European Journal of Operational Research 51 (1991) 405-411 405 North-Holland

Theory and Methodology

Cooperative games arising from network flow problems

Edward C. Rosenthal Department of Management Science and Operations Management, School of Business and Management, Temple University, Philadelphia, PA 19122, USA

Abstract: We give a new class of mathematical programming games without side payments that have nonempty cores. In these games, players control nodes or arcs in a network and participate in optimizing an objective function while attempting to share the costs or returns. Disjoint paths in the network correspond to alternative production routes or streams of jobs that are concurrently processed and yield costs or returns along the way; the problem of fair cost allocation or profit sharing motivates the formulation of a cooperative game. Our results extend work on games with side payments, and are relevant for situations in which the various players involved have dissimilar and nonlinear utilities for the generated returns. For some games of this type efficient algorithms exist to allocate costs or revenues among the players.

Keywords: Games, networks

1. Introduction

Network flow problems, in general, have been of considerable interest to the operations research community for quite some time. A significant feature is that network programs frequently model projects whose cost or throughput is to be opti- mized. Disjoint paths through the network may represent alternative production routes or separate chains of activities to be concurrently processed, where the activities themselves may involve costs or profits. After optimizing the aggregate, or col- lective, objective function, an issue of great con- cern still remains: how should the costs or profits be allocated to the various activities to equitably recompense those agents who have carried them out? Under the motivation, then of cost allocation,

Received March 1989; revised August 1989

it becomes sensible to turn to cooperative game theory to provide answers to such problems.

This type of relationship between mathematical programming and cooperative game theory is well known. In the initial work in this field, Shapley and Shubik [11] considered a cooperative game induced by a problem of optimally assigning buyers to sellers of indivisible units. They showed that the problem was equivalent to the classical assignment problem and that the solution to the dual program was essentially the core point in the game. Duality was similarly exploited by Owen [7], who constructed a cooperative game from a standard LP. Other authors, including Bird [1], Granot and Huberman [3], Kalai and Zemel [4,5], Rosenthal [9], and Topkis [12] constructed games of a combinatorial nature which they modeled and solved as combinatorial optimization problems. All of the above articles treat games with side payments.

0377-2217/91/$03.50 ~'~ 1991 Elsevier Science Publishers B.V. (North-Holland)

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406 E. C. Rosenthal / Cooperative games arising from network flow problems

In this paper we extend several of the previous results to games without side payments. Our mod- els are canonical network flow problems in which we consider players to be situated at the nodes or arcs of the network. Subsets of the players, or coalitions, control subgraphs induced by their members and benefit from flows they control. We give a class of games for which the core is non- empty, that is, no subset of the players will be inclined not to cooperate with the others by break- ing off on its own. Identification of classes of games with nonempty cores has great importance in cooperative game theory, since these games exhibit incentive compatibility for all players. We also demonstrate that results on side payment games by Shapley and Shubik [11], and Kalai and Zemel [4] are specializations of our most general model, which may be thought of as a transship- ment or minimum cost flow game. The approach of Dubey and Shapley [2] differs from ours, al- though in the side payment case they reach similar results.

Extension of the transferable utility results in the literature to the nontransferable utility case in this paper has an important justification. In treat- ing cost allocation/profit sharing problems, it may well be the case that the different players involved indeed have dissimilar and nonlinear utilities for the costs or revenues generated in the joint ven- ture. As an example, one could consider the European Community after 1992, in which joint projects will frequently occur, yet due to the dis- parate economic levels and objectives in the vari- ous participant nations, their utilities may indeed be nonlinear, rendering a transferable utility solu- tion to their profit sharing unsatisfactory.

The remainder of this paper is organized as follows. In Section 2 we give notation and defini- tions. Section 3 discusses network models in which flows are conserved at the nodes, and also relates games in which players control nodes to games with players at the arcs. In Section 4 we extend the work in Section 3 to network flows which are not conserved.

