Stability of the Average Tree Solution
for Line Graph Games
Takamasa Suzuki 631047
A thesis submitted in partial fulfillment of the requirements
for the degree of Master of Science in
Mathematical Economics and Econometric Methods
Faculty of Economics and Business Administration
Tilburg University
Supervisor:
Prof. Dr. A.J.J. Talman
Second reader: Prof. Dr. P.E.M. Borm
May, 2010
Abstract
A relatively new solution concept, the average tree solution, is studied
under a special case of communication structure on cooperative games. In
a class of line graph games, where linearly aligned agents are able to communicate only with their direct neighbors, we find a necessary and sufficient
condition for the average tree solution to be stable. While the Myerson value
is sensitive to the outcome of every coalition, the average tree solution is found
insensitive to the realization among middle players on the line. When it comes
to stability, this difference in sensitivity separates these two solutions. With
examples, we show that the average tree solution is indeed a rather stable
solution for this class of games.
Key words: cooperative game, communication structure, cycle-free game,
Myerson value, average tree solution, core
i
Contents
1 Introduction
1
2 Preliminaries
3
3 The average tree solution for line graph games
3.1 The average tree solution for cycle-free graph games . . . . . . . . . .
3.2 The average tree solution for line graph games . . . . . . . . . . . . .
8
8
9
4 Stability of line graph games
11
5 Balancedness of line graph games
15
5.1 Balancedness and minimal balancedness . . . . . . . . . . . . . . . . 15
5.2 Balancedness of line graph games . . . . . . . . . . . . . . . . . . . . 18
6 Comparison with the Myerson value
20
7 Examples
26
8 Concluding remarks
30
A Proof of the Theorem 6.1, part 2
33
ii
1
Introduction
A transferable utility game implicitly assumes that each agent can directly communicate with every other agent. Or at least there is a payoff for every possible
combination of agents. But in real life, it often happens that there exists a restriction on our way to form a coalition: sometimes you have to ask someone to reach
others. For example, a student needs to contact the housing agency who is connected
to the landlords. There are some clubs that require you to know someone who is
already a member, in order to obtain the membership. Myerson (1977) describes
these situations by introducing a cooperative game with communication structure.
In cooperative games, one of the main concerns is the way how agents allocate
a realized outcome. The one-point solution firstly appeared in Myerson (1977), also
known as the Myerson value, suggests one way to find an allocation for cooperate
games with communication structure. Another one-point solution is the position
value, described by Borm et al. (1992) and characterized by Slikker (2005). In this
paper, we study the average tree solution proposed by Herings et al. (2008). In our
study we especially focus on the games with a particular form of cycle-free graph
games where each agent is located on a line and can only form a network along with
this line. This is what we call line graph structure. Ambec and Sprumont (2002)
consider a water sharing problem among a group of agents located along a river. A
cost (time) minimizing problem in an one-machine sequencing situation is studied
in Hamers (1995) among other situations. It considers a queue of agents each of
whom has a job to be done by a machine. A profit sharing problem among cities
with a bullet train station is one of the other examples of a game with line graph
structure.
Exploiting this communication structure, we find a necessary and sufficient condition for the stability of this solution called average-superadditivity. It turns out
that this condition is indeed weaker than link-convexity, introduced by Herings et
al. (2010). Compared to convexity, which is sufficient for the Myerson value to be
stable, this condition is much weaker. Also a new characterization for balancedness
to guarantee that the game has a stable outcome is given.
This paper is organized as follows. Section 2 provides preliminary concepts and
notations regarding cooperative games with communication structure. In Section
1
3 the average tree solution for line graph games is calculated explicitly. Section
4 focuses on the stability of the average tree solution and the notion of averagesuperadditivity is introduced. In Section 5 we investigate the existence of nonempty
core in line graph games, and the following Section 6 compares all conditions with
each other. In Section 7, a series of examples are presented to complete the picture.
Concluding remarks are made in Section 8.
2
2
Preliminaries
This section is devoted to introduce some notations and concepts which are needed
for the following discussion. First we bring general expressions for a game, a graph,
and a directed graph. What follows is the concept of solutions, exemplified by the
well-known core, Shapley value, Myerson value, and the position value.
A transferable utility cooperative game is expressed by a pair (N,v) with a finite,
nonempty set of players N = {1, 2, ..., n} and a characteristic function v. We denote
S as a subset of N , called a coalition. There are 2n ways to form S, including the
empty set ∅ and the grand coalition N . We denote the set of all subsets of N as 2N
and hence S ∈ 2N . A characteristic function v : 2N → R assigns a real number v(S)
to each subset S. We assume v(S) is what the players in S can get in total if they
form this coalition. The unit of v(S) can be anything which can be quantified, such
as money or utility. We call v(S) the worth of coalition S. We assume v(∅) = 0. A
game (N, v) is called zero-normalized if we also assume v({i}) = 0 for all i ∈ N .
A pair (N, L) is called a graph, where N is a finite set of nodes and L is a set of
links between nodes, i.e., L ⊆ {{i, j} |i, j ∈ N, i 6= j}. Node i and node j are able
to communicate if {i, j} ∈ L. A subset S ∈ 2N is connected in (N, L) if there is at
least one sequence (i1 , ..., is ) for each pair of distinct nodes i, j ∈ S satisfying i1 = i
and is = j with {ih , ih+1 } ∈ L for all h = 1, ..., s − 1. A graph (N, L) is cycle-free if
it doesn’t contain a cycle in its structure. A cycle in a graph is a sequence of nodes
(i1 , ..., ik+1 ) consisting of k distinctive nodes with k ≥ 3, ik+1 = i1 and {ih , ih+1 } ∈ L
for all h = 1, ..., k. We call a sequence of connected nodes in a cycle-free graph a
path. A graph (N, L) with a finite set of nodes N is called a line graph if the set
of links is characterized as L = {{1, 2} , {2, 3} , ..., {n − 1, n}}. In a line graph, each
node can communicate only with neighbor (s), the node(s) located next to it. See
Figure 1.
A pair (N, D) is called a directed graph, with a set of nodes N and a collection
of directed links D ∈ {(i, j) ∈ N × N |i, j ∈ N, i 6= j}. If a directed link (i, j) ∈ D,
i.e., there is a link from i to j, then we call player i a predecessor of j and j a
successor of i in D. We define LD (j) = {k ∈ N |(j, k) ∈ D} as the set of successors
of j ∈ N in D. A directed path is a sequence of distinct nodes (i1 , ..., ik ) with
(i1 , i2 ), ..., (ik−1 , ik ) ∈ D. For any subset of nodes S ∈ 2N , a directed subgraph
3
Figure 1: A cycle-free graph, a line graph, and a graph with cycles
(S, D(S)) is defined in line with a graph as D(S) = {(i, j) ∈ D|i, j ∈ S, i 6= j}. For
a node i in a given directed graph, we define each node which is reachable from i
through directed paths as a subordinate of i. We write the set of subordinates of i
c
as SD (i), and SD
(i) = SD (i) ∪ {i}, the union of the set of nodes reachable from i
and player i herself. A directed graph is called a tree if there is one node with no
predecessor, called the root of the graph, and the root has a unique directed path
to each other node in the graph. A connected set of nodes in a graph (N, L) is
called a network and the collection of networks in (N, L) is denoted as C L (N ). We
say a set of networks {S1 , ..., Sk } is a partition on N if it satisfies that Si ∩ Sj = ∅
for i 6= j, Si ∈ C L (N ) for each i, and N = ∪ki=1 Si . A network that can’t add any
other node under the given links in (N, L) is called a component and the collection
of it is represented by Ĉ L (N ). Observe that Ĉ L (N ) is a partition on N . From a
graph (N, L), 2n ways of subgraphs, (S, L(S)) with L(S) = {{i, j} ∈ L|i, j ∈ S},
can be taken from (N, L) by picking any set of nodes S ∈ 2N , and as a consequence,
C L(S) (S) and Ĉ L(S) (S) are derived. Without any misunderstanding, we can denote
these as C L (S) and Ĉ L (S), respectively. Again, Ĉ L (S) is a partition on S.
