three-person spanning tree games

MATHEMATICAL

PERGAMON Mathematical and Computer Modelling 29 (1999) 107-123

COMPUTER MODELLING

Three-Person Spanning Tree Games

B. ALIDAEE

Management and Marketing Department, School of Business

University of Mississippi, University, MS 38677, U.S.A.

ALIDAEElbus.olemiss.edu

(Received March 199’7; accepted October 1997)

Abstract-There are many situations where allocation of costs among the users of a minimum spanning tree network is a problem of concern. In [I], formulation of this problem as a game theoretic model, spanning tree games, has been considered. It is well known that st games have nonempty cores. Many researchers have studied other solutions related to st games. In this paper, we study three-person st games. Various properties connected to the convexity or no-convexity, and r-value is studied. A characterization of the core and geometric interpretation is given. In special cases, the nucleolus of the game is given. @ 1999 Elsevier Science Ltd. All rights reserved.

Keywords-Cooperative games in characteristic form, Spanning tree games, Core, r-value, Nu- cleolus, Shapley value.

1. INTRODUCTION

An n-person game in characteristic function form is a pair (N, v) where N = 1, . . . , n is the set

of players and v : 2N -+ W is a real valued function with v(0) = 0 called characteristic function.

The elements of 2N are coalitions, and the real number v(S) is the worth of coalition S.

A game is called superadditive (with 2 it is called subadditive), if

v(S) + v(T) I v(S u T), forS,TCN and SnT=0;

weakly superadditive (with 2 it is called weakly subadditive), if

u(S) 1 Cv(i), (ES

for all S C N;

essential, if

v(N) L c u(i); iEN

convex (with 2 it is called concave), if

v(S)+v(T) I V(SU T)+ v(S n T), for S,T & N.

For any positive integer m, let 7r : {1,2,. . . ,m} ---) {1,2,. . . , m} be an ordering of the integers

1,2,. . . ) m. Let II to be the set of all ordering of 1,2,. . . , n. For ?r E II and i E N, define

pT = {j E N : r(j) < r(i)}, j$ = {j E N : n(j) 5 n(i)}.

0895-7177/99/$ - see front matter @ 1999 Elsevier Science Ltd. All rights reserved.

PII: SOS957177(98)00181-Z Qpeset by 44-W

108 B. ALIDAEE

Given a r E II and a game (N, v), the marginal worth vector of v with respect to x is X~ CR”.

where

XT = v (p; u {i}) - v (p;) , for all i, k E N.

A game is permutationally convex, if there exists an order A E II, called PC order, such that

~(~US)-V(~) Iv(F;us>--v(g), for all i, k E N, with rr(i) 5 n(k) and all S c g.

Given a game (N, v), an important question is that if the grand coalition N is formed, how

should the value of v(N) be fairly divided among the n players? Many solutions have been sug-

gested to answer this question. Some of the well-studied solutions are presented in the following.

Let x E R” be a payoff vector where xi is payoff to player i E N. For x E 1” and a coalition

S C N, let x(S) = Cies xi. Given a game (N, v), define

imputation: I(v) = 2 E Rn : cxi = v(N),xi 1 v(i), for all i E N ,

iEN

core: Core (v) = {x E I(v) : s(S) 2 v(S), for all S C N},

Shapley vector 4(v) E In where

4i(V) = f c (v (Pl u Ii)) - v (P:)). aen

Given a vector x E I(v) and a coalition S c N, the excess of S is defined to be e(x, S) =

v(S) - x(S). For z E I(v), let e(x) E R 2” be the vector of excesses of coalitions written in

lexicographical order, i.e., in O(x) = (Or(z), . ..,&(x)) we have 6$(x) 2 ... 2 6$,.(z). Now for

x,y E I(V), we say e(x) < e(y), if

el(4 < cl(y)

or

4(x) < 4(Y), forllkli-1, and 4(x) < &(Y), for some i > 1.

Nucleolus of a game (N, v) is defined to be

NUC (v) = {Z E I(V) : O(x) 2 O(y), for all y E I(V)}.

It is well known that each game has one and only one nucleolus. Furthermore, if the core is

nonempty, then the nucleolus is a member of the core [2]. We present the following lemma that

serves to prove some results in later sections. The proof is easy and we omit it here.

LEMMA 1. Given a game (N,v), let x E I(v) if e(x,S) = e(x, N\S) for all S c N, then

x=Nuc(v).

