Set-valued Solutions for Cooperative Game with Integer Side Payments

Transcription

1 Applied Mathematical Sciences, Vol. 8, 2014, no. 11, HIKARI Ltd, Set-valued Solutions for Cooperative Game with Integer Side Payments Alexandra B. Zinchenko Department of Mathematics, Mechanics and Computer Science Southern Federal University, , Rostov-on-Don, Russia Copyright c 2014 Alexandra B. Zinchenko. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract The paper deals with cooperative side payments game with restricted transferability (distributed utility consists of indivisible units). Corebased solutions and stable sets are considered. We review known results as well as prove new theorems concerning existence conditions and relations between solution concepts. Mathematics Subject Classification: 91A12 Keywords: discrete game, core-based solutions, stable set 1 Introduction Cooperative games arise in situations in which the participants can obtain additional utility (increase profit, power, prestige, influence, reduce costs) by pooling their resources, capitals, possibilities. The most simple and popular design of cooperative game (TU game) needs the assumptions of transferable utility and side payments. However, the distributed among players utility can consists of indivisible units. If the unit is one contract, project or one stock then the fractional values are not meaningful. This paper studies cooperative side payments game with integer outcomes and demonstrates the effect of integer requirements. Such games have been named in [4] the games with modified utility. J. von Neumann and O. Morgenstern pointed out that thus a modification of utility concept would make our theory more realistic, but

2 542 Alexandra B. Zinchenko at the same time it is clear that definite difficulties must be overcome in order to carry out this program [4]. Cooperative games with integer side payments were investigated in [1], [3], [6], [7] and other works of the same authors. In [1] the definition discrete game is introduced. Next section contains the standard facts of TU game theory. The properties of core-like solutions of discrete game are described in section 3. New sufficient condition for the core existence and criterion for the coincidence of core and D-core are presented. Section 4 is devoted to stable sets of discrete game. We formulate the necessary and sufficient condition for core stability as well as the sufficient condition for stability of D-core. 2 Preliminaries A TU game G T = (N,ν) consists of players set N = {1, 2,..., n} and set function ν :2 N R, ν( ) = 0. Let GT N be the set of n-person TU games. Let also Ω = 2 N \{N, }. The cardinality of S N is denoted by S. Given x R N and S N: x(s) = i S x i, x( ) = 0. We shall write ν(i) instead of ν({i}), S \ i instead S \{i} and so on. The dual game G T =(N,ν ) of a game G T is defined by ν (S) =ν(n) ν(n \ S), S N. A game G T is called convex if ν(s)+ν(h) ν(s H)+ν(S H), S, H N, and integer if ν :2 N Z, where Z is the set of integer numbers. The sets I(G T )={x X(G T ) x i ν(i),i N} and I (G T )={x X(G T ) x i ν (i),i N} where X(G T )={x R N x(n) = ν(n)} are called the imputation set and dual imputation set of a game G T. The core C(G T ), dual core C (G T ) and D-core DC(G T ) of a game G T are the sets C(G T )={x I(G T ) x(s) ν(s),s Ω}, C (G T )={x X(G T ) x(s) ν(s),s Ω}, DC(G T )=I(G T ) \ dom(i(g T )). The stable set NM(G T ) of a game G T is defined by conditions NM(G T ) dom(nm(g T )) = (internal stability), I(G T ) \ NM(G T ) dom(nm(g T )) (external stability). The family of all stable sets of a game G T is denoted by NM(G T ). The set CC(G T )=I(G T ) I (G T )isacore cover. It is known that for a game G T GT N: C(G T ) iff G T is balanced, i.e. λ S ν(s) ν(n), λ :2 N \ R +, λ S =1,i N. S 2 N \ S 2 N \, S i C (G T ) iff the dual game G T is balanced.