A transferrable utility game combined with a communication structure characterized by a graph, called a graph game, is expressed by a triple (N, v, L). For a
graph game, a set of players S can be picked in 2n ways, but the incorporation of a
graph implies that they can form a coalition S only if S is a network. For such S,
v assigns a worth v(S). Notice that a transferrable utility cooperative game (N, v)
is assuming that every player can communicate to any other player directly. We
say that this game has a full communication structure. We assume that the grand
coalition N forms a network, that is, Ĉ L (N ) = {N }. In other words, all players are
connected to each other and can obtain v(N ) if everyone cooperates.
4
On the class of graph games, a solution is a mapping F that assigns to every
graph game (N, v, L) a set of payoff vectors. A payoff vector x = (x1 , ..., xn ) is an
n-dimentional vector which assigns payoff xi to player i ∈ N . Intuitively we can
see a payoff vector x as an allocation and xi as what player i receives. For a payoff
P
vector x, it is customary to write x(S) = i∈S xi . A solution satisfies component
efficiency if for each (N, v, L) and x ∈ F (N, v, L), it holds that x(K) = v(K) for
each K ∈ Ĉ L (N ). Since we assume Ĉ L (N ) = {N }, component efficiency boils down
to efficiency, which means that a component efficient solution leads to an allocation
where the exact amount of worth gained by the grand coalition is allocated.
The most famous solution for games with full communication is the core C(N, v),
introduced by Gillies (1953), defined by
C(N, v) = x ∈ Rn | x(N ) = v(N ), x(S) ≥ v(S), S ∈ 2N .
In words, the core is the set of weakly dominant payoff vectors that satisfy efficiency.
Once an allocation in the core is realized, players form a grand coalition and none
of the subsets of players can do better by forming a smaller coalition. This weakly
dominant property of the core, expressed by x(S) ≥ v(S) for all S ∈ 2N , is called
stability. Bondareva (1963) and Shapley (1967) find a necessary and sufficient condition for the core to be non-empty for the class of games with full communication
structure. This condition, called balancedness, is treated in Section 5. Because of
the restriction on forming networks, the core of a graph game C(N, v, L) becomes
C(N, v, L) = x ∈ Rn | x(N ) = v(N ), x(S) ≥ v(S), S ∈ C L (N ) .
Since C L (N ) ⊆ 2N , we see that C(N, v) ⊆ C(N, v, L).
The Shapley value, introduced by Shapley (1953) is one of the most famous
one-point solutions for games with full communication structure. Consider bijective
orders of the n players from N . Denote Π(N ) as the set of all n! ways of ordering.
We call each of these orderings σ ∈ Π(N ) a permutation. For each permuation
σ = (σ(1), ..., σ(n)), we calculate the marginal vector mσ (v) = (mσ1 (v), ..., mσn (v)),
5
which assigns a payoff
mσσ(k) (v) = v({σ(1), ..., σ(k)}) − v({σ(1), ..., σ(k − 1)})
to a player who is ordered at kth position, for all k = 1, ..., n. The Shapley value,
denoted as Φ(N, v), is the average of those n! marginal vectors
1 X
mσ (v).
n!
Φ(N, v) =
σ∈Π(N )
For a graph game, the Myerson value µ(N, v, L) is proposed by Myerson (1977).
He suggests a transformation of a graph game into a full communication game, called
Myerson restricted game (N, v L ), with its characteristic function
X
v L (S) =
v(K) ∀ S ∈ 2N .
K∈Ĉ L (S)
It assigns to a coalition S the sum of the worth of the components of S. The Myerson value is the Shapley value of this restricted game, that is, µ(N, v, L) = Φ(N, v L ).
Myerson (1977) proves that this solution is the only solution which satisfies component efficiency and fairness. A solution is fair if removing a line between two players
yields the same change in payoffs of both players.
Borm et al. (1992) introduce another transformation of a graph game named the
arc game. Contrary to the Myerson restricted game which assigns a worth to any
subset of players, an arc game assigns a worth to any set of links. The arc game
v
(L, rN
) corresponding to a graph game (N, v, L) assigns the worth to a set of links
as follows:
X
v
rN
(Z) =
v(T ) ∀ Z ∈ 2L ,
T ∈Ĉ Z (N )
where Ĉ Z (N ) denotes the set of components given the graph (N, Z). For the class
of zero-normalized graph games (N, v, L), the position value π(N, v, L) is defined as
πi (N, v, L) =
X1
v
Φa (L, rN
) ∀ i ∈ N,
2
a∈L
i
6
where Li = {{i, j} ∈ L | j ∈ N } denotes the set of all links of which player i ∈ N is
an endpoint. The position value is characterized in Slikker (2005).
A game (N, v) is convex if v(S ∪ T ) + v(S ∩ T ) ≥ v(S) + v(T ) for all S, T ∈ 2N .
Convex games are introduced by Shapley (1971) and the notion of convexity is due
to the characteristic that the marginal contribution, closely linked to the derivative,
is increasing with the scale. That can be seen if we write a convex game as a game
satisfying v(S ∪ {i}) − v(S) ≤ v(T ∪ {i}) − v(T ) for all S, T ∈ 2N and i ∈ N , such
that S ⊂ T ⊂ N \ {i}. The core for a convex game is the convex hull of all marginal
vectors, and hence its Shapley value, the average of all marginal vectors, is in the
core. A weaker condition for the stability of Shapley value is introduced by Inarra
and Usategui (1993) as average-convexity. A game (N, v) is average-convex if for all
P
P
S, T ∈ 2N with S ⊂ T it holds that i∈S [v(S) − v(S\{i})] ≤ i∈S [v(T ) − v(T \{i})].