Given a game (N, v), let b E IR” (upper vector) and a E Wn (lower vector) be such that

bi = v(N) - v(N - i), for all i E N,

Ri(S) = u(S) - c bk = V(S) - b(S - i), kES-i

ai = g+$W), for all i E N.

The upper and lower vectors gives, respectively, upper and lower bounds for members of the core

of a game [3]. This is given in the following lemma.

LEMMA 2. Given a game (N, v), if x E Core (v) # 0, then ai 5 xi 5 b;, for all i E N.

Tree Games 109

Let Q” to be the set of all quasi-balanced games defined by

Q” = P,v) : ai L bi, and c ai 5 v(N) 5 c bi , GN iEN

For each game (N, v) E Q”, define the r-value to be a unique point T E Rn such that -r(N) = v(N)

and ri = Xbi + (1 - X) ai, for all i E N,

where x = v(N) - a(N)

b(N) - a(N) ’

Clearly, 0 < X 5 1 and the larger the value of X is, the closer T is to the upper vector b. And the smaller the value of X is, the closer T is to the lower vector a. When X = 1, then T = b, and when X = 0, then T = a. A geometric interpretation of r-value is given in Figure 1.

=23 3 ______.-._.. 2

Figure 1.

Given a game (N, v) , define a set function g : RN + IR, called gap function, such that

s(S) = b(S) - v(S), for all S E N.

Driessen [4] introduced the gap function in order to calculate r-value efficiently.

2. SPANNING TREE GAMES

Consider a complete undirected graph G(V, E) where V is the set of nodes with IV1 = n + 1 and E is the set of edges. Associated with each edge (i, j) E E, there is a positive weight Gj, cost of connecting i to j. One of the nodes is the source and the rest of nodes are players. Let player-nodes be N = {1,2,. . . , n} and the source-node be (0). Players can receive services from the source directly, or indirectly through other players (nodes). For a set of players S c N,

110 B. ALIDAEE

let C(S) be the cost of a minimum cost spanning tree for S U (0). Such a cost function is

the characteristic function of the cost game (N, C), called minimum cost spanning tree (mcst)

game associated with G(V, E). Let cci = Q, for all i E N, and given K E IP such that

Ki 1 ci, for all i E N, then the mcst game (N, C) can easily be transformed to a profit game

(N,v) by defining v(S) = K(S) - C(S), f or all S c N. We call the game (N, V) spanning tree

(st) game associated with the graph G(V, E). The problem of cost allocation among the users

of a minimum cost spanning tree network was considered by Claus and Kleitman [5]. Bird [l]

formulated it as a game theoretic model. Further analysis of mcst games have been considered by

Granot and Huberman [6-81, Meggido 191, and Aart and Driessen [lo]. Rosenthal [ll] generalized

it to minimum spanning forest games. Relationship of these games with permutationally convex

games have been studied by Granot and Huberman [7] and Alidaee [12]. The following result is

due to Bird (11.

PROPOSITION 1. An st game (N, V) is superadditive, and an most game (N, C) is subadditive.

Consider a vector x E Core (v), Jet y E Rn such that yi = Ki - xi for all i E N. CJearJy, y is in the core of mcst game (N, C). Note that y E Core (C) iff it satisfies the following conditions:

Y(S) I C(S)7 for aJJ S c N,

Y(N) = C(N)>

Yi L 0, for all i E N.

Thus, if x E Core (v), we can easily find the corresponding y E Core (C) and vice versa. The

following result is due to Granot and Huberman [7_j.

THEOREM 1. Every minimum cost spanning tree of G(V, E) generates a vector z E IP, which is

in Core (v). Furthermore, it is a vertex of the core. For a node representing prayer i, the cost of

edge on the path to the source starting at the node is equal to yi and xi = Ki - yi, for all i E N.

I

I

I H = {x&: igN%=v(N)}

Figure 2.

EXAMPLE. Consider the graph given in Figure 2. The mcst game (N, C) and corresponding st

game (N, v) associated with the graph G are defined as follows:

C(1) = 0.5, C(2) = 1.5, C(3) = 1.2, C(12) = 1.6, C(13) = 1.7, C(23) = 2.2, C(123) = 2.6,

y = (0.5,1.1,1) is a vertex of Core (C).

