Myerson (1977) considers graph-restricted games in which connected components in a graph ..... Prentice-Hall, Englewood Cliffs, NJ. Rosenthal, E. C.: 1985 ...
EDWARD
C. ROSENTHAL
COMMUNICATION
AND ITS COST
IN G R A P H - R E S T R I C T E D
GAMES
A B S T R A C T . We consider cooperative games which are modified by subordination to communication networks. Two players i and j will be able to cooperate only if they pay for the cost w,j of their communication link (i, j ) . Coalitions of players are treated similarly and a new characteristic function form game is developed thus. We also examine incentive for players to cooperate in such situations as well as a related index of a player's communicative strength.
Keywords: games, graphs, communication.
1. I N T R O D U C T I O N
There has been a small, but growing literature in each of the two, thus far divergent fields of study we try to merge in this paper. On one hand, several authors have studied cooperative games played on networks. These efforts began with work by Claus and Kleitman (1973), and were continued by Bird (1976), Megiddo (1978), and Granot and Huberman (1981), on minimum cost spanning tree games. Other recent work on related issues has appeared by Granot and Huberman (1982), who studied permutationally convex games, and by Granot and Granot (1986) and Rosenthal (1987a), who examined spanning forest games. All of the above references fall into the broader category of mathematical programming games, in which a group of players operates collectively in optimizing an objective function subject to a set of constraints to which the players are tied. For examples of the latter type see Shapley and Shubik (1972), Owen (1975), Dubey and Shapley (1984), Topkis (1983), Kalai and Zemel (1982a, 1982b), and Rosenthal (1985). In the other arena, certain authors have begun to show interest in how issues of communication among the players affects cooperative games. Myerson (1977) considers graph-restricted games in which connected components in a graph partition the player set. The characteristic function is additive over these components. Owen (1986) studies graphrestricted games developed in the same manner and shows that the Theory and Decision 25 (1988) 275-286. 9 1988 by Kluwer Academic Publishers.
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E D W A R D C. R O S E N T H A L
Shapley value is easily computable when the communication graph is a tree. G r o f m a n and Owen (1982) apply an extension of the Banzhaf index to the problem of measuring degree of centrality in social r~tworks. Rosenthal (1987b) considers graph-restricted games where arc weights indicate strengths of player bonds and considers the Shapley value of games thereby modified. What we shall examine herein are similar graph-restricted games in which the arc weights w0 indicate costs of opening communication between i and j. Once in full communication among all members, a coalition may then take advantage of any benefit superadditivity in the characteristic function will yield. In the next section we illustrate the construction of such games. Section 3 offers a look at incentive for players to cooperate, given such structures, and Section 4 considers a related notion, a graph-theoretic measure of a player's communicative strength.
2. A C O M M U N I C A T I O N
FOREST GAME
Let N = ll ..... n] be a finite set of players, and let v : 2 N ~ R be a characteristic function which is such that (1)
v(~)=0,
and
(2)
v(SuT)>~v(S)+v(T),
SnT=f~.
We shall term the pair (N;v) a cooperative n-person game. The real number v(S) is assumed to represent the economic possibilities open to coalition S c_N, regardless of the actions of the complement coalition
N-S. We define an undirected graph (or, simply, a graph) G = (N,A) to be a set N of nodes together with a set A of arcs which is a collection of unordered pairs (i,j), i , j ~ N . Arcs ( i , j ) e A with i=j are called loops; if more than one arc exists in G with the same node pair, we say G has parallel arcs. A graph G ' = (N',A') is a subgraph of G if N' c_N, A' c_A, and all arcs ( i , j ) e A ' are such that i , j e N ' . For a subset S c N o f nodes, we define a subgraph GS=(S,AS), called the subgraph induced by S, in which AS= [i,j e S: ( i , j ) c A I. Now let weights wij be present on all arcs
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( i , j ) c A . We call a graph G together with such weights a network. To ease notation, for So_N, let w ( G S ) = ~ (i,j)~AS wij. We say a subgraph G' spans G if all nodes in G are included in G ' . Consider a sequence of nodes Iil, i2 ..... i,,]. We say that a path is a sequence of arcs (ij, ij+ 1) for j = 1..... n - 1. A cycle is a path whose initial node and terminal node coincide. A graph G = (N,A) is connected if, for every pair of nodes i , j ~ N , there exists a path between i a n d j . We now define a connected, acyclic subgraph of a graph G as a tree and we call an acyclic subgraph of G a forest. Note that each component of a forest is a tree, a component being a maximal connected subgraph. Now let G = (N,A) be a graph without loops or parallel arcs, defined on the player set N o f a game. Let weights wij>O be present on all arcs (i,j) c A . As a model of a communication structure, we will say that i and j are able to communicate if and only if the cost wij is provided to include the link (i,j). Our aim is to show how the economic potential of a coalition is affected by its underlying communication structure. To do this we shall transform the game (N;v) into a new game (N;u) in which we will allow players in coalition S c_N to cooperate and achieve their full worth v(S) only if the subgraph G s is connected. That is to say, the players must be in communication with each other in order to realize their economic possibilities. In addition to this, we shall see that the players also must be willing to pay for the cost of the communication network a m o n g them to be built; clearly the network will be a minimal one. To formally define our graph-restricted communication game, given the original game (N;v) and the graph G=(N,A) with weights wij>O as above, ( i , j ) e A , let G ' be a subgraph of G s, and then let
v(S)-w(G') (3)
if G ' is connected and spans S
v' ( s ) =
~V(Ck)--w(G')
Ck components of G '
for all S c_N. Now let (4)
u ( S ) = m a x [v'(S), 0]
It is straightforward to show that u is superadditive.