They show that average-convexity is sufficient for the Shapley value to be in the
core. A game (N, v) is called super-additive if v(S ∪ T ) ≥ v(S) + v(T ) for all
S, T ∈ 2N such that S ∩ T = ∅. We see that a convex game is always a superadditive game. Super-additivity is not sufficient to make sure that the Shapley
value can be found in the core because the core of a super-additive game may be
empty. A graph game (N, v, L) is convex if corresponding Myerson restricted game
(N, v L ) is convex. Similarly, we call a graph game (N, v, L) average-convex if (N, v L )
is average-convex. A superadditive graph game (N, v, L) is defined in the same way,
but it can be more explicit as a graph game which has a characteristic function
satisfying v(S ∪ T ) ≥ v(S) + v(T ) for all S, T ∈ C L (N ) with S ∩ T = ∅ and
S ∪ T ∈ C L (N ). Since the set of restrictions imposed on the stability of a graph
game (N, v, L) is a subset of the ones on the stability of the corresponding Myerson
restricted game (N, v L ), we see that C(N, v L ) ⊆ C(N, v, L). Therefore, the Myerson
value of a graph game is in the core if the graph game is (average-)convex.
7
3
The average tree solution for line graph games
In this section, one of the most recent solution concepts, the average tree solution, is
explained. First, we pay attention to the average tree solution for general cycle-free
games, elaborated by Herings et al. (2008). Next, we narrow the class of graphs to
line graphs. Due to its simple structure, a line graph gives us an easy form of the
average tree solution.
3.1
The average tree solution for cycle-free graph games
Given a cycle-free graph (N, L), picking up one of the n nodes as a root creates a
directed graph. Obviously this is a tree. We denote T (i) as the tree obtained from
the graph (N, L) by setting the node i ∈ N as the root. We see that LT (i) (j) ⊆
ST (i) (i). Given a cycle-free graph game (N, v, L), we find one tree T (i) for each
i ∈ N . For each i ∈ N , we assign a tree solution tij , the payoff for player j ∈ N ,
given by
X
tij (N, v, L) = v(STc (i) (j)) −
v(STc (i) (k)).
k∈LT (i) (j)
This payoff is the difference between the worth of the coalition consisting of j and
all of his subordinates and the sum of the worth of each component which is yielded
by every successor of j with her subordinates. Tree solutions may change depending
on trees, and the average tree solution, AT (N, v, L), is defined as the average of all
n tree solutions,
ATj (N, v, L) =
1X i
t (N, v, L),
n i∈N j
j ∈ N.
The average tree solution is the average of n specific marginal contributions of the
game (N, v, L) chosen in such a way that it restricts σ, permutations to decide the
order, to ones that are induced from these trees derived from the graph. On the
other hand, the Myerson value is the average of all n! marginal vectors.
As we mentioned earlier, the Myerson value is the only solution that satisfies
component efficiency and fairness. For cycle-free graph games, the average tree
solution is characterized as the only solution with efficiency and component fairness
8
as its properties. Component fairness, introduced by Herings et al. (2008), is defined
as
1 X
1 X
x
(N,
v,
L)−x
(N,
v,
L\{i,j})
=
x
(N,
v,
L)−x
(N,
v,
L\{i,j})
h
h
h
h
|N i |
|N j |
i
j
h∈N
h∈N
for each link {i, j} ∈ L. For a cycle-free game, breaking a link separates the grand
network into two disconnected components. Here we denote those two separate
components as N i and N j , with i ∈ N i and j ∈ N j . Component fairness states
that the average loss of players of both components should be the same. Another
characterization of this solution is done by Mishra and Talman (2010) through the
contrast with the Shapley value.
3.2
The average tree solution for line graph games
Now we focus on line graph games (N, v, L) with L = {{1, 2} , {2, 3} , ..., {n − 1, n}}.
In a line graph, each node can communicate only with neighbor(s). For our convenience, we always express a line graph as a horizontal line. And we always assign a
number to each node indicating their relative position from the left, starting with
1 as the node which is located at the left end of this line. We denote Li and Ri
for each node i ∈ N as the set of nodes that is located left to i and right to i,
repectively. Said differently, Li = {1, ..., i − 1} and Ri = {i + 1, ..., n}. Notice that
Li ∪ {i} ∪ Ri = N . Li (and Ri ) is by itself a network and hence we also use it as a
coalition. These sets are visualized in Figure 2.
For the calculation of the average tree solution for a line graph game (N, v, L),
let us consider tij , the payoff to player j when i is chosen as the root. It can only
take three values as following:


if i ∈ Lj
 v(Rj ∪ {j}) − v(Rj )
i
tj =
v(N ) − v(Lj ) − v(Rj ) if i = j


v(Lj ∪ {j}) − v(Lj )
if i ∈ Rj .
Because of the structure of line graph games, it makes no difference on tree solutions
for player j as long as the root is located left to him. This is expressed in the first
line and occurs j − 1 times. Tree solutions give always the same payoff to him as
9
Li
Ri


1
2
i 1
i
i 1
n
Figure 2: A line graph
far as the root is chosen to locate to his right too. This is the case at the bottom
line and occurs n − j times. Player j also has one occasion to be the root, and the
corresponding tree solution is shown in the middle line. The average tree solution
for player j ∈ N , denoted briefly as ATj , is then expressed as
ATj =
1
1
v(N )+ (j −1)v(Rj ∪{j})−jv(Rj )+(n−j)v(Lj ∪{j})−(n−j +1)v(Lj ) .
n
n
The egalitarian allocation and some correction terms form this solution. Because
L1 = Rn = ∅, we can simplify terms for player 1 and player n. We call these two
players the end players. The average tree solutions for the end players become
1
1
AT1 = v(N ) − v(N \{1}) − (n − 1)v({1}) ,
n
n
1
1
ATn = v(N ) − v(N \{n}) − (n − 1)v({n}) .
n
n
Notice that for the two end players, the correction term is likely to be negative when
individual values for them are not too big. In a line graph, if joining an end player
to others doesn’t increase the total outcome very much and he can’t create much of
value by himself either, the average tree solution leads to an allocation in which he
receives a payoff lower than the average.
10
4
Stability of line graph games
For one-point solutions of cooperative games, stability is commonly checked. In case
of a line graph game (N, v, L), stability of a payoff vector x means x(N ) = v(N ) for
the grand coalition N and x(S) ≥ v(S) for all networks S ∈ C L (N ), as we already
mentioned. Stability is the same as the property that the solution lies in the core.
One of the one-point solutions which has the stability property is the nucleolus,
see Schmeidler (1969). The Myerson value lies in the core if the graph game is
convex, while Herings et al. (2008) show that superadditivity, a milder condition
than convexity, is sufficient for cycle-free games to ensure that the average tree
solution finds itself being in the core. In this section, we relax this condition for the
average tree solution of line graph games to be in the core.
First, we start with finding the explicit expression for AT (S), the sum of the
average tree solutions of players forming a network S, in a line graph game (N, v, L).
Players can form a network if they are connected, and hence we can determine the
. We denote Sab with a = 1, ..., n, b = 1, ..., n
number of possible networks as n(n+1)
2
and a ≤ b as the network that consists of the players from the one at the ath
position up to the player locating at the bth position. In short, Sab = {a, ..., b}.
