Values for graph-restricted games with coalition structure Anna - - PowerPoint PPT Presentation

Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users

Values for graph-restricted games with coalition structure

Anna Khmelnitskaya

• St. Petersburg Institute for Economics and Mathematics,

Russian Academy of Sciences

University of Eastern Piedmont, Alessandria April 14, 2008

Anna Khmelnitskaya Values for GR-games with CS

Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users Anna Khmelnitskaya Values for GR-games with CS

Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users

Aumann and Drèze (1974), Owen (1977)

Anna Khmelnitskaya Values for GR-games with CS

Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users

Myerson (1977)

Anna Khmelnitskaya Values for GR-games with CS

Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users

Vázquez-Brage, García-Jurado, and Carreras (1996)

Anna Khmelnitskaya Values for GR-games with CS

Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users

model of the paper

Anna Khmelnitskaya Values for GR-games with CS

Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users

case of the coincidence of both models

Anna Khmelnitskaya Values for GR-games with CS

Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users

model of the paper

Anna Khmelnitskaya Values for GR-games with CS

Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users

N1 N2 Nk Nm

sharing an international river among multiple users without international firms Anna Khmelnitskaya Values for GR-games with CS

Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users

1

Preliminaries TU games Games with coalition structure Games with cooperation structure

2

Graph games with coalition structure

3

CE G-values for games with cooperation structure The Myerson value The position value The average tree solution Values for line-graph games Uniform approach to CE G-values

4

PG-values CE values Stability Distribution of Harsanyi dividends

5

Generalization to games with level structure

6

Sharing a river with multiple users

Anna Khmelnitskaya Values for GR-games with CS

Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users TU games Games with coalition structure Games with cooperation structure

A cooperative TU game is a pair N, v where N = {1, . . . , n} is a finite set of n ≥ 2 players, v : 2N → I R, 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. (GN = I R2n−1 of vectors {v(S)} S⊆N

S=∅ )

A subgame of a game v is a game v|T with a player set T ⊂ N, T = ∅, and v|T(S) = v(S) for all S ⊆ T. A game v is superadditive if v(S ∪ T) ≥ v(S) + v(T) for all S, T ⊆ N such that S ∩ T = ∅.

Anna Khmelnitskaya Values for GR-games with CS

Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users TU games Games with coalition structure Games with cooperation structure

The unanimity games {uT} T⊆N

T=∅ ,

uT(S) =

• 1,

T ⊆ S, 0, T ⊆ S, for all S ⊆ N, create a basis for GN, i.e., every game v ∈ GN can be uniquely presented in the linear form v =

• T⊆N

T=∅

λv

T uT,

where λv

T =

• S⊆T

(−1)t−s v(S), for all T ⊆ N, T = ∅. λv

T is called the dividend of the coalition T in the game v.

v(S) =

• T⊆S

T=∅

λv

T,

for all S ⊆ N, S = ∅.

Anna Khmelnitskaya Values for GR-games with CS

Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users TU games Games with coalition structure Games with cooperation structure

For a permutation π: N → N, let πi = {j ∈ N | π(j) ≤ π(i)} be the set

• f players with rank number not greater than the rank number of i,

including i itself. The marginal contribution vector mπ(v) ∈ I Rn of a game v and a permutation π is given by mπ

i (v) = v(πi) − v(πi\i),

i ∈ N. By u we denote the permutation with natural ordering from 1 to n, i.e., u(i) = i, i ∈ N, and by l the permutation with reverse ordering n, n − 1, . . . , 1, i.e., l(i) = n + 1 − i, i ∈ N.

Anna Khmelnitskaya Values for GR-games with CS

Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users TU games Games with coalition structure Games with cooperation structure

For any G ⊆ GN, a value on G is a mapping ξ : G → I Rn, the real number ξi(v) is the payoff to player i in the game v. The Shapley value of a game v Shi(v) =

• T⊆N

T∋i

λv

T

t , for all i ∈ N,

• r

Shi(v) = 1 n!

• π∈Π

mπ

i (v),

for all i ∈ N. The core of a game v C(v) = {x ∈ I Rn | x(N) = v(N), x(S) ≥ v(S), for all S ⊆ N}. A value ξ is stable if for any superadditive game v ∈ GN, ξ(v) ∈ C(v).

Anna Khmelnitskaya Values for GR-games with CS

Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users TU games Games with coalition structure Games with cooperation structure

A coalition structure or a system of a priori unions on N is a partition P ={N1, ..., Nm}, N1 ∪ ... ∪ Nm =N, Ni ∩ Nj =∅, i =j. A pair v, P presents a game with coalition structure (P-game). GP

N is the set of all games with coalition structure with fixed N.

