Monge extensions of cooperation and communication structures

European Journal of Operational Research 206 (2010) 104–110

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European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor

Stochastics and Statistics

Monge extensions of cooperation and communication structures U. Faigle a,*, M. Grabisch b, M. Heyne a a b

Mathematisches Institut/ZAIK, Universität zu Köln, 50931 Köln, Germany Université Paris I, Centre d’Economie de la Sorbonne, 75013 Paris, France

a r t i c l e

i n f o

Article history: Received 23 June 2009 Accepted 22 January 2010 Available online 2 February 2010 Keywords: Communication structure Convex game Cooperation structure Monge extension Lovász extension Marginal value Ranking Shapley value Supermodularity Weber set

a b s t r a c t Cooperation structures without any a priori assumptions on the combinatorial structure of feasible coalitions are studied and a general theory for marginal values, cores and convexity is established. The theory is based on the notion of a Monge extension of a general characteristic function, which is equivalent to the Lovász extension in the special situation of a classical cooperative game. It is shown that convexity of a cooperation structure is tantamount to the equality of the associated core and Weber set. Extending Myerson’s graph model for game theoretic communication, general communication structures are introduced and it is shown that a notion of supermodularity exists for this class that characterizes convexity and properly extends Shapley’s convexity model for classical cooperative games. Ó 2010 Elsevier B.V. All rights reserved.

1. Introduction The classical model of cooperative games assumes that every subset of a set N of agents may form a coalition to execute the game. However, many situations require a more refined model in which only a restricted collection F of subsets describes feasible cooperation. In Myerson’s [24] communication graph model, for example, only those sets of agents are feasible for communication that induce connected subgraphs. Other examples arise from models where N is (partially) ordered by some dominance or preference relation (e.g., Derks and Gilles [8], Faigle and Kern [13,14], Gilles et al. [16], Grabisch and Lange [17], Hsiao and Raghavan [18]). The latter model was further relaxed and studied by Algaba et al. [3], Bilbao et al. [2,5] to combinatorial coalition structures of socalled antimatroids, convex geometries and augmenting systems, and by Lange and Grabisch to regular set systems [21]. All these generalized models for cooperation rely on their particular combinatorial structure for the definition of Shapley-type values, Weber sets and cores. Indeed, it appears difficult to reasonably define a notion of a ‘‘marginal value” for cooperation models without special structural properties. Moreover, it seems to be impossible to

* Corresponding author. E-mail addresses: [email protected] (U. Faigle), [email protected] (M. Grabisch), [email protected] (M. Heyne). 0377-2217/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2010.01.043

extend the concept of supermodular characteristic functions, and hence of convex games, to coalition systems that are not closed under union and intersection. On the other hand, a natural notion for the core of a general cooperation structure exists as a certain convex set in the Euclidean parameter space RN (Faigle [11]), which suggests to study general cooperation from the point of view of real convex analysis. For the classical model, such an approach was indicated by Lovász [22] (see also Algaba et al. [4]). It is the purpose of our present investigation to show that Lovász’ construction is actually a special case of a quite general construction that is meaningful for arbitrary cooperation structures. The key in our analysis is the relaxation of the notion of a cooperative game to cooperative game instances with given bounds on the activity levels of individual agents. We obtain game instances by a straightforward rule that goes back to Monge [23] and corresponds to the well-known north–west corner rule for the construction of feasible solutions for transportation problems (Section 3). Our rule yields the Monge extension of the characteristic function v of the underlying cooperation struc^ : RN ! R. Convexity properties of arbitrary ture to a function v cooperation structures can thus be studied via their Monge extensions. The Monge algorithm furthermore implies a natural ranking notion for agents and thus a framework for marginal vectors, Weber

U. Faigle et al. / European Journal of Operational Research 206 (2010) 104–110

sets and Shapley values (Section 4). In a far-reaching extension of the classical results we find that the Monge extension of a cooperation structure is concave (a.k.a. convex down) if and only if its core and Weber set coincide (Theorem 5.2). In Section 6, we introduce communication structures as a particular class of cooperation structures that are union-closed in a weak sense and hence include Myerson’s communication graph model as a special case. These structures have been studied as union-stable structures before (see, e.g., Algaba et al. [1]). We show that a meaningful notion of ‘‘supermodularity” exists for this class and characterizes convexity (Theorem 6.2). Hence convex communication structures generalize in particular Shapley’s [27] convex cooperative games. Moreover, we show that our general model of convexity implies the notion of convexity introduced by Bilbao and Ordóñez [6] for games on so-called augmenting systems, which form a subclass of communication structures. We always assume that the characteristic function of a cooperation structure describes the gain a feasible coalition may achieve. As in the classical case, our cores may equally well be interpreted as arising from associated cost games. However, we will not explore the latter model in detail here. 2. Cooperation structures Let N ¼ f1; . . . ; ng be a finite set of players. A cooperation structure on N is a pair C ¼ ðF; v Þ, where F is a family of non-empty subsets of N and v : F ! Rþ is a non-negative valuation on F. We refer to a set F 2 F as a feasible coalition of C. In the case F ¼ 2N n f;g, i.e., when each non-empty subset of N constitutes a feasible coalition, we say that C is a classical cooperative game. Remark. Strictly speaking, a classical cooperative game may include coalitions F with negative value v ðFÞ < 0. Modifying v to  with a valuation v

