On minimax and Pareto optimal security payoffs in multicriteria games

J. Math. Anal. Appl. 457 (2018) 1634–1648 Contents lists available at ScienceDirect Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa On minimax and Pareto optimal security payoffs in multicriteria games Justo Puerto a,∗, Federico Perea b aIMUS, Universidad de Sevilla, Spain bInstituto Tecnológico de Informática, Universitat Politècnica de València, Spain a r t i c l e i n f o a b s t r a c t Article history: Received 20 July 2016 Available online 5 January 2017 Submitted by A. Barbagallo Keywords: Multicriteria games Payoffs Characterization In this paper, we characterize minimax and Pareto-optimal security payoff vectors for general multicriteria zero-sum matrix games, using properties similar to the ones that have been used in the single criterion case. Our results show that these two solution concepts are rather similar, since they can be characterized with nearly the same sets of properties. Their main difference is the form of consistency that each solution concept satisfies. We also prove that both solution concepts can transform into each other, in their corresponding domains. © 2017 Elsevier Inc. All rights reserved. 1. Introduction The standard two-person zero-sum game was introduced by von Neumann [14]. Ever since then, extensive research has been devoted to these games. Even nowadays the interest on these games has not decreased, see e.g. [13], who analyze how players’ information affect their payoffs. We here elaborate on the Multicriteria Zero-Sum Matrix (MZSM) games, which allow us to handle simultaneous confrontation of two rational agents in several scenarios. The first game theoretical characterizations of minimax values in MZSM games is due to Shapley [20]. In its introductory note, F.D. Rigby remarks the importance of these games: “The topic of games with vector payoffs is one which could be expected to attract attention on the basis of its intrinsic interest”. Nevertheless, the development of the theory of multicriteria games has not been as successful as expected. There is a number of references in the literature as we will see along the paper, although its intrinsic difficulty, mainly due to the lack of total orderings among players’ payoffs, has diminished the interest of researchers. In spite of that, the goal of further developing the analysis of multicriteria games should not be forgotten. In fact, each competitive situation that can be modeled as a scalar zero-sum game has its counterpart as a multicriteria zero-sum game, when more than one scenario has to be compared *Corresponding author. E-mail addresses: [email protected] (J. Puerto), p[email protected]v.es (F. Perea). http://dx.doi.org/10.1016/j.jmaa.2017.01.002 0022-247X/© 2017 Elsevier Inc. All rights reserved. J. Puerto, F. Perea / J. Math. Anal. Appl. 457 (2018) 1634–1648 1635 simultaneously. Moreover, different scenarios do not need to have the same set (number) of strategies, which makes the analysis even more challenging. Thus, conflicting interests appear not only between different decision-makers but also within each individual, due to the different criteria they may have. For instance, the production policies of two firms which are competing for a market can be seen as a scalar game. However, when they compete simultaneously in several markets and the returns in each one of them cannot be aggregated, the multicriteria approach naturally leads to a multicriteria game. The main criticism made to this approach is its difficulty to be applied, because in most cases the solutions (values) are not unique. Therefore, new solution concepts have been proposed and compared with the existing ones. When dealing with scalar zero sum matrix games, a natural question one wants to answer is the following: “If I play a strategy, what is the worst (to me) that the other player may do? To anticipate this, I will play the strategy that makes my worst case scenario as pleasant as possible.” This question leads to a well accepted solution concept for scalar zero sum matrix games: the minimax solution, also known