoordinate function p i : X → [0 ,
1] we call payoff function of i -th player. For i ∈ { , . . . , n } we denote by X − i = Q j = i X j the set of strategy combinations whichplayers other than i could choose. For x ∈ X the corresponding point in X − i wedenote by x − i . In contrast to standard game theory, beliefs of i -th player aboutopponents behaviour are represented by non-additive measures (or capacities) on X − i .Let ∗ be a continuous t-norm. For i ∈ { , . . . , n } we consider the expected payofffunction P ∗ i : X i × M X − i → [0 ,
1] defined as follows P ∗ i ( x i , ν ) = R ∨∗ X − i p x i i dν wherethe function p x i i : X − i → [0 ,
1] is defined by the formula p x i i ( x − i ) = p i ( x i , x − i ), x i ∈ X i and ν ∈ M X − i .We are going to prove continuity of P ∗ i . We will need some notations and atechnical lemma from [24]. Let f : X × Y → [0 ,
1] be a function. Consider any x ∈ X and t ∈ [0 , f x ≤ t = { y ∈ Y | f ( x, y ) ≤ t } . We also will useanalogous notations f x ≥ t , f x
Let f : X × Y → [0 , be a continuous function on the product X × Y of compacta X and Y . Then for each x ∈ X , t ∈ [0 , and δ > there existsan open neighborhood O of x such that f z ≤ t ⊂ f x
The map P ∗ i is continuous.Proof. Consider any x ∈ X i and ν ∈ M X − i and put P ∗ i ( x, ν ) = s ∈ [0 , f = p i . Consider any δ >
0. Since t-norm ∗ is uniformly continuous, we can choose ε > | r ∗ l − p ∗ t | < δ/ r , l , p , t ∈ [0 ,
1] such that | r − p | < ε and | l − t | < ε . Choose n ∈ N such that 1 /n ≤ ε and put t i = i/n for i ∈ { , . . . , n } .By Lemma 3 applied to the continuous function f : X i × X − i → [0 ,
1] we canchoose a neighborhood O of x such that for each z ∈ O we have f z ≥ t i ⊂ f x ≥ t i − for each i ∈ { , . . . , n } . Put V = { ν ∈ M ( X − i ) | ν ( f x ≥ t i ) < ν ( f x ≥ t i ) + ε } for each i ∈ { , . . . , n } , then V is a neighborhood of ν .Consider any ( z, ν ) ∈ O × V and t ∈ [0 , i ∈ { , . . . , n − } such that t ∈ [ t i , t i +1 ]. Since t i ≤ t and | t i − t | < ε , we have ν ( f z ≥ t ) ∗ t ≤ ν ( f z ≥ t i ) ∗ t i + δ/ i = 0, we have ν ( f z ≥ t ) ∗ t < δ/ ≤ s + δ/ i >
0. Since z ∈ O and ∗ is monotone, we have ν ( f z ≥ t i ) ∗ t i + δ/ ≤ ν ( f x ≥ t i − ) ∗ t i − + δ/ ν ( f x ≥ t i − ) ≤ ν ( f x ≥ t i − ), we have ν ( f x ≥ t i − ) ∗ t i − + δ/ ≤ ν ( f x ≥ t i − ) ∗ t i − + δ/ ≤ s + δ/ ν ( f x ≥ t i − ) > ν ( f x ≥ t i − ), we have | ν ( f x ≥ t i − ) − ν ( f x ≥ t i − ) | , because ν ∈ V . Thenwe obtain ν ( f x ≥ t i − ) ∗ t i − + δ/ ≤ ν ( f x ≥ t i − ) ∗ t i − + δ/ ≤ s + δ/ ν ( f z ≥ t ) ∗ t ≤ s + δ/ t ∈ [0 , P i ( z, ν ) < s + δ for each ( z, ν ) ∈ O × V .Choose d ∈ [0 ,
1] such that ν ( f x ≤ d ) ∗ d = max { µ ( f x ≤ t ) ∗ t | t ∈ [0 , } = s .By Lemma 3 there exists a neighborhood O of x such that for each z ∈ O wehave f z ≤ d − ε ⊂ f x
4. So, we obtain that P ∗ i ( z, ν ) > s − δ for each( z, ν ) ∈ O × V and the map P ∗ i is continuous. (cid:3) For ν i ∈ M ( X − i ) denote by R ∗ i ( ν i ) = { x ∈ X i | P ∗ i ( x, ν i ) = max { P ∗ i ( z, ν i ) | z ∈ X i }} the best response correspondence of player i given belief ν i . The set R i ( ν i ) is welldefined and compact by Lemma 4.A belief system ( ν , . . . , ν n ), where ν i ∈ M ( X − i ), is called an equilibrium underuncertainty respect ∗ with fuzzy payoff if for all i we have ν i ( X − i \ Q j = i R ∗ j ( ν j )) = 0.The main goal of this paper is to prove the existence of such equilibrium wherecorresponding belief system consist of possibility measures. Since the space M ∪ X has no natural linear convex structure, we will use some another natural convexitystructure on the space of capacities described in the next section.5. A convexity on the space of capacities