2. Notation and definitions

2.1. Game theory

Let P = ( 1 , 2 . . . . . p} be a set of players. Let R e be the p-dimensional Euclidean space, and let

R e + = { x ~ R P : x>~0} and R++P - - { x ~ R P : x > 0}. A p-person game without side payments (an NTU game) is a pair (P; V) such that the func- tion V: 2 e ~ R P satisfies:

V ( S ) is closed, nonempty, and convex; (2.1)

i f x ~ V(S) and y,~<x, for a l l i e S ,

then y ~ V(S) ; (2.2)

if S(~ T = ~ , then V( S ) ~ V ( T ) c V( S U T) ;

(2.3)

let B be a compact subset of Re;

X ~ V(P) if and only if x ~< y for some y E B.

(2.4)

We call V the characteristic function, and we call S _c p a coalition. Let int(X) be the interior of X. An imputation for an NTU game (P; V) is a point x such that x ~ V(P) , x eli int(V(P)), and x i f~ int(V({i})) for all i ~ P. An important solu- tion concept for NTU games is known as the core; vectors x ~ R e are in the core of an NTU game if and only if

x ~ V ( P ) , (2.5)

and

there is no S _ P, S 4= ~(, and y E V(S)

such that y~ > x, for all i ~ S. (2.6)

Let P = {1 . . . . , p } be a set of elements, and let C = { S l . . . . . S,~ } be a collection of subsets Sj of P, j = 1 . . . . . m. We say that C is balanced if there exist weights (2,1 . . . . . X,~) such that for all i ~ P, Z j : ~ s ) k = 1 wi th0~<~j~<l , j = l . . . . . m.

An NTU game (P; V) is balanced if the inclu- sion f ' ) j~ IV(S j ) c V (P) holds for any balanced collection C.

In light of the fact that the core of an NTU game corresponds to those imputations which are undominated, and because the core may in general be empty, the following theorem is important.

Theorem 1 (Scarf [10]). A sufficient condition for an NTU game (P; V) to have a nonempty core is that it is balanced.

2.2. Network flows

Let G = (N, A) be a directed graph. We call G together with a vector of capacities c ~ RA+ a

E. C. Rosenthal / Cooperative games arising from network flow problems 407

network. Let x ~ R A be a vector of f lows on the arcs A of the network; if 0 ~ x e <. c e for all e ~ A, we say x is feasible• For any subset E _c A of arcs, we let x ( E ) = E e ~ E X e .

We say that for any node v ~ N, the arcs leav- ing v are denoted by 8(v ) = (e : e = (v, y), y N }, and the arcs entering v are denoted by 6 ' ( v ) = { e : e = ( y , v), y ~ N } . We call arcs e such that (e = (v, v) for v ~ N loops, and arcs e 1 and e 2 such that e 1, e 2 = (u, v), u, v ~ Nparal le l arcs. F o r G = ( N , A ) a n d a r c e = ( u , v ) ~ A , t h e t a i l o f e, t (e ) , is u and the head of e, h(e) , is v. The graph G N' = ( N ' , A ' ) for N ' _ c N when A ' is the set of all (u, v ) ~ A such that u, v ~ N ' is called the subgraph induced by N ' . Similarly, the graph G A ' = ( N ', A ' ) for A ' c A , when N ' is the set of all nodes v ~ N such that v = t ( e ) or h ( e ) for all e ~ A' , is called the subgraph induced by A'.

Finally, we will denote a utility function for a player i ~ N by u~: R --+ R, and we will say that u is concave on the domain F R if for every xp x 2 ~ F a n d e v e r y a , 0 ~ < a ~ l ,

U(OtX 1 + (1 -- O~)X2) > @bl(X,) "+ (1 - ~ ) u ( x 2 ) •

(2.7)

which satisfy y, ~ u,(x~) for all i ~ S, where x ( S ) is any feasible flow in the subnetwork G s, and where each player has a utility function u,: R --+ R+. To ensure that (P ; V) is well defined, we need to impose

u, is concave and cont inuous for all i ~ P. (3.2)

Condit ion (3.2) forces each V ( S ) to be convex. For a proof see Owen [8]. It is s traightforward to check the other conditions for a well-defined game.

Let us call (P ; V) the N T U f low game with conservat ion .