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EDWARD
C. R O S E N T H A L
We now give an example of the relationship between functions v and b/.
Example 1. Consider the "divide the dollar" game (N;v) where N = I1,2,3] and v is such that v(N) = 10, v(/1,2]) = v([l,3]) = v(I2,3]) = 10, v(~l~) = v(~2~) = v(~3/) = v(Q) = 0 , and subordinate this game to the network below in which w/j is given on all arcs:
u([1]) = u([2/) = u([3]) = u ( Z ) = 0. u([1,Z])=max ~0+0, 1 0 - 5 ] = 5 . u([1,31)=max ~0+0, 1 0 - 11 = 9 . u(I2,3]) = 0 . u(~l,2,3])=max 1 0 + 0 + 0 , 0 + 0 , 5 + 0 , 9 + 0 , 1 0 - 6 ] = 9 . In this game the absence of communication between players 2 and 3 and the relatively inexpensive link between players 1 and 3 will force a significant change in the characteristic function. It is evident that the transformation u captures the economic as well as the communicative aspects of the situation. It should be clear that the structure G ' that arises for a coalition S will be a minimum weight forest. If the weights wij are sufficiently small, then G ' will be a minimum weight spanning tree for S. What remains for us to do is to state an efficient algorithm that will, for all S c_N, obtain the forest G ' and then the characteristic function u(S). First, for S c_N, let T s denote the minimum cost spanning tree on G s. Very efficient algorithms exist to compute T s (Papadimitriou and Steiglitz 1982). Consider the deletion of an arc (i,j) from a tree. As is well
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known, deletion of one arc ( i , j ) f r o m a tree separates the tree into two subtrees. Call these T s and Tj-s. For GI = T s,
compute v'l (S).
For Gz= TiSw Tjs, compute v'2(S). If V'l (S) < v'2 (S), then arc ( i , j ) is termed useful. For a given graph F, let E = [(i,j)eF: (i,j) is useful]. For each e eE, let de= v'2(S)-V'l (S). We define the arc which maximizes de over e E E to be the most useful arc for F. This yields the kernel of an algorithm for obtaining u(S), given S c_N. We will start with tree T s and delete, one at a time, all useful arcs, until we are left with a minimal communication structure that maximizes v' (S). A L G O R I T H M to obtain, for So_N, G' and u(S). Step 0: Step 1: Step 2:
Let k = 0 and find the minimum cost spanning tree T s of GS. L e t L o = T s , l e t F k = T s, and l e t D = T s. Let k = k + l . Let F k = IFk_ 1 - D I w L k _ I. C o m p u t e v' (S) = ~
v(Cm) - w(Ftc)
m
Step 3:
Step 4:
where Cm are the components of F k. Find most useful arc ( i , j ) in F~. Call D the component that (i, j ) is contained in. F o r m new components F s and Fjs by the deletion of (i,j) f r o m D. Let L k =FSuF~k and go to Step 2, else STOP. No more useful arcs exist. Therefore F k is minimal. Let G ' = F k and then find u(S) from Step 3 and (4).
P R O P O S I T I O N 1. The above algorithm terminates with the correct forest F k. and characteristic function u(S). Proof. It is clear that the algorithm terminates. Suppose that u(S) does not maximize v'(S) over all G'. Let Fk be the graph generated by the algorithm and let G* be the optimal structure, Fie :~ G *. Either G * = T s, or G* is a forest of T s.
280
EDWARD C. ROSENTHAL
Since F~4: G * either (i)
there exists ( i , j ) ~ G * such that (i,j)~F~,
(ii)
there exists ( i , j ) e F k such that ( i , j ) r
or
Looking at condition (i), if we apply the algorithm to G*, ( i , j ) is found to be useful, since G* is a subset of T s and the algorithm scans all its arcs. This contradicts the optimality of G*. Now we have G* c_Fk, and we consider (ii), noting that the algorithm terminated with F k. Since (i,j) postulated by (ii) is not useful, we must have v ~ . (S) < v ~ (S), a contradiction. [] What the above proposition is tantamount to saying is that, in our algorithm, it is sufficient to remove one edge at a time in obtaining F a n d u(S). As we don't need to consider removal, then, of subsets of arcs, of which there are an exponential number, we are assured of having an efficient algorithm with which to find u(S). All that remains for us to do is to attempt to characterize the graphrestricted games (N;u) in some way. First let us define the core of a game (N;v) as the set of all vectors x which satisfy
(5)
2 Xi "~"v(N)
and
i~/V
(6)
~ x i~
v(S)
for all S c_N.
t~S
It is natural to investigate the relationship between the core of a game (N;v) and the core of an associated graph-restricted game (N;u). The following examples show that no obvious relationship holds between the two.
Example 2. Core (N;v) is not empty but core (N;u) is empty. Let N=[1,2,3~. Let v(Ill) = v(I2]) = v([31) = v ( ~ ) = 0; v([1,2~) = v([1,31) = v([2,3]) = 2; v(N) = 3. Clearly x = ( 1 , 1 , 1 ) is the unique core point. Now let G=(N,A) be as below:
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A N D ITS C O S T
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Let 0~
