1. Introduction
In this note, we generalize the 
value [1] [2] for efficient values for TU (transferable utility) games with a cooperation structure (graph on the set of players; CO games and CO value) like the Myerson value [3] , the position value [4] , and the average tree solution [5] . This value determines payoffs in weighted hierarchical games. Modelling hierarchies in games of cooperative game theory is one step to analyze firms in the framework of cooperative game theory. A first step for this was done by Kalai and Samet [6] using ordered partitions and weights to model hierarchies. Another approach comes from Winter [7] using a sequence of bargaining components [8] to model hierarchies. These values can model levels in the sense of Lazear and Rosen [9] , Carmichael [10] , and Prendergast [11] . The problem of both approaches is that they could not model a clear manager-subordinate structure. Such a structure is a main element of hierarchies in firms [12] [13] . Another approach to model hierarchies might be games with an undirected graph/network on the set of players [3] -[5] . The problem is: all players are symmetrical in the graph; all players are in one level. The paper by van den Brink [14] uses a directed graph/permission structure on the set of players to analyze the effects of hierarchies on players’ payoffs. The basic idea is that a player needs approval from all predecessors. The approach was outlined in Gilles, Owen, and van den Brink [15] . The axiomatization is carried out by van den Brink and Gilles [16] .
In the modeling of van den Brink [14] , single dominance relationships, i.e. the influence of a predecessor on the direct successors, are equally strong. It is, however, quite a plausible assumption that the relations among the players can have different levels of strength (e.g. they account for different leadership styles). This possibility is especially necessary when mapping corporate hierarchies. With this in mind, Casajus, Hiller and Wiese [1] [2] introduce the 
value. The authors use a weighted directed graph to model hierarchies. The basic idea for the 
value includes two elements. In order to create the output, all players work symmetrically together. As a first step, the result generated is distributed to the players according to the Shapley value [17] . In a second step, the weighted hierarchy reallocates a certain fraction of these payoffs. A player 
on the lowest hierarchical level gives the fraction 
from his Shapley payoff to his direct predecessor 
Player 
gives the fraction 
from all (gross) payoffs he receives (his Shapley payoff and the fractions from his successors) to his direct predecessor, etc. The weighted hierarchy has only allocation effects.
This is our starting point. We modify the 
value in order to model the coordinating tasks of hierarchies. Within the framework of cooperative game theory, this is a new development starting from the papers done by Gilles, Owen, and van den Brink [15] , van den Brink and Gilles [16] , van den Brink [19] , and van den Brink [14] .
The paper is outlined as follows. In Section 3, we generalize the 
value. Now the players do not work symmetrically together in order to create the output (first step). The problem of coordination of players is taken into account. The player who coordinates the other players is honored, if the coordination yields cooperation benefits. This approach is based on Hiller [18] . Section 4 concludes the paper. The paper starts with some preliminaries.
2. Preliminaries
A TU game is a pair 
is the set of players. The coalitional function 
assigns every subset 
of 
a certain worth 
reflecting the economic abilities of 
i.e. 
such that
A game 
is called symmetric if a function 
exists such that 
for all nonempty sets
, where 
denotes the cardinality of 
A symmetric game is called monotone if 
for all 
is increasing.
A TU value is an operator 
that assigns pay off vectors to all games
. One important value is the Shapley value. In order to calculate the players’ payoffs, rank orders 
on 
are used. They are written as
where 
is the first player in the order, 
the second player etc. The set of the orders is denoted by
; 
rank order sexist. The set of players before 
in rank order 
and player 
is called 
Thus, given a player 
and a rank order
, the marginal contribution of 
is defined by
The Shapley value is the average of the marginal contributions taken over all rank orders of the players [17] :
(1)
In this note, we use one example to demonstrate the calculations.
Example 1 We assume a game with 
and
(2)
This game is symmetric and monotone. The 
are calculated by:
(3)
As in Gilles, Owen, and van den Brink [15] , van den Brink and Gilles [16] , van den Brink [19] , van den Brink [14] , and Casajus, Hiller and Wiese [1] [2] , the permission structure is a mapping 
It maps to each player 
those players that are direct successors of
. 
can be interpreted as directed graph [20] . 
identifies the direct successors of 
with 
The players in 
are called direct predecessors of
; 
As usual in the literature, it is assumed that the hierarchy has a tree structure [21] [22] . These structures satisfy two conditions:
there is one player
, such that 
and 
and
for every player 
we have
.
In a tree structure, a path 
in 
from 
to 
is a sequence of players 
with
, 
and 
for all
. A path can be interpreted as a chain of commands between 
and
, whereby 
is a predecessor of
. The set of successors of 
is denoted by
. Analogously, we denote the set of
’s predecessors by
.
Beyond this hierarchy
, weighted relations between players are considered. The vector 
maps to every player 
a weight
, 
For 
we have 
If a vector maps all players the same weight
, except 
i.e. 
for all 
we also denote the vector by
. With 
we denote a vector that maps to all 
the weight zero, 
The players 
retain their initial weights, 
To simplify your notation, we write 
instead of
.