as the value of the game. The first characterization in terms of axioms of the value of a zero-sum matrix game is due to Vilkas [22]. Later, Tijs [21] addressed the same problem, whereas more recently Norde and Voorneveld [17] and Carpente et al. [4] provided different axiomatizations for the value of standard zero-sum matrix games. Hart et al. [12] give a Bayesian foundation of minimax values. In a natural extension to more than one criterion, players have to play the same strategy in all the different scenarios. One of the most attractive alternatives to minimax payoff vectors is the concept of Pareto-optimal security level vectors (POSLV). Pareto-optimal security strategies (POSS) were introduced by Ghose and Prasad [11] as a solution concept in multicriteria zero-sum matrix games, extending the idea of security level strategies to more than one criterion, where players are allowed to play different strategies in each component. Ghose [10] characterized this solution concept as minimax strategies in a serial weighted zero-sum game, whereas Voorneveld [23] gave an alternative characterization of these strategies as minimax strategies of an amalgamated game (see [1] for the concept of amalgamation of games). Alternatively, Fernández and Puerto [6] provided a way to jointly determine POSS and their corresponding set of payoffs by solving a certain multiobjective linear program (see also [5,7,8]). Although there exist axiomatic foundations for the minimax value of one-criteria zero-sum games, there is no characterization for the set of minimax values in the multicriteria version. On the other hand, there are several axiomatizations of Nash equilibrium strategies (see e.g. Peleg and Tijs [19], Peleg et al. [18] and Norde et al. [16]). Some of these carry over to multicriteria games, as shown in Borm et al. [2] and Voorneveld et al. [24]. Nevertheless, these are axiomatizations for strategy sets rather than for payoffs, they use different sets of axioms, and they do not clearly show the relationship between the two solutions. In standard single criterion games, minimax and optimal security payoffs coincide. Nevertheless, whenever we have multiple criteria, which in turns may imply to have a different set (and number) of strategies in each component game, these two concepts differ. This fact raises the question of which are the common roots and which are the main differences between these two solution concepts. We will answer this question by providing characterizations that show the common properties and the differences between them. In this paper we characterize extensions of minimax and Pareto-optimal security payoffs to general multicriteria zero-sum matrix games. Our approach uses classical properties in game theory and decision theory (objectivity, column dominance, column elimination, row dominance, row elimination and consistency). The contribution of this paper is twofold: –we characterize solution sets rather than single value solutions which makes the analysis more involved. In this regard, we are interested in finding the largest set of payoffs compatible with the axioms proposed. This property is rather important when dealing with set-valued functions (correspondences) since it ensures that this is the largest object (solution concept) satisfying the required game theoretic properties. This approach is not new and has been already used among others by Gerard-Varet and Zamir [9] for characterizing the ‘Reasonable set of outcomes’ and Calleja et al. [3] for the ‘Aggregate-monotonic core’. 