v ðFÞ ¼ v ðFÞ þ j  jFj

ðF # NÞ;

where j > 0 is a suitably large constant, however, any classical game is seen to be essentially equivalent to a non-negative classical game. The next example may serve as a motivation for leaving the classical context. (It will be taken up in Section 6.) Example 2.1 (Myerson games [24]). Let G ¼ ðN; EÞ be a graph with node set N and edge set E with the interpretation that x; y 2 N may ‘‘communicate” if fx; yg 2 E. One is interested in the family F of those non-empty subsets F # N that induce a connected subgraph of G and hence allow communication paths among all members of F to be established. v ðFÞ describes the value of the communication within the connected subgraph with node set F. Throughout the paper F ¼ fF 1 ; . . . ; F m g so that

we

index

the

coalitions

in

(I1) F i  F j ) i < j. In some parts of the paper, we will suppose that in the sense

F # F0 )

v is monotone

v ðFÞ 6 v ðF 0 Þ:

If monotonicity holds, we can (and will) assume in addition that the indexing of coalitions also satisfies the property (I2)

v ðF 1 Þ P    P v ðF m Þ.

105

2.1. Game instances with activity bounds Let c 2 RN be a fixed parameter vector. A c-feasible game instance is a parameter vector y 2 RF such that yF P 0 holds for all F–N and

X

aj ðyÞ ¼

yF 6 c j

for all j 2 N:

F3j

We interpret yF as the activity level of the coalition F 2 F (i.e., the activity contribution of each j 2 F relative to F) in the cooperation effort. So aj ðyÞ measures the total activity of the player j with respect to y, and the vector c plays the role of an activity bound. The value of the game instance y is the parameter

yðv Þ ¼

X

yF v ðFÞ:

F2F þ   Writing yF ¼ yþ F  yF , where yF ¼ maxf0; yF g and yF ¼ maxf0; yF g, we note

yF ¼ yþF P 0 for all F–N: In the case N 2 F, we may view rðyÞ ¼ y N  v ðNÞ as the setup cost for the game instance y and the numbers yþ F v ðFÞ as the values generated by the coalitions F 2 F at the activity levels yþ F . The players j thus respect the activity bounds

06

X

yþF 6 cj þ yN :

F3j

In the following, we will allow for setup costs and therefore assume  N 2 F (and thus F 1 ¼ N in the listing F ¼ fF 1 ; . . . ; F m g), unless stated otherwise.

3. Monge extensions Assuming N 2 F, we turn our attention to the construction of cfeasible game instances y in the context ðF; v Þ according to a generalized north–west corner rule for transportation problems. We therefore refer to these game instances as being Monge. For the description of the algorithm, we use the notation

FðXÞ :¼ fF 2 FjF # Xg for any X # N: 3.1. The Monge algorithm We construct sequences l; p and a vector y 2 Rm as follows for any given c 2 RN . As usual, if l; l0 are sequences, ll0 denotes the concatenation of the two sequences, and  denotes the empty sequence. Monge Algorithm (MA): (0) Set X ¼ N; l ¼ ; p ¼  and yi ¼ 0 for all i ¼ 1; . . . ; m. Set cj ¼ cj for all j ¼ 1; . . . ; n. (1) Select the F s 2 FðXÞ with the smallest index s and the smallest p 2 F s with cp ¼ minfct jt 2 F s g. l ½ls; p ½pp; ys cp ; X ½X n p; Update (2) Update ct ½ct  cp  for all t 2 F s . (3) If FðXÞ ¼ ; then output ðl; p; yÞ and stop; Otherwise goto (1). Let ðl; p; yÞ be the output of the Monge algorithm and assume l ¼ i1 . . . ik (with i1 ¼ 1). Setting

M ¼ MðlÞ :¼ fM 1 ; . . . ; M k g ¼ fF i1 ; . . . ; F ik g ði:e:; M s ¼ F is Þ;