Tree Games 111

Let Ki = ci for i = 1,2,3, then

V(i) = 0, for i = 1,2,3, ~(12) = 0.4, ~(13) = 0, ~(23) = 0.5, C(123) = 0.6,and

5 = (0,0.4,0.2) is a vertex of Core (v).

It is well known that the st games are permutationally convex, and a PC order can be found

by a labeling algorithm, for example by Prim’s algorithm [7]. The algorithm is presented in the

following.

LABELING ALGORITHM FOR GENERATING A PC ORDER.

Step 1. Let N* = (0) and E’ = 0.

Step 2. If N* = (0) U N, stop, Else

Step 3. Choose a and b such that a E N*, b E N\N* and c& is minimal among

{czy : XEN*,~EN\N*}.

Step 4. Let N* +- N* U {b}, E’ + E* U {(a, b)}.

Step 5. Go to Step 2.

Upon termination, the set E* c N* x N* is the edge set of minimum cost spanning tree T

generated by the algorithm. The ordering of the players that N* is generated is a PC order. The

following theorem is due to Granot and Huberman [7], however, we here give an easier proof.

THEOREM 2. Let (N, V) be an st game, then it is a PC game.

PROOF. It appears in the Appendix.

In the rest of the paper, we concentrate on three-person st games and we characterize the

solutions of the game according to the value of E defined by E = C( 123) + cl + cz + cs - (C( 12) +

C(13) + C(23)). Th ree different cases are considered as follows:

(i) E = 0, (ii) E > 0,

(iii) E < 0.

Before we formally discuss the three-person st games, we define some terms with respect to

mcst games (N, C) that will help in further analysis. All terms with a bar,“-“, is related to the

cost game (N, C). Given an mcst game (N, C), let

& = C(N) - C(N -i), for all i E N,

?ii = syti.. {C(S) - qs - i)} ) for all i E N,

g:WN +W, such that for all S C N, q(S) = C(S) - a(S).

Given K E W” such that Ki 2 ci for i E N, then for an mcst game (N, C) and the corresponding

st game (N, TJ), the following are true:

bi = Ki - bi, for i E N,

ai = Ki - ai, for i E N,

ai + ?ii = bi + 6i, for all i E N,

s(S) = S(S), for all S C N,

TV = Xbi + (1 - X)ai, for i E N,

vi = x& + (1 - X)&, for i E N,

ri = Ki -TV, for i E N,

x = v(N) - a(N) = z(N) - C(N)

b(N) - a(N) Z(N) -z(N) *

The following lemma follows easily from Lemma 2.

LEMMA 3. For an mcst game (N, C) if y E Core (C) # 0, then & 5 yi 5 T& for all i E N.

112 B. ALIDAEE

3. THREE-PERSON st GAMES

In the following, we assume N = 1,2,3 and that C(N) # cl + c2 + ~3. In the case where the

equality holds, we have a trivial situation thus not considered in the further analysis.

PROPOSITION 2. Given an st game if E = 0, then

(i) the game is convex,

(ii) ai = Ki - ci, then i& = G, for all i E N,

(iii) T(V) = 4(v) = NW (v), (iv) X = l/2.


PROPOSITION 3. Given an st game if E < 0, then

(i) the game is convex,

(ii) ?ii = ci, then ai = Ki - i&, for ail i E N, (iii) X < l/2.


PROPOSITION 4. Given an st game, if E > 0 and C(N - i) = cj + ck for some i E N, then

(i) the game is not convex, (ii) X = l/2,

(iii) if Ki = ci for all i E N, then Nut (v) = z, where

xk = w(N) - ig(N), for i # j # k.


PROPOSITION 5. Given an st game, if E > 0 and C(ij) # ci + cj, for all i,j E N and i # j,

suppose the vertices have been labeled according to the order defined by the greedy algorithm, then we have the following.

(i) If cl I c2 I ~3, then l/2 < X. Furthermore, if c3 + ~12 > Q + ~13, then l/2 < X 5 2/3.

(ii) If cl I c3 < ~2, then l/2 < X. firthermore, if q + ~23 > c2 + ~13, or if ~23 5 c13, then l/2 < X 5 213.


4. CORE OF THREE-PERSON st GAMES

In this section, we characterize the core of the st games. It is well known that imputation of

all weakly superadditive games can be characterized as follows [2].