Consider a compactum X . There exists a natural lattice structure on M X defined as follows ν ∨ µ ( A ) = max { ν ( A ) , µ ( A ) } and ν ∧ µ ( A ) = min { ν ( A ) , µ ( A ) } foreach closed subset A ⊂ X and ν , µ ∈ M X (see for instance Theorem 7.1 in [4]).Thelattice
M X has a greatest element and a a least element defined as µ X ( A ) = 1 foreach A = ∅ , µ X ( ∅ ) = 0 and µ X ( A ) = 0 for each A = X , µ X ( X ) = 1.Following [34] we call a family C ⊂ F ( X ) a convexity on a compactum X if C isstable for intersection and contains X and the empty set. Elements of C are called C -convex (or simply convex). Let us remark that this definition is different from thedefinition of convexity in [29] where the abstract convexity theory is covered fromthe axioms to applications in different areas. We consider here only closed convexsets and we do not consider the condition that union of each nested subfamily of C is in C . In fact considered in this paper convexities have such property, but we donot need it for our purposes. So, we use the simpler definition from [34].A convexity C on X is called T (normal) if for each disjoint C , C ∈ C thereexist S , S ∈ C such that S ∪ S = X , C ∩ S = ∅ and C ∩ S = ∅ .The main goal of this section is to choose an appropriate normal convexity onthe compactum M ∪ X . There is no natural linear structure on M ∪ X , so, we cannot use classical linear convexity. The idempotent max-plus convexity on the spaceof homogeneous possibility (or necessity) capacities was considered in [13]. Sincewe deal with fuzzy integrals defined by the maximum operation and a t-norm ∗ , itseems to be natural to consider the idempotent max- ∗ convexity. But we do notknow if such convexity is normal. So, we consider here a coarser convexity on M ∪ X used in [24] which follows from a general categorical approach developed in [22] and[23].For ν , µ ∈ M X we denote [ ν, µ ] = { α ∈ M X | ν ∧ µ ≤ α ≤ ν ∨ µ } . It is easyto see that [ ν, µ ] is a closed subset of M X . We consider on the compactum M ∪ X the convexity C ∪ X = { C ∩ M ∪ X | C ∈ C X } . It is easy to see that a closed subset A of M ∪ X belongs to the family C ∪ X if and only if [ V A, W A ] ∩ M ∪ X = A . Let usremark that W A ∈ M ∪ X but we can not state it about V A .The following lemma was proved in [24]. Lemma 5. [24]
The convexity C X is normal. QUILIBRIUM UNDER UNCERTAINTY WITH FUZZY PAYOFF 7 The main result
By a multimap (set-valued map) of a set X into a set Y we mean a map F : X → Y . We use the notation F : X ⊸ Y . If X and Y are topological spaces, thena multimap F : X ⊸ Y is called upper semi-continuous (USC) provided for eachopen set O ⊂ Y the set { x ∈ X | F ( x ) ⊂ O } is open in X . It is well-known thata multimap between compacta X and Y with closed values is USC iff its graph isclosed in X × Y .Let F : X ⊸ X be a multimap. We say that a point x ∈ X is a fixed point of F if x ∈ F ( x ). The following counterpart of Kakutani theorem for abstract convexityis a partial case of Theorem 3 from [34]. Theorem 1. [34]
Let C be a normal convexity on a compactum X such that allconvex sets are connected and F : X ⊸ X is a USC multimap with values in C \{∅} .Then F has a fixed point. The convexity C ∪ X is normal. We need also connectedness to apply the previoustheorem. A stronger statement was proved in [24]. Lemma 6. [24]
Each element of the convexity C ∪ X is path connected. We use definitions and notations from Section 4. The following theorem is ageneralization of Theorem 3 from [24] and arguments in the proofs are the same.But, for sake of completeness we give here a complete proof.