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we find

hv ; yi ¼

m X

yi v ðF i Þ ¼

k X

yis v ðM s Þ:

s¼1

i¼1

Notice that the selection rule (1) and the update rule (2) in MA guarantee yi P 0 for all F i –N. So y yields indeed a game instance. Moreover, we have for all j 2 N,

X F i 3j

yi



¼ cj

if j occurs in

6 cj

otherwise:

p;

With the interpretation yF i ¼ yi for i ¼ 1; . . . ; m, the Monge algorithm thus generates a c-feasible game instance. The output sequence p of MA is not necessarily a permutation of N, i.e., not every j 2 N may occur in p. However, we observe that p is representative for F in the following sense: Lemma 3.1. Let ðl; p ¼ p1 . . . pk ; yÞ be the output of the Monge algorithm for some c 2 Rn . Then F \ fp1 ; . . . ; pk g – ; holds for all F 2 F. Example 3.1. Let N ¼ f1; 2; 3; 4; 5g and F ¼ f12345; 2345; 1345; 124; 234; 345; 12; 35; 2; 5g, where ‘‘12345” stands for {1,2,3,4,5} etc. (see Fig. 1). For any c 2 RN with c4 < c3 < c2 < c1 < c5 , the algorithm will produce the sequences l ¼ ð1; 7; 8; 10Þ (corresponding to the coalitions 12345, 12, 35 and 5), p ¼ ð4; 2; 3; 5Þ and the vector

3.1.2. Rankings The output p ¼ p1 . . . pk of the Monge algorithm provides a ranking of the players of N: Sequentially pick a representative p of a feasible coalition F s of maximal possible value v ðF s Þ and discard the coalitions already represented from further consideration. 3.2. The extension function Notice that the output ðl; p; yÞ of the Monge algorithm is uniquely determined by the input c 2 Rn , provided the indexing of coalitions in F is fixed. So MA yields a well-defined function

c 2 Rn # v^ ðcÞ :¼ hv ; yi 2 R: We call v^ : Rn ! R the Monge extension of the valuation and justify the terminology as follows.

v:F!R

^ ð1F Þ ¼ v ðFÞ holds for all F 2 F, where 1F 2 f0; 1gN is Lemma 3.2. v the incidence vector of F # N (with components ð1F Þj ¼ 1 if and only if j 2 F). Proof. Take F 2 F and consider c ¼ 1F . Since F 2 F and all elements corresponding to zeroes of c are selected first, M s ¼ F at some step s. So

v^ ð1F Þ ¼ v ðMs Þ ¼ v ðFÞ follows by the definition of y.

h

F

y ¼ ðc4 ; 0; 0; 0; 0; 0; c2  c4 ; c3  c4 ; c5  c3 Þ 2 R : 3.1.1. The greedy algorithm If v is monotone and the coalitions are indexed according to the rules ðI1 Þ and ðI2 Þ, the Monge algorithm may be viewed as a greedy algorithm for the construction of a game instance: Sequentially pick a feasible coalition F s of maximal value v ðF s Þ and assign to the vari~s without violating the individual able ys the maximal possible value y activity bounds cj . Viewed as a greedy algorithm, the Monge algorithm is also ~ (the so-called meaningful in the case N R F. The output vector y ‘‘greedy solution”) will be feasible for the linear program

maxhv ; yi ¼

X

v ðFÞyF

F2F

s:t:

X

yF 6 cp ;

^ correRemark. In the case F ¼ 2N n ;, the Monge extension v sponds to the extension introduced by Lovász [22] for the set function v, which equals the discrete Choquet integral [7] when v is monotone. The authors show in a companion paper [12] how the Choquet integral extends to arbitrary set families F via the Monge algorithm.

4. Core and weber set Let ðF; v Þ be a cooperation structure with a monotone valuation v. We define the core of C ¼ ðF; v Þ as the closed convex set

coreðv Þ :¼ fx 2 RN jhc; xi P v^ ðcÞ; 8c 2 RN g # RN :

8p 2 N:

F3p

~ will be non-negative for any (non-negative) input Moreover, y c P 0.