THEOREM 3. Let (N,v) be a weakly superadditive game, then I(V) # 0 and I(w) is a compact

convex polyhedron with extreme points given as

Zzi = (V(l), V(2), . . . ,v(n)) + for all i E N,

where ei E Wn is a vector with ith components equal to 1 and all other components equal to zero. Since st games are weakly super additive games, then we can use Theorem 3 to write all points

of the core as a convex combination of the vertices. This is given in the following proposition.

PROPOSITION 6. If (N,v) is an st game, then

zECore(v), iffO<Qi< ci + C(N - i) - C(N)

Cl+Cz+C3-C(N) ’ for all i E N and (~1 + (~2 + cy3 = 1 where

x = Q~Z’ + a2z2 + CQJ~, and zi is defined by the above theorem for i E N.

TreeGames 113


PROPOSITION 7. Given an st game, if C(ij) = ci + cj for some i,j E N, then Q = (Q~,cQ,cY~)

corresponding to all vertices of the core are as follows:

( s(i) gG> mml- g(k) ’ “‘“‘> ( 0 dd, g(j)

‘go --

> 9(k) ’


The image of the core in a-space is shown in Figure 3.

( g(i) -,O,l-$), g(k) (O,O, 1).

Figure 3.

PROPOSITION 8. Given an st game (IV, v), then cr = (q, CV~, cys) corresponding to all vertices of the core are as follows:

where D = cl + c2 + c3 - C(N).

114 B. ALIDAEE

PROOF. It is similar to the proof of Proposition 7. The image of the core in a-space is shown in Figure 4.

Figure 4.

5. CONCLUSION

We have studied the three-person spanning tree games. Various properties connected to the convexity or no-convexity, and r-value was studied. A characterization of the core and geometric interpretation was given. In special cases, the nucleolus of the game was given. Relationship of these games with convex games for the case when E > 0 needs to be discussed in further research. It is an interesting question to be yet considered, if and how the results of this paper can be applied to a more general case, i.e., n-person st games.

APPENDIX

PROOF OF THEOREM 2. Suppose in finding the minimum cost spanning tree T, we used Prim’s algorithm and without loss of generality, let the final set N’ i E N”, let [i] = (1,. . . ,i}. Let j < k in N*, then we have

be numbered 1,2,. . . , n. For any

Tree Games 115

We want to show that for the order 1,2,. . , n, the condition of permutationally convexity holds. Consider a coalition 5’ c N\[k] and suppose in finding an mcst on the nodes b] u S u (0)

by Prim’s algorithm, the labeling was {1,2,. . . , j,pj+l, . . . ,~j+~}, where b] = {1,2,. . . , j} and S = {pj+i, . . . ,JI.~+~}, then we have

Suppose also that in finding an mcst on the nodes [k] USU (0) by Prim’s algorithm, the labeling was {I,&. . . , j, . . . , k,p~+l, . . . ,~k+~}, where [k] = {1,2,. . . , k} and S = {pk+~, . . . ,~k+~), then we have

CWJ~) =C([W+ pE {iE(l, ~~k,,,~){cid}. l=Pk+l ’ ..’ ’ ’ ’ ’

Combining the two equalities, we can conclude that

cm1 u s> - Cw4 5 cu.d u 9 - c M).

This proves the permutationally convexity of the game (N, v).

PROOF OF PROPOSITION 2. Since E = 0, then

ci + cj + Ck - C(N) = C(ij) f C(ik) + C(jk) - 2C(N), for i # j # k.

It follows from subadditivity of the game (N, C) that

-2C(N) + C(ij) + C(ik) + C(jk) = ci + cj + ck - C(N) > q + C(N - i) - C(N),

-X(N) + C(N - i) + C(N - j) + C(N - k) > q + C(N - i) - C(N),

C(N-j)+C(N-k)>ci+C(N), for all i E N and i # j # k.

This proves that the game (N, w) is convex. Driessen and Tijs [13) h ave shown that given a convex game, then ai = v(i), for all i E N.

This proves part (ii) of the theorem, i.e., ai = Ki - ci and & = cir for all i E N. Driessen and Tijs also showed that for any three-person game, if

v(N) + v(1) + v(2) + w(3) = 412) + 413) + v(23), (I)

then r(w) = 4(u) = Nut (v). Note that condition E = 0 is equivalent to (l), thus the result in part (iii) follows. On the other hand, from & = Q, for i E N, we have

x = E(N) - C(N) = ci+cj+crC(N)

B(N) -'i;(N) ci + cj + ck + C(ij) + C(ik) + C(jk) - 3C(N) ’

ci+cj+ck-C(N) 1

A = 2 {ci + cj + ck - C(N)} = 2’

PROOF OF PROPOSITION 3. Consider an i E N, since E < 0 then

C(ij) + C(ik) + C(jk) - 2C(N) > ci + cj + C& - C(N) 2 Ci + C(N - i) - C(N).