1636 J. Puerto, F. Perea / J. Math. Anal. Appl. 457 (2018) 1634–1648 –we use a new consistency property that deals with the persistence of any solution payoff of a multicriteria game, with any given dimension on the space of criteria, on some lower dimension multicriteria game. The difference between this new consistency property and the traditional one, is the way in which solutions for a game with kcriteria transform into solutions for a game with k−1 criteria. Extended minimax payoff vectors, in a multicriteria game with k-criteria, can be converted to extended minimax payoff vectors in a new (k−1)-criteria game, that makes a convex combination of two of the original payoff matrices. However, Pareto-optimal security payoff vectors become solutions of a game with k−1 criteria, but with an amalgamation of strategies from two of the previous matrices. This difference is crucial and distinguishes the two solution concepts. The rest of the paper is organized as follows. In Section 2we present the basic definitions of the multicriteria games, in which we will define the two solution concepts that are the aim of this paper. In Section 3 we formally define the minimax solution concept, and characterize it using a number of axioms. Section 4is devoted to the definition and characterization of the set of Pareto-optimal security level vectors. Section 5 is devoted to analyzing the pairwise logical independence of the axioms used in the characterizations presented. Next, in Section 6, we analyze the relationships between the two solution concepts. The paper ends with some conclusions, drawn from the results in the paper. 2. Multicriteria games Let us begin by recalling the concept of scalar two-person zero-sum games. Definition 1. A scalar two-person zero sum game is characterized by a payoff matrix A ∈Rm×n, such that if player I (the row player, who wants to minimize payoffs) plays his strategy iand player II (the column player, who wants to maximize payoffs) plays his strategy j, then the row player’s payoff is aij and the column player’s is −aij, for i =1, ..., m, j=1, ..., n. The set of pure strategies of player I consists of m strategies, indexed by i, and the set of pure strategies of player II consists of nstrategies, indexed by j. Furthermore, the set of (mixed) strategies for player I is Sm:= {s∈Rm: m  i=1 si=1; si≥0,i=1, ..., m}, whereas the set of (mixed) strategies for player II is Sn. In the sequel, at times we will make use of the set S> m, defined as the vectors in Smwith all its components strictly positive. Note that if player I plays x ∈Smand player II plays y∈Sm, the payoff of player I is xtAy ∈R, whereas player II gets the opposite. The value of the game Ais defined as val(A) := maxy∈Snminx∈SmxtAy = minx∈Smmaxy∈SnxtAy, which always exists (see [14]). The traditional multicriteria approach assumes that the payoffs of the players are vectors instead of scalars. The next definition recalls this class of games, which we here denote as D1. Definition 2. Consider an array of matrices A =(A(1), ..., A(k)) ∈Rm×n×k, with A() ∈Rm×n∀  = 1, ..., k. If player I plays x ∈Smand player II plays y∈Sn, the expected payoff for player I is xtAy = (xtA(1)y, ..., xtA(k)y) ∈Rk, and player II gets the opposite. In summary, D1=∪n,m,k∈NRm×n×k. Note that a game in D1is uniquely characterized by its payoff matrix A, since the strategy sets only depend on the dimensions of this matrix. J. Puerto, F. Perea / J. Math. Anal. Appl. 457 (2018) 1634–1648 1637 Note as well that, in games in D1, each player plays the same strategy in the kpayoff matrices. The following class of games, denoted by D2, allows the column player to play different strategies in the different scenarios. Definition 3. Consider an array of matrices A =(A(1), ..., A(k)) with A() ∈Rm×nfor  =1, ..., k(note that there could be a different number of available strategies for the second player in the different payoff matrices). Let Gk=m,n1,...,nk∈NRm×n1×... ×Rm×nkbe the set of all such k-criteria matrices. Define the strategy spaces for players I and II as Smand S(n1,...,nk), where S(n1,...,nk):= {y=(y(1),...,y(k)) ∈Rn1×...×Rnk:n j=1 y()j=1; y()j≥0, j=1,...,n ,=1...k}. If player I plays x ∈Sm, and player II plays y∈S(n1,...,nk), the payoff of player I is xtAy =(xtA(1)y(1),...,x tA(k)y(k)),(1) and player II gets the opposite. In summary, D2=∪k∈NGk. Note that a game in D2is uniquely characterized by its payoff matrix A, since the strategy sets only depend on the dimensions of the matrix. For the sake of notation we will refer to Rm×n1×...