We next give a direct characterization of the core which exhibits coreðv Þ as a non-negative and bounded polyhedron. As usual, we P employ the notation xðSÞ :¼ h1S ; xi ¼ j2S xj for any x 2 RN and S # N. Theorem 4.1. Assume F 3 N and v monotone. Then one has

coreðv Þ ¼ fx 2 RNþ jxðNÞ ¼ v ðNÞ; xðFÞ P v ðFÞ; 8F 2 Fg:

12345

124

2345

1345

Proof. Let Pðv Þ ¼ fx 2 RNþ jxðNÞ ¼ v ðNÞ; xðFÞ P v ðFÞ; 8F 2 Fg and consider any x 2 coreðv Þ. Since v is non-negative by monotonicity, v^ ðcÞ P 0 holds for every c P 0. Letting c ¼ 1j be the jth unit vector in RN , we obtain

234

345

xj ¼ h1j ; xi P v^ ð1j Þ P 0 for all j 2 N:

35

Moreover, v^ ð1N Þ ¼ v ðNÞ and v^ ð1N Þ ¼ v ðNÞ immediately yields xðNÞ ¼ h1N ; xi ¼ v ðNÞ. In view of v^ ð1F Þ ¼ v ðFÞ (Lemma 3.2), we thus conclude x 2 Pðv Þ. To prove the converse, observe that any z 2 Pðv Þ is a feasible solution for the linear program

12

2

ð1Þ

5

minhc; xi s:t: xðNÞ ¼ v ðNÞ; xP0

xðFÞ P v ðFÞ;

8F 2 F:

 be the output of the Monge algorithm with respect to c. Then y  Let y is a feasible solution for the dual linear program

maxhv ; yi s:t: y

Fig. 1. Example of a family of feasible coalitions, ordered by inclusion.

X F3j

yF 6 cj ;

8j 2 N;

yF P 0 8F 2 F n fNg:

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i ¼ v^ ðcÞ and hence z 2 coreðv Þ follows from linear So hc; zi P hv ; y programming duality. h Remark. Theorem 4.1 shows that coreðv Þ coincides with the notion of the positive core for ‘‘cooperative games with restricted cooperation” introduced in Faigle [11].

4.2. Weber set We associate with the cooperation structure C ¼ ðF; v Þ the convex hull Wðv Þ of all marginal vectors xp , i.e.,

Wðv Þ :¼ convfxp jp 2 Pg and call the polytope Wðv Þ # RN the Weber set of C. Theorem 4.2. Assume N 2 F. Then coreðv Þ # Wðv Þ.

4.1. Marginal vectors To study marginal vectors relative to the cooperation structure

C ¼ ðF; v Þ, consider the output ðl ¼ i1 . . . ik ; p ¼ p1 . . . pk ; yÞ of the Monge algorithm with respect to the input c. Note that l and y can be reconstructed from the knowledge of the ranking sequence p ¼ p1 . . . pk (given the fixed linear arrangement F ¼ fF 1 ; . . . ; F m gÞ. We let P denote the collection of all possible ranking sequences. Recalling the notation MðlÞ ¼ fM 1 ; . . . ; M k g, consider the ðp; lÞ-incidence matrix R ¼ ½rst  2 f0; 1gkk with the coefficients

rst ¼



otherwise:

R is (lower) triangular with diagonal elements r ss ¼ 1 and hence  (resp. v  ) and c denote the restriction of y (resp. v) invertible. Let y to l and of c to p. Then we have

¼c Ry

v^ ðcÞ ¼ hv ; yi:

and

Putting xT :¼ v T R1 , we therefore obtain

v^ ðcÞ ¼ v T y ¼ xT Ry ¼ xT c ¼ hc; xi:

ð2Þ

We extend  x to the vector xp 2 RN by setting xpp ¼  xp if p occurs p in p and x ¼ 0 otherwise. xp is the marginal vector of C ¼ ðF; v Þ associated with c 2 RN . Lemma 4.1. The marginal vector xp can be computed as follows: (0) xppk ¼ v ðMk Þ; P (1) xpps ¼ v ðMs Þ  GMs v ðGÞ, for s ¼ 1; . . . ; k  1(where G  M s means that G is a maximal member of the family Ms ðlÞ ¼ fG0 2 MðlÞ n fM s gjG0 Ms ; g). Moreover, xp ðMt Þ ¼ v ðMt Þ holds for t ¼ 1; . . . ; k. Proof. (1) follows immediately from the relation

xpps ¼ v ðM s Þ 

hc; zi < hc; xi for all marginal vectors x: But then the marginal vector xp 2 Wðv Þ associated with c would yield a contradiction:

hc; zÞ P v^ ðcÞ ¼ hc; xp i: 

1 if ps 2 M t ; 0

Proof. Suppose that the claim of the theorem were false and a vector z 2 coreðv Þ n Wðv Þ existed. Since Wðv Þ is a closed convex set, we could now separate z from Wðv Þ by a hyperplane, i.e., there would be a parameter vector c 2 RN such that

X fxpt jt > s; pt 2 Ms g ðs ¼ k  1; k  2; . . . ; 1Þ: 

Remark. For the classical case F ¼ 2N n ;, Theorem 4.2 is due to Weber [29]. 4.2.1. Shapley value It appears natural to define the ‘‘Shapley value” Uðv Þ of a cooperation structure as the average of its marginal vectors:

Uðv Þ :¼

1 X p x 2 Wðv Þ; jPj p2P

ð3Þ

where P is the collection of all possible rankings p produced by the Monge algorithm. In the classical case C ¼ ð2N n ;; v Þ; Uðv Þ coincides with the value introduced by Shapley [26].