Theorem 2.
Let ⋆ and ∗ be two continuous t-norms. There exists ( µ , . . . , µ n ) ∈ M ∪ ( X ) × · · · × M ∪ ( X n ) such that ( µ ∗ , . . . , µ ∗ n ) is an equilibrium under uncertaintyrespect ⋆ with fuzzy payoff, where µ ∗ i = ⊛ j = i µ j .Proof. For each i ∈ { , . . . , n } consider a multimap γ i : Q nj =1 M ∪ ( X j ) ⊸ M ∪ ( X i )defined as follows γ i ( µ , . . . µ n ) = { µ ∈ M ∪ ( X i ) | µ ( X i \ R ⋆i ( µ ∗ i )) = 0 } . It followsfrom the definition of topology on M ∪ ( X i ) that γ i ( µ , . . . µ n ) is a closed subsetof M ∪ ( X i ) for each ( µ , . . . µ n ) ∈ Q nj =1 M ∪ ( X j ). Consider ν ∈ M ∪ ( X i ) definedas follows ν ( A ) = 1 if A ∩ R i ( µ ∗ i ) = ∅ and ν ( A ) = 0 otherwise. Then we have γ i ( µ , . . . µ n ) = [ µ X i , ν ] ∩ M ∪ ( X i ), hence γ i ( µ , . . . µ n ) ∈ C ∪ X i .Define a multimap γ : Q nj =1 M ∪ ( X j ) ⊸ Q nj =1 M ∪ ( X j ) by the formula γ ( µ , . . . ,µ n ) = Q ni =1 γ i ( µ , . . . , µ n ). Let us show that γ is USC. Consider any pair ( µ, ν ) ∈ Q nj =1 M ∪ ( X j ) × Q nj =1 M ∪ ( X j ) such that ν / ∈ γ ( µ ). Then there exists i ∈ { , . . . , n } and a compactum K ⊂ X i \ R ⋆i ( µ ∗ i ) such that ν i ( K ) >
0. Put O ν { α ∈ Q nj =1 M ( X j ) | α i ( K ) > } . Then O ν is an open neighborhood of ν . It follows from Lemma 4 andcontinuity of tensor product that there exists an open neighborhood O µ of µ suchthat for each α ∈ O µ we have R ⋆i ( α ∗ i ) ∩ K = ∅ . Hence for each ( α, β ) ∈ O µ × O ν we have β / ∈ γ ( α ) and γ is USC.We consider on Q nj =1 M ∪ ( X j ) the family C = { Q ni =1 C i | C i ∈ C ∪ X i } . It iseasy to see that C forms a normal convexity on a compactum Q nj =1 M ∪ ( X j ) suchthat all convex sets are connected. Then by Theorem 1 γ has a fixed point µ =( µ , . . . µ n ) ∈ Q nj =1 M ∪ ( X j ). Let us show that ( µ ∗ , . . . , µ ∗ n ) is an equilibrium underuncertainty respect ⋆ . Consider any i ∈ { , . . . , n } . Then µ i ( X i \ R ⋆i ( µ ∗ i )) = 0. Wehave by Lemma 2 µ ∗ i ( Q j = i X i \ Q j = i R ⋆j ( µ ∗ j )) = 0. (cid:3) We can define on M ∩ X a convexity C ∩ X = { C ∩ M ∩ X | C ∈ C X } . The home-omorphism κX : M ∪ X → M ∩ X is an isomorphism of convex structures C ∪ X and C ∩ X (more precisely C ∈ C ∪ X iff κX ( C ) ∈ C ∩ X ). Hence, using the same argumentsas before, we obtain the following theorem. TARAS RADUL
Theorem 3.