×Rm×nkas Rm×(n1,...,nk). The following example illustrates this class of games. Example 1. Let F1and F2be two companies that form a duopoly competing in the same sector. The payoffs of this game will be monetary benefit and public image, measured in units that are not easily quantifiable economically. Company F1has to face one decision: whether or not to invest in advertising (strategies Yand N, respectively). Company F2has to face two decisions: whether or not to invest in advertising (strategies Yand N, respectively), and what to do about the polluting emissions its factory produces. Three different strategies are possible in this scenario: increase, leave as it is, decrease (denoted by I, L, D). Both companies know that if the two of them invest in advertising or none of them does, then their monetary benefit does not increase nor decrease. On the contrary, if one of them invests and the other does not, the company investing will have one extra unit of benefit and the other company one less unit of benefit. If F2increases the polluting emissions and F1invests in publicity, F1will use this fact in its campaign, and the public image of F2will deteriorate, and that of F1will improve, both by two units. In case F1 would not invest in advertising, the extra emissions will be somehow found out and the public image of F2 will deteriorate, and that of F1will improve, both by 1 unit. If F2decreases its emissions and F1does not invest in publicity, the public image of F2will improve, and that of F1will deteriorate, both by one unit. An advertising campaign of F1will compensate this fact, and the improvement/deterioration in public image will be of 0.5 units only. This situation can be modeled as a game in D2, with m =k=n1=2, n2=3and the payoff matrix A =(A(1), A(2)), where A(1) and A(2) are: A=(A(1),A(2)) = 0−1 10 ,−200.5 −10 1 . Note that, in order to be consistent with the fact that player I is the minimizer and player II is the maximizer, the payoffs described before have been multiplied times −1when building matrix A. To illustrate, two possible such strategies are: 1638 J. Puerto, F. Perea / J. Math. Anal. Appl. 457 (2018) 1634–1648 1. x =(1, 0) and y= ((0, 1), (0, 1, 0)) are two pure strategies, that yield a payoff for player I equal to (0, 0) (player II gets the opposite). 2. x =(0.5, 0.5) and y= ((0.25, 0.75), (0.5, 0, 0.5)) are two mixed strategies, that yield a payoff for player I equal to (−0.25, −0.375). Both for D1and D2, the scalar zero-sum game defined by the matrix A(),  =1 ...k, will be called the -component game or -scenario of the corresponding multicriteria game A =(A(1), ..., A(k)). In the sequel, the transpose operator twill be omitted when its use is clear. Before finishing this section, we introduce some notation that will be used in the rest of the paper. –Bi·refers to the i-th row of B, and B·jrefers to the j-th column of B, for any matrix B∈Rn×m, n, m ∈N. –Given two vectors a, b ∈Rk, a ≤bif and only if a≤bfor  =1, ..., k, and a bif a ≤band a =b. –MIN(MAX) stands for the set of minimal (maximal) elements with respect to the componentwise order of Rk. So, for any S∈Rk, MIN(S) ={s ∈S: there is no t ∈S−{s} :ti≤si∀ i =1, ..., k} (MAX is defined analogously). 3. Minimax in D1 The multicriteria extension of the concept of minimax payoff looses some of the interesting properties shown in the scalar case: uniqueness and coincidence with security payoffs. In spite of that, it is still possible to establish the existence of such strategies under rather standard hypothesis. For instance, if the strategy set is compact, the set of minimax payoff vectors is non-empty, see [15]. The rationale behind minimax strategies is that each player uses the same strategy, in all the k-component games, looking for all the non-dominated vector valued payoffs. This rationale is possible in the domain D1, and so we shall consider multicriteria games defined on this domain to analyze multicriteria minimax payoff vectors. Therefore, define as w1(x)the worst payoffs player I can get if he/she plays strategy x ∈Sm, w1(x):=MAX{(xA(1)y,...,xA(k)y),y∈Sn}.