Cores of Cooperative Games, Superdifferentials of Functions, and the Minkowski Difference of Sets

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1 Journal of Mathematical Analysis and Applications 247, doi: jmaa , available online at on Cores of Cooperative Games, uperdifferentials of Functions, and the Minkowski Difference of ets Vladimir I. Danilov and Gleb A. Koshevoy Central Institute of Economics and Mathematics, Russian Academy of ciences, ahimoskii Prospect 47, Moscow , Russia ubmitted by L. Berkoitz Received February 17, 1998 Let be a cooperative Ž TU. game and 1 2 be a decomposition of as a difference of two convex games and. Then the core CŽ. 1 2 of the game has a similar decomposition CŽ. CŽ. CŽ. 1 2, where denotes the Minkowski difference. We prove such a decomposition as a consequence of two claims: the core of a game is equal to the superdifferential of its continuation, known as the Choquet integral, and the superdifferential of a difference of two concave functions equals the Minkowski difference of corresponding superdifferentials Academic Press Key Words: totally monotone game; support function; inverse Mobius transform; Minkowski sum. 1. ITRODUCTIO The core of an n-person cooperative game is the set of all feasible payments that cannot be improved upon by any coalition of players. The core is a compact polyhedron, possibly empty. The aim of this paper is to shed light on the structure of the core. We show how the core is built from some elementary objects, simplexes. Any game is a linear combination of simple games. The core of a simple game is the unit simplex in the corresponding subspace. If a game is a linear combination of simple games with nonnegative coefficients Žsuch games are called totally monotone games., then its core equals the same linear combination of the simplexes. um of sets is understood as the Minkowski sum X00 $35.00 Copyright 2000 by Academic Press All rights of reproduction in any form reserved.

2 2 DAILOV AD KOHEVOY Our main result is an extension of this assertion to any games. Let be a cooperative game. It can be represented Ž the inverse Mobius transform. in a form A A A B B B, with some nonnegative coefficients A B,, where denotes a simple game with a coalition A, i.e., Ž. A A 1 if A and 0 otherwise. Then the core CŽ. has a similar representation Ž A A. Ž B B. C A C B C. Here, and are the Minkowski sum and the Minkowski difference; CŽ. A is the core of a simple game A. The following example exhibits the result. EXAMPLE. Consider the following three-persons game: Ž. 1, if. The game has the following representation via simple games:. 1,24 1,34 2,34 1,2,34 The core of the game is the triangle CŽ. Ž x, x, x x1 x2 x3 2, 0 x 14. The core of the game is the simplex Ž triangle. i 1, 2, 34 CŽ. Ž x, x, x. x x x 1, x 04 1, 2, i. The core of the game is a simplex Ž segment with vertices 1 and 1. CŽ. i, j4 i j i, j4 Ž x, x, x. x x 1, x, x 0, x i j i j 1, 2, 34i, j4 The Minkowski sum of three segments C Ž 1, 24. C Ž 1, 34. C Ž 2, 34. equals the hexagon whose vertices are the coordinate permutations of the vector Ž 2, 1, 0.. The Minkowski sum of two triangles CŽ. CŽ. 1, 2, 34 is equal to the same hexagon. o, there holds CŽ. CŽ. CŽ. 1, 2, 34 1, 24 CŽ. CŽ., and we get the following decomposition of the core 1, 34 2, 34 CŽ. CŽ 1,24. CŽ 1,34. CŽ 2,34. CŽ 1,2,34.. Figure 1 illustrates this example. Our main result is a consequence of two important claims. First ŽPro- position 3., the core of a game equals the superdifferential of one of its continuations, known as the Choquet integral. econd Ž Lemma 1., the superdifferential of the difference of two concave functions equals the Minkowski difference of the corresponding superdifferentials Ža kind of reversion of the MoreauRockafellar rule.. In view of received representation of the core, we can say that the core as the solution concept possesses a kind of additivity property, similar to that the hapley vector possesses. otations: V is a finite dimensional vector space; V * is its dual. A homogeneous function f: V * is a function such that fž p. fž p., for any p V * and 0. 1,...,n 4. 1,, denotes the vector in whose ith coordinate equals 1 if i and 0 otherwise. The accordance 1,, allows to identify the lattice 2 with the set of vertices of the unit cube 0, 14. Given a linear function x Ž.*, x xž 1. x denotes the value of x in the corresponding vertex i i