A P-value is an operator ξ : GP

N → I

Rn that assigns a vector of payoffs to any game with coalition structure. For a game with coalition structure v, P, following Owen we define a quotient game vP on the player set M = {1, . . . , m}: vP(Q) = v(

• k∈Q

Nk), for all Q ⊆ M. For any payoff x ∈ I Rn we denote xP =

• x(Nk)
• k∈M ∈ I

Rm. Notice that v, {N} coincides the game v itself. N = {{1}, . . . , {n}} For i ∈ N let k(i) be the index such that i ∈ Nk(i).

Anna Khmelnitskaya Values for GR-games with CS

Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users TU games Games with coalition structure Games with cooperation structure

A cooperation structure on N is specified by an undirected graph without loops L, L ⊆ Lc = { {i, j} | i, j ∈ N, i = j}, where Lc is the complete graph on N while an unordered pair {i, j} is a link (eage) between players i, j ∈ N. A pair v, L constitutes a game with cooperation structure or, in other terms, a graph game (G-game). GL

N is the set of all games with cooperation structure with fixed N.

A G-value is an operator ξ : GL

N → I

Rn that assigns a vector of payoffs to any game with cooperation structure. A subgraph of L on S ⊆ N is the graph L|S = {{i, j} ∈ L | i, j ∈ S}. CL(S) is the set of all connected subcoalitions of S. S/L is the set of all components (maximally connected) of S ⊆ N. (S/L)i is the component of S containing player i ∈ S.

Anna Khmelnitskaya Values for GR-games with CS

Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users TU games Games with coalition structure Games with cooperation structure

A payoff vector x ∈ I Rn is component efficient if x(C) = v(C), for every component C ∈ N/L. For a game with cooperation structure v, L, following Myerson we assume that cooperation possible only among connected players and consider a restricted game vL vL(S) =

• C∈S/L

v(C), for all S ⊆ N. The core C(v, L) of a graph game v, L is defined as a set of component efficient payoff vectors that are not dominated by any connected coalition, i.e., C(v, L)={x ∈ I Rn |x(C)=v(C), ∀C ∈N/L, and x(T)≥v(T), ∀T ∈CL(N)}. C(v, L) = C(vL).

Anna Khmelnitskaya Values for GR-games with CS

Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users TU games Games with coalition structure Games with cooperation structure

An undirected graph L is cycle-free if it contains no cycles. A sequence of nodes {i1, . . . , ik+1} ⊆ N is a cycle in L if (i) k ≥ 2, (ii) ih = il, h, l = 1, . . . , k, h = l, (iii) ik+1 = i1, and (iv) {ih, ih+1} ∈ L, h = 1, . . . , k. An undirected connected cycle-free graph is called a tree. A directed graph L is a collection of directed links. If a directed link (i, j) ∈ L, then j is successor of i and i is a predecessor of j. j = i is a subordinate of i if there is a sequence of directed links (ih, ih+1) ∈ L, h = 1, . . . , k, such that i1 = i and ik+1 = j. By FL(i) we denote the set of all successors of i in L, by SL(i) the set

• f all subordinates of i in L, and ¯

SL(i) = SL(i) ∪ i.

Anna Khmelnitskaya Values for GR-games with CS

Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users TU games Games with coalition structure Games with cooperation structure

A directed graph L is a rooted tree if there is one node in N, called the root, having no predecessors in L and there is a unique sequence of directed links in L from this node to any other node in N. A line-graph is a directed graph that contains links only between subsequent nodes. Without loss of generality we may assume that in a line-graph L nodes are ordered according to the natural order from 1 to n, i.e., L ⊆ {(i, i + 1) | i = 1, . . . , n − 1}.

Anna Khmelnitskaya Values for GR-games with CS

Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users

A graph game with coalition structure, or shortly PG-game, is given by a tuple v, P, LM, {LNk }k∈M. For simplicity of notation we denote graphs LNk within a priori unions Nk, k ∈ M, by Lk, a cooperation structure by LM, {Lk}k∈M or even simply by LP, and for a PG-game write v, P, LP.

Anna Khmelnitskaya Values for GR-games with CS

Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users

A graph game with coalition structure, or shortly PG-game, is given by a tuple v, P, LM, {LNk }k∈M. For simplicity of notation we denote graphs LNk within a priori unions Nk, k ∈ M, by Lk, a cooperation structure by LM, {Lk}k∈M or even simply by LP, and for a PG-game write v, P, LP. The primary is a coalition structure and a cooperation structure is introduced above the given coalition structure. The cooperation structure LP = LM, {Lk}k∈M is specified by means of graphs of two types – a graph LM connecting a priori unions as single elements, and graphs Lk within a priori unions Nk, k ∈ M, connecting single players.