3 Set-valued solutions for cooperative games with integer side payments 543 If DC(G T ) then DC(G T )=C(G T )iff ν(s)+ ν(i) ν(n), S N, (1) i N\S and, moreover, DC(G T )=C(G T ) where G T =(N,ν ), ν (S) =min{ν(s),ν(n) ν(i)} for all S N. (2) i N\S C(G T ) DC(G T ). If C(G T ) DC(G T ) then C(G T )=. DC(G T ) NM(G T ) for any NM(G T ) NM(G T ). If DC(G T ) NM(G T ) then DC(G T ) is the unique stable set. On the domain of convex game NM(G T )={C(G T )}. 3 Cores for discrete game A discrete game G D differs from G T that ν is an integer-valued function and the outcomes are integers. Since G T can be rewritten as NTU game (N,V T ) with V T (S) ={x R S x(s) ν(s)}, S N, we have G D =(N,V D ), where V D (S) =V T (S) Z S. Let GD N be the set of n-person discrete games. The most NTU results cannot be applied to G D because V D (S) Z S. But the possibility of side payments in G D allows to extend the notions from TU game theory that have not been defined for NTU game. The operator Ψ : GD N GN T will be used to compare TU and discrete game solutions, i.e. Ψ(G D ) is an integer TU game corresponding to G D. The dual game of a game G D is G D =(N,V D ), where VD (S) ={x ZS x(s) ν (S)}, S N. Imputation set I(G D ), dual imputation set I (G D ), core C(G D ), dual core C (G D ) and core cover CC(G D ) of a game G D are the intersections between corresponding sets of TU game Ψ(G D ) and integer lattice Z N : I(G D )=I(Ψ(G D )) Z N, I (G D )=I (Ψ(G D )) Z N, C(G D )=C(Ψ(G D )) Z N, C (G D )=C (Ψ(G D )) Z N, CC(G D )=CC(Ψ(G D )) Z N. D-core of a game G D consists of all undominated imputations DC(G D )=I(G D ) \ dom(i(g D )). Notice that the intersection between D-core of TU game Ψ(G D ) and integer lattice cannot be equal to D-core of discrete game G D [6]. It follows straightforwardly from above definitions and the statements proved in [1], [6], [7] that for a game G D G N D : CC(G D ) iff CC(Ψ(G D )).

4 544 Alexandra B. Zinchenko If ν is a convex function then C(G D ). If C(G D ) then the game Ψ(G D ) is balanced. C (G D )=C(G D ). If C (G D ) then the game Ψ(G D ) is balanced. Balancedness of game G T determined by (2) is not necessary for the nonemptiness of DC(G D ). A necessary nonemptiness condition for DC(G D ) is balancedness of TU game G T =(N, ν), where ν(n) =ν(n), ν(s) =ν(s) S +1,S Ω. There exist discrete games such that CC(G D ) DC(G D ). C(G D ) DC(G D ). But even convexity ν (convex function satisfies (1)) is not sufficient for the equality C(G D )=DC(G D ). There exist discrete games where DC(G D ) C(G D ) and C(G D ). In [1] the set of all undominated imputations was called the core of discrete game. Theorem 3.3 (below) shows that, in fact, these sets coincide only on very special subclass of G N D. Lemma 3.1. Let G D G N D, x I(G D), y I(G D ) and x y. Then exists such nonzero δ Z N that x = y + δ and δ(n) =0. Proof. From x y it follows δ i 0 for some i N. Further, δ Z N because x, y Z N. The condition x(n) =y(n) implies δ(n) =0. Lemma 3.2. Let G D G N D, x I(G D), y I(G D ) and y S x via some coalition S N. Then x(s) ν(s) S and 2 S n 1. Proof. If y S x then y(s) ν(s) and y i >x i, i S. Since x, y Z N then y i x i +1, i S. So ν(s) x(s) + S. If S = {i} (S = N) then x i ν(i) 1(x(N) ν(n) n) which is contradictory to x I(G D ). Theorem 3.3. For G D G N D : C(G D)=DC(G D ) iff C(G D )=I(G D ). Proof. C(G D )=I(G D ) implies C(G D )=DC(G D ) because C(G D ) DC(G D ) I(G D ). Conversely, let C(G D )=DC(G D ). If I(G D )= then C(G D )=I(G D ). Suppose I(G D ) and C(G D ) I(G D ). Determine x,y such that d(x,y )=min{d(x, y) (x, y) I}, (3) where I = {(x, y) x I(G D ) \ C(G D ), y C(G D )}, d(x, y) = i N x i y i and α denotes the absolute value of number α. By lemma 3.1, x i = y i + δ i, δ i Z, i N, δ(n) =0. (4) If δ k 2 for some k N, then the following two cases should be considered. Case (a): δ k 2. Then there exists δ l 1 with l N \k and d(x,y )= i N δ i > 2. Define h Z N by: h k = x k 1, h l = x l +1, h i = x i otherwise.