3 CORE AD MIKOWKI DIFFERECE 3 FIG. 1. of the unit cube. x Ž. x 14 is the unit simplex in Ž.*. A a a A 4,, A V. 2. MIKOWKI UM AD MIKOWKI DIFFERECE Consider a finite dimensional vector space V. It is possible to define some operations on the set of its subsets, P. The Minkowski sum of sets A and B P is defined by the rule A B a ba A, b B 4. This operation is associative and commutative, A 04 A, and A. The Minkowski sum of convex sets is a convex set. Ž P,. is a semigroup, but it is not a group: there is no inverse operation to the Minkowski sum. However, it is possible to define a partial inversion by the rule A B z V z B A 4. Ž 1.

4 4 DAILOV AD KOHEVOY This operation is said to be the Minkowski difference. ŽMatheron 4 defined the difference of sets slightly different from Eq. Ž.. 1. Although Minkowski difference is a partial inversion of Minkowski sum, it possesses some properties of difference: A B is equal to the maximal set C for which C B A. The equality Ž A B. B A holds iff there exists a set C with A C B. In such a case, the set B is said to be the Minkowski summand to A. If A is a convex set, then the Minkowski difference A B is a convex set for any B. The Minkowski difference is a monotone operation, i.e., from B C follows A B A C for any A. There also holds A Ž B C. Ž A B. C. Ž 2. In fact, the set Z A Ž B C. is the maximal set for which Z ŽB C. A holds. The set Z Ž A B. C is the maximal set for which Z C A B. In view of the inclusion Z C B Ž A B. B A, there holds Z Z. On the other hand, due to associativity and commutativity of the Minkowski sum, there holds Ž Z C. B A. Therefore, the inclusion Z C A B holds. Due to the maximality Z with respect to the inclusion Z C A B, we get Z Z. That yields the equality. There is a close relation between convex sets and concave functions: let A be a nonempty compact in V, and define a function Ž A;. on the dual space V * by the rule Ž A; p. inf pž x., p V *. Ž 3. xa The function Ž A;. is said to be the support function of A. In the literature, usually the function s: V *, sž A; p. sup pž x. x A is said to be the support function. For us, it will be convenient to use Eq. Ž. 3. s and are related via the equality Ž p. sž p.. The support function is homogeneous and concave. On the other hand, every homogeneous concave function f: V * is the support function of a nonempty compact in V. uch a set can be explicitly determined. Given a homogeneous concave function f on V *, a set Ž f. in V of the form 4 Ž f. x V xž p. fž p. p V * Ž 4. is said to be the superdifferential of f. Obviously, the set Ž f. is a convex compact. The operations and are dual. The exact statement is the following well-known theorem, which is a form of the Minkowski duality in convex analysis Žsee, for example, Rockafellar. 5.

