Preliminaries
Tree-type values for cycle-free digraph games
The web values for cycle-free digraph games
Sharing a river with multiple sources, a delta and islands
Tree-type values for cycle-free directed graph games
Anna Khmelnitskaya†,
Dolf Talman‡
† St. Petersburg Institute for Economics and Mathematics,
Russian Academy of Sciences
‡ Tilburg University
Universitá del Piemonte Orientale, Alessandria
January 17, 2011
Anna Khmelnitskaya, Dolf Talman
Tree-type values for CF DG-games
Preliminaries
Tree-type values for cycle-free digraph games
The web values for cycle-free digraph games
Sharing a river with multiple sources, a delta and islands
Outline
1
Preliminaries
TU games
Games with cooperation structure
2
Tree-type values for cycle-free digraph games
Tree connectedness
Efficiency and stability
Sink connectedness
3
The web values for cycle-free digraph games
4
Sharing a river with multiple sources, a delta and islands
Anna Khmelnitskaya, Dolf Talman
Tree-type values for CF DG-games
Preliminaries
Tree-type values for cycle-free digraph games
The web values for cycle-free digraph games
Sharing a river with multiple sources, a delta and islands
TU games
Games with cooperation structure
A cooperative TU game is a pair hN, v i where
N = {1, . . . , n} is a finite set of n ≥ 2 players,
v : 2N → IR, v (∅) = 0, is a characteristic function.
A subset S ⊆ N (or S ∈ 2N ) of s players is a coalition,
v (S) presents the worth of the coalition S.
GN is the class of TU games with a fixed player set N.
n
(GN = IR2 −1 of vectors {v (S)} S⊆N )
S6=∅
A game v is superadditive if v (S ∪ T ) ≥ v (S) + v (T ) for all S, T ⊆ N such that
S ∩ T = ∅.
A game v is convex if v (S) + v (T ) ≤ v (S ∪ T ) + v (S ∩ T ) for all S, T ⊆ N.
Anna Khmelnitskaya, Dolf Talman
Tree-type values for CF DG-games
Preliminaries
Tree-type values for cycle-free digraph games
The web values for cycle-free digraph games
Sharing a river with multiple sources, a delta and islands
TU games
Games with cooperation structure
For any G ⊆ GN , a value on G is a mapping ξ : G → IRN ,
the real number ξi (v ) is the payoff to the player i in the game v .
The core of a game v
C(v ) = {x ∈ IRN | x(N) = v (N), x(S) ≥ v (S), for all S ⊆ N}.
The weak core of a game v
C̃(v ) = {x ∈ IRN | x(N) ≤ v (N), x(S) ≥ v (S), for all S $ N}.
A value ξ is stable if for any game v ∈ GN with a nonempty core C(v ), ξ(v ) ∈ C(v ).
A value ξ is weakly stable if for any game v ∈ GN with a nonempty weak core C̃(v ),
ξ(v ) ∈ C̃(v ).
Anna Khmelnitskaya, Dolf Talman
Tree-type values for CF DG-games
Preliminaries
Tree-type values for cycle-free digraph games
The web values for cycle-free digraph games
Sharing a river with multiple sources, a delta and islands
TU games
Games with cooperation structure
A cooperation structure on N is specified by a graph Γ without loops, directed or
undirected.
A pair hv , Γi constitutes a game with cooperation structure or, in other terms, a graph
game or a digraph game, when we want to emphasize that Γ is directed, or simply
Γ-game.
GNΓ is the set of all games with cooperation structure with fixed N.
A Γ-value is an operator ξ : GNΓ → IRN that assigns a vector of payoffs to any game with
cooperation structure.
Anna Khmelnitskaya, Dolf Talman
Tree-type values for CF DG-games
Preliminaries
Tree-type values for cycle-free digraph games
The web values for cycle-free digraph games
Sharing a river with multiple sources, a delta and islands
TU games
Games with cooperation structure
An undirected graph is Γ ⊆ ΓcN = { {i, j} | i, j ∈ N, i 6= j},
where an unordered pair {i, j} is a link between i, j ∈ N.
A directed graph (digraph) Γ is a collection of directed links.
In a directed link (i, j) ∈ Γ, i is a superior of j and j is subordinate of i
All players with the same superior in Γ are brothers.
In a digraph Γ, i 6= j is a predecessor of j and j is successor of i, if there is a sequence
of directed links (ih , ih+1 ) ∈ Γ, h = 1, . . . , k , s.t. i1 = i and ik +1 = j.
PΓ (i) is the set of all predecessors of i in Γ
OΓ (i) is the set of all superiors of i in Γ
FΓ (i) is the set of all subordinates of i in Γ
SΓ (i) the set of all successors of i in Γ
BΓ (i) the set of all brothers of i in Γ
P̄Γ (i) = PΓ (i) ∪ i,
S̄Γ (i) = SΓ (i) ∪ i,
B̄Γ (i) = BΓ (i) ∪ i.
Γ0
Any ⊂ Γ is a subgraph of Γ. A subgraph of Γ on S ⊆ N is the graph
Γ|S = {{i, j} ∈ Γ | i, j ∈ S} or Γ|S = {(i, j) ∈ Γ | i, j ∈ S} for digraph Γ.
For any subgraph Γ0 ∈ Γ, S(Γ0 ) ⊆ N is the coalition equal to the set of nodes in Γ0 , i.e.,
S(Γ0 ) = {i ∈ N | ∃j ∈ N : {i, j} ∈ Γ0 } if Γ is undirected, and
S(Γ0 ) = {i ∈ N | ∃j ∈ N : {(i, j), (j, i)} ∩ Γ0 6= ∅} for a directed graph Γ.
Anna Khmelnitskaya, Dolf Talman
Tree-type values for CF DG-games
Preliminaries
Tree-type values for cycle-free digraph games
The web values for cycle-free digraph games
Sharing a river with multiple sources, a delta and islands
TU games
Games with cooperation structure
A directed graph Γ on N is a (rooted) tree if there is one node in N, called the root,
having no predecessors in Γ and there is a unique sequence of directed links in Γ from
this node to any other node in N.
A directed graph Γ is a sink tree, if the directed graph, composed by the same links as
Γ but with opposite orientation, is a rooted tree.
A directed graph Γ is a (rooted/sink) forest if it is composed by a number of disjoint
(rooted/sink) trees.
A line-graph is a directed graph that contains links only between subsequent nodes.
For a digraph Γ, a subgraph T is a subtree of Γ, if T is a tree on S(T ).
A subtree T of a digraph Γ is a full subtree, if it contains the root together with all its
successors in Γ, or else S(T ) = S̄Γ (r (T )).
A full subtree T of Γ is a maximal subtree if its root is a root of Γ, i.e., OΓ (r (T )) = ∅.