Since the end players have a different expression for the average tree solution, we
n−r
n
), and AT (Sl+1
). They are the sum of the average
consider AT (S1l ), AT (Sn−r+1
tree solutions among the first l players, among the last r players, and among those
players who are between l and n − r + 1, respectively. Some calculations lead to
1
l
AT (S1l ) = v(N ) − lv(N \S1l ) − (n − l)v(S1l ) , 1 ≤ l ≤ n,
n
n
1
r
n
n
n
AT (Sn−r+1
) = v(N ) − rv(N \Sn−r+1
) − (n − r)v(Sn−r+1
) , 1 ≤ r ≤ n,
n
n
n−r
AT (Sl+1
)=
n−l−r
1
v(N ) + lv(N \S1l ) − (n − l)v(S1l )
n
n
1
n
n
+ rv(N \Sn−r+1
) − (n − r)v(Sn−r+1
) , l, r ≥ 1, l + r ≤ n − 1.
n
11
From above, we see that those correction terms from the egalitarian outcome for
network with end players go to the coalition in the middle. Similarly we can check
that the average tree solution indeed satisfies efficiency, by observing
n−r
n
AT (N ) = AT (S1l ) + AT (Sn−r+1
) + AT (Sl+1
) = v(N )
for any l, r ≥ 1 with l + r ≤ n − 1.
Next, we find a condition on the characteristic function in line graph games
such that the average tree solution satisfies stability. For a network S ∈ C L (N ),
let us denote LS and RS in a similar manner as Li and Ri . That is, for a network S = {l + 1, ..., n − r} with l, r ≥ 1 and l + r ≤ n − 1, LS = {1, ..., l} and
RS = {n − r + 1, ..., n}. If 1 ∈ S, then LS = ∅. Similarly if n ∈ S, then RS = ∅. LS
and RS are networks by themselves. We now introduce average-surperadditivity.
Definition 4.1 A line graph game (N, v, L) satisfies average-superadditivity if it
holds for any S ∈ C L (N ) that
0≤
|LS |
|S|
v(N ) − v(S) − v(LS ) − v(RS ) +
v(S ∪ RS ) − v(S) − v(RS )
n
n
|RS |
+
v(S ∪ LS ) − v(S) − v(LS ) .
n
Average-superadditivity is a convex combination of three superadditive relationships. A superadditive line graph game obviously satisfies this condition, since values in every bracket are nonnegative. Rearranging terms yields another expression
of this condition
v(S) ≤
|LS |
|S|
v(N ) − v(LS ) − v(RS ) +
v(S ∪ RS ) − v(RS )
n
n
|RS |
+
v(LS ∪ S) − v(LS )
∀ S ∈ C L (N ).
n
We immediately proceed to our first theorem.
Theorem 4.2 For a line graph game (N, v, L), the average tree solution of the game
is an element of the core if and only if this game satisfies average-superadditivity.
12
Proof ” ⇐ ” We consider three cases in calculating AT (S), such as cases when S
has no end player, one end player, and two end players. For each case we check if
AT (S) ≥ v(S) holds.
The network with two end players is the grand coalition and from efficiency,
AT (N ) = v(N ).
For networks S with one end player, without loss of generality we take some
coalition S = {l + 1, ..., n} with 1 ≤ l ≤ n − 1, which has the last player as its
member. Correspondingly RS = ∅ and LS = {1, ..., l}. Using the previous result,
we compute
AT (S) − v(S) = AT (N ) − AT (N \S) − v(S)
= v(N ) − AT (LS ) − v(S)
l
n−l
n−l
v(N ) + v(S) −
v(LS ) − v(S)
=
n
n
n
n−l
=
v(N ) − v(S) − v(LS ) .
n
By putting RS = ∅ into the average-superadditivity condition above, we verify that
this is nonnegative, and hence AT (S) ≥ v(S) for coalitions with one end player.
For the no end player cases, take some S = {l + 1, ..., n − r}, with l, r ≥ 1 and
l + r ≤ n − 1. Then we find
n−r
AT (S) = AT (Sl+1
)
1
n−l−r
v(N ) + lv(S ∪ RS ) − (n − l)v(LS )
=
n
n
1
+ rv(LS ∪ S) − (n − r)v(RS )
n
|LS |
|S|
=
v(N ) − v(LS ) − v(RS ) +
v(S ∪ RS ) − v(RS )
n
n
|RS |
+
v(LS ∪ S) − v(LS ) ,
n
which is the same expression as the right hand side of the rearranged averagesuperadditiviy. Hence AT (S) ≥ v(S) holds. These three cases cover all networks in
a line graph game, and we verify in each case AT (S) ≥ v(S).
13
” ⇒ ” Suppose it doesn’t hold. Then there is at least one network S with
v(S) >
|LS |
|S|
v(N ) − v(LS ) − v(RS ) +
v(S ∪ RS ) − v(RS )
n
n
|RS |
v(LS ∪ S) − v(LS ) ,
+
n
while the average tree solution is in the core. Since the total payoff of the average
tree solution for this network is exactly the right hand side of this formula, this
means v(S) > AT (S). This contradicts that the average tree solution is in the core.
Combining ” ⇐ ” and ” ⇒ ” concludes the proof.
Attention must be paid that we condition on the average tree solution being
in the core. Otherwise there are two cases: the average tree solution is outside of
the non-empty core, or the core is empty. Now our concern is the condition on which
the core of a line graph game is non-empty.
14
5
Balancedness of line graph games
In the previous section, we find a sufficient and necessary condition for stability
of the average tree solution, that is, the average tree solution to be in the core.
However, sometimes the core itself can be empty. In this section we investigate the
condition for stability of line graph games, i.e., the core to be non-empty. As we
shortly mentioned in Section 2, balancedness plays an important role and it is to
this topic that we now turn.
5.1
Balancedness and minimal balancedness
For the condition of non-empty core of a game, the concept of balancedness is essential. A map λ : 2N \{∅} → [0, ∞) is called balanced on N if
X
λ (S) eS = eN
S∈2N \{∅}
where, for each S ∈ 2N \{∅}, eS ∈ RN is such that
(
eSi =
1 if i ∈ S
0 otherwise
for all i ∈ N. Now consider a full communication graph with a set of nodes N . The
collection of coalitions B = {S1 , ..., Sk }, Sj ∈ 2N \{∅} , j = 1, ..., k, is balanced on N
if there exist k positive numbers λ (S1 ) , ..., λ (Sk ), we call them balancing weights,
P
such that kj=1 λ (Sj ) eSj = eN . A balanced game (N, v) is a game which satisP
fies kj=1 λ (Sj ) v (Sj ) ≤ v (N ) for every balanced collection B = {S1 , ..., Sk } with
balancing weights λ (S1 ) , ..., λ (Sk ) on N . Bondareva (1963) and Shapley (1967)
independently prove a non-empty condition for the core of a game (N, v) in terms
of balancedness.
Theorem 5.1 (Bondareva and Shapley) The core of a game (N, v) is non-empty if
and only if this game is balanced.
15
Proof Consider the following linear programming problem:
min
X
s.t.
X
xi
i∈N
xi ≥ v (Sj ) ∀Sj ∈ 2N \{∅}.
i∈Sj
The core is non-empty if the value of it is v(N ). Now consider the following linear
programming problem:
max
X
λ (S) v (S)
S∈2N \{∅}
s.t.
