Decision making via generalized Bajraktarevi\'c means

We define decision-making functions which arise from studying the multidimensional generalization of the weighted Bajraktarevi\'c means. It allows a nonlinear approach to optimization problems. These functions admit several interesting (from the point of view of decision-making) properties, for example, delegativity (which states that each subgroup of decision-makers can aggregate their decisions and efforts), casuativity (each decision affects the final outcome except two trivial cases) and convexity-type properties. Beyond establishing the most important properties of such means, we solve their equality problem, we introduce a notion of synergy and characterize the null-synergy decision-making functions of this type.


Introduction
In game theory the mathematical models for decision-making create challenging and important problems binding computer science, economy, mathematics and psychology.In one of such models there is a set D of all possible decisions and a finite number n of players (decision makers) with their individual nonnegative weights (efforts) and decisions.Obviously, the sum of all weights cannot be zero and thus (as a vector) it belongs to the set W n := [0, +∞) n \ {(0, . . ., 0)}.The issue is to aggregate all the individual decisions with the corresponding weights to one (common) decision.For this purpose, we need the notion of an aggregation function on D, which is defined to be a map For instance, when the set D of decisions is a convex subset of a linear space X, then the weighted arithmetic mean A, which is defined as is a well-known aggregation function.Further examples for an aggregation function are as follows (see for example [11]): (1) The Primacy Effect D P E : W(D) → D is defined by D P E (x, λ) := x i λ i = 0 and λ j = 0 for all j ∈ {1, . . ., i − 1} .
(3) The First Dominating Decision D F DD : W(D) → D is given by D F DD (x, λ) := x i λ i = max(λ) and λ j < λ i for all j ∈ {1, . . ., i − 1} .All functions listed in ( 1)-( 4) are reflexive, eliminative, nullhomogeneous in the weights but not symmetric (see the relevant definitions below).Furthermore, they are all conservative (or selective), which means that the aggregated decision is always one of the individual ones.For a detailed study of (nonweighted) conservative aggregation functions, we refer the reader to the recent study by Couceiro-Devillet-Marichal [4] and Devillet-Kiss-Marichal [9].In many settings, D is an infinite set which often refers to the position of the players in a space before the game.An aggregation function unites the positions of all the players into one.An individual nonnegative weight measures the impact of the decision of the corresponding players to the final outcome.In order to introduce plausible and natural properties for aggregation functions, we introduce the concept of decision-making functions on an arbitrary set D. For this aim, we adopt the notion of weighted means (which were defined on an interval) from the paper [16] to our more general setting.An aggregation function M : W(D) → D is called a decision-making function (on D) if it satisfies the following five conditions: (i) M is reflexive: For all x ∈ D and λ ∈ R + , we have M(x, λ) = x.
(ii) M is nullhomogeneous in the weights: For all n ∈ N, (x 1 , . . ., x n ) ∈ D n , (λ 1 , . . ., λ n ) ∈ W n , and t ∈ R + , we have and for all permutations σ of {1, . . ., n}, we have (iv) M is eliminative or neglective: For all n ≥ 2, (x 1 , . . ., x n ) ∈ D n and (λ 1 , . . ., λ n ) ∈ W n with λ 1 = 0, we have We also introduce the concept of the effort function, which is aiming to aggregate the individual weights (efforts) into one positive number: A function E : W(D) → R + is called an effort function (on D) if it satisfies the following five conditions: (i) E is reflexive in the weights: For all x ∈ D and λ ∈ R + , we have E(x, λ) = λ.
(ii) E is homogeneous in the weights: For all n ∈ N, (x 1 , . . ., x n ) ∈ D n , (λ 1 , . . ., λ n ) ∈ W n , and t ∈ R + , we have One can easily see that the map α : is an effort function, which we call the arithmetic effort function.
The symmetry property of decision-making and effort functions means that there is no distinction between players and also their order is irrelevant for the decision.This property has a far-reaching consequences especially for conservative functions, as it determines the anty-symmetric preference relation on D by x ≻ y : ⇐⇒ x = M((x, y), (1, 1)) (cf.[8] for details).It was proved experimentally that this relation cannot be generalized to multivariable choice; this phenomena is known as a decoy effect (see for example Huber-Payne-Puto [12]).
The nullhomogeneity of M and the homogeneity of E in the weights states that if the weights are scaled by the same factor, then the decision remains unchanged and the effort is scaled by the same factor.The meaning of the elimination principle is that players with zero weight do not affect the decision and the effort.One can easily check that the arithmetic mean is a decision-making function over any convex subset of a linear space.
Similarly, the properties of effort functions and applying the delegativity of E yield which completes the proof of the lemma.
The aim of this paper is to present a construction of a broad class of decision-making and effort functions which arise from studying so-called weighted Bajraktarević means.Despite of the analytical background of this paper, we are convinced that our construction could provide useful models for game theoretical and for decision-making problems.