Anna Khmelnitskaya, Dolf Talman
Tree-type values for CF DG-games
Preliminaries
Tree-type values for cycle-free digraph games
The web values for cycle-free digraph games
Sharing a river with multiple sources, a delta and islands
TU games
Games with cooperation structure
1
2
10
4
3
5
6
7
8
9
directed graph
Anna Khmelnitskaya, Dolf Talman
Tree-type values for CF DG-games
Preliminaries
Tree-type values for cycle-free digraph games
The web values for cycle-free digraph games
Sharing a river with multiple sources, a delta and islands
TU games
Games with cooperation structure
1
2
10
4
3
5
6
7
8
9
roots: 1, 2
Anna Khmelnitskaya, Dolf Talman
Tree-type values for CF DG-games
Preliminaries
Tree-type values for cycle-free digraph games
The web values for cycle-free digraph games
Sharing a river with multiple sources, a delta and islands
TU games
Games with cooperation structure
1
2
10
4
3
5
6
7
8
9
root 1 together with all his successors, S̄Γ (1)
Anna Khmelnitskaya, Dolf Talman
Tree-type values for CF DG-games
Preliminaries
Tree-type values for cycle-free digraph games
The web values for cycle-free digraph games
Sharing a river with multiple sources, a delta and islands
TU games
Games with cooperation structure
1
2
10
4
3
5
6
7
8
9
subgraph Γ|S̄
Γ (1)
Anna Khmelnitskaya, Dolf Talman
Tree-type values for CF DG-games
Preliminaries
Tree-type values for cycle-free digraph games
The web values for cycle-free digraph games
Sharing a river with multiple sources, a delta and islands
TU games
Games with cooperation structure
1
2
10
4
3
5
6
7
8
9
maximal subtree
Anna Khmelnitskaya, Dolf Talman
Tree-type values for CF DG-games
Preliminaries
Tree-type values for cycle-free digraph games
The web values for cycle-free digraph games
Sharing a river with multiple sources, a delta and islands
TU games
Games with cooperation structure
1
2
10
4
3
5
6
7
8
9
maximal subtree
Anna Khmelnitskaya, Dolf Talman
Tree-type values for CF DG-games
Preliminaries
Tree-type values for cycle-free digraph games
The web values for cycle-free digraph games
Sharing a river with multiple sources, a delta and islands
TU games
Games with cooperation structure
In a graph Γ on N a sequence of different nodes (i1 , . . . , ik ) is a path in Γ from node i1
to node ik , if k ≥ 2 and for h = 1, . . . , k −1, it holds that {ih , ih+1 } ∈ Γ for undirected Γ
and if for h = 1, . . . , k −1 it holds that {(ih , ih+1 ), (ih+1 , ih )} ∩ Γ 6= ∅ when Γ is directed.
In a directed graph Γ on N a sequence of different nodes (i1 , . . . , ik ) is a directed path
in Γ from node i1 to node ik , if for h = 1, . . . , k −1 it holds that (ih , ih+1 ) ∈ Γ.
In a graph Γ on N a sequence of different nodes (i1 , . . . , ik ), k ≥ 3, is a cycle in Γ, if it is
a path and {ik , i1 } ∈ Γ for undirected Γ and {(ik , i1 ), (i1 , ik )} ∩ Γ 6= ∅ when Γ is directed.
In a digraph Γ a sequence of different nodes (i1 , . . . , ik ), k ≥ 2, is a directed cycle in Γ,
if it is a directed path and (ik , i1 ) ∈ Γ.
The latter definition is due to while in a digraph there might be cycles of length 2, in an
undirected graph a cycle of length 2 is not well defined.
An undirected graph Γ is cycle-free, if it contains no cycles.
A digraph Γ on N is cycle-free, if it contains no directed cycles, i.e., no node is a
successor of itself.
A digraph Γ on N is strongly cycle-free, if it is cycle-free and contains no cycles.
Notice that both a rooted tree and a sink tree, and in particular a line-graph, are
strongly cycle-free.
Anna Khmelnitskaya, Dolf Talman
Tree-type values for CF DG-games
Preliminaries
Tree-type values for cycle-free digraph games
The web values for cycle-free digraph games
Sharing a river with multiple sources, a delta and islands
TU games
Games with cooperation structure
1
2
10
4
3
5
6
7
8
9
cycle-free digraph but not strongly cycle-free – (4, 7, 8, 6) is a cycle but not a directed cycle
Anna Khmelnitskaya, Dolf Talman
Tree-type values for CF DG-games
Preliminaries
Tree-type values for cycle-free digraph games
The web values for cycle-free digraph games
Sharing a river with multiple sources, a delta and islands
TU games
Games with cooperation structure
1
2
10
3
4
5
6
7
8
9
strongly cycle-free graph
Anna Khmelnitskaya, Dolf Talman
Tree-type values for CF DG-games
Preliminaries
Tree-type values for cycle-free digraph games
The web values for cycle-free digraph games
Sharing a river with multiple sources, a delta and islands
TU games
Games with cooperation structure
Given a graph Γ on N, two nodes i, j ∈ N are connected, if there is a path from i to j.
Graph Γ on N is connected, if any two nodes in N are connected.
Given a graph Γ on N, a coalition S ⊆ N is connected, if the subgraph Γ|S is connected.
For a given graph Γ and a coalition S ⊆ N,
by C Γ (S) denote the set of all connected subcoalitions of S,
by S/Γ the set of maximally connected subcoalitions, called components of S,
by (S/Γ)i the component of S containing player i ∈ S.
Notice that S/Γ always provides a partition of S.
For a Γ-game hv , Γi ∈ GNΓ , a payoff vector x ∈ IRN is component efficient, if for every
component C ∈ N/Γ, it holds that x(C) = v (C).
Anna Khmelnitskaya, Dolf Talman
Tree-type values for CF DG-games
Preliminaries
Tree-type values for cycle-free digraph games
The web values for cycle-free digraph games
Sharing a river with multiple sources, a delta and islands
Tree connectedness
Efficiency and stability
Sink connectedness
In what follows we assume that the cooperation structure on the player set N is
specified by a cycle-free directed graph, not necessarily strongly cycle-free.
1
2
10
4
3
5
6
7
8
Anna Khmelnitskaya, Dolf Talman
9
Tree-type values for CF DG-games
Preliminaries
Tree-type values for cycle-free digraph games
The web values for cycle-free digraph games
Sharing a river with multiple sources, a delta and islands
Tree connectedness
Efficiency and stability
Sink connectedness
For a directed link there are two different interpretations:
• a link is directed to indicate which player initiated the communication, but it also
represents a fully developed communication link, =⇒ following Myerson all
connected coalitions are productive;
• a directed link represents the only one-way communication situation, =⇒ not
every connected coalition might be productive.
We abide by the second interpretation and consider different options for creation of
productive coalitions.
There are different scenarios possible for controlling cooperation in case of directed
communication.
In a cycle-free digraph Γ there is at least one node having no superior.
A node without superior, i.e., any root i ∈ RΓ (N), can be seen as a topman of the
communication structure given by Γ.
Anna Khmelnitskaya, Dolf Talman
Tree-type values for CF DG-games
Preliminaries
Tree-type values for cycle-free digraph games
The web values for cycle-free digraph games
Sharing a river with multiple sources, a delta and islands
Tree connectedness
Efficiency and stability
Sink connectedness
For a directed link there are two different interpretations:
• a link is directed to indicate which player initiated the communication, but it also
represents a fully developed communication link, =⇒ following Myerson all
connected coalitions are productive;
• a directed link represents the only one-way communication situation, =⇒ not
every connected coalition might be productive.
We abide by the second interpretation and consider different options for creation of
productive coalitions.
There are different scenarios possible for controlling cooperation in case of directed
communication.
In a cycle-free digraph Γ there is at least one node having no superior.
A node without superior, i.e., any root i ∈ RΓ (N), can be seen as a topman of the
communication structure given by Γ.