116 B. ALIDAEE

The second inequality follows from the subadditivity of the game (N, C). Now, it follows that the game (N, V) is convex, thus oi = v(i) and Ei = ci for i E N. Furthermore, we have

E(N) -6(N) = cl + c2 + c3 + C(12) + C(13) + C(23) - 3C(N)

> 2 {Cl + c2 + q - C(N)} = 2 {Z(N) - C(N)}. (2)

Since we have assumed C(N) # cl + c2 + cs and since the game (N, C) is subadditive, then cl + c2 + cs > C(N). Now, it follows from (2) that X < l/2.

PROOF OF PROPOSITION 4. Let C(N - i) = C(jk) = cj + ck, then using E > 0, we have

Ci + Cj + Ck + C(N) = Ci + C(jk) + C(N) > C(ij) + C(ik) + C(jk). (3)

Then, we have q + C(N) > C(ij) + C(ik) which proves the game (N,v) is not convex. Further- more, from (3), we have

ci + C(jk) - C(N) > C(ij) + C(ik) + C(jk) - 2C(N). (4)

Since for any i E N, we have

Ei = 6, + &Ils {C(S) - 5(S)},

and since for all coalitions of size ISI = 2, we have

g(S) = g(N) = -2C(N) + C(N - i) + C(N - j) + C(N - k),

then

zi = C(N) - C(N - i) + min{ci - C(N) + C(N - i),

- 2C(N) + C(N - i) + C(N - j) + C(N - k)}, for all i E N. (5)

Now, since i is chosen such that C(N - i) = cj + ck, then it follows from (4),(5) that

i& = C(N) - C(N - i) - 2C(N) + C(N - i) + C(N - j) + C(N - k),

Ei = -C(N) + C(N - j) + C(N - k) = -C(N) + C(ik) + C(ij).

It follows from subadditivity of the game (N, C) that

ck + c(ij> 2 C(N), then ck + C(ij) - C(N) 2 0.

Therefore, we have

Cj + C(N - j) - C(N) 5 Cj + C(N - j) - C(N) + {ck + C(ij) - C(N)},

-2C(N) + cj + ck + C(ij) + C(ik) = -2C(N) + C(jk) + C(ij) + C(ik), (6) cj + C(N - j) - C(N) < -2C(N) + C(jk) + C(ij) + C(ik),

Similarly, we get

ck + C(N - k) - C(N) 5 -2C(N) + C(ij) + C(ik) + C(jk). (7)

Now, it follows from (5)-(7) that

and then

Ej = Cj and ?ik =ck,

a(N) = -C(N) + C(ij) + C(ik) + C(jk).

Tree Games 117

Hence,

x = E(N) - C(N) = -C(N) + C(ij) + C(ik) + cj + Ck - C(N)

E(N) - i;(N) -C(N) + C(ij) + C(ik) + cj + ck - 3C(N) + C(ij) + C(ik) + C(jk) ’

-2C(N) + C(ij) + C(ik) + C(jk) 1

x = -4C(N) + 2 {C(ij) + C(ik) + C(jk)} = 2’

To prove (iii), note that Ki = ci, then v(i) = 0, for i = 1,2,3, and since C(jk) = cj + ck, then v(jk) = 0. It follows from E > 0, i.e., from

Ci + Cj + ck + C(N) > C(ij) + C(ik) + C(jk)

that

g(i) = -C(N) + C(jk) + ci > -2C(N) + C(ij) + C(ik) + C(jk) = g(N).

Furthermore, by subadditivity of (N, C), we have

(8)

Cj + C(ik) 2 C(N), that implies g(j) 2 0.

Similarly, we have g(k) 2 0.