A weighted hierarchical game is a tuple 
A weighted hierarchical value is an operator 
The 
value is one weighted hierarchical value. All players 
with 
respectively all players in the path
, get a fraction of
’s Shapley payoff. Any player 
gets from
’s Shapley payoff the fraction [1] [2] :
(4)
With this formula, the wH-payoff of player 
is determined by [1] [2] :
(5)
Example 2 In addition to example 1, we assume 
and 
Figure 1 illustrates the weighted hierarchy. For the players’ 
payoffs, we have:
(6)
Casajus, Hiller and Wiese [1] [2] present eight axioms for the axiomatization of the 
value. They can be categorized into two groups. The first four axioms—additivity (A), efficiency (E), weak symmetry (WS), weak null player (WN)—are modified Shapley axioms. They ensure that the Shapley payoffs results if 
The last four axioms—gross net (GN), splitting (SP), isolation (IS), independency (ID)—control the reallocation from the bottom to the top of the hierarchy.
Axiom 3 (A) For all coalitional functions 
and
,
(7)
A players’ payoff in the game 
should be the sum of the payoffs he achieved in two separate games 
and 
Axiom 4 (E) It holds
(8)
The worth 
is distributed among the players.
Definition 5 (S) Players 
are called symmetric, if 
for all 
I.e., both players 
and 
have the same marginal contributions to coalitions that do not contain them.
Axiom 6 (WS) If 
are symmetric then
(9)
This axiom says that symmetric players get the same payoffs if all weights are zero.
Definition 7 (N) Player 
is a null player, if 
for all 
Axiom 8 (WN) If 
is a null player then
If all weights are zero, a null player gets the payoff zero.
Axiom 9 (GN) For the relation between gross and net payoffs, the equation
(10)
holds.
The gross payoff 
is the value that 
would receive if he did not have to pay anything to his direct predecessor. If this value is multiplied with
, the net pay off results.
Axiom 10 (SP) For the relation between player
’s pay off and player
’s payoff, 
, the following equation holds:
(11)
This axiom says that reducing the weight between 
and 
to zero (
is changed to zero), reduces the gross payoff of player 
(left-hand side of equation) in the same way as it raises the payoff of player 
(right-hand side).
Axiom 11 (IS) For any player 
holds:
(12)
This axioms states that player 
get the same payoff in two cases. In the first case, his weight is zero and the weights of his direct successors are zero too. In the second case, all players’ weights are zero.
Axiom 12 (ID) If a player participates in two weighted hierachical games 
and 
with 
for all 
then
(13)
The games 
and 
have the same set of players
, the same coalitional function 
and the same hierarchy 
Further, the weight of 
and the weights of all players 
are the same in both games. The axiom states that 
s payoff should be the same in both games.
Now we will introduce some preliminaries for CO games (for a literature review on CO games see Slikker and van den Nouweland [23] ). This games account for coordination tasks of managers in firms. The set of possible pair wise links between players is called 
whereat 
and 
respectively (or 
and 
respectively) is the direct link between 
and
. A cooperation structure on 
is a graph 
with 
A CO game 
is a game 
together with a cooperation structure
. A CO value is an operator 
that assigns payoff vectors to all CO games. The most popular CO values are the Myerson value [3] , the position value [4] , and the average tree solution [5] . From 
we construct 
in the following way: 
for all 
The tree structure is inherited from 
to 
Hence, we have a CO game 
Example 13 In our example, we have 
We will exemplify the CO generalization of the 
value with the Myerson value. To calculate the Myerson payoffs, we need some more preliminaries. The graph partitions the set of players into components 
This partition is denoted by 
Connected (directly or indirectly) players are in one component. Each player is in one component; 
denotes the component containing player
. Now, we can introduce the restricted coalitional function 
[3] :
(14)
The worth of a coalition 
is the sum of the worth of its components. In the case of 
we have
for 
The Myerson payoff of player
is determined by:
(15)
Example 14 In our example we have
(16)
Hence, we obtain:
(17)
Since player 3 coordinates the other players, he gets a higher payoff.
3. Generalization for CO Games
In this section we generalize the 
value for CO games. First, we will introduce an axiom that is used to simplify the original axiomatization [18] :
Axiom 15 (zero weights invariance L, ZWIL) If 
then 
for all 
This axiom says that the players get the payoff
, if all weights are zero. 
could be any efficient CO value. Using axiom ZWIL, we have Corollary 16 The 
value is the unique weighted hierarchical value satisfying GN, SP, IS, ID, and ZWIL.
The formula to calculate the 
payoff of player 
is:
(18)
Hence it is now possible to model coordinating tasks within the 
framework.
Using the 
value for
, we get the 
payoffs:
(19)
For our example, we obtain:
Example 17 In our example, we get the following 
payoffs:
(20)
For our example, Table 1 shows the players’ payoffs for the 
value and the 
value. It is easy to see that the 
value rewards player 
since he coordinates players 1 and 2.
4. Conclusion
In this paper, we generalized the 
value for efficient CO values. The new value 
takes into account the problem of coordination of players—one main task for managers in firms. Hence, the 
approach is another step in modelling firms and hierarchies. In the first step, the worth 
is divided among the players with respect to their productivity and their coordination tasks. In the second step, the weighted hierarchy reallocates a certain fraction of these payoffs. We hope that this value is an appropriate value for analyzing hierarchies in firms.