Anna Khmelnitskaya, Dolf Talman
Tree-type values for CF DG-games
Preliminaries
Tree-type values for cycle-free digraph games
The web values for cycle-free digraph games
Sharing a river with multiple sources, a delta and islands
Tree connectedness
Efficiency and stability
Sink connectedness
1
2
10
4
3
5
6
7
8
9
roots: 1, 2
Anna Khmelnitskaya, Dolf Talman
Tree-type values for CF DG-games
Preliminaries
Tree-type values for cycle-free digraph games
The web values for cycle-free digraph games
Sharing a river with multiple sources, a delta and islands
Tree connectedness
Efficiency and stability
Sink connectedness
Assume that each player may be controlled only by his predecessors and that nobody
accepts that his former subordinate becomes his equal partner if a coalition forms.
This entails the assumption that the only productive coalitions are the so-called tree
connected, or simply t-connected, coalitions that are being connected coalitions
S ∈ C Γ(N) meet also the condition that for every root r ∈ RΓ (S) it holds that r ∈
/ SΓ (r 0 )
for another root r 0 ∈ RΓ (S).
It is not difficult to see that the latter condition guarantees that every t-connected
coalition inherits the subordination of players prescribed by Γ in N.
Obviously, every component C ∈ N/Γ is t-connected;
For every i ∈ N the full successors set S̄Γ (i) is t-connected.
In what follows for a given cycle-free digraph Γ and a coalition S ⊆ N,
by CtΓ (S) denote the set of all t-connected subcoalitions of S,
by [S/Γ]t the set of maximally t-connected subcoalitions of S, called t-connected
components of S,
by [S/Γ]ti the t-connected component of S containing player i ∈ S.
Anna Khmelnitskaya, Dolf Talman
Tree-type values for CF DG-games
Preliminaries
Tree-type values for cycle-free digraph games
The web values for cycle-free digraph games
Sharing a river with multiple sources, a delta and islands
Tree connectedness
Efficiency and stability
Sink connectedness
1
2
10
4
3
5
6
7
8
9
S = {5, 6, 7, 8} = S̄Γ (5) is t-connected
Anna Khmelnitskaya, Dolf Talman
Tree-type values for CF DG-games
Preliminaries
Tree-type values for cycle-free digraph games
The web values for cycle-free digraph games
Sharing a river with multiple sources, a delta and islands
Tree connectedness
Efficiency and stability
Sink connectedness
1
2
10
4
3
5
6
7
8
9
S = {6, 7, 8} is t-connected
Anna Khmelnitskaya, Dolf Talman
Tree-type values for CF DG-games
Preliminaries
Tree-type values for cycle-free digraph games
The web values for cycle-free digraph games
Sharing a river with multiple sources, a delta and islands
Tree connectedness
Efficiency and stability
Sink connectedness
1
2
10
4
3
5
6
7
8
9
S = {3, 6, 7, 8} is not t-connected since RΓ (S) = {3, 6, 7} and 6, 7 ∈ SΓ (3)
Anna Khmelnitskaya, Dolf Talman
Tree-type values for CF DG-games
Preliminaries
Tree-type values for cycle-free digraph games
The web values for cycle-free digraph games
Sharing a river with multiple sources, a delta and islands
Tree connectedness
Efficiency and stability
Sink connectedness
For efficiency of a value we require that each topman realizes together with all his
successors the total worth they possess, what generates the first property a value may
satisfy, the so-called maximal-tree efficiency.
A value ξ on GNΓ is maximal-tree efficient (MTE) if, for every cycle-free digraph game
hv , Γi, for all i ∈ RΓ (N) it holds that
X
ξj (v , Γ) = v (S̄Γ (i)).
j∈S̄Γ (i)
MTE generalizes the usual definition of efficiency for a tree. In a tree with only one
topman, MTE just says that the total payoff should be equal to v (N).
Still, MTE is not the productive component efficiency.
Different from the Myerson undirected communication model we suppose that not
every productive component is able to realize its exact capacity but only those with the
tree structure.
Indeed, one worker can work in two firms and so, the two managers of these firms and
the worker create a productive coalition. Yet, it is impossible to guarantee the efficiency
of this coalition because there is no communication link between managers.
A value ξ on GNΓ is successor equivalent (SE) if, for every cycle-free digraph game
hv , Γi, for all (i, j) ∈ Γ, for every k ∈ S̄Γ (j) it holds that
ξk (v , Γ\(i, j)) = ξk (v , Γ).
Anna Khmelnitskaya, Dolf Talman
Tree-type values for CF DG-games
Preliminaries
Tree-type values for cycle-free digraph games
The web values for cycle-free digraph games
Sharing a river with multiple sources, a delta and islands
Tree connectedness
Efficiency and stability
Sink connectedness
For efficiency of a value we require that each topman realizes together with all his
successors the total worth they possess, what generates the first property a value may
satisfy, the so-called maximal-tree efficiency.
A value ξ on GNΓ is maximal-tree efficient (MTE) if, for every cycle-free digraph game
hv , Γi, for all i ∈ RΓ (N) it holds that
X
ξj (v , Γ) = v (S̄Γ (i)).
j∈S̄Γ (i)
MTE generalizes the usual definition of efficiency for a tree. In a tree with only one
topman, MTE just says that the total payoff should be equal to v (N).
Still, MTE is not the productive component efficiency.
Different from the Myerson undirected communication model we suppose that not
every productive component is able to realize its exact capacity but only those with the
tree structure.
Indeed, one worker can work in two firms and so, the two managers of these firms and
the worker create a productive coalition. Yet, it is impossible to guarantee the efficiency
of this coalition because there is no communication link between managers.
A value ξ on GNΓ is successor equivalent (SE) if, for every cycle-free digraph game
hv , Γi, for all (i, j) ∈ Γ, for every k ∈ S̄Γ (j) it holds that
ξk (v , Γ\(i, j)) = ξk (v , Γ).
Anna Khmelnitskaya, Dolf Talman
Tree-type values for CF DG-games
Preliminaries
Tree-type values for cycle-free digraph games
The web values for cycle-free digraph games
Sharing a river with multiple sources, a delta and islands
Tree connectedness
Efficiency and stability
Sink connectedness
For efficiency of a value we require that each topman realizes together with all his
successors the total worth they possess, what generates the first property a value may
satisfy, the so-called maximal-tree efficiency.
A value ξ on GNΓ is maximal-tree efficient (MTE) if, for every cycle-free digraph game
hv , Γi, for all i ∈ RΓ (N) it holds that
X
ξj (v , Γ) = v (S̄Γ (i)).
j∈S̄Γ (i)
MTE generalizes the usual definition of efficiency for a tree. In a tree with only one
topman, MTE just says that the total payoff should be equal to v (N).
Still, MTE is not the productive component efficiency.
Different from the Myerson undirected communication model we suppose that not
every productive component is able to realize its exact capacity but only those with the
tree structure.
Indeed, one worker can work in two firms and so, the two managers of these firms and
the worker create a productive coalition. Yet, it is impossible to guarantee the efficiency
of this coalition because there is no communication link between managers.
A value ξ on GNΓ is successor equivalent (SE) if, for every cycle-free digraph game
hv , Γi, for all (i, j) ∈ Γ, for every k ∈ S̄Γ (j) it holds that
ξk (v , Γ\(i, j)) = ξk (v , Γ).
Anna Khmelnitskaya, Dolf Talman
Tree-type values for CF DG-games
Preliminaries
Tree-type values for cycle-free digraph games
The web values for cycle-free digraph games
Sharing a river with multiple sources, a delta and islands
Tree connectedness
Efficiency and stability
Sink connectedness
For efficiency of a value we require that each topman realizes together with all his
successors the total worth they possess, what generates the first property a value may
satisfy, the so-called maximal-tree efficiency.