(2) Now, among those strategies that give componentwise maximal payoffs in the above problems, player I will choose the strategies with the best payoff. Note that best (worst) for player I must be understood as finding the set of minimal (maximal) elements in the componentwise ordering of Rk. Definition 4. The set of extended minimax payoffs of the game A ∈D 1is given by: MINMAXk(A(1),...,A(k)) = MIN{∪x∈Smw1(x)},(3) where kis the number of different criteria. Similarly, one can define maximin payoffs. According to the above expression, the extended minimax payoff vectors are the non-dominated vectors obtained from the solutions to all the vector valued maximization problems (2), for all x ∈Sm. Clearly, if B∈Rm×pthen MINMAX1(B) =val(B). After defining the extended minimax payoff vectors, the concept of extended minimax strategy naturally follows. Definition 5. An extended minimax strategy of player I is any strategy x ∈Smattaining an extended minimax payoff vector. Similarly, one can define extended maximin strategies of player II. J. Puerto, F. Perea / J. Math. Anal. Appl. 457 (2018) 1634–1648 1639 The first result about extended minimax payoff vectors in multicriteria games was given by Shapley [20], who provides a simple way for finding them by solving zero-sum scalar games with payoff matrix A(α) =k =1 αA(), a positive linear combination of the matrices A(),  =1, ..., k. Theorem 1 (Adapted from [20]). Let z∗be an extended minimax value for the multicriteria game with matrix A =(A(1), ..., A(k)). Then, there exists α∈S> ksuch that k =1 αz∗ =val (k =1 αA()). Conversely, let z∗(α)be the minimax value of (k =1 αA()), then there exists an extended minimax payoff vector z∗∈MINMAXk(A(1), ..., A(k)), satisfying k =1 αz∗ =z∗(α). 3.1. A characterization of the set of extended minimax payoff vectors We begin this section by introducing the axioms that will characterize the set of minimax payoff vectors. Let {fk}k≥1be a family of point-to-set maps (correspondences) defined as fk:∪n,m∈NRm×n×k−→ 2Rk A=(A(1),...,A(k)) −→ fk(A). The axioms needed are: A0 Objectivity. For any z∈Rk, fk(z) =z. A1 Monotonicity. For any A, A∈Rm×n×ksuch that A≤A, fk(A) ⊆fk(A) +Rk −. A2 Column dominance. Let Ac()be the matrix that results from Aafter adding to the matrix A()a new column which is dominated by a convex combination of its columns. Then fk(Ac()) =fk(A). A3 Column elimination. Let A−c()be the matrix that results from Aafter removing column c() from the matrix A(). Then fk(A−c()) ⊆fk(A) +Rk −. A4 Row dominance. Let Arbe the matrix that results from Aafter adding a new row which is dominated by a convex combination of its rows. Then fk(Ar) =fk(A). A5 Row elimination. Let A−rbe the matrix that results from Aafter removing row r. Then fk(A−r) ⊆ fk(A) +Rk +. A6 Consistency.For any k≥2, if z∈fk(A)then there exists 0 <α<1, such that (αz1+(1−α)z2,z 3,...,z k)∈fk−1(M(αA(1),(1 −α)A(2)),A(3),...,A(k)), where M(αA(1), (1 −α)A(2)) is a matrix with mrows, labeled i =1, ..., mand n1×n2columns, labeled c =(c(1), c(2)) with c() ∈{1, ..., n}for each  =1, 2. The entry in row iand column c =(c(1), c(2)) of M(αA(1), (1 −α)A(2)) equals αA(1)ic(1) +(1 −α)A(2)ic(2), see [23]. A7 Linear consistency. For any k≥2and Asuch that n=n ∀ , if z∈fk(A)then there exists 0 <α<1 such that (αz1+(1 −α)z2, z3, ..., zk) ∈fk−1((αA(1) +(1 −α)A(2)), A(3), ..., A(k)). Objectivity establishes the evaluation in a trivial situation where the game has kcriteria and both players have exactly one action available. Monotonicity states that the set of solution payoff vectors should not decrease, in the componentwise order of Rk, when all the payoff matrices weakly increase. Column (row) dominance states that the set of solution payoff vectors should not change if player II (I) can no longer choose an action, in some of the component games, which is worse for him/her than a combination of some other actions. Column elimination states that the set of solution payoff vectors can not increase their values, in the componentwise order of Rk, when some action of player II in some of the component games is removed. Row elimination states that, when removing an action of player I, the new set of solution payoff vectors must be dominated by the original one. Consistency states that any solution