Observability and conical convexity
A subset S of a linear space X over R is called a ray if S = R x := R + x := {λx | λ ∈ R + } holds for some nonzero element x ∈ X.If y ∈ R x , then the unique positive numbers λ for which y = λx holds will be denoted by [y : x].
A set S is called a cone if it is the union of rays of X.One can see that S is a cone precisely if it is closed under multiplication by positive scalars.The cone generated by the set S -denoted by cone(S) -is the smallest cone containing S. It is clear that cone(S) = R + S = x∈S R x .
A subset S of X over R is called observable (from the origin) if the rays generated by two distinct elements of S are disjoint.Note that for an observable subset S, every ray contained in cone(S) intersects S at exactly one point.Moreover, the family {R x | x ∈ S} is a partition of cone(S).Due to this fact, for every observable set S one can define the projection along rays (briefly the ray projection) π S : cone(S) → S as follows: For every x ∈ S and y ∈ R x , we have π S (y) = x.As a matter of fact, observability of S is not only sufficient but also necessary to define such map.
We say that a function f : D → X is observable if it is injective and has an observable image.Analogously to the previous setup, we define π f := π f (D) .The extended inverse of f , denoted by f (−1) : cone(f (D)) → D is defined by We say that a subset S of X is conically convex if cone(S) is a convex set and convex hull of S does not contain the origin.Note that conical convexity is a weaker property than convexity: every convex set which does not contain zero is conically convex, the converse implication is not true.
Hereafter an observable function f : D → X such that f (D) is conically convex is called admissible.
In the next result we give a characterization of admissibility for functions with 2dimensional range.
Theorem 2.1.Let f = (f 1 , f 2 ) : I → R 2 be continuous.Then f is admissible if and only if, for all distinct elements x, y ∈ I, f (x) and f (y) are linearly independent, that is, Proof.Assume first that f is admissible but (2.1) is not valid for some x, y ∈ I with x = y.Then the two vectors f (x) and f (y) are linearly dependent.Thus, there exist In the case when αβ < 0, we get that the rays generated by f (x) and f (y) are the same, which contradicts the observability.
which contradicts the property that the convex hull of f (I) does not contain the origin.Assume now that (2.1) holds for all (x, y) ∈ ∆(I) := {(x, y) | x, y ∈ I, x < y}.This immediately shows that f (I) is observable.
Since ∆(I) is a convex set and f is continuous, therefore the determinant is either positive on ∆(I) or negative on ∆(I).We may assume that it is positive everywhere on ∆(I).Then the pair (f 1 , f 2 ) is a so-called Chebyshev system on I.According to [3, Theorem 2], there exist two real constants α and β such that αf 1 + βf 2 is positive over the interior of I and thus it is nonnegative over I.This implies that the curve f (I) is contained in the closed half plain Consider now the unit circle and project the curve f (I) into it.Then, by the continuity of the projection mapping, the projection of the curve is an arc of the unit circle which is contained in P .It is obvious that the conical hull of the curve f (I) and the conical hull of the arc are identical.On the other hand, the property that the arc is contained in the half plain P , implies that its conical hull is convex.This shows the conical hull of f (I) is also convex.
The origin cannot be contained in the conical hull of f (I) because then there were two distinct points of f (I) belonging to the boundary line {(u, v) ∈ R 2 | αu + βv = 0} such that the origin were in the segment connecting them.This contradicts the property that every two distinct points of f (I) are linearly independent.