Theorem 2. The N T U f low game with conservation has a nonempty core.

Proof. We show the game is balanced. Let { S l . . . . . S,, ) be a balanced collection with

corresponding weights (X l , . . %,.). Let y ~ f ' l" • , .1=|

V(Sj) . For each coalition Sj _c P, there is a feasi- ble flow (possibly zero) x ( S / ) for which y, ~< u,(x ,(Sg)) for all i ~ Sa. For each player i, let

2 i = E l : i ~ S ,~k j x i (S i ) •

Lemma I. Flow 2 --- ( 2 i ), i ~ P, is feasible in G.

3. NTU games with conservation of flow

3.1• The model and existence proof for the core

Consider the network consisting of a directed graph G = (N, A) without loops, and capacity vector c ~ Ra+. In addition, distinguish two vertices s, t ~ N, a source and sink, respectively, such that

{e : e = ( y , s ) , y ~ N } =~J

and

{e: e = ( t , v) , y G N } =fJ.

Let the system (3.1) represent the set of feasible flows x from s to t, viz.,

x ( g ( v ) ) - x ( 6 ' ( v ) ) = 0 for all v ~ N - {s, t } ,

0 ~< x ~< c. (3.1)

An N T U game arises as follows. Let the set P = { 1, 2 . . . . . p } of players coincide with the arc set A of the network. For each coalition S ___ P, define V ( S ) to be the set of all y = (Yl . . . . . Yp)

Proof of Lemma 1. It is sufficient to prove

0 <~ ~, ~ c, for all i ~ A, (3.3)

and

E E L =o fora, eN-(s,t}. i~8(v) i~8 ' (v )

(3.4)

Proof of (3.3) is straightforward. Notice that (3.4) indicates conservat ion of flow

at all intermediate nodes. Consider any such v N - { s , t ) and flows x , (S j ) . Each x(S]) is feasi- ble in the SFinduced subnetwork, which implies conservation at v, i.e.,

E x,(s,)- Z x,(S,)=0 iGS(v) i~8 ' (v )

=x, E E x,(S,)=0 i~8(v) i~3'(v)

E E Xix,(S )=o i~8(v) i~8 ' ( v )

= E E X,x,(Si) j: iGS i i~8(v)

408 E. C. Rosenthal / Cooperative games arising from network flow problems

- E E j: i~S~ iES'(o)

= E Z iE6(v) i~6"(v)

i.e., ~ is feasible, proving Lemma 1.

By concavity of each u,,

u,(Z) E j: i~S I

Since Ei:i~s,~i = 1, we obtain y, <~ u , ( ~ ) ~ y V(P).

The game is therefore balanced and, by appli- cation of Theorem 1, has a nonempty core.

3.2. Incorporation of an objective function

We generalize the maximum flow game of Kalai and Zemel [4]. Let G = ( N , A), c ~ R ~ , and s, t ~ N be given as before, again let p = A , and consider the mathematical program

max ~. x i 1~6(s)

subject to

- x ( 8 ' ( v ) ) = 0

O<~x<~c.

for all v ~ N - {s, t } ,

(3 .5)

In this case the players who control the arc set A are collectively involved in maximizing the s, t- flow. For subset S_c A of players, define V(S) to be the maximum s,t-flow in the subgraph G s. Loosely speaking, only those coalitions S such that G s contains an s , t -path have V ( S ) > 0. To ensure that maximal flows in the subnetworks G s are indeed in the best interests of the individual players, we must restrict their utility functions as follows:

each u, is monotonical ly nondecreasing. (3.6)

Condit ion (3.6) may be taken as an axiom of utility theory. Note that (3.2) and (3.6), taken together, force each u i to achieve a maximum when the arc controlled by i is saturated. Without this condition, maximal arc flows would not yield core points.

e

G ' G

For each S ~ P, then, as before, define V(S) to be the set of all y = (Y l , . . . , Yp) which satisfy Yi <~ u,(xi) for all i ~ S, where x ( S ) is a feasible flow in G s, and where each players has a utility function ug as before. We call the game (P ; V), arising from (3.5) as described, the maximum flow NT U game.