A value ξ on GNΓ is maximal-tree efficient (MTE) if, for every cycle-free digraph game
hv , Γi, for all i ∈ RΓ (N) it holds that
X
ξj (v , Γ) = v (S̄Γ (i)).
j∈S̄Γ (i)
MTE generalizes the usual definition of efficiency for a tree. In a tree with only one
topman, MTE just says that the total payoff should be equal to v (N).
Still, MTE is not the productive component efficiency.
Different from the Myerson undirected communication model we suppose that not
every productive component is able to realize its exact capacity but only those with the
tree structure.
Indeed, one worker can work in two firms and so, the two managers of these firms and
the worker create a productive coalition. Yet, it is impossible to guarantee the efficiency
of this coalition because there is no communication link between managers.
A value ξ on GNΓ is successor equivalent (SE) if, for every cycle-free digraph game
hv , Γi, for all (i, j) ∈ Γ, for every k ∈ S̄Γ (j) it holds that
ξk (v , Γ\(i, j)) = ξk (v , Γ).
Anna Khmelnitskaya, Dolf Talman
Tree-type values for CF DG-games
Preliminaries
Tree-type values for cycle-free digraph games
The web values for cycle-free digraph games
Sharing a river with multiple sources, a delta and islands
Tree connectedness
Efficiency and stability
Sink connectedness
A value ξ on GNΓ is full-tree efficient (FTE) if, for every cycle-free digraph game hv , Γi,
for all i ∈ N it holds that
X
ξj (v , Γ) = v (S̄Γ (i)).
j∈S̄Γ (i)
Proposition
On the class of cycle-free digraph games GNΓ MTE and SE together imply FTE.
Anna Khmelnitskaya, Dolf Talman
Tree-type values for CF DG-games
Preliminaries
Tree-type values for cycle-free digraph games
The web values for cycle-free digraph games
Sharing a river with multiple sources, a delta and islands
Tree connectedness
Efficiency and stability
Sink connectedness
A value ξ on GNΓ is full-tree efficient (FTE) if, for every cycle-free digraph game hv , Γi,
for all i ∈ N it holds that
X
ξj (v , Γ) = v (S̄Γ (i)).
j∈S̄Γ (i)
Proposition
On the class of cycle-free digraph games GNΓ MTE and SE together imply FTE.
Anna Khmelnitskaya, Dolf Talman
Tree-type values for CF DG-games
Preliminaries
Tree-type values for cycle-free digraph games
The web values for cycle-free digraph games
Sharing a river with multiple sources, a delta and islands
Tree connectedness
Efficiency and stability
Sink connectedness
Theorem
On the class of cycle-free digraph games there is a unique value t, the so-called tree
value, that satisfies MTE and SE, and for any cycle-free digraph game hv , Γi, t(v , Γ)
(i) meets the recursive equality
X
ti (v , Γ) = v (S̄Γ (i)) −
tj (v , Γ),
for all i ∈ N;
j∈SΓ (i)
(ii) admits the explicit formula representation in the form
X
ti (v , Γ) = v (S̄Γ (i)) −
κi (j)v (S̄Γ (j)),
for all i ∈ N,
j∈SΓ (i)
where for all i ∈ N, j ∈ SΓ (i)
κi (j) =
n−2
X
(−1)r κri (j),
r =0
and
ih ∈
κri (j) is defined as a number
SΓ (ih−1 ), h = 1, . . . , r + 1.
of tuples (i0 , . . . , ir +1 ) such that i0 = i, ir +1 = j,
Corollary (i) provides a simple recursive algorithm for computing the tree value going
upstream from the leaves of the given graph.
Anna Khmelnitskaya, Dolf Talman
Tree-type values for CF DG-games
Preliminaries
Tree-type values for cycle-free digraph games
The web values for cycle-free digraph games
Sharing a river with multiple sources, a delta and islands
Tree connectedness
Efficiency and stability
Sink connectedness
Inessential links may be eliminated.
A directed link (i, j) ∈ Γ for which there exists a directed path ~p in Γ from i to j such that
~p 6= (i, j) is inessential, otherwise (i, j) is an essential link. A directed path ~p is a proper
path, if it contains only essential links.
For any digraph Γ on N and i ∈ N the set SΓ (i) of all successors of i can be partitioned
into three disjoint subsets
SΓ (i) = FΓ∗ (i) ∪ SΓ1 (i) ∪ SΓ2 (i),
FΓ∗ (i = {j ∈ FΓ (i) | (i, j) is essential} is the set of all proper subordinates of i in Γ,
κi (j) = 1 for all i ∈ N, j ∈ F`∗ (i);
SΓ1 (i) consists of j ∈ SΓ (i) \ FΓ∗ (i) for which all paths from i to j can be partitioned into
a number of separate groups, might be only one group, such that all paths in the same
group have at least one common node different from i and j and paths from different
groups do not intersect between i and j,
the number of these groups of paths d̃ i (j) is the proper in-degree of j with respect to i
κi (j) = −d̃ i (j) + 1 for all i ∈ N, j ∈ SΓ1 (i);
Remark that κi (j) = 0, if |OΓi (j)| = 1;
SΓ2 (i) = SΓ (i) \ FΓ∗ (i) ∪ SΓ1 (i) .
Anna Khmelnitskaya, Dolf Talman
Tree-type values for CF DG-games
Preliminaries
Tree-type values for cycle-free digraph games
The web values for cycle-free digraph games
Sharing a river with multiple sources, a delta and islands
Tree connectedness
Efficiency and stability
Sink connectedness
Inessential links may be eliminated.
A directed link (i, j) ∈ Γ for which there exists a directed path ~p in Γ from i to j such that
~p 6= (i, j) is inessential, otherwise (i, j) is an essential link. A directed path ~p is a proper
path, if it contains only essential links.
For any digraph Γ on N and i ∈ N the set SΓ (i) of all successors of i can be partitioned
into three disjoint subsets
SΓ (i) = FΓ∗ (i) ∪ SΓ1 (i) ∪ SΓ2 (i),
FΓ∗ (i = {j ∈ FΓ (i) | (i, j) is essential} is the set of all proper subordinates of i in Γ,
κi (j) = 1 for all i ∈ N, j ∈ F`∗ (i);
SΓ1 (i) consists of j ∈ SΓ (i) \ FΓ∗ (i) for which all paths from i to j can be partitioned into
a number of separate groups, might be only one group, such that all paths in the same
group have at least one common node different from i and j and paths from different
groups do not intersect between i and j,
the number of these groups of paths d̃ i (j) is the proper in-degree of j with respect to i
κi (j) = −d̃ i (j) + 1 for all i ∈ N, j ∈ SΓ1 (i);
Remark that κi (j) = 0, if |OΓi (j)| = 1;
SΓ2 (i) = SΓ (i) \ FΓ∗ (i) ∪ SΓ1 (i) .
Anna Khmelnitskaya, Dolf Talman
Tree-type values for CF DG-games
Preliminaries
Tree-type values for cycle-free digraph games
The web values for cycle-free digraph games
Sharing a river with multiple sources, a delta and islands
Tree connectedness
Efficiency and stability
Sink connectedness
Theorem
For every cycle-free digraph game hv , Γi the tree value t admits the equivalent
representation in the form
X
ti (v , Γ) = v (S̄Γ (i)) −
v (S̄Γ (j)) +
j∈FΓ∗ (i)
+
X
X
(d̃ i (j) − 1)v (S̄Γ (j)) −
κM
i (j)v (S̄Γ (j)),
for all i ∈ N.
j∈SΓ2 (i)
j∈SΓ1 (i)
If the consideration is restricted to only strongly cycle-free digraph games then the
above representation reduces to
X
v (S̄Γ (j)), for all i ∈ N.
ti (v , Γ) = v (S̄Γ (i)) −
j∈FΓ (i)
Anna Khmelnitskaya, Dolf Talman
Tree-type values for CF DG-games
Preliminaries
Tree-type values for cycle-free digraph games
The web values for cycle-free digraph games
Sharing a river with multiple sources, a delta and islands
Tree connectedness
Efficiency and stability
Sink connectedness
In particular, for rooted-forest digraph games the tree value coincides with the tree
value first introduced under the name of the hierarchical outcome in Demange (2004)
where it was also shown that under the mild condition of superadditivity, the tree value
belongs to the core of the restricted game.