Since, we also have

g(N) = Cj + ck + C(ik) + C(jk) - 2C(N) = g(j) + g(k) 2 g(j) = Cj + C(ik) - C(N),

then it follows that

g(N) L g(j), and similarly we have g(N) L g(k). (9)

Now using (B),(9), we have

Now, for all S C N, we calculate excesses with the vector x where

xi = w(N) - ;g(N), Xj = ig(j)y xk = i!?(k), for i # j # k,

e(x, N) = 0,

e(x,j) = -ig(jL

e(x, ik) = ci + ck - C(ik) - ig(k) - v(N) + ig(N)

= Ci + Ck - C(ik) - f (Ci - C(N) + C(ij)) - (Q + Cj + ck - C(N))

+ f (-2C(N) + C(jk) + C(ik) + C(jk))

= f (cj + C(ik) - C(N)) = -is(j) = e(x, j).

In a similar way, we compute that

e(x, k) = is(k) = e(x,ij)

118 B. ALIDAEE

and

e(z,jk) = w(jk) - x(jk) = -cc(jk) = -f (g(j) + g(k))

= -f (cj + C(ik) - C(N) + ck + C(ij) - C(N)) = -2C(N) + C(jk) + C(ij) + C(ik)

= -;gm

e(2, i) = g(i) + +g(N).

Since we have

e(z, N) 2 e(x, j) = e(z, ik) 2 -ig(N) = e(z, jk) 2 -g(i) + &N) = +, 4

and

e(x,N) 2 e(z, k) = e(s,ij) 2 -ig(N) = e(z,jk) L -g(i) + tg(IV) = e(z,i).

It follows from Lemma 1 that 2 = J/UC(V).

PROOF OF PROPOSITION 5. We will present a proof of part (i) and similar proof follows for part (ii). The condition E > 0 is equivalent to

C(12) + C(B) + C(23) < C(N) + cl + cz + c3. (10)

Without loss of generality, we can suppose that we have labeled the nodes according to the order defined by the greedy algorithm. Of course, this order is a PC order. Let i = 1, j = 2, and k = 3 which means cl 5 cs, ci 5 cg, and suppose cz I cs. Now, we have

C ([.A U S) - C (lil) I C (14 U S) - C ([iI), C(123) - C(12) 5 C(13) - C(l), (11)

C(N) + Cl 5 C(12) + C(13).

It follows from (10) and (11) that

C(N) + cl + C(12) + C(13) + C(23) < C(12) + C(13) + C(N) + cl + c2 + c3,

then C(23) < c2 + c3.

Since c2 L: c3 and C(ij) # ci + cj, for all i, j E N, then

~2 + C23 = C(23) < ~2 + ~3,

then we have

C23 < c3.

Now, we calculate 7ii, for all i E N. It follows from (11) that

(12)

cl - C(N) + C(23) 5 C(12) + C(13) + C(23) - 2C(N),

thus,

NOW, there are two cases to be considered (other cases are ruled out due to C(ij) # ci + cj, for all i, j E N, and the inequality (12)).

Tree Games 119

CASE 1. C(N) = cl + ~12 + ~23. It follows from this equality that ~23 5 ~13, then we have

~2 + C(N) = ~1 + ~2 + ~12 + ~23 = C(12) + C(23),

c2 + C(N) + C(13) = C(12) + C(13) + C(23),

c2 - C(N) + C(13) = C(12) + C(13) + C(23) - 2C(N),

then we have ~2 = ~2. Furthermore, we have

C(N) + c3 5 c3 +cl +cl2 + c23.

Since cl I c2 5 cs and C(ij) # ci + cj, for all i, j E N, it follows that

c(23) + C(13) = c2 + C23 + cl + ~13.

Now, there are two subcases to be considered.

CASE 1.i. cg + ~12 5 cp + cis. Using this inequality and (13) and (14), it follows that

(13)

(14)

c3 + C(N) I C(23) + C(13),

and c3 + C(12) - C(N) 5 -2C(N) + C(12) + C(13) + C(23).

Then ?is = ca. Thus,

x = V) - C(N) = c2 + c3 - cl2 - c23

qv)-6(N) 2C2 +C3 +Cl3 - 2C12 - 2C23'

l-X= c2 + cl3 - cl2 - c23

2c2 + c3 + c13 - 292 - 2023’

Suppose that X < l/2, then cg 5 cl3 that implies C(13) = cl +cs, which is a contradiction, hence l/2 < A.

CASE l.ii. ca + cl2 I cs + ~13. Now, we have

E3 = C(N) - C(12) + {-2C(N) + C(12) + C(13) + C(23)) = c7, + cl3 - ~12,

and

A=

l-X=

Suppose that X 5 l/2, then hence X > l/2.