5. Convexity We say that cooperation structure C ¼ ðF; v Þ is convex (or sim^ : RN ! R is a concave ply, that v is convex) if its Monge extension v (a.k.a. convex down) function, i.e., satisfies for all parameter vectors c; d 2 RN and real scalars 0 < t < 1,

tv^ ðcÞ þ ð1  tÞv^ ðdÞ 6 v^ ðtc þ ð1  tÞdÞ: Theorem 5.1. Assume N 2 F and v is monotone. Then C ¼ ðF; v Þ is convex if and only if for all c 2 RN ,

In the case N 2 F, we have M 1 ¼ N and observe (from Lemma 4.1) that xp ðNÞ ¼ v ðM1 Þ ¼ v ðNÞ holds for any marginal vector xp . Note furthermore that C admits only a finite number of marginal vectors (since P is finite).

v^ ðcÞ ¼ minfhc; xijx 2

Example 4.1. Let us take again the communication structure of Example 3.1. Then the corresponding ðp; lÞ-incidence matrix is

^ Proof. It is straightforward to check in the Monge algorithm that v is positively homogeneous in the sense

2

1

0

0

0

3

1 0 1 1 and one obtains the solution x ¼ ½ x4 ; x2 ;  x3 ; x5  of the system

þ x2 x2

þ x3

þ x5

¼ ¼

x3

þ x5

¼

x5

¼

v ð12345Þ v ð12Þ v ð35Þ v ð5Þ

as x ¼ ½v ð12345Þ  v ð12Þ  v ð35Þ; v ð12Þ; v ð35Þ  v ð5Þ; v ð5Þ.

core ðv Þg

¼ max hv ; yijyF P 0; 8F 2 F n fNg;

X

) y F 6 c j ; 8j 2 N :

F3j

v^ ðkcÞ ¼ kv^ ðcÞ

61 1 0 07 7 6 R¼6 7 41 0 1 05

x4

(

for all c 2 RN and real scalars k P 0:

A well-known result from convex analysis (see, e.g., Rockafellar [25]) therefore asserts that the concavity of v^ is equivalent with v^ being the lower support function of its core, which is the first equality claimed. The second equality follows from linear programming duality with respect to the core representation (1) of Theorem 4.1. h For the proof of an alternative characterization in Theorem 5.2, we need a technical fact. Lemma 5.1. Let p ¼ p1 . . . pk 2 P be an arbitrary ranking sequence. Then there exists some ~c 2 RN such that the Monge algorithm ~Þ with the properties: produces the output ðl; p; y

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~Ms > 0 for all s ¼ 1; . . . ; k. (i) y P ~ ~ (ii) F3j yF < c j for each j R p.

6. Communication structures

Proof. Let c 2 RN be a parameter vector so that the Monge algorithm produces the output ðl; p; yÞ. We now modify c to a weighting ~c 2 RN as follows. We choose some c0 > maxfjcp jjp 2 Ng and replace each cp by c0p ¼ cp þ c0 P 0. Relative to c0 , the Monge algorithm then clearly produces the output ðl; p; y0 Þ with y0 P 0. Each component c0ps with ps 2 p is now replaced by ~cps ¼ c0ps þ 2s for s ¼ 1; . . . ; k. Each of the remaining components c0j with j R p is replaced by a large positive constant K 0 (e.g., K > n  c0 þ 2n ). It is straightforward to verify that ~c produces the same ranking sequence p as c0 and therefore as c. Moreover, the latter modification ensures property (ii) to hold while the former modification guarantees (i). h Theorem 5.2. Assume N 2 F and v monotone. Then C ¼ ðF; v Þ is convex if and only if xp 2 coreðv Þ holds for each marginal vector xp , i.e., if and only if

coreðv Þ ¼ Wðv Þ: Proof. Assume first Wðv Þ # coreðv Þ and consider an arbitrary c 2 RN with associated marginal vector xp 2 Wðv Þ. Then we have

v^ ðcÞ P minfhc; xijx 2 coreðv Þg ( ¼ max hv ; yijyF P 0; 8F 2 F n N;

X

) yF 6 cj ; 8j 2 N

P v^ ðcÞ:

F3j

So equality holds throughout and exhibits C as convex by Theorem 5.1. Conversely, consider the marginal vector xp . By Lemma 5.1, xp ~Þ relative to some input arises from the MA-output ðl; p ¼ p1 . . . pk ; y c such that ~Ms > 0 for all s ¼ 1; . . . ; k. (i) y P ~ (ii) F3j yF < c j for each j R p. ~ is an optimal solution for the linear program If C is convex, y

( max hv ; yijyF P 0; 8F 2 F n N;

X

) yF 6 cj ; 8j 2 N :

F3j

Let ~x be an optimal solution for the dual linear program

minfhc; xijx P 0; xðNÞ ¼ v ðNÞ; xðFÞ P v ðFÞ; 8F 2 Fg: ~ must satisfy the complementary slackness Being optimal, ~x and y conditions:

~Ms > 0 ) ~xðMs Þ ¼ v ðMs Þ ðM s –NÞ; y X ~xj > 0 ) ~F ¼ c j : y F3j

By (ii), the latter conditions imply x~j ¼ 0 if j R p. Because ~xðNÞ ¼ v ðNÞ is true for the core vectors ~ x, we conclude from (i) and the former conditions that ~ x is identical to the marginal vector xp , which means xp 2 coreðv Þ in particular. Since coreðv Þ is a convex subset of RN , we therefore find in view of Theorem 4.2:

Wðv Þ ¼ convfxp jp 2 Pg # coreðv Þ # Wðv Þ: 

We say that the cooperation structure C ¼ ðF; v Þ (with possibly N R F) is a communication structure if F is weakly union-closed, i.e., satisfies (WU) F [ F 0 2 F for all F; F 0 2 F with F \ F 0 – ;. Note that the set systems F with property (WU) coincide with the union-stable systems investigated by Algaba et al. [1]. Our terminology is motivated by the interpretation that the player in a feasible set F 2 F can communicate and that the communication spreads to the players in F 0 2 F as well if there exists some player p 2 F \ F0. Example 6.1. Let G ¼ ðN; EÞ a graph with node set N and edge set E. Consider any non-empty node sets F and F 0 that induce connected subgraphs of G. Then F [ F 0 induces a connected subgraph if F \ F 0 – ; holds. So the Myerson games on graphs (cf. Example 2.1) form a special class of communication structures. As an immediate consequence of (WU), the maximal feasible coalitions of a communication structure are pairwise disjoint. Hence a communication structure naturally decomposes into pairwise disjoint communication structures, each of them exhibiting a unique maximal feasible coalition (see also Algaba et al. [1]). Without loss of generality, we therefore assume N 2 F in our subsequent analysis of communication structures. A special case of a communication structure is given when F is closed under arbitrary unions. Examples arise from cooperative games under precedence constraints (Faigle and Kern [13]), games with permission structure (Gilles et al. [16]), or antimatroids, which are the complements of discrete convex geometries (see, e.g., Korte et al. [20]). In view of F0 ¼ 2N , every classical cooperative game can be understood as a union-closed communication structure. Remark. Algaba et al. [1] have proposed a ‘‘Myerson value” for union-stable structures as the (classical) Shapley value of an associated classical cooperative game. This value, however, does not coincide with the Shapley value (3) that arises naturally from the Monge algorithm for this class. The notion of games on regular set systems introduced by Lange and Grabisch (see [21], where a Shapley-like value is proposed) is also closely related to Myerson games. Example 6.2. A communication structure ðF; v Þ is an augmenting system in the sense of Bilbao [5] if it satisfies for all F; G 2 F with F # G,

G n F – ; ) F [ fig 2 F for some i 2 G n F: The class of union-closed augmenting systems is exactly the class of antimatroids. 6.1. Greedy communication structures We want to characterize convex communication structures (with N 2 F). To this end, we relax the definition and call an arbitrary communication structure C ¼ ðF ¼ fF 1 ; . . . ; F m g; v Þ greedy if the Monge algorithm (viewed as a greedy algorithm) is guaranteed to produce an optimal solution for the linear program

maxhv ; yi s:t: yP0

Remark. For the special case of classical cooperative games, Theorem 5.1 was observed by Schmeidler [28], while Theorem 5.2 is due to Lovász [22].

X

yF 6 cp ;

8p 2 N

ð4Þ

F3p

for any (non-negative) c P 0. Hence a convex communication structure ðF; v Þ is necessarily greedy (cf. Theorem 5.1).