Let ⋆ and ∗ be two continuous t-norms. There exists ( µ , . . . , µ n ) ∈ M ∩ ( X ) × · · · × M ∩ ( X n ) such that ( µ ∗ , . . . , µ ∗ n ) is an equilibrium under uncertaintywith fuzzy payoff respect ⋆ , where µ ∗ i = ⊛ j = i µ j . Some examples
As we remark before, our result is a generalisation of results obtained in [25]. Inthis section we consider some examples which demonstrate that this generalizationis not trivial and distinguish our approach between approaches considered before.
Example 1.
Consider the 2-person game in pure strategies u : { a, b }×{ a, b } → R ,where u ( a, a ) = 1 / , u ( a, b ) = 0 , u ( b, a ) = 0 , u ( b, b ) = 1 / and u ( a, a ) = 0 , u ( a, b ) = 1 / , u ( b, a ) = 1 / , u ( b, b ) = 0 . Choose ν ∈ M ∪ ( { a, b } ) defined by itsdensity [ ν ]( a ) = 1 and [ ν ]( b ) = 1 / . Let us show that the pair ( ν, ν ) is an equilibriumunder uncertainty respect t-norm ∧ (by ∧ we denote the minimum operation on [0 , .We have P ∧ ( a, ν ) = max { ν ( f t ) ∧ t | t ∈ [0 , } = 1 ∧ / / where f ( a ) = 1 / and f ( b ) = 0 . We also have P ∧ ( b, ν ) = max { ν ( g t ) ∧ t | t ∈ [0 , } = 1 / ∧ / / where g ( a ) = 0 and g ( b ) = 1 / . Hence R ∧ ( ν ) = { a, b } . Analogously we can checkthat R ∧ ( ν ) = { a, b } . Hence ( ν, ν ) is an equilibrium under uncertainty respect ∧ .Now, consider another well-known continuous t-norm · (multiplication). Thenwe have P · ( a, ν ) = max { ν ( f t ) · t | t ∈ [0 , } = 1 · / / and P · ( b, ν ) =max { ν ( g t ) · t | t ∈ [0 , } = 1 / · / / . Hence R · ( ν ) = { a } and ( ν, ν ) is notan equilibrium under uncertainty respect · .This example demonstrate that equilibrium depends on t-norm, so, our general-ization is not trivial. Another approach to equilibrium is considered in [25] where players are allowedto play their mixed non-additive strategies expressed by capacities and expectedpayoff functions are represented by fuzzy integral. Let us describe this constructionin more detail.We consider a game u : Z = Q ni =1 Z i → [0 , n with compact Hausdorff spacesof pure strategies Z , . . . , Z n and continuous payoff functions u i : Q ni =1 Z i → [0 , ⋆ and ∗ be two t-norms. We will extend the game u : Z = Q ni =1 Z i → [0 , n toa game in mixed strategies eu : Q ni =1 M ∪ Z i → [0 , n using the integral generatedby t-norm ⋆ and the tensor product generated by t-norm ∗ .We define expected payoff functions eu i : Q nj =1 M ∪ Z j → [0 ,
1] by the formula eu i ( ν , . . . , ν n ) = Z ∨ ⋆X u i d ( ν ⊛ · · · ⊛ ν n )for ( ν , . . . , ν n ) ∈ Q nj =1 M ∪ Z j . We are looking for Nash equilibrium for such game.There exists a trivial solution of the problem of existence of Nash equilibrium.We can consider the natural order on M ∪ Z i . Then each M ∪ Z i contains the greatestelement µ i defined by the formula µ i ( A ) = ( , A = ∅ , , A = ∅ for A ∈ F ( Z i ). It is easy to see that ( µ , . . . , µ n ) is a Nash equilibrium point. (Infact, the main attention in [25] is paid to the model, when the goal of each player isto minimize his expected payoff function. Since M ∪ Z i does not contain the smallestelement, existence of Nash equilibrium is not trivial for such games.) The followingexample shows that it is not the case for the equilibrium under uncertainty. Example 2.