Generalized Bajraktarević means and their properties
Given an admissible function f : D → X, we can define the (weighted) generalized Bajraktarević mean where x = (x 1 , . . ., x n ) ∈ D n are the entries and λ = (λ 1 , . . ., λ n ) ∈ W n are the corresponding weights.We also define which we call the effort function associated to B f .Intuitively D is the space of all possible decisions, x is a vector of all considered decisions, and λ is the effort of players invested in the option.In this way B f (x, λ) is the decision derived from the possible decisions and their weights.Lemma 3.1.Let f : D → X be an admissible function.Then, for all n ∈ N and (x, λ) ∈ D n × W n , there exists exactly one pair (u, η) Proof.As f (D) is conically convex, we know that Thus there exists η ∈ R + such that Moreover, as f (D) is observable, the constant η is uniquely determined by x and λ.Now, as f is injective, there exists exactly one u ∈ D such that which is trivially equivalent to (3.1).The last assertion simply follows from the definition of B f (x, λ) and β f (x, λ).Now B f is the equivalent (or aggregated) decision, β f is the equivalent (or aggregated) effort.In fact β f , can be considered as the amount of goods we need to invest into a single decision to be irrelevant between the diversed an the aggregated situation.This property is much more transparent in view of Theorem 4.1.
Theorem 3.2.Let f : D → X be an admissible function.Then B f is a decision-making function and β f is an effort function on D.
Proof.Fix n ∈ N and (x, λ) ∈ D n × W n .In view of Lemma 3.1 both B f (x, λ) and β f (x, λ) depends on its arguments implicitly via the sum n k=1 λ k f (x k ).This immediately implies that B f and β f are both symmetric, reductive and eliminative.The nullhomogeneity of B f is obvious in view of the definition of f (−1) = f −1 • π f .The reflexivity properties of B f and β f are immediate consequences of the equalities Now we only need to verify the homogeneity of β f in the weights.To this end, take additionally t ∈ R + .By the definition of β f and the nullhomogeneity of B f , we have which completes the proof.
is admissible and we have the equalities Proof.First we show that f = (f 1 , f 2 ) is admissible.In view of Theorem 2.1, it is sufficient to show that (2.1) holds for all distinct elements x, y of I. Indeed, which is nonzero by our assumptions.
To prove the identity (3.2), let n ∈ N and a pair (x, λ) ∈ I n × W n be fixed.As f (B f (x, λ)) and n i=1 λ i f (x i ) are on the same ray, we have Using that f 1 /f 2 is invertible and applying its inverse side-by-side, we obtain the desired equality (3.2).The equality A classical particular case of the above theorem is when f 1 and f 2 are power functions on I = R + : f 1 (x) = x p and f 2 (x) = x q , where p, q ∈ R with p = q.Then (3.4) which are called the Gini mean and the Gini effort function of parameter (p, q) where p = q (cf.[10]).For the case p = q, let f 1 (x) := x p ln(x) and f 2 (x) := x p .Then we have Then, these equalites and Lemma 3.1 imply which then yields (1.1).
The following result could be derived from the above theorem, but we shall provide a direct and short proof for it.
The latter corollary can be also rewritten in the matrix form.
where "•" stands for the coordinate-wise multiplication of the elements of W m .

4.2.
Casuativity.We say that a decision making function M on D is casuative if for all (x, λ) ∈ W(D) and for all pair (y, µ) ∈ D × (0, +∞), we have Casuativity is somehow opposite to conservativity.Namely there holds the following easy-to-see lemma.It is also reasonable to define weak casuativity in the case when the (⇐) implication of (4.3) holds.Observe that there are number of weakly casuative decision making functions which are not casuative, for example decision making functions which are induced by a preference relation.Proof.As B f is delegative it is sufficient to show that (4.3) holds for all (x, λ) ∈ D × R + .The (⇐) part is then trivial.To prove the converse implication let x, y ∈ D and λ, µ ∈ (0, +∞) such that B f (x, y), (λ, µ) = B f (x, λ) = x.Thus, by Lemma 3.1, there exists η ∈ R + such that λf (x)+µf (y) = ηf (x).Consequently, as µ = 0 we get f (y) = η−λ µ f (x).Now admissibility of f implies 0 / ∈ conv f (X) and therefore η−λ µ > 0. Then observability of f yields y = x.This property plays an important role in the definition of effort function.Proof.Let (x, λ) ∈ W(D), set m := B f (x, λ) = B g (x, λ) and assume that d := β g (x, λ) − β f (x, λ) = 0. Without loss of generality, we may assume that d > 0.