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We call the valuation v : F ! Rþ strongly monotone if it satisfies for any F 2 F and pairwise disjoint feasible sets G1 ; . . . ; Gf 2 FðFÞ the inequality f X

v ðG‘ Þ

6

i ¼ hv ; y

1 ½v ðF [ F 0 Þ þ v ðFÞ þ v ðF 0 Þ þ v ðF \ F 0 Þ: 2

~ can only be optimal if the supermodular inequality holds. h So y We now investigate sufficient conditions and first recall that the support of a vector y 2 RF is defined as the set

v ðFÞ:

‘¼1

Note that f ¼ 1 exhibits every strongly monotone monotone in the usual sense.

v

to be also

Example 6.3. Assume that F is closed under taking arbitrary unions. Then v : F ! Rþ is strongly monotone if and only if v is monotone and superadditive. Lemma 6.1. If the communication structure ðF; v Þ is greedy, then v is necessarily strongly monotone. Pf

Proof. Take F 2 F and suppose that v ðFÞ < ‘¼1 v ðG‘ Þ holds. Take c ¼ 1F and y the output of the Monge algorithm, whose only nonzero component is yF ¼ 1. Then y0 defined by

suppðyÞ ¼ fF 2 FjyF –0g: Lemma 6.3. Assuming that ðF ¼ fF 1 ; . . . ; F m g; v Þ is a supermodular communication structure, let y be the lexicographically maximal optimal solution for the linear program (4). Then suppðy Þ is a nested family, i.e., one has for any F i ; F j 2 suppðy Þ with i < j,

either F i \ F j ¼ ; or F i F j : Proof. Suppose F i ; F j 2 suppðy Þ are intersecting and F s ¼ F i [ F j . If  with the s < i were true, we could modify y to the vector y components

y0G :¼ yG ¼ 0 otherwise;

8  yG þ e > > >  < yG þ e G ¼ y  > y > Ge > :  yG

is feasible but hv ; y i > hv ; yi, a contradiction to the fact that y is optimal for (4). h

and obtain a feasible solution that is lexicographically strictly larger than y . Moreover,

y0F :¼ 0; y0G1 ¼   

¼ y0Gf :¼ yF ¼ 1; 0

For the next definition, it is convenient to augment the family F to F0 ¼ fF 1 ; . . . ; F m ; F mþ1 g with F mþ1 ¼ ; and to set v ð;Þ ¼ 0. Let F; F 0 2 F be intersecting, i.e., F \ F 0 – ;. Then F [ F 0 2 F follows from the weak union property (WU), while F \ F 0 R F may be possible. Nevertheless, (WU) implies that the maximal sets in the family F0 ðF \ F 0 Þ ¼ FðF \ F 0 Þ [ f;g are pairwise disjoint. So we arrive at the well-defined parameter 0

v ðF \ F Þ :¼

X

0

fv ðGÞjG 2 F0 ðF \ F Þ maximalg:

ð5Þ

Example 6.4. Assume that F is closed under arbitrary unions. Then for any F; F 0 2 F, there is a unique maximal feasible set

F ^ F 0 ¼ [fG 2 F0 jG # F \ F 0 g 2 F0 and v ðF \ F 0 Þ ¼ v ðF ^ F 0 Þ follows for any intersecting F; F 0 2 F. We now say that the communication structure ðF; v Þ is supermodular (or simply, v is supermodular) if for any intersecting feasible sets F; F 0 2 F the following inequality holds:

v ðF [ F 0 Þ þ v ðF \ F 0 Þ P v ðFÞ þ v ðF 0 Þ; 0

ð6Þ 0

where v ðF \ F Þ is understood as in (5) if F \ F R F holds. Lemma 6.2. If the communication structure ðF; v Þ is greedy, then v is necessarily supermodular.

if G ¼ F s ; if G is maximal in; FðF i \ F j Þ; if G ¼ F i or G ¼ F j ; otherwise

  i ¼ hv ; y i þ e v ðF i [ F j Þ þ v ðF i \ F j Þ  v ðF i Þ  v ðF j Þ : hv ; y  must be optimal and we arSupermodularity of v implies that also y rive at a contradiction to the choice of y . So s ¼ i and hence F i F j must hold. h Theorem 6.1. The communication structure C ¼ ðF ¼ fF 1 ; . . . ; F m g; v Þ is greedy if and only if the valuation v : F ! Rþ is strongly monotone and supermodular. Proof. The necessity of the conditions follows from Lemmas 6.1 and 6.2. We prove sufficiency by induction on the number jFj of feasible coalitions. Let y be the greedy solution and denote by y the (with respect to the index order of F) lexicographically maximal optimal solution.CLAIM: yF 1 ¼ yF 1 . Now yF 1 P yF 1 is a direct consequence of the Monge algorithm. So it suffices to show that strict dominance yF 1 > yF 1 is impossible. Recalling from Lemma 6.3 that suppðy Þ is a nested family, let fG1 ; G2 ; . . . ; Gk g be the collection of (inclusion-wise) maximal proper subsets of F 1 in the support So the Gi s are pairwise disjoint. Let suppðyast Þ. e ¼ minfyF 1  yF 1 ; yG1 ; . . . ; yGk g P 0 and define y by