X
λ (S) eS = eN , λ (S) ≥ 0 ∀S ∈ 2N \{∅}.
S∈2N \{∅}
This is the dual of the previous linear programming problem. Clearly, the game is
balanced if and only if the value of the dual is v(N ). And by the duality theorem,
the value of the dual is identical to the value of the primal. Hence the core is nonempty if and only if the game is balanced.
We observe that the union of balanced collections is also a balanced collection.
This leads to the notion of minimal balanced collection. A minimal balanced collection of coalitions is a balanced collection of which no subcollection is balanced.
Indeed, every balanced collection is a union of minimal balanced collections. The
other characteristic of a minimal balanced collection is its uniquely determined balancing weights. Also, the balanced maps form a convex set. This means that the
maximum of the dual program is achieved at one of the extreme points, which is
corresponding to some minimal balanced collection. From those facts the following
is derived.
Theorem 5.2 (Bondareva and Shapley) The core of a game (N, v) is non-empty if
and only if this game is balanced for every minimal balanced collection of coalitions.
Proof See, for example, Owen (1982).
16
games with full communication
line graph games
n=2
balanced collection
3
{1,2},{1}{2},{1,2}{1}{2}
min. bal. collection
2
{1,2},{1}{2}
balanced collection
3
{1,2},{1}{2},{1,2}{1}{2}
min. bal. collection
2
{1,2},{1}{2}
n=3
43
6
{1,2,3},{1}{2,3},{2}{1,3},
{3}{1,2},{1}{2}{3},{1,2}{1,3}{2,3}
15
4
{1,2,3},{1}{2,3},
{1,2}{3},{1}{2}{3}
n=4
many(> 2000)
42
many(> 100)
8
{1,2,3,4},{1}{2,3,4},{1,2}{3,4},
{1,2,3}{4},{1}{2}{3,4},{1}{2,3}{4},
{1,2}{3}{4},{1}{2}{3}{4}
Table 1: Balanced collections
For a graph game (N, v, L), a change has to be made on the set of possible
coalitions S from 2N to C L (N ). Consequently, a balanced graph game (N, v, L) is
a game which satisfies
X
λ (S) v (S) ≤ v(N )
S∈C L (N )\{∅}
for every balanced map λ on N . Now consider a minimal balanced collection
of coalitions B = {S1 , ..., Sk }, Sj ∈ 2N \{∅}, j = 1, ..., k with balancing weights
λ (S1 ) , ..., λ (Sk ). Suppose, due to the communication structure, Sh ∈ B is not
a network. Then the set of coalitions {S1 , ...Sh−1 , T1 , ..., Tg , Sh+1 , ..., Sk }, where
Td ∈ C L (Sh ) and {T1 , ..., Tg } is a partition on Sh for 1 ≤ d ≤ g, is balanced with
the same balancing weights but λ(T1 ) = λ(T2 ) = ... = λ(Tg ) = λ(Sh ). If this is still
a minimal balanced collection of coalitions, then its balancing weights are the ones
just described due to the uniqueness. The bottom line is that since the coalition
which could have formed without the communication restriction (Sh ) spreads into
a partitoned set of smaller coalitions ({T1 , ..., Tg }), it is sufficient to check the balancedness for the minimal balanced collections of connected coalitions.
Corollary 5.3 The core of a graph game (N, v, L) is non-empty if and only if this
game is balanced for every minimal balanced collection of connected coalitions.
The numbers and sets of balanced collections and minimal balanced collections
for small n are shown in Table 1 with slight abuse of notation. For two-person
17
games, both the full and line graph communication structures identically bring three
balanced collections ({{1, 2}} , {{1} , {2}} and {{1} , {2} , {1, 2}}) and two minimal
balanced collections ({{1, 2}} and {{1} , {2}}). Adding more players increases the
numbers of these collections in both games, but because of the limitation on forming
coalitions, those numbers increase less in line graph games. For n = 4, the number of minimal balanced collections becomes 42 for games with full communication
structure, which is provided in Shapley (1967) (we add the grand coalition to the
41 sets shown in the table in the paper), while we find 8 counterparts for line graph
games. Finding minimal balanced collections reduces the conditions to consider for
non-empty core to a great extent, and this is our next focus.
5.2
Balancedness of line graph games
Taking a glance on the minimal balanced collections of coalitions in the previous section suggests that the minimal balanced collections for an n-person line graph game
are the sets of partitioned players on N . To prove this, we observe the following first.
Lemma 5.4 Let B = {S1 , ..., Sk } be a balanced collection of connected coalitions on a line graph (N, L) and for some l, 1 ≤ l ≤ k, let Sl = {al , ..., bl } with
1 ≤ al ≤ bl < n. Then there is a coalition Sm ∈ B which starts from bl + 1, i.e.,
Sm = {bl + 1, ..., bm } with bl < bm .
Proof Suppose Sm doesn’t exist. Then whenever the node bl + 1 appears in some
coalition, there is the node bl as well. Since B is a balanced collection, the sum of
balancing weights is one for the node bl + 1. Then the sum of balancing weights exceeds one for the node bl because of the coalition Sl with positive balancing weight,
which contradicts that this is a balanced set.
This observation leads to the next theorem.
Theorem 5.5 For a line graph (N, L), a balanced collection of coalitions is minimal
balanced if and only if it is a partitioned set of nodes on this line graph.
18
Proof ” ⇐ ” A partitioned set is a minimal balanced collection.
” ⇒ ” Without loss of generality, let us denote a minimal balanced collection in lexicographic order {{a1 , ..., b1 } , {a2 , ..., b2 } ..., {ak , ..., bk }}, with 1 ≤ a1 ≤ a2 ≤ ... ≤ ak .
Clearly, a1 = 1. Consider the coalition {1, ..., b1 } and denote as S1 . If b1 = n, then
S1 = N . Trivially, the grand coalition is both a minimal balanced and a partitioned collection. Next, suppose b1 < n. Lemma 5.4 says that there is at least one
coalition that starts from b1 + 1. Take this coalition as S2 = {b1 + 1, ..., bS2 } with
b1 + 1 ≤ bS2 . Again, if bS2 < n, from Lemma 5.4, we can find S3 = {bS2 + 1, ..., bS3 }
with bS2 + 1 ≤ bS3 . Repeating this until we find Sl = bSl−1 + 1, ..., bSl with
bSl−1 + 1 ≤ bSl = n. Then the set of coalitions {S1 , ..., Sl } is minimally balanced in
N and should be identical to the set {{a1 , ..., b1 } , {a2 , ..., b2 } , ..., {ak , ..., bk }}. This
means k = l, bi + 1 = ai+1 for 1 ≤ i ≤ k − 1, and bk = n. In other words, this is a
partitioned set of nodes.
Combining ” ⇐ ” and ” ⇒ ” concludes that every minimal balanced collection is a
partitioned set in N for an line graph (N, L) and conversely.
Notice that each balancing weight for a partitioned set is unity. Consequently, a
necessary and sufficient condition for stability of line graph games is stated in the
following theorem.