Convexity induced by admissible functions
For an admissible function f Observe that if I ⊂ R is an interval and f = (f 1 , f 2 ) : I → R 2 satisfies conditions of Theorem 3.3 then the mean-value property for a Bajraktarević mean can be written as B f (x, λ) ∈ J for every subinterval J ⊆ I, n ∈ N and (x, λ) ∈ J n × W n .This yields that every subinterval of I is f -convex.Furthermore, in view of continuity of f , the converse implication is also valid.
The first assertion characterizes f -convexity in terms of standard convexity.Proof.Assume first that S is f -convex.It suffices to show that for every n ∈ N and a pair (x, λ) ∈ S n × W n we have ξ : Conversely, if cone(f (S)) is convex then, as it is positively homogeneous we obtain r := n i=1 λ i f (x i ) ∈ cone(f (S)) for every n ∈ N and (x, λ) ∈ S n × W n .By the definition of projection we obtain π f (r) ∈ f (S) and therefore Now we show that f -convex sets admit two very important properties of convex sets.Lemma 5.2.Let f : D → X be an admissible function.Then the class of f -convex subsets of D is closed with respect to intersection and chain union.
Proof.Let S be an arbitrary family of f -convex subsets of D. Fix n ∈ N, and (x, λ) ∈ S n × W n .Then for every S ∈ S we have x ∈ S n and, in view f -convexity of S, we also have B f (x, λ) ∈ S. Therefore B f (x, λ) ∈ S which shows that S is f -convex.Now take a chain Q of f -convex sets.Take n ∈ N and (x, λ) ∈ ( Q) n × W n arbitrarily.Then, by the chain property of Q, there exists Applying this lemma, for every admissible function f : D → X and S ⊂ D we define f -convex hull of S as the smallest f -convex subset of D containing S and denote it by conv f (S).In the next lemma we show that, similarly to the ordinary convex hull, this definition can be also expressed as a set of all possible combinations of elements in S.
Lemma 5.3.Let f : D → X be an admissible function and S ⊆ D. Then Proof.Denote the set on the right-hand-side of the latter equality by T .The inclusion T ⊆ conv f (S) is the obvious implication of f -convexity of the hull.To prove the converse inclusion, we need to show that T is f -convex.
Let m ∈ N, y = (y 1 , . . ., y m ) ∈ T m and ν ∈ W m be arbitrary.By the definition of T , in view of reduction priciple and Corollary 4.3 there exists n ∈ N, a vector x ∈ S n and Λ = (λ (1) ) for all i ∈ {1, . . ., m}.Now, Corollary 4.3 implies that there exists µ ∈ R m + such that (4.2) holds.In particular for which shows that T is f -convex, indeed.
We now establish the fundamental relationship between the f -convex hull in D and the standard convex hull in X.
Proof.Observe that for every y ∈ conv f (S) there exists n ∈ N and (x, λ) ∈ S n × W n such that y = B f (x, λ).Then Conversely, for every y ∈ f (−1) conv(f (S)) there exists n ∈ N and (x, λ) . Then, by the definition of B f , we obtain y = B f (x, λ) ∈ conv f (S) which completes the proof.