5 CORE AD MIKOWKI DIFFERECE 5 THEOREM 1 Ž Duality.. Ž a. Let f be a homogeneous concae function on V *. Then Ž f. and f Ž Ž f.;.. Ž b. Let A be a nonempty conex compact in V. Then ŽŽ A;.. A. The part Ž. a can be slightly generalized: let f be a homogeneous function on V *. Let fˆ be its concavification, ˆfŽ p. inf xž p., Ž 5. xf ˆ where x is a linear function on V *, i.e., x V. In other words, f is the function whose ordinate set is the convex hull of the ordinate set of f. Because of this, the formula i i ˆfŽ p. sup fž p. Ž 6. holds, where the supremum is taken over all possible representations p as a convex combination: p ii p i, i 0, ii 1. However, we should take care of the possibility that inf and sup may not achieve: in the first case, there is no linear function x such that x f; in the second case, there is a convex combination 0 p such that fž p. i i i i i i 0. In such a case, we set fˆ. Define the superdifferential of a homogeneous function f by Eq. Ž. 4. Obviously, Ž f. if and only if fˆ. PROPOITIO 1. Let f be a homogeneous function on V * and fˆ. Then Ž f. Ž fˆ. and Ž Ž f.. f. ˆ Proof. As fˆ f, there holds Ž fˆ. Ž f.. On the other hand, let x Ž f., i.e., xž p. fž p. for any p V *. By the formula Ž 5., we have xž p. fˆž p. and hence x Ž f.. Due to concavity of f, ˆ we conclude Ž Theorem 1. the equality Ž Ž f.. Ž Ž fˆ.. f. ˆ Q.E.D. Ž. The following properties of the support function see, for example, 2, 5 will be of use. 1. Ž A;. Ž A;., Ž A B;. Ž A;. Ž B;.. 3. A B Ž A;. Ž B;.; with convex B, there holds A B Ž A;. Ž B;.. As A B equals the maximal set C with C B A, we conclude that Ž A B. is the minimal homogeneous concave function f such that f Ž B. Ž A.. In view of Proposition 1, this yields that Ž A B. is the concavification of Ž A. Ž B.. This establishes the following. i

6 6 DAILOV AD KOHEVOY PROPOITIO 2. Let A and B be nonempty conex compacts. Then Ž A B. ŽŽ A. Ž B.. and A B ŽŽ A. Ž B... Let us deduce some other consequences from these propositions. LEMMA 1. Let 1 and 2 be concae homogeneous functions. Then Ž a. Ž 1 2. Ž 1. Ž 2., Ž 7. Ž b. Ž 1 2. Ž 1. Ž 2.. Ž 8. Proof. Ž. a This is known as the MoreauRockafellar rule Ž. 5. Ž b. Because of concavity of and, we have Ž A , 2 Ž B. for some nonempty compacts A, B. Then, by Proposition 2, there holds Ž. Ž A. Ž B. A B. Ž By the duality theorem, A ŽŽ A.. Ž. and B ŽŽ B.. Ž This and Eq. Ž. 9 yield Eq. Ž. 8. Q.E.D. COROLLARY 1. In the case of conex compacts A, B, C, the following properties hold: A Ž A C. C, Ž 10. Ž A C. Ž B C. A B. Ž 11. COROLLARY 2. Gien sets A and B, the set B is the Minkowski summand to A if and only if the difference Ž A;. Ž B;. is a concae function, and hence there holds Ž A B;. Ž A;. Ž B;.. COROLLARY 3. Let V be n-dimensional ector space. Let A, B be conex compacts in V. Then A B if and only if, for any p 0,..., pn V * with n p 0, holds i0 i n i i i0 Ž A; p. Ž B; p. 0. Ž 12. Proof. From Propositions 1 and 2 follows that A B if and only if ŽŽ A. Ž B... The latter is equivalent to the existence of the convex combination 0 ii pi such that i Ž A; p. Ž B; p. Ž A; p. Ž B; p. 0. i i i i i i i i

7 CORE AD MIKOWKI DIFFERECE 7 By the Caratheodory theorem, it is possible to restrict ourselves to convex combinations of at most n 1 vectors. Q.E.D. 3. GAME AD CORE We recall that a cooperative TU game is a function : 2, satisfying 0. Elements of the set 1,...,n4 are interpreted as players; subsets of are coalitions; the number Ž. is the gain of the coalition. The core CŽ. of a game is the set of all feasible payments that cannot be improved upon by any coalition of players. The core CŽ. is the following subset in Ž.* 4 C x * x, x,. 13 The core is a compact convex polyhedron, possibly empty, of dimension at most n 1. The definition of the core of a cooperative game recalls the definition of the superdifferential of a homogeneous function given in the previous section with V Ž.*. Consider a game as a function on the linear space. Precisely, it is defined only on a part of the space: on vertices of the unit cube 0, 14. Given coalition, assign the value to the vector 1, whose ith coordinate equals 1 if i and 0 otherwise. We also assign the value Ž. to the vector 1. In order to get the core of a cooperative game as the superdifferential of a homogeneous function, we shall, in a proper way, continue a game to a function on the whole. It occurs that a proper continuation exists and is known as the Choquet integral. ŽFor axiomatization of Choquet integral, see, for example, chmeidler. 6. The continuation is based on a nice decomposition of into cones. These cones are constructed by chains of coalitions. A decreasing sequence of coalitions W 4 0 k is said to be a chain. It will be convenient always to assume 0. To the chain W we relate the cone spanned by the line 1 and rays 1, i 1,...,k, i ½ i 5 ConŽ W. i 1 0, i 0, i 1,...,k. i Cones that correspond to maximal chains have the full dimension. There are n! full dimensional cones. The space is equal to the union of all