Theorem 5.6 The core of a line graph game (N, v, L) is nonempty if and only if
P
the inequality kj=1 v(Sj ) ≤ v(N ) holds for every partitioned collection {S1 , ..., Sk }.
Proof It follows immediately from Theorem 5.5.
According to this finding, the number of conditions we have to care for the
non-empty core of n-person line graph game is 2n−1 − 1 (the grand coalition, also
one of the minimal balanced collections of coalitions, will not create a condition).
This seems to be a much smaller number than that for a full communication nperson game. As we have already seen, even for n = 4, the difference in this number
is as substantial as 7 and 41.
19
6
Comparison with the Myerson value
Previous literatures give conditions on the average tree solution to be in the core
for more general games. Talman and Yamamoto (2008) give a condition for cyclefree games with the notion of satellites. The link-convexity condition introduced by
Herings et al. (2010) applies for all graph games. Here we compare those conditions
on line graph games. Other conditions to be considered are convexity, averageconvexity, superadditivity, average-superadditivity, and balancedness. First we start
with stating those conditions for the class of line graph games (N, v, L).
• Convexity: For a line graph game (N, v, L), convexity is defined as
v L (S ∪ T ) + v L (S ∩ T ) ≥ v L (S) + v L (T ) ∀ S, T ∈ 2N .
As mentioned before, convex games imply that the Myerson value is in the
core and hence the core is non-empty.
• Average-convexity: An average-convex line graph game (N, v, L) satisfies
X
X
[v L (S) − v L (S\{i})] ≤
[v L (T ) − v L (T \{i})] ∀ S, T ∈ 2N with S ⊂ T.
i∈S
i∈S
In Section 2, we see that µ(N, v, L) = Φ(N, v L ) ∈ C(N, v L ) ⊂ C(N, v, L) for
an average-convex game due to the result of Inarra and Usategui (1993). This
implies that the Myerson value of an average-convex graph game is in the core
and the core is non-empty.
• Superadditivity: A line graph game (N, v, L) is satisfying superadditivity if
v(S ∪ T ) ≥ v(S) + v(T ) ∀ S, T ∈ C L (N ), S ∩ T = ∅ and S ∪ T ∈ C L (N ).
That is, if two disjoint networks are connected, they will jointly get at least the
same as the sum of what they can have if they operate separately. Herings et
al. (2008) show that superadditivity is sufficient for the average tree solution of
a cycle-free graph game to be in the core, therefore this condition guarantees
a nonempty core for line graph games.
20
• Satellite-convexity: Talman and Yamamoto (2008) introduce the concept
of satellite for cycle-free graph games. For a cycle-free graph game (N, v, L),
satellites for a network S ∈ C L (N ) are defined as the components after taking
away nodes in S from the graph (N, L). Borrowing the notation from the paper, we define the set of edges that connect one node in S and another one outside of S as δ(S). In other words, δ(S) = {e ∈ L|e = {i, j} , i ∈ S, j ∈ N \S}.
Also denote T S (a) as a satellite, the largest network which connects to S
through an edge a ∈ δ(S). Then they propose the condition regarding the
satellites, we call it satellite-convexity for convenience. A cycle-free graph
game (N, v, L) is called satellite-convex if
1. v(N ) ≥ v(S) +
P
v(T S (e)) ,
P
e∈δ(S)
2. v(N \T S (a)) ≥ v(S) +
e∈δ(S)\{a}
v(T S (e)) ∀ S ∈ C L (N ), a ∈ δ(S).
This means that for each network S and satellites associated with S, the sum
of the worth of those subsets must not exceed the value of the grand network,
and also for every subgame in which one of the satellites is removed from the
graph, this relationship should hold. In their paper, it is shown that for a
satellite-convex cycle-free graph game, AT (N, v, L) ∈ C(N, v, L). They also
show that this condition is weaker than superadditivity.
For a line graph game, picking any network S generates at most two satellites.
Following the previous notation, they are LS and RS . Then satellite-convexity
for a line graph game (N, v, L), shortly (SCLG), becomes
1. v(N ) ≥ v(S) + v(LS ) + v(RS ),
2. v(S ∪ LS ) ≥ v(S) + v(LS ),
3. v(S ∪ RS ) ≥ v(S) + v(RS ) ∀ S ∈ C L (N ).
• Link-convexity: Herings et al. (2010) introduce the concept of link-convexity.
A game with communication structure (N, v, L) is called link-convex if
v(S) + v(T ) ≤ v(S ∪ T ) +
X
K∈C L (S∩T )
for any S, T ∈ C L (N ) that satisfy
21
v(K)
– S\T, T \S, and (S\T ) ∪ (T \S) are nonempty networks,
– N \S or N \T is a network.
For a line graph game, making (S\T )∪(T \S) a nonempty network asks that S
and T are adjacent to each other. Consequently S ∩ T = ∅. It is also required
that at least one of S and T includes an end player. All in all, we modify the
link-convexity for a line graph game (N, v, L), shortly (LCLG), as
v(S) + v(T ) ≤ v(S ∪ T )
for any S, T ∈ C L (N ) that satisfy
– S and T are adjacent to each other,
– at least one of S and T includes an end player.
They also show that for any link-convex graph game, AT (N, v, L) ∈ C(N, v, L).
• Average-superadditivity: A necessary and sufficient condition for a line
graph game that its average tree solution is in the core. For a line graph game
(N, v, L) the condition is
0≤
|LS |
|S|
v (N ) − v (S) − v (LS ) − v (RS ) +
v (S ∪ RS ) − v (S) − v (RS )
n
n
|RS |
+
v (S ∪ LS ) − v (S) − v (LS ) ∀ S ∈ C L (N ).
n
• Balancedness: A necessary and sufficient condition for non-empty core of a
game. For a line graph game (N, v, L) this becomes
k
X
v(Sj ) ≤ v(N ) for each partitioned set {S1 , ..., Sk } on N .
j=1
Those conditions lead to the following theorem.
Theorem 6.1 On the class of line graph games the following properties hold:
1. Every convex game is average-convex.
22
2. Every average-convex game is superadditive.
3. Every superadditive game is link-convex.
4. Link-convexity coincides with satellite-convexity.
5. Every link-convex game is average-superadditive.
6. Every average-superadditive game is balanced.
Proof
1. Obvious.
2. See Appendix.
3. Both conditions require v(S) + v(T ) ≤ v(S ∪ T ), but for different subsets S
and T . Any S and T which satisfies the first condition of link-convexity is already
a subset of those S and T that should be considered for superadditivity. The second condition restricts this subset even further, and hence superadditive line graph
games are link-convex.
4. ” ⇐ ” Suppose it does not hold. Then there exist adjacent sets S, T ∈ C L (N )
with T containing {1} or {n} such that v(S) + v(T ) > v(S + T ) while satisfying
(SCLG). So T = LS or T = RS . Obviously there are no such S and T because
(SCLG) requires super-additive relations for S and LS and for S and RS . Hence
(SCLG) implies (LCLG).