Theorem 3. The maximum flow N T U game has a nonempty core.

(Proof is in the Appendix.) Theorem 3 is a generalization of a result of

Kalai and Zemel [4], extending a max imum flow game with side payments to one without side payments.

3.3. "Edge player' and "vertex player" games

Often the natural way of construct ing a mathematical p rogramming game may involve let- ting P = N, not A. We now demonst ra te simple t ransformations by which we may consider N T U games where players control flow through vertices, not arcs. We call these games vertex player games and arc player games, respectively.

First we introduce graph homeomorphism. Consider two directed graphs G = (N, A) and G ' = ( N ' , A ' ) . Call the replacement of e = (a , c), a, c ~ N, by the series e 1, e 2 in which e 1 = (a , b), e 2 = (b, c), b ~ N, an insertion and call the con- verse operat ion a shrinking. We say that G and G" are homeomorphic if they are isomorphic after a finite number of insertions a n d / o r shrinkings.

To create an equivalent vertex player game from an arc player game, we construct the follow- ing homeomorph ic graph: insert a vertex into ev- ery arc (Figure 1).

To convert f rom vertex player to arc player games: replace all v ~ N by a pair of new vertices vl, v 2 E N with an arc e ' = (v 1, v2). Let 6 ( v 2 ) = 6(v) , and 8(v l ) = 8 ' ( v ) (Figure 2).

In both transformations, the new player set is only a subset of the vertices or arcs. The families of games are isomorphic in that they preserve the same player set and the same characteristic func- tion. However, Kalai and Zemel [4] have observed

Q Figure 1.

e I e 2 Q,Q

E. C. Rosenthal / Cooperative games arising from network flow problems 409

Figure 2.

~T

>

that such games, where the players constitute a proper subset of the vertices or arcs, in general have nonempty cores. Therefore, a claim about arc player games will not automatical ly hold for vertex player games, which is why we treat the latter separately in the next section.

4. NTU games without conservation of flow

Herein we discuss vertex player, not arc player, games, We extend our model of Section 3 to include cases in which flow is not conserved. The constraint system has a constant r ight-hand side and incorporates the combinatorial structures of min imum cost circulation problems, transship- ment problems, and assignment problems.

Once more let G = (N, A) be a directed graph without loops together with c ~ RA+.

Consider the following system of flows x ~ RA:

x ( 6 ( v ) ) - x ( 8 ' ( v ) ) < ~ b , , fora l i v ~ N

O<~x<~c. (4.1)

We define an N T U game without conservation as follows. Let P coincide with the vertex set N. For each S _c P define the feasible set V(S) to be the set of all y = (Yl . . . . . yp) arising from feasible (possibly zero) flows x ( S ) , satisfying y¢, <~

u v ( x ( ~ ( v ) ) ) for all v ~ S, where x ( S ) is any feasi- ble flow in G s, and each player v has a utility funct ion uo" R ---, R+ which is concave and con- tinuous. We call the N T U game arising in this fashion the N T U game without conservation.

the min imum cost circulation problem is a gener- alization of the maximum flow problem. More- over, it is well known that the min imum cost circulation problem, the t ransshipment problem, and the assignment problem are all equivalent in the sense that any one can be t ransformed into another in polynomial time. To interpret the con- straints (4.1) as t ransshipment or min imum cost flow problems, b,, = 0 ~ v is a t ransshipment vertex; b , , > 0 ~ v is a supply vertex; b , . < 0 ~ v is a demand vertex. If we assign a cost a,/ to each arc e = (i, j ) in G, we obtain the classical mini- mum cost flow problem by minimizing ax subject to (4.1). To see how a side payment version may be reached from our N T U models, we let the utility u,, at each vertex v equal the sum of flow leaving v, i.e., u , . ( x (~ (v ) ) ) = x ( ~ ( v ) ) . Then create a side payment version by stipulating, for S _c P, that

v(s)= E u,,= E x(8(v)). t '~S v~S

Optimizing the objective function gives a Pareto- optimal solution (a core point; p roof that the side payment version is balanced is along the lines of Owen [8], pp. 294-295).