8 8 DAILOV AD KOHEVOY cones and relative interiors of different cones do not intersect. Let p. Consider p as a function on and let c0 c1 ck be its ordered values and set 4 i p c. j i j Then p c 1 Ž c c. 0 j1 j j1 1 j belongs to the relative interior of 4 k Con W, where W. The expression p c 1 Ž 0 k 0 j1 cj c. j1 1, where cj cj1 for all j, is called the canonical form of p. j Define the continuation of the game on by the rule 0 j j1 Ž j. k Ž p. c Ž. Ž c c., Ž 14. where p is written in the canonical form. From this follows that the equality Ž 1. Ž j. holds for every j, j j 0,...,k, Ž 1. Ž., and is a linear function on each cone ConŽ W.. The following proposition establishes the importance of this continuation operation. PROPOITIO 3. Let be a game and let be its continuation on. Then j1 CŽ. Ž.. Proof. The inclusion CŽ. Ž. holds due to the fact that any linear inequality in the definition of CŽ. arises also in the definition of Ž.. Reversely, let x CŽ.. Consider an inequality xž p. Ž p. in the defini- tion Ž.. Let p belong to a cone ConŽ W.. Functions x and are linear on this cone. The inequality holds at the generators 1, 1 j, and the equality holds at 1. Therefore, the inequality still holds in p. Q.E.D. o, we have shown that the core of a cooperative game coincides with the superdifferential of the homogeneous function. In view of this and the duality theorem, the following definition looks natural. DEFIITIO 1. A game is said to be the conex if is a concave function. We call such games convex, and not concave Žwhich may seem more logical., because later it comes out that such games are convex due to the hapley Ž definition. Exploiting the relation between cores of Ž TU. games and superdifferentials of homogeneous function, we receive the following.

9 CORE AD MIKOWKI DIFFERECE 9 PROPOITIO 4. Let and be conex games. Then 1 2 Ž a. CŽ 1 2. CŽ 1. CŽ 2., Ž 15. Ž b. CŽ 1 2. CŽ 1. CŽ 2.. Ž 16. Proof. Because of Lemma 1 and Proposition 3, and due to the linearity of the operation, the following sequence of equalities holds Ž a. CŽ. Ž Ž Ž. Ž. Ž. C C, Ž b. CŽ 1 2. Ž 1 2. Ž 17. Ž. Ž. CŽ. CŽ Q.E.D. 4. DECOMPOITIO OF GAME AD CORE It is well known Ž8. that any game can be represented as a linear combination of simple games. Here, we show that the core possesses a similar decomposition via the cores of simple games. With any nonempty coalition A is related the simple game Žuna- nimity game. that has payoffs Ž. 1 if A, and Ž. A A A 0 otherwise. The core of a simple game is the set CŽ. A A A, where x xž A A A is the face of the unit simplex with vertices in the set A. The set of games is a linear space. The set of simple games is the basis of this linear space: any game is a linear combination Žwith possibly. negative coefficients of simple games. In fact, let : 2 be a function such that 0. Ž uch functions are called mass functions.. Define a game of the form From Eq. 18 follows that Ž.. Ž 18. Ž T. Ž T., T. Ž 19. T