” ⇒ ” Suppose it does not hold. Satisfying (LCLG) leads to v(S∪LS ) ≥ v(S)+v(LS )
and v(S ∪ RS ) ≥ v(S) + v(RS ) because S and LS (also S and RS ) are adjacent while LS (also RS ) has an end player in it. Now consider there exists a
way to divide all players on a line graph into three subsets LS , S and RS such
that v(N ) < v(S) + v(LS ) + v(RS ), while satisfying (LCLG). This is not possible because, again putting it in mind that both LS and RS have an end player,
v(S) + v(LS ) + v(RS ) ≤ v(S ∪ LS ) + v(RS ) ≤ v (S ∪ LS ) ∪ RS = v(N ). Hence
(LCLG) implies (SCLG).
23
5. Suppose a line graph game (N, v, L) is link-convex. Then every term in the
right hand side of the average-superadditivity condition is nonnegative. Hence for
any satellite-convex line graph game, the average-superadditivity condition holds.
6. Suppose it does not hold. Then there is some partition {S1 , ..., Sk } such that
Pk
j=1 v(Sj ) > v(N ) while satisfying the condition for average-superadditivity. If
P
k = 1, S1 is the grand coalition and kj=1 v(Sj ) = v(N ). Therefore balancedness is
satisfied. If k = 2, the two sets can be denoted as S and LS with 1 ∈ LS . Then
{v(N ) − v(S) − v(LS )}. This
the average-superadditivity condition becomes 0 ≤ |S|
n
is equivalent to v(S) + v(LS ) ≤ v(N ) and the balancedness condition is satisfied.
If k ≥ 3, let {S1 , ..., Sk } be ordered such that for every h = 1, ..., k it holds that
S
Sk
LSh = h−1
i=1 Si and RSh =
i=h+1 Si . From the average-superadditivity condition,
we have
|Sh |
v(N ) − v(LSh ) − v(RSh )
n
|LS |
+ h v(Sh ∪ RSh ) − v(RSh )
n
|RSh |
v(LSh ∪ Sh ) − v(LSh )
+
n
!
!!
h−1
k
[
[
|Sh |
=
v(N ) − v
Si − v
Si
n
i=1
i=h+1
!
!!
k
k
[
[
|S1 | + ... + |Sh−1 |
+
v
Si − v
Si
n
i=h
i=h+1
!
!!
h
h−1
[
[
|Sh+1 | + ... + |Sk |
+
v
Si − v
Si
.
n
i=1
i=1
v(Sh ) ≤
24
Summing up these inequalities of Sh with 2 ≤ h ≤ k − 1 cancels out several terms
and yields
v(S2 ) + ... + v(Sk−1 ) ≤ v(N ) − v (S1 ) − v (Sk )
|S1 |
v (S1 ) + v (N \S1 ) − v (N )
+
n
|Sk |
+
v (Sk ) + v (N \Sk ) − v (N ) ,
n
followed by
k
X
v (Si ) − v (N )
i=1
|Sk |
|S1 |
v (S1 ) + v (N \S1 ) − v (N ) +
v (Sk ) + v (N \Sk ) − v (N )
n
n
≤ 0,
≤
since both terms are nonpositive due to the result when k = 2. This contradicts the
assumption that the left hand side is positive and hence such a partition doesn’t
exist.
Therefore we can also conclude the following.
Corollary 6.2 Any average-superadditive line graph game (N, v, L) has a nonempty core and its average tree solution lies in it.
25
7
Examples
Previously we see the strictness of conditions by means of ordering. As the last
part of our study, we try to quantify the strictness in terms of the number of inequalities allowed to be ignored. Direct comparison with respect to the number is
possible for some conditions (convexity, superadditivity, and link-convexity) since
they have the same form of inequality (v(S) + v(T ) ≤ v(S ∪ T )) to be satisfied for
different set of coalitions. Also we present counter examples to show that the ordering we obtained in the Theorem 6.1 is only one-way and not the other way round.
Example 7.1 Convex line graph game
S
v(S)
1
0
2
0
3
0
4
0
12
1
23
1
34
1
123
3
234
3
1234
10
Consider the line graph game above. This game is convex, and joining to bigger line associates a larger marginal contribution. The corresponding solutions are
µ = ( 25
, 35 , 35 , 25 ), π = ( 19
, 41 , 41 , 19 ) and AT = ( 21
, 39 , 39 , 21 ). They are close to
12 12 12 12
12 12 12 12
12 12 12 12
each other, but the position value gives a higher payoff to middle players and the
Myerson value gives a lower payoff to them. Verify that all of them are in the core.
Example 7.2 Average-convex but not convex line graph game
S
v(S)
1
0
2
0
3
0
4
0
12
1
23
1
34
1
123
6
234
6
1234
10
We increase the worth associated with the two three-person networks. This game
doesn’t satisfy convexity (take S = {1, 2, 3} and T = {2, 3, 4}) but average-convexity.
19 41 41 19
We calculate µ = ( 12
, 12 , 12 , 12 ), π = ( 16
, 44 , 44 , 16 ) and AT = ( 12
, 48 , 48 , 12 ). Com12 12 12 12
12 12 12 12
pared with the previous example, all of these values allcate less to the end players.
The magnitudes of reduction differ, and the average tree solution reduces the value
of the end players most. We see that all of them are in the core.
26
Example 7.3 Superadditive but not average-convex line graph game
S
v(S)
1
0
2
0
3
0
4
0
12
1
23
1
34
1
123
9
234
9
1234
10
Following the previous example, we increase the same worth further so that averageconvexity doesn’t hold anymore (take S = {1, 2, 3} and T = {N }), but this game
3 57 57 3
, 12 , 12 , 12 )
still satisfies superadditivity. The average tree solution becomes AT = ( 12
and is still in the core. The Myerson value and the position value coincide, µ = π =
( 13
, 47 , 47 , 13 ), and this is not in the core ( 107
= µ(123) = π(123) < v(123) = 9). In
12 12 12 12
12
general, given an n-person line graph game, we need to check
n−1
X
k=1
k+
n−2
X
k+
k=1
n−3
X
n−(n−1)
X
k + ... +
k=1
k=
n−1
X
1
k=1
k=1
2
k(k + 1) =
n
(n − 1)(n + 1)
6
inequalities for superadditivity. For 4-person game, this number is 10. For convexity,
n(n − 1) +
n−2
X
l(n − l − 1) =
l=1
n
(n − 1)(n + 4)
6
inequalities have to be added further. For n = 4, a total of 26 inequalities have to
hold for a convex line graph game.
Example 7.4 Link-convex but not superadditive line graph game
S
v(S)
1
0
2
0
3
0
4
0
12 23
1 -100
34
1
123
9
234
9
1234
10
Suppose the two players in the middle have a bad match, leading to a very low
outcome if a network is made only by them. Example 7.3 represents this kind of situation with v(23) = −100. Now superadditivity doesn’t hold because v(2) + v(3) >
v(23). We see this game still satisfies link-convexity. In fact, since neither player 2
27
nor player 3 is an end player, v(2) + v(3) ≤ v(23) is not necessary to be considered
when it comes to link-convexity. The average tree solution is the same as that of the
previous example, AT = ( 14 , 19
, 19 , 1 ), and still in the core. The Myerson value again
4 4 4
, − 29 , − 29 , 19
), which yields negative
coincides with the position value, µ = π = ( 19
2
2
entries and obviously this is not in the core. The average tree solution is stable
against bad matches between middle players while the other two solutions are not.