Equality of generalized Bajraktarević means
Let us now state our main result which characterizes the equality of two generalized Bajraktarević means.In the one-dimensional case this is a classical result due to Aczél-Daróczy [2] and Daróczy-Páles [7].Theorem 6.1.Let X and Y be linear spaces, D be an arbitrary set and f : D → X, g : D → Y be admissible functions.Then B f = B g if and only if g = A • f for some linear map A : X → Y .Unverified alternative proof.Take n ∈ N, a pair (x, λ) ∈ D n × W n , and denote briefly y := B f (x, λ) ∈ D. By Lemma 3.1, applying A side-by-side in (3.1) and using the equality g = A • f twice, we get Now the converse implication in Lemma 3.1 implies y = B g (x, λ).As x and λ are arbitrary, we obtain the equality B f = B g .
To prove the converse implication assume that B f = B g .Then by Corollary 4.6 we have β f = β g .We shall use to following claim several times in the proof: Proof.For λ ≡ 0 the statement is trivial.From now on assume that λ is a nonzero vector and define (6.1) Then λ + and λ − are disjointly supported with nonnegative entries, and By the first equality we have Therefore as λ is nonzero and 0 / ∈ conv f (D), we obtain that both λ − , λ + ∈ W n .Furthermore by the definition m : The second implication is analogous.
Denote the linear span of f (D) and g(D) by X 0 and Y 0 , respectively.Let H f ⊆ f (D) be a Hamel base for X 0 .Then one can choose a system of elements We are now going to show that H g := {g(x γ ) | γ ∈ Γ} is a Hamel base for Y 0 .Indeed, for every collection of pairwise-distinct elements γ 1 , . . ., γ d ∈ Γ, the system {f (x γ 1 ), . . ., f (x γ d )} is linearly independent and thus, by our Claim, so is {g(x γ 1 ), . . ., g(x γ d )}.
To show that it is a Hamel base for Y 0 , we have to prove that H g is also a generating system.If not, then there exists an element x * ∈ D such that H g ∪ {g(x * )} is linearly independent.Repeating the same argument (by interchanging the roles of f and g) it follows that H f ∪{f (x * )} is linearly independent, which contradicts that H f is a generating system.
As H f and H g are Hamel bases for X 0 and Y 0 , respectively, there exists a unique linear mapping A : X 0 → Y 0 such that (6.2) Our aim is to extend the latter equality to all x ∈ D. To this end, take x ∈ D \ D Γ arbitrarily.Using that H f is a Hamel base for X 0 , we can find elements γ 1 , . . ., γ d ∈ Γ and nonzero real numbers λ 1 , . . ., λ d such that Then d i=1 λ i f (x γ i ) − f (x) = 0, and by our claim d i=1 λ i g(x γ i ) − g(x) = 0. Finally we obtain Therefore (6.2) holds for all x ∈ D which completes the proof.

Synergy
Before we introduce the notion of synergy let us present some interpretation of the aggregated effort.The initial issue of coalitions in decision making theory (and, more general, theory of cooperation in games) is the problem how to measure the coalition quality.Intiuitively, synergy is the difference between the aggregated effort and the sum of the individual efforts, i.e., the arithmetic effort.For the detailed study of synergy, we refer the reader to [17] and references therein.
Obviously, each party wants to be in a coalition.However A prefers C than B (as s AC ≥ s AB ) but both B and C prefer to make a coalition with each other (as s BC ≥ s AB and s BC ≥ s AC ).Consequently the coalition BC is the unique Nash equilibrium.Situation II: (λ 1 , λ 2 , λ 3 ) = (55, 30, 15): Then A does not want to make a coalition with either B or C (as the synergy is negative).Similarly neither B nor C wants to make a coalition with A (this essentially follows from the real situation).The coalition BC is irrelevant (which refers to the zero synergy).
Obviously, as it was announced, the examples above are instrumental to dominant decision systems only.For more complicated decision making systems, we need to define the synergy in a different way.In general, the synergy depends on the players' decisions.There are essentially two important assertions: (1) zero synergy refers to the situation when aggregation is irrelevant to the rest of the system; (2) positive synergy should be profitable from the point of view of a decision making system.
In our model, for an effort function In other words, σ E measures the difference between the given effort and the arithmetic effort, which is the sum of individual efforts.If f : D → X is an admissible function and E = β f , then σ E will simply be denoted as σ f .Furthermore, in view of Corollary 4.6, we can see that the synergy depends only on the mean B f , that is, the equality B f = B g implies σ f = σ g .Therefore, we can define σ B f := σ f .This property has an important interpretation in the theory of coalitional games.The case when the synergy is negative corresponds to the situation when there appear some distractions in the cooperation (see Example 7.1.II).The case of positive synergy refers to the situation when the aggregated effort of the group is greater then sum of efforts of the individuals (see Example 7.1.I).