WJ) - C(N) = 2c2 + cl3 - 2c12 - c23

E(N) -6(N) 3~2 + 2~13 - 3~12 - 2c23 ’

c2 + cl3 - cl2 - c23

3~2 + 2~13 - 392 - 2c2s.

c2 5 cl2 that implies C(12) = cl + ~2, which is a contradiction,

Suppose that X > 2/3, then ~23 5 cl3 that implies C(N) # cl + cl2 + ~23, which is a contradiction, hence l/2 < X 5 213.

CASE 2. C(N) = cl + cl2 + ~13. In this case, we have cl3 5 ~23, then

c2 + C(N) = c2 +c1 +c12 + Cl3,

C(12) + C(23) = ~1 + ~12 + ~2 + ~23.

Now, it follows from (15),(16) and cl3 5 ~23 that

0.2 + C(N) I C(12) + C(23),

(15)

(16)

c2 + C(13) - C(N) 5 C(12) + C(13) + C(23) - 2C(N).

120 B. ALIDAEE

Furthermore, we have

c3 + C(N) = c3 + Cl + Cl2 + cr3,

C(l3) + C(23) = cl + ~13 + ~2 + ~23.

(17)

(18)

Now, there are two cases to be considered.

CASE 2.i. c3 + ~12 5 c2 + ~23. It follows from this that

z3 = C(N) - (312) + (c3 + C(12) - C(N)} = cg.

Therefore, we have

x = mv) - C(N) = c2 + c3 - Cl2 - Cl3

E(N) -6(N) 2C2 + C3 + C23 - 2c12 - 2c13 ’

l-X= c2 + c23 - cl2 - cl3

2c2 + c3 + c23 - 2C12 - kr3 *

Suppose that A I l/2, then ~3 2 ~23 that implies C(23) = cs + cs, which is a contradiction, hence, l/2 < A.

CASE 2.ii. c3 + cl2 > c2 + ~23. Now, we have

Es = C(N) - C(12) + {-2C(N) + C(12) + C(13) + C(23)) = ~2 + ~23 - ~12,

and

x = E(N) - W) = 2C2 + C23 - 2C12 - Cl3

B(N) -6(N) 3~2 + 2c23 - 3~12 - 2~13 ’

l-X= c2 + c23 - cl2 - cl3

3c2 + 2c23 - 3cr2 - 2~13 ’

Suppose that X 5 l/2, then cs 5 cl2 that implies C(12) = cr + ~2, which is a contradiction, hence, X > l/2.

Suppose that X > 2/3, then crs 5 ~23 that implies C(N) # cl + cl2 + ~13, which is a contradiction, hence, l/2 < X 5 213.

PROOF OF PROPOSITION 6. Since st games are weakly superadditive, then it follows from The-

orem 3 that extreme points of the imputation are as follows:

Let z E I(V), then

9 = (v(N) - V(2) - V(3), v(2), V(3)) ,

z2 = (V(1) w(N) - V(1) - V(3), w(3)) I ,

z3 = (V(1) 42),v(N) - V(1) - V(2)). ,

2 = cxylzr + (Y222 + 0323,

Thus, components of z E W3 are

for 0 I CQ 5 1 all i E N and CY~ + ~2 + a3 = 1.

zr = err (v(N) -v(2) - v(3)) + W(1) + aav(l),

22 = CQ742) + (Y2 (u(N) - V(1) -43)) + Cr3+),

53 = CqV(3) + (rzV(3) + Q3 (u(N) - V(1) - V(2)).

Note, that we can write these in compact form as follows:

xi = (Yi (w(N) - v(i) - v(j) - v(k)) + v(i), for all i E {1,2,3} and j,k # i,

Tree Games 121

where ai+aj +cQ = 1.

Expressed in terms of the cost game (N, C), we have

xi = ~ri (K(N) - C(N) - K(N) + ci + Cj + ck) + Ki - Ci,

or

Xi = CI!~ (Ci f Cj + ck - C(N)) + Ki - Cir i = 1,2,3, O<c~<lando1+~~3+~3=1.

For x to be in the core of (N, v), we need

x(S) L 4S), for all S C N.

Since x E I(V), then the core will be characterized by

4s) 2 4% for all ISI = 2.

This means

X(ij) = Xi + Xj = (W + Crj) (Ci + Cj + ck - C(N)) + K(ij) - Ci - Cj 2 K(ij) - C(ij),

(1 - ak) (ci + Cj + ck - C(N)) - Ci - Cj 2 C(ij),

Ck - C(N) + C(ij) 2 &k (Ci + Cj + ck - C(N)).