10 10 DAILOV AD KOHEVOY The transformation from the games to the mass functions, defined by the rule T 1 T, 20 T is said to be the inerse Mobius transform. By the method of inclusion and exclusion Ž3., there holds. Consider games with nonnegative Mobius inverse. Any such game is said to be the totally monotone game. From the following lemma and that the sum of convex games is a convex game, we get that any totally monotone game is convex in accord with Definition 1. A LEMMA 2. Gien A, the simple game A is conex, i.e., the function is concae. Proof. Let be a simple game, A. how that A By Eq. 14 there holds A p min p i. 21 ia k A p c0 A cj cj1 AŽ j., 22 j1 where c c c are ordered values of p, i p c k j i j, j 1,...,k. Let c min p. Then A if j a and hence Ž. a i A i j A j 1, j a, and A if j a and hence Ž. j A j 0, j a. Therefore, we have a AŽ p. c0 Ž cj cj1. ca min p i. j1 ia Given A, the function min Ž coordinate. functions. i A pi is concave as the minimum of linear Q.E.D. From this lemma and Proposition 4a., we get the structure of the core of a totally monotone game. COROLLARY 4. Let be a totally monotone game and be its Mobius inerse. Then CŽ. Ž..

11 CORE AD MIKOWKI DIFFERECE 11 Let now be an arbitrary game and two games be its Mobius inverse. Define Ž., Ž 23., Ž. 0 and Ž.. Ž 24., Ž. 0 Games and are totally monotone games and. Define the following two sets: the upper core and the lower core C Ž. CŽ. Ž., Ž 25., Ž. 0 C Ž. CŽ. Ž.. Ž 26., Ž. 0 The sum over empty set equals, by convention, 0 4. Because of Proposition 4 and Corollary 4, we get our main result. THEOREM 2. Let be a game. Then there holds CŽ. C Ž. C Ž.. Ž 27. Remark. Vasil ev 10 studied a set HŽ. C Ž. ŽC Ž.., which looks similar to Eq. Ž 27., but it is different. For totally monotone games ŽC Ž. 0., the set HŽ. coincides with the core, but this coincidence holds only for totally monotone games. In 10, Vasil ev proved that there exists a convex game with CŽ. HŽ.. This can be easily seen from Proposition 4: given a coalition A, define the following game w, w Ž B. A A 1 if B A and 0 otherwise. The game w is convex Žw Ž p. A A max p. and there holds Cw a A a A A. Let now be a game and be its Mobius inverse. Define the following game Ž. w, Ž 28., Ž. 0 and set, where is defined by Eq. Ž 23.. The game is the convex game as sum of convex games. By Proposition 4, there holds

12 12 DAILOV AD KOHEVOY CŽ. CŽ. CŽ.. Because of Cw, we have CŽ. CŽ. C Ž.. That yields the equality CŽ. HŽ.. To summarize, we can say that the core as the solution concept possesses a kind of additivity: given a game being represented in a form Ž A. Ž B., with some nonnegative coefficients Ž A A B B mass functions. and, the core CŽ. has a similar representation CŽ. Ž A. CŽ. Ž B. CŽ.. A A B B 5. GAME WITH OEMPTY CORE Let us start with a characterization of convex games. PROPOITIO 5. Let : 2 be a game. The following statements are equialent. Ž. i A game is conex, i.e., the function is a homogeneous concae function on. Ž ii. is a supermodular function on the lattice 2, i.e., for any, T, there holds Ž. Ž T. Ž T. Ž T.. Ž iii. Ž p. ŽCŽ., p. for any p. Ž iv. There holds C Ž. CŽ. C Ž.. Proof. The equivalence Ž iv. Ž iii. follows from Theorem 2 and Proposition 2. The equivalence Ž. i Ž iii. follows from Proposition 3 and Theorem 1. Ž. i Ž ii.. Let and T be subsets of. Then there exists a maximal chain W that contains sets T,, and T. Due to the definition of, there is a linear function, say y, that coincides with on the cone ConŽ W.. Because of concavity of, there holds Ž 1 T. Ž T. yž 1 T.. o, we have Ž T. Ž T. yž 1. yž 1. yž 1. yž 1. T T T Ž. Ž T.. Ž ii. Ž i.. The concavity of will be established if we show that for any p, there exists a linear function x such that x and xž p. Ž p.. We define such a function x explicitly. Let p. Pick a full-dimensional cone ConŽ W. to which p belongs. Let W 4,,. et xž 1. Ž. and xž 1. 0 n 0 n j j, j 1,...,n. We complete the definition by linearity on.in particular, we have xž p. c xž 1. Ž c c. xž 1. c Ž. 0 j1 j j1 j 0