outcome of a game 1640 J. Puerto, F. Perea / J. Math. Anal. Appl. 457 (2018) 1634–1648 with a given dimension, in the criteria space, can be converted into a solution outcome of a new game with lower dimension of an amalgamated game ‘à la’ Borm et al. [1]. Linear consistency states that any solution outcome provided by this correspondence, with a given dimension in the criteria space, can be converted into a solution outcome of this correspondence with lower dimension, by considering a convex combination of two of the original criteria. The next result is a characterization of the set of extended minimax payoff vectors of any general multicriteria zero-sum game. Theorem 2. The set of extended minimax payoff vectors MINMAXkis the largest (w.r.t. inclusion) map on D1, the set of multicriteria zero sum games, that satisfies objectivity, monotonicity, column dominance for k=1, row dominance and linear consistency. Proof. First of all, we check that MINMAXksatisfies the properties. A0 – Objectivity: It is clear that MINMAXksatisfies A0. A1 – Monotonicity: Since x ≥0, then xA() ≥x¯ A()for all  =1, ..., k. Hence, for all y∈Sn, (xA(1)y, ..., xA(k)y) ≥(xA(1)y, ..., xA(k)y), and the property follows. A2 – Column dominance for k=1: It is clear that MINMAX1satisfies column dominance for k=1, since MINMAX1(B) =val(B), and it is known that the value function, val(·), of a matrix game satisfies this property, see [4]. A4 – Row dominance: Let Ar=A(1) ... A(k) b1... b k, b=(b1 , ..., bn ) ∈Rn,  =1, ..., k, such that b= m i=1 αiAi·(), with m i=1 αi=1and αi≥0. Take x=(x1, ..., xm, xm+1) ∈Sm+1. Then, for any y∈Sn xA() by= m  i=1  j∈I (xi+αixm+1)aij ()yj=ˆxA()y, ∀=1,...,k, where ˆx=((x1+α1xm+1), ..., (xn+αnxm+1)) ∈Sm. Hence, MINMAXk(Ar) =MINMAXk(A). A7 – Linear Consistency: Assume A7 is not satisfied. Therefore, there exists z=(z1, ..., zk) ∈MINMAXk(A) such that for all 0 <α<1, (αz1+(1 −α)z2, z3, ..., zk) /∈MINMAXk−1([αA(1) +(1 −α)A(2)], A(3), ..., A(k)). Hence by Theorem 1, there is no (β1, β3, ..., βk) ∈S> k−1such that β1αz1+β1(1 −α)z2+β3z3+...+βkzk∈val(β1αA(1) + β1(1 −α)A(2) + β3A(3) + ...+βkA(k)). However, this contradicts that zis a minimax payoff vector, because by Theorem 1, there must exist (λ1, ..., λk) ∈S> k, such that k  =1 λz=val(λ1A(1) + ...+λkA(k)). The above contradiction proves that MINMAXksatisfies A7. To finish the proof, it is enough to show that if {gk}k≥1is an arbitrary family of point-to-set maps that satisfy A0, A1, A2 for k=1, A4 and A7, then for any general multicriteria game given by a matrix A =(A(1), ..., A(k)), with A() ∈Rm×nfor all  =1, ..., k, we get gk(A) ⊆MINMAXk(A) ∀ k≥1. For k=1, the axioms A0, A1, A2 and A4 characterize the val(·) function of a matrix game (see Carpente et al. [4, Theorem 2]). Therefore, g1(·) =val(·) =MINMAX1(·). J. Puerto, F. Perea / J. Math. Anal. Appl. 457 (2018) 1634–1648 1641 Let z∈gk(A), k≥2. Apply A7 (k−1)-times to conclude that there exists α∈Rk, k =1 α=1, α>0, such that k  =1 αz=val( k  =1 αA()). Then, by Theorem 1, zis an extended minimax payoff vector of the game defined by A. Hence, gk(A)⊆MINMAXk(A)∀A, k ≥1.2 The above theorem also implies that MINMAXkis the largest map on D1that satisfies Linear Consistency (A7), and that coincides with the value function on standard single criterion matrix games. Another characterization of extended minimax payoff vectors using a different set of properties is possible. The rationale is to alternatively characterize the val(·) function and then to apply the consistency construction. This is possible based on Carpente et al. [4, Theorem 3]. Theorem 3. The set of extended minimax payoff vectors MINMAXkis the largest (w.r.t. inclusion) map on D1that satisfies objectivity, column dominance, row dominance, column elimination, row elimination and linear consistency. The proof runs similarly to that of Theorem 2, but using [4, Theorem 3] instead of [4, Theorem 2]. 