If q + cj + ck = C(N), we have the trivial case. Thus for nontrivial case, we have Q + cj + ck - C(N) > 0. Now, it follows that

Qlk < ck+C(N-k)--c(N)

- cl+C3+C3-C(N) ’

Also for all i E N, we have ci + C(iV - i) - C(N) 5 cl + c2 + q - C(N), then

O<a. < ~+C(~-4-CW) <1

-

‘- C1+C3+C3-C(N) - ’ (17)

hence x E Core (v), if and only if (~1 + CXY:! + (~3 = 1 and ok for k = 1,2,3 satisfies (17).

PROOF OF PROPOSITION 7. Since ci + cj = C(ij), then Q + cj + ck - C(ij) = g(k). Moreover, we have

g(i) + g(j) = bi + bj - v(i) - v(j) = Ci + Cj - 2C(N) + C(ik) + C(jk) = g(N).

It follows from Theorem 3 that

g(I) xEcore(v), iff&+aj+ak=l and O<al~- g(k) ’

I E {i,j,k}, or

122 B. ALIDAEE

Now, intersection of the simplex ai+c~tark = 1 and the cube (18) represents all CYS corresponding to the core. Thus, vertices of the core can be found as follows.

On the plane cyj = g(j)/g(k), we have

g(j) f&+(Yk=l-- g(i) g(k)

and O_<q_<- g(k) ’

I_ WI --<cQ+$$ g(k) -

hence, the two vertices are

Qi = 0,

Similarly, on the plane CY~ = g(i)/g(k), we have

g(i) -- ai - g(k)’

O~~~<Z@ l-g(N)<ak<l_s(j) and a +ak=l g(i) s(k) ’ g(k) - - SW

j _P s(k) ’


g(i) Qi = s(lc)y 'Yj = 0, g(i) ak=l--- dW'

s(i) @ = dk)'

g(j) Qj=go' ak=+$.

Now on the plane cr( = 0, we have Cyj + ck’k = 1, then

Qi = 0,


cui =o, g(j) (%=9(k),

ai =O, Qj "0,

On the plane aj = 0, we have cyi + CY~ = 1, then


CYi = 0, “j = 0,

g(i) Qi=So’ O!j = 0,

This proves the proposition.

Tree Games 123

REFERENCES

1. C.G. Bird, On cost allocation for a spanning tree: A game theoretic approach, Network 8, 335-350 (1976). 2. J. Hosenmiiller, The Theory of Games and Markets, North-Holland, Amsterdam, (1981). 3. S. Tijs, An axiomatization of the T-value, Mathematical Social Sciences 13, 177-181 (1987). 4. T. Driessen, Contribution to the theory of cooperative games: The T-value and K-convex games, Ph.D.

Dissertation, University of Nijemegan, Nijmegan, The Netherlands, (1985). 5. A. Claus and D.J. Kleitman, Cost allocation for spanning tree, Network 3, 289-304 (1973). 6. D. Granot and G. Huberman, Minimum cost spanning tree games, Mathematical Progmmming 21, 1-18

(1981). 7. D. Granot and G. Huberman, On the core and nucleolus of minimum cost spanning tree games, Mathematical

Programming, 29, 323-347 (1984). 8. D. Granot and G. Huberman, The relationship between convex games and minimum cost spanning tree

games, SIAM Journal of Algebraic and Discrete Method 3, 288-292 (1982). 9. N. Meggido, Computational complexity and the game theory approach to cost allocation for a tree, Mathe-

matics of Operations Research 3, 189-196 (1978). 10. H. Aart and T. Driessen, The irreducible core of a minimum cost spanning tree game, ZOR Method and

Models of Operations Research 38, 163-173 (1993). 11. E.C. Rosenthal, The minimum cost spanning forest game, Economics Letters 23, 355-357 (1987). 12. B. Alidaee, Proof of permutationally convexity of MCSF games, ZOR Method and Models of Operations

Research 39, 243-248 (1994). 13. T. Driessen and S. Tijs, The T-value, the core and semiconvex games, Int. J. Game Theory 14, 229-248

(1985). 14. I. Curie1 and S. Tijs, Minimarge and maximarge operators, J. Optimization Theory and Applications,

277-287 (71).

three-person spanning tree games

Documents