For a 4-person game, v(2) + v(3) ≤ v(23) is indeed the only inequality we can drop
from the ones for superadditivity while satisfying link-convexity. For an n-person
game with n > 3, such networks consist of n − 3 sets of 2-person networks, n − 4
sets of 3-person networks etc., and one (n − 2)-person network. By adding up those
numbers, we find the total number of networks without any end players as
n−3
X
n−(n−1)
k + ... +
k=1
X
k=1
1
k = (n − 3)(n − 2)(n − 1).
6
Hence the number of inequalities to be satisfied for link-convexity is the difference
between this number and the number of inequalities we need for superadditivity,
n−1
X
k+
n−2
X
k = (n − 1)2 .
k=1
k=1
As we see, this number is 9 = 10 − 1 for n = 4.
Example 7.5 Average-superadditive but not link convex line graph game
S
v(S)
1
0
2
2
3
0
4
0
12 23
1 -100
34
1
123
9
234
9
1234
10
Here player 2 can create some worth by himself, which is higher than what he can
yield with player 1. Notice that this game is not zero-normalized game1 . It is
not a link-convex game due to v(2) + v(1) > v(12). The average tree solution is
1
For not zero-normalized cycle-free graph game, the position value can be defined in several
ways. Here and in the next example we calculate the position value of the strategically equivalent
zero-normalized game.
28
still unchanged and stays at AT = ( 14 , 19
, 19 , 1 ), because v(2) doesn’t appear in its
4 4 4
calculation. Observe that this allocation is again in the core. The Myerson value
µ = ( 55
, − 23
, − 29
, 57 ) and the position value π = ( 54
, − 21
, − 30
, 57 ) allocates slightly
6
6
6 6
6
6
6 6
more to player 2 at the expenses of neighbors, comparing to the previous example.
The payoff to player 4 is unchanged in both cases. Still these allocations are out
of the core. Although average-superadditivity is a weaker condition than previous
ones, since this is the convex combination of some superadditive relationships, we
can’t state how many inequalities can be dropped from those which are needed for
link-convexity. It is rather the condition on the magnitude. Some of the inequalities
don’t need to hold if others are satisfied with great margin.
Example 7.6 Balanced but not average-superadditive line graph game
S
v(S)
1
0
2
8
3
0
4
0
12 23
1 -100
34
1
123
9
234
9
1234
10
Now player 2 can enjoy his high output by himself if he doesn’t form a network with
anyone. This game has nonempty core but in the core, player 2 should receive at
least 8. For example, (0, 8, 1, 1) is an element of the core. With the same argument
, 19 , 1 ). This alloas before, the average tree solution is unchanged at AT = ( 41 , 19
4 4 4
cation can’t attract player 2 and is not a stable allocation. The robustness of the
average tree solution against the changes in the worth of middle players now puts
, − 11
, − 35
, 57 ) and the position
itself out of the core. The Myerson value is µ = ( 49
6
6
6 6
value π = ( 45
, − 36 , − 39
, 57 ). As we see already, we need 2n−1−1 inequalities for
6
6 6
the balancedness. This number is 7 for 4-person line graph games. But we cannot
just compare the number of inequalities here, because many of these inequalities
for balancedness are different from the superadditive inequality, i.e., more than two
elements are involved.
29
8
Concluding remarks
In this paper, the average tree solution is studied under a particular type of communication structure, namely line graphs. One of our findings is the concept of
average-superadditivity, a sufficient and necessity condition for this class of games
that their average tree solutions are in the core. This condition is less restrictive than
link-convexity, which is weaker than superadditivity. Through examples, we see that
the average tree solution is insensitive against the worth generated by networks of
middle players, while the Myerson value and the position value take those values into
account. This leads to the average tree solution stable against lower worth formed
by middle players, but meantime it is not stable against too high worth formed by
middle players. We also show that balancedness, a necessary and sufficient condition for non-emptiness of the core, is weaker than average-superadditivity. Finally
we prove that a collection of networks is minimal balanced if and only if it is a
partitioned set.
A major limit of this study is the specific communication structure of line graph
games. Further extensions to other class of graph games should be made. One of
the concerns is its stability. For instance, in our examples, the average tree solution
is always in the core when the Myerson value and the positon value are in the core.
We left it unanswered whether this holds for any (line) graph game and this would
be an interesting study.
30
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32
A
Proof of the Theorem 6.1, part 2
Proof by induction on the size of S with S ⊂ T , S, T ∈ C L (N ).
Step (1) |S| = 1 : Denote S = {i}. Without loss of generality, suppose superadditivity doesn’t hold on Q, {i} , Q ∪ {i} ∈ C L (N ), that is,
v(Q ∪ {i}) − v(Q) − v({i}) < 0.
Then average-convexity doesn’t hold for S = {i} and T = Q ∪ {i} since
X
[v L (T ) − v L (T \{j})] −
j∈S
X
[v L (S) − v L (S\{j})] = v(Q ∪ {i}) − v(Q) − v({i}) < 0.
j∈S
Step (2) Suppose superadditivity holds for every S with |S| < k.
Step (3) |S| = k : Denote S = {i1 , ..., ik } with ih = ih−1 + 1 for h = 1, ..., k. Without
loss of generality, suppose superadditivity doesn’t hold on Q, S, Q ∪ S ∈ C L (N ) with
Q ⊆ LS and Q ∩ S = ∅, that is,
v(Q ∪ S) − v(Q) − v(S) < 0.
(∗)
Then average-convexity doesn’t hold for S = {i1 , ..., ik } and T = Q ∪ S since
X
[v L (T ) − v L (T \{i})] −
X
[v L (S) − v L (S\{i})]
i∈S
i∈S
=
L
[v(T ) − v (T \{i1 })] − [v(S) − v L (S\{i1 })]
+[v(T ) − v L (T \{i2 })] − [v(S) − v L (S\{i2 })]
...
+[v(T ) − v L (T \{ik })] − [v(S) − v L (S\{ik })]
=
[v(Q ∪ S) − v(Q)
− v(S\{i1 })
] − [v(S)
− v(S\{i1 })
]
+[v(Q ∪ S) − v(Q ∪ {i1 }) − v(S\{i1 , i2 }) ] − [v(S) − v({i1 }) − v(S\{i1 , i2 }) ]
...
+[v(Q ∪ S) − v(Q ∪ {i1 , ..., ik−1 })
33
] − [v(S) − v({i1 , ..., ik−1 })
]
<(∗)
v(Q) − v(Q)
+v(Q) − v(Q ∪ {i1 }) + v({i1 })
...
+v(Q) − v(Q ∪ {i1 , ..., ik−1 }) + v({i1 , ..., ik−1 })
≤
0,
due to the superadditive assumption up to |S| < k. This concludes that whenever
a line graph game is average-convex, it must be superadditive.
34