4. POSS in D2 This solution concept is independent of the notion of equilibrium, so that the opponents are only taken into account to establish the security levels for one’s own payoff. Therefore, this notion does not require to play the same strategy in all the scalar -component games A(), and thus is defined in the class D2. Definition 6. Every strategy x ∈Smdefines security levels v I(x)as the payoffs with respect to each criterion, when II bets to maximize the criteria [10]. Hence, v I(x)= max y∈Sn xA()y,  =1,...,k, (4) and the security levels are k-tuples denoted by vI(x)=(v1 I(x),...,vk I(x)).(5) We will use the notation vI(x, A)to specify such matrix A, whenever this is needed to avoid confusion. It must be noted that for a given strategy xfor player I, the security levels vI(x)might be obtained by different strategies of player II. In [11] the concept of Pareto optimal security strategy (POSS) is defined in D1as follows. It should be noted that the definition on D2is the same. Definition 7. A strategy x∗∈Smis a Pareto-optimal security strategy for I if and only if there is no x ∈Sm such that vI(x∗) vI(x). Similarly, one can define POSS for II. The set of Pareto-optimal security level vectors is the set of payoffs that can be attained by POSS, and will be denoted by VPOSSk(A(1), ..., A(k)), thus VPOSS k(A(1),...,A(k)) = {z∈Rk:z=vI(x)forsomex∈Smand x∈Smsuch that vI(x)vI(x)}.(6) 1642 J. Puerto, F. Perea / J. Math. Anal. Appl. 457 (2018) 1634–1648 The reader should observe that there are extended minimax values that cannot be obtained by Paretooptimal security level vectors. The reason is that POSS attain inferior payoffs by allowing the column player to change their strategies in the different criteria games. This fact will be made clearer in Section 6. The following theorem provides a way to determine all POSS and their security level vectors. Theorem 4. ([6, Theorem 3.1]) A strategy x∗∈Smis a POSS and v∗=(v∗ 1, ..., v∗ k)is its security level vector if and only if (v∗, x∗)is an efficient solution to the problem min (v1,...,v k), s.t. xA()≤(v,...,v ),=1,...,k, n i=1 xi=1, x≥0,v∈Rk. Equivalently, Voorneveld [23, Theorem 3.1] characterize POSS strategies as minimax strategies of particular classes of scalar games. Theorem 5. ([23, Theorem 3.1]) A strategy x∗∈Smis a POSS for player I in the multicriteria matrix game A =(A(1), ..., A(k)) if and only if there exists a vector α∈S> ksuch that x∗is a minimax strategy in the scalar matrix game M(α1A(1), ..., αkA(k)) being M(α1A(1), ..., αkA(k)) a matrix with mrows, labeled i =1, ..., mand nkcolumns, labeled c =(c(1), ..., c(k)) with c() ∈{1, ..., n}for each  =1, ..., k. The entry in row iand column c =(c(1), ..., c(k)) of M(αA)equals k =1 αA()ic(). The reader may note that in the scalar case, VPOSS1(B) =MINMAX1(B) =val(B), and Theorems 4 and 5coincide. 4.1. A characterization of Pareto-optimal security payoffs In this section, we characterize the set of Pareto-optimal security payoffs defined on a general multicriteria two-person zero-sum game in D2. Thus, we try to identify a map (solution concept) that associates to any array of kmatrices with the same number of rows a set of vectors in Rk, k≥1. Using the set of properties introduced in Section 3.1, we can obtain a characterization of the entire set of Pareto-optimal security payoff vectors as the maximal (in the inclusion sense) point-to-set map that satisfies A0, A1, A2, A4and A6. Theorem 6. The set of Pareto-optimal security level vectors VPOSS kis the largest (w.r.t. inclusion) map on D2that satisfies objectivity, monotonicity, column dominance, row dominance and consistency. Proof. The proof is similar to Theorem 2, but using Theorem 5 instead of Theorem 1. First of all, we check that VPOSS ksatisfies the properties of objectivity, monotonicity, column dominance, row dominance and consistency. A0 – Objectivity: Clearly, VPOSS ksatisfies A0. A1 – Monotonicity: The security level vectors for the strategy xwith respect to Aare: (maxy∈Sn1xA(1)y, ..., maxy∈SnkxA(k)) = (maxy∈ext{Sn1}xA(1)y, ..., maxy∈ext{Snk}xA(k)) =(max{xA(1)},...,max{xA(k)}), where max{xA()}denotes the maximum component of the vector xA(). Note that this is true, because ext{Sn}(the set of extreme points of Sn) consists of the vectors whose ith component is one and the rest is zero, for i =1, ..., n. Analogously,