Acknowledgment

The author would like to thank an anonymous referee for several comments that considerably improved an earlier version.

Theorem 4, The N T U game without conservation has a nonempty core. Appendix

Proof. We show the game is balanced, and pro- ceed as in the proof of Theorem 2. The reader is again referred to the Appendix.

Theorem 4 is a generalization of Theorem 2 in the same way that, in combinatorial optimization,

In this Appendix we provide proofs of Theo- rems 3 and 4.

Proof of Theorem 3. Let x be the max imum flow from s to t. Define a cut to be a minimal set of arcs which, when deleted from the network, will

410 E.C. Rosenthal / Cooperative games arising from network flow problems

disconnect s and t. Let s ~ X and t ~ II, where X ~ Y = ,O, X t.) Y = N. Define the capacity of a cut to be the sum of the capaci t ies of the arcs d i rec ted from X to Y, which we term ' fo rward . ' We call a cut K of m in imum capac i ty a minimum cut. F r o m s t anda rd ne twork flow theory, we know

that the forward arcs in a m i n i m u m cut are sa tura ted by the flow x (Lawler [6]).

Cons ider the coal i t ions S for which V ( S ) n P R + + = 0, for all such S, there exists at least one

p layer i cont ro l l ing an arc in K. F r o m condi t ions (3.2) and (3.6), we know that the i-th c o m p o n e n t of V ( S ) is maximized.

Now, let y be an imputa t ion der ived from the m ax i mum flow by let t ing the p layers i who con- trol the arc set K receive their m a x i m u m utility. If there exists an impu ta t ion z such that z, >y , , we ob ta in a con t rad ic t ion to the flow x being maxi- mal. Therefore, y is a core imputa t ion .

Proof of Theorem 4. Take any ba lanced col lect ion of vertex p layer sets Sj, j = 1 . . . . . m, with corre- spond ing weights ~ = (3k I . . . . . ~ , , ) , for which Y~/ . . . . s A, = 1 for all u ~ N .

Note that e ~ 6 (v ) for a unique v ~ N. Hence, we use the no ta t ion e ~ Sj to denote v ~ Sj for the

vertex v = t (e) . Therefore,

j: y e s I j: eESj

Let x j be any feasible flow in the subgraph induced by Sj.

Let y ~ fqj~=1V(Sj). For each coal i t ion Sj ___ P, there is a feasible (poss ib ly zero) flow x j for which y~ <~ u ,~(xq6(v)) ) .

Let Xe = ~j : e ~ S,)~jX~" We show £ = (Xe), e ~ A, is feasible, for which it suffices to demons t r a t e

0 ~ 2 e ~< c e, (4.2)

and

~ ( 6 ( v ) ) - . ~ ( 8 ' ( v ) ) ~ < b,, for all v ~ N. (4.3)

It is s t ra ight forward to verify (4.2). To show (4.3), we have by assumpt ion feasibil-

ity of each xL Therefore, x J ( 8 ( v ) ) - xJ (6 ' ( v ) )~< b,, for all j = 1 . . . . . m. Al though different coali- t ions Sj use di f ferent arcs, if they have vertices in

common, we know {j: e ~ S j } = { j : v~Sj } for v = t(e) . Now, let

a j = x Y ( 8 ( v ) ) - x J ( 8 ' ( v ) ) , j = l . . . . . m,

and let a = ( a j ) , j = 1 . . . . . m. Since E )~j = 1, then aT), ~< max aj <~ b U. Therefore , .~ as cons t ruc ted is feasible in G. This impl ies that u ~ ( . ~ ( 8 ( v ) ) ) ~ V(P) , i.e., is a feasible imputa t ion .

But

eeS(v) j: v@S~

j: eeSj ee8(v)

as each )~j factors out over the flows for each coal i t ion Sj

> E buy E x:, J: e~Sj e~8(v)

by concavi ty of u v, and hence

u v ( x ( ~ ( v ) ) ) >~Yv as E X j = 1

so that y ~ V(P). Therefore the game is ba l anced and, by Scarf 's

theorem, has a n o n e m p t y core.

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