Example 7.2. Given a manifold
with a parametrization f : R 2 → S given by f (x, y) = (x, y, 1 + x 2 + y 2 ).Then S is observable, f is admissible and the Bajraktarević-type mean B f : W(R 2 ) → R 2 is of the following form (here and below n ∈ N is fixed, x, y ∈ R n , and λ ∈ W n ) : Consequently λf (x) + µf (y) and ᾱf (x) + µf (y) are on the same ray, i.e., there exists a constant C > 0 such that As y = x, the admissibility implies that f (y) and f (x) are linearly independent.Consequently, the above equality implies C = 1 and λ = ᾱ.Thus, which implies that f (D) is a convex set.Finally, applying f −1 side-by-side we get that (7.1) holds.
7.2.Gini means.We are now going to calculate the sign of the synergy for Gini means.Before we go into the details, we recall a few properties of this family.First, it is easy to observe that G p,q = G q,p for all p, q ∈ R. Furthermore, in a case q = 0, the Gini mean G p,0 equals the p-th Power mean (in particular it is associative).These means are monotone with respect to their parameters, more precisely, for all p, q, r, s ∈ R, we have that G p,q ≤ G r,s if and only if min(p, q) ≤ min(r, s) and max(p, q) ≤ max(r, s) (cf.[6]).Finally, a Gini mean G p,q is monotone as a mean (in each of its argument) if and only if pq ≤ 0 (see [13,14]).We show that the sign of pq is also important in characterizing the sign of their synergy.
Proposition 7.4.Sign of the synergy of the Gini mean G p,q coincides with that of −pq.
In the last case, when p = q = 0, we have But the inequality on the right-hand-side holds for all p ∈ R \ {0}, which completes the proof.
Remark.Observe that Gini mean has a positive synergy if the graph of γ p,q is hyperbolic and negative for parabolic graphs.In the case of hyperboloid in Example 7.2 the synergy was also positive.

. Aggregation-type properties 4 . 1 .
Delegativity.In this subsection we establish the delegativity of the generalized Bajraktarević means and the corresponding effort functions, moreover, we show that these two maps are associated to each other.Theorem 4.1.Let f : D → X be an admissible function.Then both B f and β f are delegative and associated.Proof.Fix m ∈ N and (y, µ) ∈ D m × W m .For an arbitrary (x, λ) ∈ D n × W n , according to the Lemma 3.1, we have

Proposition 4 . 5 .
Generalized Bajraktarević mean B f is casuative for every admissible function f : D → X.

Corollary 4 . 6 .
Let f : D → X and g : D → Y be two admissible function such that B f = B g .Then β f = β g .

Proposition 5 . 1 .
Let f : D → X be an admissible function.Then S ⊆ D is f -convex if and only if cone(f (S)) is convex.

Proposition 5 . 4 .
Let f : D → X be an admissible function and S ⊆ D. Then (5.1)

Example 7 . 1 .
In a toy model we have three parties in a parliament with a total number of 100 votes and three parties: Party A (λ 1 votes), Party B (λ 2 votes), Party C (λ 3 votes).Assume that λ 1 ≥ λ 2 ≥ λ 3 .The are three possible coalitions AB, AC and BC.From the point of view of the dominant decision system (for example D F DD ) each coalition above 50 votes, is equivalent to the same number, the smallest majority which is 51.Thus we have for all x ∈ D 2 and i, j ∈ {1, 2, 3} with i = j, α(λ) = 100 n i=1 λ i ≥ 51; n i=1 λ i n i=1 λ i ≤ 50.Now we could compare the sum of weights with the equivalent weight in two cases: Situation I: (λ 1 , λ 2 , λ 3 ) = (45, 35, 20):