F 1 ¼ yF þ e; y 1 Gi ¼ yG  e; y i

i ¼ 1; . . . ; k;

G ¼ yG otherwise: y  is a feasible solution and satisfies Then y

0

Proof. Let F; F 2 F be intersecting. Then the supermodular inequality is trivial if F F 0 or F 0 F holds. We thus assume that neither is the case and consider the non-negative parameter vector ~ for (4) yields c ¼ 1F[F 0 þ 1F\F 0 . The greedy solution y

~i ¼ v ðF [ F 0 Þ þ v ðF \ F 0 Þ: hv ; y  2 RF with the components On the other hand, the vector y

8 0 0 > < 1=2 if G 2 fF [ F ; F; F g;  yG ¼ 1=2 if G maximal in FðF \ F 0 Þ; > : 0 otherwise; is also a feasible solution with the objective value

i ¼ hv ; y i þ e hv ; y

v ðNÞ 

‘ X

!

v ðGi Þ

P hv ; y i

i¼1

 is optimal and lexicoby the strong monotonicity of v. Hence also y  ¼ y and hence e ¼ 0, as graphically maximal, which implies y claimed. To finish the proof, consider the representative p1 2 F 1 chosen by the Monge algorithm. Because of yF 1 ¼ yF 1 ¼ cp1 ¼ minfcp jp 2 F 1 g, we find:

yF ¼ yF

holds for all F 2 F with p1 2 F: 0

0

ð7Þ

Let F ¼ fF 2 Fjp1 R Fg. Then F is weakly union-closed. Moreover, the Monge algorithm produces the value

110

X

U. Faigle et al. / European Journal of Operational Research 206 (2010) 104–110

yF v ðFÞ 6

F2F0

X

yF v ðFÞ

F2F0

on F0 . On the other hand, jF0 j 6 jFj  1 holds. So we know by induction that the Monge algorithm is optimal on F0 . Taking (7) into account, we therefore conclude

X F2F

yF v ðFÞ 6

X

yF v ðFÞ 6

F2F

X

Remark. We do not know of a characterization of general convex cooperation structures ðF; v Þ in terms of an appropriately generalized notion of ‘‘supermodularity”. (For some sufficient conditions, see, e.g., Faigle and Peis [15].) Acknowledgment

yF v ðFÞ

F2F

and hence optimality of y. h

The authors are grateful to the reviewers for their careful reading of the manuscript, which improved the presentation.

Remark. Theorem 6.1 generalizes Theorem 4 in [12].

References

6.2. Convex communication structures Theorem 6.1 allows us to characterize convex communication structures as follows. Theorem 6.2. Assume N 2 F. Then the communication structure

C ¼ ðF; v Þ is convex if and only if v is strongly monotone and supermodular. Proof. If C is convex, then C is greedy. Hence (by Theorem 6.1) the conditions are necessary. Conversely, we show that a greedy communication structure is convex if N 2 F holds. It suffices to argue that the greedy algorithm is optimal for the linear program

maxhv ; yi s:t: yF P 0 8F–N;

X

yF 6 cp ; 8p 2 N:

ð8Þ

F3p

Indeed, let C 0 be a large constant and modify c to the vector ~ is c with components cj ¼ cj þ C > 0. Then the greedy solution y optimal relative to c. On the other hand we have

hc; yi ¼ hc; yi  C v ðNÞ for each feasible solution y for (8). Since C v ðNÞ is constant, we con~ is also optimal for c. h clude that y Corollary 6.1. cf. [6]Let ðF; v Þ be an augmenting system with a monotone characteristic function v : F0 ! R. Then ðF; v Þ is convex if and only if v is supermodular. Proof. Any monotone function v on an augmenting system F is necessarily strongly monotone. h Remark. Convexity of augmenting systems is defined without reference to any Monge-type extensions and relative to a different model for Weber sets in [6]. Our Corollary 6.1 shows that the two notions of convexity coincide on this special class. Classical cooperative games. Assume F0 ¼ 2N , i.e., ðF; v Þ is a classical cooperative game. Then F0 is the union- and intersection-stable Boolean lattice of all subsets of N with the operations F ^ F 0 ¼ F \ F 0 and F _ F 0 ¼ F [ F 0 . In this context, the supermodular inequality (6) is the defining property for ”convex games” in the sense of Shapley [27]. The equivalence of v being supermodular and coreðv Þ containing all marginal vectors was first realized in the classical context by Edmonds [9] (see also Faigle [10] and Ichiishi [19]). It is easy to see that a classical non-negative supermodular function is necessarily monotone.

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