Consider the 2-person game in pure strategies u : { a, b }×{ a, b } → R ,where u ( a, a ) = 1 , u ( a, b ) = 0 , u ( b, a ) = 1 / , u ( b, b ) = 1 / and u ( a, a ) = 0 , QUILIBRIUM UNDER UNCERTAINTY WITH FUZZY PAYOFF 9 u ( a, b ) = 1 , u ( b, a ) = 1 / , u ( b, b ) = 1 / . Then the capacity ν ∈ M ∪ ( { a, b } ) defined by its density [ ν ]( a ) = 1 and [ ν ]( b ) = 1 is the greatest element of M ∪ ( { a, b } ) .Evidently the pair ( ν, ν ) is an Nash equilibrium for the game with expected payofffunctions eu i (we take ∧ as both t-norms in the defining of payoff ).On the other hand we have P ∧ ( a, ν ) = max { ν ( f t ) ∧ t | t ∈ [0 , } = 1 ∧ where f ( a ) = 1 and f ( b ) = 0 . We also have P ∧ ( b, ν ) = max { ν ( g t ) ∧ t | t ∈ [0 , } =1 / ∧ / where g ( a ) = 1 / and g ( b ) = 1 / . Hence R ∧ ( ν ) = { a } . Hence ( ν, ν ) is not an equilibrium under uncertainty respect ∧ . However we have the following proposition.
Proposition 1.
Let ( f, g ) : X × Y → [0 , be a game in pure strategies withcompact X and Y and continuous payoff functions f and g . If a pair of capacities ( ν, µ ) ∈ M ∪ ( X ) × M ∪ ( Y ) is an equilibrium under uncertainty respect a t-norm ⋆ ,then ( ν, µ ) is a point of Nash equilibrium of the corresponding game in capacitieswith payoff functions ef and eg defined by fuzzy integral respect ⋆ and tensor productgenerated by any t-norm ∗ .Proof. Suppose the contrary ( ν, µ ) is not a point of Nash equilibrium. We can as-sume, without loss of generality, that there exists ν ′ ∈ M ∪ ( X ) such that ef ( ν ′ , µ ) >ef ( ν, µ ). Let ν be the maximal element of M ∪ ( X ) (it means that ν ( A ) = 1 foreach closed non-empty subset A of X or equivalently [ ν ]( x ) = 1 for each x ∈ X ).It is easy to see that ef ( ν ′ , µ ) ≤ f ( ν , µ ), hence we have ef ( ν , µ ) > ef ( ν, µ ). Wealso have ef ( ν , µ ) = max { ν ⊛ µ ( f t ) ⋆ t | t ∈ [0 , } = max { max { [ ν ]( x ) ∗ [ µ ]( y ) | ( x, y ) ∈ f t } ⋆ t | t ∈ [0 , } = max { max { [ µ ]( y ) | ( x, y ) ∈ f t } ⋆ t | t ∈ [0 , } . Hencethere exist t ∈ [0 ,
1] and ( x , y ) ∈ f t such that [ µ ]( y ) ⋆ t > ef ( ν, µ ). Since ν ∈ M ∪ ( X ), there exists x ∈ X such that [ ν ]( x ) = 1. Then we have ef ( ν, µ ) =max { max { [ ν ]( x ) ∗ [ µ ]( y ) | ( x, y ) ∈ f t } ⋆ t | t ∈ [0 , } ≥ max { max { [ µ ]( y ) | ( x , y ) ∈ f t } ⋆ t | t ∈ [0 , } = P ⋆ ( x , µ ) and P ⋆ ( x , µ ) ≥ [ µ ]( y ) ⋆ t > ef ( ν, µ ) ≥ P ⋆ ( x , µ ).Hence x / ∈ R ⋆ ( µ ). Since ν ( { x } ) = 1, we obtain a contradiction. (cid:3) References [1] C.D.Aliprantis, M.Florencano, R.Tourky
General equilibrium analisis in ordered topologicalvector spaces,
J. Math. Econom. 40 (2004) 247–269.[2] T.Banakh, T.Radul
F-Dugundji spaces, F-Milutin spaces and absolute F-valued retracts,
Topology Appl. (2015), 34–50.[3] W.Briec, Ch.Horvath
Nash points, Ku Fan inequality and equilibria of abstract economies inMax-Plus and B -convexity, J. Math. Anal. Appl. (2008), 188–199.[4] G.L.OBrien, W.Verwaat
How subadditive are subadditive capacities?,
Comment. Math. Univ.Carolinae (1994), 311–324.[5] A. Chateauneuf, M. Grabisch, A. Rico, Modeling attitudes toward uncertainty through the useof the Sugeno integral , Journal of Mathematical Economics (2008) 1084–1099.[6] J.Dow, S.Werlang, Nash equilibrium under Knightian uncertainty: breaking down backwardinduction ,J Econ. Theory (1994) 205–224.[7] D.Dubois, H.Fargier, A.Rico, Sugeno Integrals and the Commutation Problem , (2018) In:15th International Conference on Modeling Decisions for Artificial Intelligence (MDAI 2018), 15October 2018 - 18 October 2018 (Palma de Mallorca, Spain).[8] D.Dubois, H.Prade, R.Sabbadin,