Anna Khmelnitskaya Values for GR-games with CS

Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users

A graph game with coalition structure, or shortly PG-game, is given by a tuple v, P, LM, {LNk }k∈M. For simplicity of notation we denote graphs LNk within a priori unions Nk, k ∈ M, by Lk, a cooperation structure by LM, {Lk}k∈M or even simply by LP, and for a PG-game write v, P, LP. The primary is a coalition structure and a cooperation structure is introduced above the given coalition structure. The cooperation structure LP = LM, {Lk}k∈M is specified by means of graphs of two types – a graph LM connecting a priori unions as single elements, and graphs Lk within a priori unions Nk, k ∈ M, connecting single players. Denote by GPL

N

the set of all PG-games v, P, LP with v ∈ GN. A PG-value is an operator ξ : GPL

N

→ I Rn that associates with each PG-game v, P, LP a vector of payoffs ξ(v, P, LP) ∈ I Rn.

Anna Khmelnitskaya Values for GR-games with CS

Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users

Remark A PG-game v, P, LP can be considered, in particular, with the trivial coalition structure P, i.e., when P = N or P = {N}. If P = N, then M = N, LM = LN, and all graphs Lk = ∅, k ∈ M, if P = {N} then M = {1}, LM = ∅, and L1 = LN. Thus, both PG-games v, N, LN and v, {N}, L{N} reduces to a G-game v, LN.

Anna Khmelnitskaya Values for GR-games with CS

Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users

For a given PG-game v, P, LP, LP = LM, {Lk}k∈M, one can consider graph games within a priori unions vk, Lk, vk =v|Nk , k ∈M. Moreover, given a coalition structure one can consider a quotient

• game. However, a quotient game relevant to a PG-game should take

into account the limited cooperation within a priori unions, and thus it differs from the classical one of Owen. For a given PG-game v, P, LP, LP = LM, {Lk}k∈M, the related quotient game vPL we define as vPL(Q) =      vLk

k (Nk),

Q = {k}, v(

• k∈Q

Nk), |Q| > 1, for all Q ⊆ M. Next, one can naturally consider a graph quotient game vPL, LM.

Anna Khmelnitskaya Values for GR-games with CS

Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users

Furthermore, let ξ be a G-value. For a PG-game v, P, LP with cooperation structure LP = LM, {Lk}k∈M suitable for application of ξ to the corresponding graph quotient game vPL, LM, along with a subgame vk within the a priori union Nk, k ∈ M, one can also consider a ξk-game vξ

k within the a priori union Nk, k ∈ M, defined as

vξ

k (S) =

• ξk(vPL, LM),

S = Nk, v(S), S = Nk, for all S ⊆ Nk, where ξk(vPL, LM) is the payoff to the union Nk in the graph quotient game vPL, LM with respect to G-value ξ. In particular, for any x ∈ I Rm, a xk-game vx

k within the a priori union

Nk, k ∈ MP, is defined by vx

k (S) =

• xk,

S = Nk, v(S), S = Nk, for all S ⊆ Nk. In this context one can consider graph games vξ

k , Lk, k ∈ M, as well.

Anna Khmelnitskaya Values for GR-games with CS

Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users

The core C(v, P, LP) of a PG-game is a set of payoff vectors that are (i) component efficient both in the graph quotient game vPL, LM and in all graph games within a priori unions vk, Lk, k ∈ M, containing more than one player, (ii) not dominated by any connected coalition: C(v, P, LP) =

• x ∈ I

Rn |

• xP(K) = vPL(K), ∀K ∈ M/LM
• &
• xP(Q) ≥ vPL(Q), ∀Q ∈ CLM(M)
• &
• x(C) = v(C), ∀C ∈ Nk/Lk, C = Nk
• &
• x(S) ≥ v(S), ∀S ∈ CLk (Nk)
• , ∀k ∈M : nk > 1
• .

Anna Khmelnitskaya Values for GR-games with CS

Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users

Proposition x ∈ C(v, P, LP) ⇐ ⇒

• xP ∈ C(vPL, LM)
• &
• xNk ∈ C(vxP

k

, Lk), ∀k ∈ M : nk >1

• Anna Khmelnitskaya

Values for GR-games with CS

Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users The Myerson value The position value The average tree solution Values for line-graph games Uniform approach to CE G-values

A G-value ξ is component efficient (CE) if, for any graph game v, L, for all C ∈ N/L,

• i∈C

ξi(v, L) = v(C).

Anna Khmelnitskaya Values for GR-games with CS

Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users The Myerson value The position value The average tree solution Values for line-graph games Uniform approach to CE G-values

A G-value ξ is component efficient (CE) if, for any graph game v, L, for all C ∈ N/L,

• i∈C

ξi(v, L) = v(C). For arbitrary undirected graphs the Myerson value (Myerson, 1977) µi(v, L) = Shi(vL), for all i ∈ N, is characterized via component efficiency and fairness. A G-value ξ is fair (F) if, for any graph game v, L, for every link {i, j} ∈ L, ξi(v, L) − ξi(v, L\{i, j}) = ξj(v, L) − ξj(v, L\{i, j}).