More recently, the tree value for trees was used as a basic element in the construction
of the average tree solution for undirected cycle-free graph games in Herings, van der
Laan, and Talman (2008).
In Khmelnitskaya (2010) it is shown that the tree value on the class of rooted-forest
digraph games is characterized via component efficiency (CE) and successor
equivalence (SE); moreover, it is also shown that the class of rooted-forest digraph
games is the maximal subclass in the class of strongly cycle-free digraph games where
this axiomatization holds true.
Recall that for a rooted-tree digraph game every connected component is a tree.
Therefore, on the class of rooted-forest digraph games every connected component is
productive and MTE coincides with CE.
Anna Khmelnitskaya, Dolf Talman
Tree-type values for CF DG-games
Preliminaries
Tree-type values for cycle-free digraph games
The web values for cycle-free digraph games
Sharing a river with multiple sources, a delta and islands
Tree connectedness
Efficiency and stability
Sink connectedness
The tree value assigns to every player the payoff equal to the worth of his full
successors set minus the worths of all full successors sets of his proper subordinates
plus or minus the worths of all full successors sets of any other of his successors that
are subtracted or added more than once.
In fact a player receives what he contributes when he joins his successors when only
the full successors sets, that are the only efficient productive coalitions, are counted.
Besides, the right sides of both formulas being considered with respect not to
coalitional worths but to players in these coalitions contain only player i when taking
into account all pluses and minuses.
Anna Khmelnitskaya, Dolf Talman
Tree-type values for CF DG-games
Preliminaries
Tree-type values for cycle-free digraph games
The web values for cycle-free digraph games
Sharing a river with multiple sources, a delta and islands
Tree connectedness
Efficiency and stability
Sink connectedness
Example 1
The tree value for a 10-person game with the graph structure given on the picture.
v (13456789, 10)−v (356789)−v (46789)−
v (246789, 10)−v (46789)
1
−v (689, 10)+2v (689)+v (78)−v (8)
2
−v (689, 10)+v (689)
v (689, 10)−v (689)
10
v (356789)−v (56789) 3
4
v (46789)−v (689)−v (78)+v (8)
v (56789)−v (689)−v (78)+v (8) 5
6
v (78)−v (8)
v (689)−v (8)−v (9)
7
8
v (8)
Anna Khmelnitskaya, Dolf Talman
9
v (9)
Tree-type values for CF DG-games
Preliminaries
Tree-type values for cycle-free digraph games
The web values for cycle-free digraph games
Sharing a river with multiple sources, a delta and islands
Tree connectedness
Efficiency and stability
Sink connectedness
Example 2
The tree value for a 10-person game with the strongly cycle-free graph structure given
on the picture.
v (13456789) − v (356789) − v (4689)
v (10, 356789) − v (356789)
2
v (24689) − v (4689)
10
v (356789) − v (56789)
3
v (56789) − v (7) − v (689)
v (7)
1
4
v (4689) − v (689)
6
v (689) − v (8) − v (9)
5
7
v (8)
Anna Khmelnitskaya, Dolf Talman
8
9
Tree-type values for CF DG-games
v (9)
Preliminaries
Tree-type values for cycle-free digraph games
The web values for cycle-free digraph games
Sharing a river with multiple sources, a delta and islands
Tree connectedness
Efficiency and stability
Sink connectedness
It turns out that the tree value not only meets FTE but FTE alone uniquely defines the
tree value on the class of cycle-free digraph games.
Theorem
On the class of cycle-free digraph games the tree value is the unique value that
satisfies FTE.
Corollary
FTE on the class of cycle-free digraph games implies not only MTE but SE as well.
Anna Khmelnitskaya, Dolf Talman
Tree-type values for CF DG-games
Preliminaries
Tree-type values for cycle-free digraph games
The web values for cycle-free digraph games
Sharing a river with multiple sources, a delta and islands
Tree connectedness
Efficiency and stability
Sink connectedness
Theorem
For any cycle-free digraph game hv , Γi, for all t-connected coalitions S ⊆ N,
X
X
X
X
κS (i)−1 v (S̄Γ (i)) −
κS (i)v (S̄Γ (i));
ti (v , Γ) =
v (S̄Γ (i)) −
i∈S
i∈RΓ (S)
i∈S\RΓ (S)
i∈S̄Γ (S)\S
if the consideration is restricted to only strongly cycle-free digraph games
X
X
X
X
ti (v , Γ) =
v (S̄Γ (i))−
dS (i)−1 v (S̄Γ (i))−
dS (i)v (S̄Γ (i)),
i∈S
i∈RΓ (S)
i∈S\RΓ (S)
where
S̄Γ (S) =
i∈RΓ (S̄Γ (S)\S)
[
S̄Γ (i),
i∈RΓ (S)
κS (i) =
X
κj (i),
for all i ∈ S̄Γ (S),
j∈P̄Γ (j)∩SΓ (S)
dS (i) = |OΓ (i) ∩ S̄Γ (S)|
Anna Khmelnitskaya, Dolf Talman
for all j ∈ SΓ (S).
Tree-type values for CF DG-games
Preliminaries
Tree-type values for cycle-free digraph games
The web values for cycle-free digraph games
Sharing a river with multiple sources, a delta and islands
Tree connectedness
Efficiency and stability
Sink connectedness
The overall efficiency for arbitrary cycle-free digraph game hv , Γi is given by
X
X
X
ti (v , Γ) =
v (S̄Γ (i)) −
κN (i)−1 v (S̄Γ (i));
i∈N
i∈RΓ (N)
i∈N\RΓ (N)
if the consideration is restricted to only strongly cycle-free digraph games
X
X
X
ti (v , Γ) =
v (S̄Γ (i)) −
dΓ (i)−1 v (S̄Γ (i)),
i∈N
i∈RΓ (N)
(1)
i∈N\RΓ (N)
where dΓ (i) = |OΓ (i)| is the in-degree of i in digraph Γ.
The Myerson component efficiency entails the equality
X
X
ξi (v , Γ) =
v (C).
i∈N
C∈N/Γ
For strongly cycle-free rooted-forest digraph games (1) reduces to (2):
X
X
X
ti (v , Γ) =
v (S̄Γ (i)) =
v (C).
i∈N
i∈RΓ (N)
Anna Khmelnitskaya, Dolf Talman
C∈N/Γ
Tree-type values for CF DG-games
(2)
Preliminaries
Tree-type values for cycle-free digraph games
The web values for cycle-free digraph games
Sharing a river with multiple sources, a delta and islands
Tree connectedness
Efficiency and stability
Sink connectedness
The overall efficiency for arbitrary cycle-free digraph game hv , Γi is given by
X
X
X
ti (v , Γ) =
v (S̄Γ (i)) −
κN (i)−1 v (S̄Γ (i));
i∈N
i∈RΓ (N)
i∈N\RΓ (N)
if the consideration is restricted to only strongly cycle-free digraph games
X
X
X
ti (v , Γ) =
v (S̄Γ (i)) −
dΓ (i)−1 v (S̄Γ (i)),
i∈N
i∈RΓ (N)
(1)
i∈N\RΓ (N)
where dΓ (i) = |OΓ (i)| is the in-degree of i in digraph Γ.