5 Set-valued solutions for cooperative games with integer side payments 545 Obviously, h I(G D ), h x and h y. If h C(G D )(h/ C(G D )) then d(x,h)=2(d(h, y )=d(x,y ) 2) contradicting (3). Case (b): δ k 2. Then δ r 1 for some r N \ k. Similarly as above we can obtain a pair of elements of I with distance less than d(x,y ). Thus δ i {0, 1, 1} for all i N. Let J + = {j N δ j = 1} and J = {j N δ j = 1}. Evidently J + = J. From (4) and x y follows that J + 1. Suppose J + > 1. Take k J + and l J. By increasing x l by 1 and respectively decreasing x k we again can derive the contradiction of (3). The equality J + = J = 1 and (4) imply x (S) y (S) 1 for all S Ω. Since y C(G D ) then x (S) ν(s) 1, S Ω. If 2 S n 1 then we have x (S) >ν(s) S. In view of lemma 3.2, x / dom(i(g D )). We obtain x DC(G D ) and x I(G D ) \ C(G D ). This contradicts DC(G D )= C(G D ). For any integer TU game the nonempty sets I(G T ) and I (G T ) are integer polytops. The cores of permutationally convex (in particular, convex) games, big boss games, games satisfy the CoMa-property, T -simplex ( S N) and dual simplex games contain a marginal vector ore an extreme point of imputation set ore dual imputation set. Therefore, the characterizations of such games give sufficient conditions for the nonemptiness of core of discrete game. Allow us to present the extended condition. Theorem 3.4. Let G D G N D. Let also k N, H (2N \ N) \ k are fixed and Ω = {S Ω S {i} for i H, S N \ i for i (N \ H) \ k}. If the following condition ν(s) ν(i)+ i S H ν(s) ν(n) holds then C(G D ). i H\S ν (i), S Ω, k / S, ν(i) ν (i), S Ω, k S. i S\H i (N\H)\S Proof. It was proved in [5] that the payoff vector x k H RN with ν(i), i H, (x k H ) i = ν (i), i (N \ H) \ k, ν(n) ν(j) ν (j), i = k, j H j (N\H)\k belongs to the core of TU game (N,v) satisfying (5). Thus x k H C(Ψ(G D)). Since x k H ZN we have x k H C(G D). Notice that x k H ext(i(ψ(g D))) for H = N \ k, x k H ext(i (Ψ(G D ))) for H = and x k H ext(cc(ψ(g D))) otherwise. (5)

6 546 Alexandra B. Zinchenko 4 Stable sets for discrete game The stable set NM(G D ) for discrete game G D is defined by conditions NM(G D ) dom(nm(g D )) =, I(G D ) \ NM(G D ) dom(nm(g D )). From properties of TU game stable sets and the statements proved in [7] follows that for a game G D G N D : If NM(Ψ(G D )) Z N then NM(Ψ(G D )) NM(G D ). It is possible that for all NM(Ψ(G D )) NM(Ψ(G D )): NM(Ψ(G D )) Z N / NM(G D ). There exist games determined by convex function ν where C(G D ) / NM(G D ). DC(G D ) NM(G D ) for all NM(G D ) NM(G D ). If DC(G D ) NM(G D ) then DC(G D ) is the unique stable set. The next theorem extends the result that each TU convex game has a unique stable set coinciding with D-core. Theorem 4.1. Let G D G N D and ν is a convex function. Then NM(G D)= {DC(G D )}. Proof. Under made assumption C(G D ). Also DC(G D ) because C(G D ) DC(G D ). Evidently the theorem holds for a two-person game. Assume that n 3. Then Ω. Since DC(G D ) satisfies the internal stability we need only to show that I(G D ) \ DC(G D ) dom(dc(g D )). This is true if DC(G D )=I(G D ). Suppose DC(G D ) I(G D ) and take y I(G D )\DC(G D ). By lemma 3.2, there exists S Ω such that y(s) ν(s) S. This implies ν(s) y(s) S where Δ S is the fractional part of ν(s) y(s). Define S Δ S 1, (6) y(l) H = arg max{ν(l) Δ L }. (7) L Ω L From y I(G D ), (6) and (7) follow 2 H n 1. Since a convex TU game is exact then there is t C(Ψ(G D )) with t(h) =ν(h). Therefore, C(Ψ(G D )) have the proper face F H = {r C(Ψ(G D )) r(h) =ν(h)}. Due to integrality of polytope C(Ψ(G D )) there is z C(G D ) F H. (8)