Qualitative decision theory with Sugeno integrals , arxiv.org1301.7372 (2014)[9] D. Dubois, J.-L. Marichal, H. Prade, M. Roubens, R. Sabbadin,
The use of the discreteSugeno integral in decision making: a survey , Internat. J. Uncertainty, Fuzziness Knowledge-Based Systems (5) (2001) 539-561.[10] D. Dubois, H. Prade, Fuzzy logics and the generalized modus ponens revisited. , Cyberneticsand Systems (1984) 293-331.[11] J.Eichberger, D.Kelsey, Non-additive beliefs and strategic equilibria , Games Econ Behav (2000) 183–215.[12] V.V.Fedorchuk, V.V.Filippov, General topology. Fundamental constructions, Moscow, 1988. [13] T. Flaminio, L. Godo, E. Marchioni, Geometrical aspects of possibility measures on finitedomain MV-clans , Soft Comput (2000) 1863–1873.[14] I.Gilboa, Expected utility with purely subjective non-additive probabilities , J. of MathematicalEconomics (1987) 65–88.[15] D.Glycopantis, A.Muir, Nash equilibria with Knightian uncertainty; the case of capacities ,Econ. Theory (2008) 147–159.[16] H.Hosni, E.Marchioni, Possibilistic randomisation in strategic-form games , InternationalJournal of Approximate Reasoning (2019) 204–225.[17] E.P.Klement, R.Mesiar and E.Pap.
Triangular Norms . Dordrecht: Kluwer. 2000.[18] R.Kozhan, M.Zarichnyi,
Nash equilibria for games in capacities , Econ. Theory (2008)321–331.[19] Q.Luo, KKM and Nash equilibria type theorems in topological ordered spaces , J. Math. Anal.Appl. (2001) 262–269.[20] M.Marinacci,
Ambiguous Games , Games and Economic Behavior (2000) 191–219.[21] O.R.Nykyforchyn, M.M.Zarichnyi, Capacity functor in the category of compacta , Mat.Sb. (2008) 3–26.[22] T.Radul,
Convexities generated by L-monads , Applied Categorical Structures (2011) 729–739.[23] T.Radul, Nash equilibrium for binary convexities , Topological Methods in Nonlinear Analysis (2016) 555–564.[24] T.Radul, Equilibrium under uncertainty with Sugeno payoff , Fuzzy Sets and sytems (2018) 64–70.[25] T.Radul,
Games in possibility capacities with payoff expressed by fuzzy integral , Fuzzy Setsand systems (submitted).[26] A. Rico, M. Grabisch, Ch. Labreuchea, A. Chateauneuf
Preference modeling on totally or-dered sets by the Sugeno integral , Discrete Applied Mathematics (2005) 113–124.[27] D.Schmeidler,
Subjective probability and expected utility without additivity , Econometrica (1989) 571–587.[28] F. Suarez, Familias de integrales difusas y medidas de entropia relacionadas , Thesis, Univer-sidad de Oviedo, Oviedo (1983).[29] M.van de Vel,
Theory of convex strutures , North-Holland, 1993.[30] X. Vives,
Nash equilibrium with strategic complementarities , J. Math. Econom. 19 (1990)305–321.[31] Zhenyuan Wang, George J.Klir
Generalized measure theory , Springer, New York, 2009.[32] S. Weber
Decomposable measures and integrals for archimedean t-conorms , J. Math. Anal.Appl. (1984), 114–138.[33] S. Weber
Two integrals and some modified versions - Critical remarks , Fuzzy Sets and Sys-tems (1986), 97–105.[34] A.Wieczorek The Kakutani property and the fixed point property of topological spaces withabstract convexity,
J. Math. Anal. Appl.168