The Myerson component efficiency entails the equality
X
X
ξi (v , Γ) =
v (C).
i∈N
C∈N/Γ
For strongly cycle-free rooted-forest digraph games (1) reduces to (2):
X
X
X
ti (v , Γ) =
v (S̄Γ (i)) =
v (C).
i∈N
i∈RΓ (N)
Anna Khmelnitskaya, Dolf Talman
C∈N/Γ
Tree-type values for CF DG-games
(2)
Preliminaries
Tree-type values for cycle-free digraph games
The web values for cycle-free digraph games
Sharing a river with multiple sources, a delta and islands
Tree connectedness
Efficiency and stability
Sink connectedness
The overall efficiency for arbitrary cycle-free digraph game hv , Γi is given by
X
X
X
ti (v , Γ) =
v (S̄Γ (i)) −
κN (i)−1 v (S̄Γ (i));
i∈N
i∈RΓ (N)
i∈N\RΓ (N)
if the consideration is restricted to only strongly cycle-free digraph games
X
X
X
ti (v , Γ) =
v (S̄Γ (i)) −
dΓ (i)−1 v (S̄Γ (i)),
i∈N
i∈RΓ (N)
(1)
i∈N\RΓ (N)
where dΓ (i) = |OΓ (i)| is the in-degree of i in digraph Γ.
The Myerson component efficiency entails the equality
X
X
ξi (v , Γ) =
v (C).
i∈N
C∈N/Γ
For strongly cycle-free rooted-forest digraph games (1) reduces to (2):
X
X
X
ti (v , Γ) =
v (S̄Γ (i)) =
v (C).
i∈N
i∈RΓ (N)
Anna Khmelnitskaya, Dolf Talman
C∈N/Γ
Tree-type values for CF DG-games
(2)
Preliminaries
Tree-type values for cycle-free digraph games
The web values for cycle-free digraph games
Sharing a river with multiple sources, a delta and islands
Tree connectedness
Efficiency and stability
Sink connectedness
For a cycle-free digraph game hv , Γi ∈ GNΓ , we define the t-core C t (v , Γ) as the set of
component efficient payoff vectors that are not dominated by any t-connected coalition,
C t (v , Γ) = {x ∈ IRN | x(C) = v (C), ∀C ∈ N/Γ; x(S) ≥ v (S), ∀S ∈ CtΓ (N)},
while the weak t-core C̃ t (v , Γ) is the set of weakly component efficient payoff vectors
that are not dominated by any t-connected coalition,
C̃ t (v , Γ) = {x ∈ IRN | x(C) ≤ v (C), ∀C ∈ N/Γ; x(S) ≥ v (S), ∀S ∈ CtΓ (N)}.
Theorem
The tree value on the subclass of superadditive rooted-forest digraph games is stable.
However, for a superadditive game the requirement on the digraph to be a rooted forest
is non-reducible.
Anna Khmelnitskaya, Dolf Talman
Tree-type values for CF DG-games
Preliminaries
Tree-type values for cycle-free digraph games
The web values for cycle-free digraph games
Sharing a river with multiple sources, a delta and islands
Tree connectedness
Efficiency and stability
Sink connectedness
For a cycle-free digraph game hv , Γi ∈ GNΓ , we define the t-core C t (v , Γ) as the set of
component efficient payoff vectors that are not dominated by any t-connected coalition,
C t (v , Γ) = {x ∈ IRN | x(C) = v (C), ∀C ∈ N/Γ; x(S) ≥ v (S), ∀S ∈ CtΓ (N)},
while the weak t-core C̃ t (v , Γ) is the set of weakly component efficient payoff vectors
that are not dominated by any t-connected coalition,
C̃ t (v , Γ) = {x ∈ IRN | x(C) ≤ v (C), ∀C ∈ N/Γ; x(S) ≥ v (S), ∀S ∈ CtΓ (N)}.
Theorem
The tree value on the subclass of superadditive rooted-forest digraph games is stable.
However, for a superadditive game the requirement on the digraph to be a rooted forest
is non-reducible.
Anna Khmelnitskaya, Dolf Talman
Tree-type values for CF DG-games
Preliminaries
Tree-type values for cycle-free digraph games
The web values for cycle-free digraph games
Sharing a river with multiple sources, a delta and islands
Tree connectedness
Efficiency and stability
Sink connectedness
For a cycle-free digraph game hv , Γi ∈ GNΓ , we define the t-core C t (v , Γ) as the set of
component efficient payoff vectors that are not dominated by any t-connected coalition,
C t (v , Γ) = {x ∈ IRN | x(C) = v (C), ∀C ∈ N/Γ; x(S) ≥ v (S), ∀S ∈ CtΓ (N)},
while the weak t-core C̃ t (v , Γ) is the set of weakly component efficient payoff vectors
that are not dominated by any t-connected coalition,
C̃ t (v , Γ) = {x ∈ IRN | x(C) ≤ v (C), ∀C ∈ N/Γ; x(S) ≥ v (S), ∀S ∈ CtΓ (N)}.
Theorem
The tree value on the subclass of superadditive rooted-forest digraph games is stable.
However, for a superadditive game the requirement on the digraph to be a rooted forest
is non-reducible.
Anna Khmelnitskaya, Dolf Talman
Tree-type values for CF DG-games
Preliminaries
Tree-type values for cycle-free digraph games
The web values for cycle-free digraph games
Sharing a river with multiple sources, a delta and islands
Tree connectedness
Efficiency and stability
Sink connectedness
Example
The tree value of the 4-person cycle-free digraph game hv , Γi
with superadditive v : v (24) = v (34) = v (234) = v (N) = 1, v (S) = 0 otherwise,
and cycle-free but not strongly cycle-free digraph Γ
1
3
2
4
does not meet the second constraint of the weak core:
t(v , Γ)=(-1,1,1,0), whence t1 (v , Γ) = −1 < 0 = v (1).
Anna Khmelnitskaya, Dolf Talman
Tree-type values for CF DG-games
Preliminaries
Tree-type values for cycle-free digraph games
The web values for cycle-free digraph games
Sharing a river with multiple sources, a delta and islands
Tree connectedness
Efficiency and stability
Sink connectedness
Example
The tree value of the 3-person cycle-free digraph game hv , Γi
with superadditive v : v (12) = v (13) = v (N) = 1, v (S) = 0 otherwise,
and strongly cycle-free digraph Γ with two roots 1 and 2.
1
2
3
violates weak efficiency:
t(v , Γ)=(1,1,0), i.e., t1 (v , Γ) + t2 (v , Γ) + t3 (v , Γ) = 2 > 1 = v (N).
Anna Khmelnitskaya, Dolf Talman
Tree-type values for CF DG-games
Preliminaries
Tree-type values for cycle-free digraph games
The web values for cycle-free digraph games
Sharing a river with multiple sources, a delta and islands
Tree connectedness
Efficiency and stability
Sink connectedness
A cycle-free digraph game hv , Γi is t-convex, if for all t-connected coalitions
T , Q ⊂ CtΓ (N) such that T is a full t-connected set, Q is a full successors set, and
T ∪ Q ∈ CtΓ (N), it holds that
v (T ) + v (Q) ≤ v (T ∪ Q) + v (T ∩ Q).