7 Set-valued solutions for cooperative games with integer side payments 547 Let l N \ H be fixed. Determine x Z N by x i = y i + ν(h) y(h) H Δ H, i H, z i + H Δ H, i = l, z i, otherwise. We have x i >y i ν(i) for i H, x i z i ν(i) for i N \ H, x(h) = ν(h) H Δ H ν(h). Consequently, x I(G D ) and x H y. To prove x DC(G D ) consider two cases. Case (a): Δ H = 0. Then x C(Ψ(G D )) follows from [2] (the proof of theorem 4.13). Hence x C(G D ) DC(G D ) holds. Case (b): Δ H > 0. For all T H with T Ω x(t H) = y(t H)+ T H ( ν(h) y(h) Δ H H ) (7) ν(t H) T H Δ T H >ν(t H) T H. If l/ T then (9) x(t ) = x(t H)+z(T \ H) =x(t H)+z(T H) z(h) If l T then (8),(9) > ν(t H) T H + ν(t H) ν(h) (by convexity of ν) ν(t ) T H >ν(t ) T. x(t )=x(t H)+z(T \ H)+ H Δ H >ν(t ) T + H Δ H >ν(t ) T. Further, for all T H = with T Ω hold: x(t )=z(t ) ν(t )ifl/ T ore x(t )=z(t )+ H Δ H >ν(t )ifl T. Finally, x(t ) >ν(t ) T, T Ω. By Lemma 3.2, x DC(G D ). So, we have proved that DC(G D ) NM(G D ). The last statement in listed at the beginning of this section completes the proof. The last theorem states that (contrary to TU game) the core of discrete game is stable iff it coincides with imputation set. Theorem 4.2. For G D G N D : C(G D) NM(G D ) iff C(G D )=I(G D ). Proof. If C(G D )=I(G D ) then C(G D )=DC(G D ). Hence, C(G D ) NM(G D ). Conversely, let C(G D ) NM(G D ). Then DC(G D ) C(G D ). The inclusion C(G D ) DC(G D ) implies C(G D )=DC(G D ). By Theorem 3.3, C(G D )=I(G D ).

8 548 Alexandra B. Zinchenko 5 Concluding remarks We have seen that the properties of core, dual core, core cover of discrete game and relations between them are closely related to ones for TU game. The allocations belonging to these sets (and also D-core) can be computed with combinatorial algorithms. However, the solution concepts defined by means of domination relation do not show a similar behavior. For instance, it is very difficult working with TU game stable sets, but all stable sets of discrete game can be found with the graph theory algorithms, because any kernel of dominance graph corresponds to NM(G D ). The existence conditions for a kernel of directed graph give ones for stable set of discrete game. References [1] M. Kh. Azamkhuzhaev, Nonemptiness conditions for cores of discrete cooperative game, Computational Mathematics and Modeling, 2(4) (1991), [2] R. Branzei, D. Dimitrov and S. Tijs, Models in cooperative game theory: crisp, fuzzy and multichoice games. Lecture notes in economics and mathematical systems, Springer-Verlag, Berlin, [3] V.V. Morozov and M.Kh. Azamkhuzhaev, About search of imputations of discrete cooperative game, Application of computing means in scientific researches and educational process, MSU, Moscow, 1991 (in Russian). [4] J. von Neumann and O. Morgenstern, Theory of Games and Economic Behavior. Princeton University Press: Princeton, NJ, [5] A.B. Zinchenko, A simple way to obtain the sufficient nonemptiness conditions for core of TU game, Contributions to game theory and management, St. Petersburg Graduate School of Management, 6 (2013), [6] A.B. Zinchenko, Core properties of discrete cooperative game, News of high schools, The North Caucasian region. Natural sciences, 2 (2009), 4-7 (in Russian). [7] A.B. Zinchenko, L.S. Oganjan and G.G. Mermelshtejn, Cooperative side payments games with restricted transferability, Contributions to game theory and management, St. Petersburg Graduate School of Management, 3 (2010), Received: December 21, 2013