13 CORE AD MIKOWKI DIFFERECE 13 Ž c c. Ž. Ž p., where p c 1 Ž c c. j1 j j1 j 0 j1 j j1 1 j. The inequality x will be followed if we will check that xž 1. Ž 1. for any. Proceed by induction. Given, consider the greatest k such that k. Then we have the inequality Ž. Ž k1. Ž k1. Ž k1.. Further, k1 k and, by induction on the number of elements, there holds Ž. x Ž.. o, we have j k1 k1 Ž k1. Ž k1. Ž k1. xž k1. xž k1. xž k1. xž.. Thus, Ž. Ž. x for any and Ž. xž.. Q.E.D. Remarks. Recall that hapley 8 called those games convex that satisfy the supermodularity property: for any, T, there holds Ž. Ž T. Ž T. Ž T.. This proposition shows the equivalence of the supermodularity to Definition 1. Using other methods, chmeidler 6 received the property Ž. i and Vasil ev 9 received the property Ž iii.. From a geometric point of view, the proposition says that a game is convex if and only if C Ž. is the Minkowski summand to C Ž.. A family B 2 of sets and a system of weights 0, B, with i 1, i, is said to be the balanced family. A game is said to be balanced if for any balanced family Ž B,., there holds B Ž. Ž.. Ž 29. Bondareva 1 and hapley 7 proved that the core is nonempty if and only if a game is balanced. Here we employ Proposition 3 and Corollary 3 to show that the nonemptyness of the core is equivalent to the fulfilment Ž 29. for balanced families that contain at most n 1 sets. THEOREM 3. Let be a game. Then the core, CŽ., is nonempty if and only if for any sets 1,...,n and numbers 1,...,n0 with n j1 j1 j 1, there holds n i j1 Ž.. Ž 30. j

14 14 DAILOV AD KOHEVOY Proof. By Proposition 3, the equality CŽ. Ž. holds. Ž. iff Ž.. The latter holds iff the convex hull of the ordinate set of does not coincide with the whole. Due to the definition of, the convex hull of the ordinate set of coincides with the cone generated by the set of vectors Ž1, Ž.., Ž.4, 1,. The latter cone does not coincide with iff Eq. Ž 30. holds. Q.E.D. REFERECE 1. O.. Bondareva, ome applications of linear programming methods to the theory of cooperative games Ž in Russian., Prob. Kibern. 10 Ž 1963., H. G. Eggleston, Convexity, Cambridge University Press, ew York, M. Hall, Combinatorial Theory, Blaisdell, Waltham, MA, G. Matheron, Random ets and Integral Geometry, Wiley, ew York, R. T. Rockafellar, Convex Analysis, Princeton Univ. Press, Princeton, J, D. chmeidler, Integral representation without additivity, Proc. AM 97 Ž 1986., L.. hapley, On balanced sets and cores, a. Res. Logistics Quart. 14 Ž 1967., L.. hapley, Cores of convex games, Int. J. Game Theory 1 Ž 1971., V. A. Vasil ev, The support function of the core of a convex cooperative game, Žin Russian., Tr. Inst. Mat. O Akad. auk. R, Optim. 21 Ž 1978., V. A. Vasil ev, About one class of imputations in cooperative games Ž in Russian., Dokl. Akad. auk. R 256 Ž.Ž ,