Theorem
The tree value on the subclass of t-convex strongly cycle-free digraph games is weakly
efficient.
Anna Khmelnitskaya, Dolf Talman
Tree-type values for CF DG-games
Preliminaries
Tree-type values for cycle-free digraph games
The web values for cycle-free digraph games
Sharing a river with multiple sources, a delta and islands
Tree connectedness
Efficiency and stability
Sink connectedness
The following example of a convex strongly cycle-free digraph game shows that even
under the assumption of convexity, which is stronger than t-convexity, one or more
constraints for not being dominated in the definition of the week core might be violated
by the tree value, i.e., the tree value is not weakly stable.
Example
Consider a 5-person cycle-free convex digraph game hv , Γi: v (N) = 10,
v (123) = v (1234) = v (1235) = 3, v (1345) = v (2345) = 2, v (S) = 0 otherwise,
and strongly cycle-free digraph Γ
1
2
1
4
5
Then t(v , Γ) = (1, 1, 0, 0, 0), whence, t1 (v , Γ) + t2 (v , Γ) + t3 (v , Γ) = 2 < 3 = v (123).
Anna Khmelnitskaya, Dolf Talman
Tree-type values for CF DG-games
Preliminaries
Tree-type values for cycle-free digraph games
The web values for cycle-free digraph games
Sharing a river with multiple sources, a delta and islands
Tree connectedness
Efficiency and stability
Sink connectedness
We consider now another scenario of controlling cooperation in case of directed
communication and assume that each player may be controlled only by his successors
and that nobody accepts that his former superior becomes his equal partner if a
coalition forms.
This entails the assumption that the only productive coalitions are the so-called sink
connected, or simply s-connected, coalitions that are being connected coalitions
S ∈ C Γ(N) meet also the condition that for every leaf l ∈ LΓ (S) it holds that l ∈
/ PΓ (l 0 ) for
another leaf l 0 ∈ LΓ (S).
Anna Khmelnitskaya, Dolf Talman
Tree-type values for CF DG-games
Preliminaries
Tree-type values for cycle-free digraph games
The web values for cycle-free digraph games
Sharing a river with multiple sources, a delta and islands
Tree connectedness
Efficiency and stability
Sink connectedness
1
2
10
4
3
5
6
7
8
9
leaves: 8, 9
Anna Khmelnitskaya, Dolf Talman
Tree-type values for CF DG-games
Preliminaries
Tree-type values for cycle-free digraph games
The web values for cycle-free digraph games
Sharing a river with multiple sources, a delta and islands
Tree connectedness
Efficiency and stability
Sink connectedness
A value ξ on GNΓ is full-sink efficient (FSE) if, for every cycle-free digraph game hv , Γi,
for all i ∈ N it holds that
X
ξj (v , Γ) = v (P̄Γ (i)).
j∈P̄Γ (i)
A value ξ on GNΓ is maximal-sink efficient (MSE) if, for every cycle-free digraph game
hv , Γi, for all i ∈ LΓ (N) it holds that
X
ξj (v , Γ) = v (P̄Γ (i)).
j∈P̄Γ (i)
A value ξ on GNΓ is predecessor equivalent (PE) if, for every cycle-free digraph game
hv , Γi, for all (i, j) ∈ Γ, for every k ∈ P̄Γ (j) it holds that
ξk (v , Γ\(i, j)) = ξk (v , Γ).
Anna Khmelnitskaya, Dolf Talman
Tree-type values for CF DG-games
Preliminaries
Tree-type values for cycle-free digraph games
The web values for cycle-free digraph games
Sharing a river with multiple sources, a delta and islands
For a digraph Γ on N, the set WΓ (i) = SΓ (i) ∪ PΓ (i) ∪ i defines the web of i ∈ N in Γ.
Given a directed graph Γ on N, a coalition M ⊂ N is a management team in Γ, if
S
WΓ (i) = N,
•
i∈M
• S̄Γ (i) ∩ P̄Γ (j) = ∅ for all i, j ∈ M, i 6= j.
Remark that some managers in M might be simply roots or sinks in Γ.
Anna Khmelnitskaya, Dolf Talman
Tree-type values for CF DG-games
Preliminaries
Tree-type values for cycle-free digraph games
The web values for cycle-free digraph games
Sharing a river with multiple sources, a delta and islands
1
2
10
4
3
5
6
7
8
9
(3,4,10) is a management team
Anna Khmelnitskaya, Dolf Talman
Tree-type values for CF DG-games
Preliminaries
Tree-type values for cycle-free digraph games
The web values for cycle-free digraph games
Sharing a river with multiple sources, a delta and islands
1
2
10
4
3
5
6
7
8
9
(6,7) is a management team
Anna Khmelnitskaya, Dolf Talman
Tree-type values for CF DG-games
Preliminaries
Tree-type values for cycle-free digraph games
The web values for cycle-free digraph games
Sharing a river with multiple sources, a delta and islands
1
2
10
4
3
5
6
7
8
9
(3,4) is not a management team
Anna Khmelnitskaya, Dolf Talman
Tree-type values for CF DG-games
Preliminaries
Tree-type values for cycle-free digraph games
The web values for cycle-free digraph games
Sharing a river with multiple sources, a delta and islands
A value ξ on GNΓ is web efficient (WE) if, for every cycle-free digraph game hv , Γi and a
given management team M in Γ, it holds that for all i ∈ M,
X
ξj (v , Γ, M) = v (WΓ (i)).
j∈WΓ (i)
A value ξ on GN` is successor equivalent (SEM) if, for every cycle-free digraph game
hv , Γi and a given management team M in Γ, for all (i, j) ∈ Γ such that i, j ∈ S̄Γ (M), for
every k ∈ S̄Γ (j) it holds that
ξk (v , Γ\(i, j), M) = ξk (v , Γ, M).
A value ξ on GNΓ is predecessor equivalent (PEM) if, for every cycle-free digraph game
hv , Γi and a given management team M in Γ, for all (i, j) ∈ Γ such that i, j ∈ P̄Γ (M), for
every k ∈ P̄Γ (i) it holds that
ξk (v , Γ\(i, j), M) = ξk (v , Γ, M).
It turns out that WE together with SEM and PEM uniquely defines a value on the class
of cycle-free digraph games.
Anna Khmelnitskaya, Dolf Talman
Tree-type values for CF DG-games
Preliminaries
Tree-type values for cycle-free digraph games
The web values for cycle-free digraph games
Sharing a river with multiple sources, a delta and islands
Sharing a river
N is a set players (users of water), located along the river.
eki ≥ 0, i ∈ N, k ∈ O(i), is the inflow of water in front of the most upstream player(s)
(in this case k = 0) or the inflow of water entering the river between neighboring
players in front of player i.
e1,5
e0,1
1
5
e0,2
e11,12
ei−1,i ei,i+1
11
3
e0,4
e10,11
e5,10
2
e0,3
ei+2,i+5
ei+1,i+2
12
i +5
i +2
6
e7,8
8
i i +1
10
i +3
13
9
4
7
e10,13
i +4
e0,7
A river with multiple sources, a delta, and several islands along the river bed
Anna Khmelnitskaya, Dolf Talman
Tree-type values for CF DG-games
Preliminaries
Tree-type values for cycle-free digraph games
The web values for cycle-free digraph games
Sharing a river with multiple sources, a delta and islands
Sharing a river
N is a set players (users of water), located along the river.
eki ≥ 0, i ∈ N, k ∈ O(i), is the inflow of water in front of the most upstream player(s)
(in this case k = 0) or the inflow of water entering the river between neighboring
players in front of player i.
e1,5
e0,1
1
5
e0,2
e11,12
ei−1,i ei,i+1
11
3
e0,4
e10,11
e5,10
2
e0,3
ei+2,i+5
ei+1,i+2
12
i +5
i +2
6
e7,8
8
i i +1
10
i +3
13
9
4
7
e10,13
i +4
e0,7
A river with multiple sources, a delta, and several islands along the river bed
Anna Khmelnitskaya, Dolf Talman
Tree-type values for CF DG-games
Preliminaries
Tree-type values for cycle-free digraph games
The web values for cycle-free digraph games
Sharing a river with multiple sources, a delta and islands
Sharing a river
N is a set players (users of water), located along the river.
eki ≥ 0, i ∈ N, k ∈ O(i), is the inflow of water in front of the most upstream player(s)
(in this case k = 0) or the inflow of water entering the river between neighboring
players in front of player i.
e1,5
e0,1
1
5
e0,2
e11,12
ei−1,i ei,i+1
11
3
e0,4
e10,11
e5,10
2
e0,3
ei+2,i+5
ei+1,i+2
12
i +5
i +2
6
e7,8
8
i i +1
10
i +3
13
9
4
7
e10,13
i +4
e0,7
A river with multiple sources, a delta, and several islands along the river bed
Anna Khmelnitskaya, Dolf Talman
Tree-type values for CF DG-games
Preliminaries
Tree-type values for cycle-free digraph games
The web values for cycle-free digraph games
Sharing a river with multiple sources, a delta and islands
Sharing a river
Following Ambec and Sprumont (2002), it is assumed that each player i ∈ N has a
quasi-linear utility function given by u i (xi , ti ) = bi (xi ) + ti where ti is a monetary
compensation to player i, xi is the amount of water allocated to player i, and
bi : IR+ → IR is a continuous nondecreasing function providing benefit bi (xi ) to player i
by the consumption of xi of water.
Moreover in case of a river with a delta it is also assumed that if there is a splitting of a
river into branches after a player i then ith player possesses some technical facilities
that allow him not only to take his own share xi∗ according chosen water distribution
plan x ∗ but also to split the rest of the water flow to the branches such that to
guarantee the realization of the water distribution plan x ∗ to all his successors.
Anna Khmelnitskaya, Dolf Talman
Tree-type values for CF DG-games
Preliminaries
Tree-type values for cycle-free digraph games
The web values for cycle-free digraph games
Sharing a river with multiple sources, a delta and islands
Sharing a river
An allocation is a pair (x, t) ∈ IRn+ × IRn of water distribution and compensation
scheme, that similar to the case of a river with a delta or with multiple sources
(Khmelnitskaya, 2010) satisfies
X
X X

xj
≤
ekj ,





j∈P̄Γ (i)
j∈P̄Γ (i) k ∈OΓ (j)
X
for all i ∈ N,
ti ≤ 0, and
X
X
X



i∈N
x
≤
e
,
j
kj


j∈PΓ (i)∪B̄Γ (i)
j∈PΓ (i)∪B̄Γ (i) k ∈OΓ (j)
with Γ being a graph presenting the river structure.
The first condition is a budget constraint. While the second one first, reflects that a
player can use only his upstream inflow of water and second, the total usage of water
by any set of all brothers in Γ cannot exceed the total upstream inflow to all of them.
In the second constraint the summation over OΓ (j) may contain more than one element
only in case of a river with multiple sources.
In the second constraint the second inequality may differ from the first one only for a
river with a delta.
In case of a river with a delta for all brothers the second inequality in the second
constraint is the same.
Anna Khmelnitskaya, Dolf Talman
Tree-type values for CF DG-games
Preliminaries
Tree-type values for cycle-free digraph games
The web values for cycle-free digraph games
Sharing a river with multiple sources, a delta and islands
Sharing a river
The optimal water distribution x ∗ ∈ IRn+ maximizes the total welfare
max
x∈IRn+
X
i∈N
bi (xi ) s.t.











X
X
ekj ,
j∈P̄Γ (i) k ∈OΓ (j)
j∈P̄Γ (i)
X
X
≤
xj
xj
X
≤
X
∀i ∈ N.
ekj ,
j∈PΓ (i)∪B̄Γ (i) k ∈OΓ (j)
j∈PΓ (i)∪B̄Γ (i)
A welfare distribution distributes the total benefits
distribution x ∗ among the players.
P
i∈N
bi (xi∗ ) of optimal water
We approach the optimization problem the same way as in Brink, Laan, and Vasil’ev
(2007) in case of line-graphs.
The problem of finding an optimal welfare distribution among users of water located
along a river with a delta or with multiple sources can be modeled as a cycle-free
digraph game.
For any pair of players, the water inflow entering the river before the upstream player
can only be allocated to the downstream player if all players between them cooperate,
otherwise any player between them can take this water for his own use.
Hence, only coalitions of consecutive players are admissible.
Anna Khmelnitskaya, Dolf Talman
Tree-type values for CF DG-games
Preliminaries
Tree-type values for cycle-free digraph games
The web values for cycle-free digraph games
Sharing a river with multiple sources, a delta and islands
Sharing a river
The characteristic function v is defined as,
X
a)
v (N) =
bi (xi∗ );
i∈N
b) for any connected coalition S ⊂ N,
v (S) =
X
bi (xiS ),
i∈N
where x S ∈ IRs solves






X
i
max
b (xi ) s.t.
s

x∈IR+


i∈S


X
X
≤
xj
j∈(PΓ (i)∪B̄Γ (i))∩S
c) for any disconnected coalition S ⊂ N,
xj
X
≤
X
j∈(PΓ (i)∪B̄Γ (i))∩S k ∈OΓ (j)
v (S) =
X
v (T ).
T ∈C Γ (S)
We refer to this game as the river game.
The river game v is superadditive.
Anna Khmelnitskaya, Dolf Talman
ekj ,
j∈P̄Γ (i)∩S k ∈OΓ (j)
j∈P̄Γ (i)∩S
X
X
Tree-type values for CF DG-games
∀i ∈ S;
ekj ,
Preliminaries
Tree-type values for cycle-free digraph games
The web values for cycle-free digraph games
Sharing a river with multiple sources, a delta and islands
Thank You!
Anna Khmelnitskaya, Dolf Talman
Tree-type values for CF DG-games
Preliminaries
Tree-type values for cycle-free digraph games
The web values for cycle-free digraph games
Sharing a river with multiple sources, a delta and islands
Bibliography
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Brink, R. van den, G. van der Laan, and V. Vasil’ev (2007), Component efficient solutions in
line-graph games with applications, Economic Theory, 33, 349–364.
Demange, G. (2004), On group stability in hierarchies and networks, Journal of Political
Economy, 112, 754–778.
Gillies, D.B. (1953), Some theorems on n-person games, Ph.D. Thesis, Princeton University.
Herings, P.J.J., G. van der Laan, and A.J.J. Talman (2008), The average tree solution for
cycle-free graph games, Games and Economic Behavior, 62, 77–92.
Khmelnitskaya, A.B. (2010), Values for rooted-tree and sink-tree digraphs games and sharing
a river, Theory and Decision, 69, 657-669.
Myerson, R.B. (1977), Graphs and cooperation in games, Mathematics of Operations
Research, 2, 225–229.
Anna Khmelnitskaya, Dolf Talman
Tree-type values for CF DG-games