Anna Khmelnitskaya Values for GR-games with CS

Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users The Myerson value The position value The average tree solution Values for line-graph games Uniform approach to CE G-values

For arbitrary undirected graphs the position value (Messen (1988), Borm, Owen, and Tijs (1992)) attributes to each player in v, L the sum of v(i) and half of the value of each link he is involved in, where the value of a link is defined as the Shapley payoff to this link in the associated link game on links of L, i.e., πi(v, L) = v(i) + 1 2

• l∈Li

Shl(L, v0

L ),

for all i ∈ N, where Li = {l ∈ L|l ∋ i}, v0 is the zero-normalization of v, i.e., for all S ⊆ N, v0(S) = v(S) −

i∈S v(i), and for any zero-normalized game

v ∈ GN and a graph L, the associated link game L, vL between links in L is defined as vL(L′) = vL′(N), for all L′ ∈ 2L.

Anna Khmelnitskaya Values for GR-games with CS

Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users The Myerson value The position value The average tree solution Values for line-graph games Uniform approach to CE G-values

Slikker (2005) characterizes the position value via component efficiency and balanced link contributions. A G-value ξ satisfies balanced link contributions (BLC) if, for any graph game v, L, for any i, j ∈ N,

• h|{ß,h}∈L
• ξj(v, L) − ξj(v, L\{i, h})
• =
• h|{æ,h}∈L
• ξi(v, L) − ξi(v, L\{j, h})
• .

Anna Khmelnitskaya Values for GR-games with CS

Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users The Myerson value The position value The average tree solution Values for line-graph games Uniform approach to CE G-values

The average tree solution (AT-solution) for cycle-free graph games (Herings, Laan, and Talman, GEB, 2007) is defined as ATj(v, L) = 1 |(N/L)j|

• i∈(N/L)j

ti

j (v, L),

for all j ∈ N, where ti

j (v, L) = v(¯

ST(i)(j)) −

• h∈FT(i)(j)

v(¯ ST(i)(h)), for all j ∈ (N/L)i, T(i) is a unique rooted tree with the root i in (N/L)i.

Anna Khmelnitskaya Values for GR-games with CS

Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users The Myerson value The position value The average tree solution Values for line-graph games Uniform approach to CE G-values

For superadditive games the average tree solution belongs to the core. The AT-solution for cycle-free graph games is characterized via component efficiency and component fairness. A G-value ξ is component fair (CF) if, for any graph game v, L, for every link {i, j} ∈ L, 1 |(N/L\{i, j})i|

• t∈(N/L\{i,j})i
• ξt(v, L)−ξt(v, L\{i, j}
• =

1 |(N/L\{i, j})j|

• t∈(N/L\{i,j})j
• ξt(v, L) − ξt(v, L\{i, j}
• .

Anna Khmelnitskaya Values for GR-games with CS

Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users The Myerson value The position value The average tree solution Values for line-graph games Uniform approach to CE G-values

Values for games with cooperation structure given by line-graphs are studied in Brink, Laan, Vasil’ev (Economic Theory, 2007). The upper equivalent solution (UE-solution) ξUE

i

(v, L) = mu

i (vL),

for all i ∈ N. The lower equivalent solution (LE-solution) ξLE

i

(v, L) = ml

i(vL),

for all i ∈ N. The equal loss solution (EL-solution) ξEL

i (v, L) = mu i (vL) + ml i(vL)

2 , for all i ∈ N. For superadditive games these three solutions belong to the core. The last three solutions are characterized via component efficiency and one of the three following fairness axioms.

Anna Khmelnitskaya Values for GR-games with CS

Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users The Myerson value The position value The average tree solution Values for line-graph games Uniform approach to CE G-values

A G-value ξ is upper equivalent (UE) if, for any line-graph game v, L, for any i = 1, . . . , n − 1, for all j = 1, . . . , i, ξj(v, L\{i, i+1}) = ξj(v, L). A G-value ξ is lower equivalent (LE) if, for any line-graph game v, L, for any i = 1, . . . , n − 1, for all j = i + 1, . . . , n, ξj(v, L\{i, i+1}) = ξj(v, L). A G-value ξ possesses the equal loss property (EL) if, for any line-graph game v, L, for any i = 1, . . . , n − 1,

i

• j=1
• ξj(v, L) − ξj(v, L\{i, i + 1})
• =

n

• j=i+1
• ξj(v, L) − ξj(v, L\{i, i + 1})
• .

Anna Khmelnitskaya Values for GR-games with CS

Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users The Myerson value The position value The average tree solution Values for line-graph games Uniform approach to CE G-values

CE + F for all G-games ⇐ ⇒ µ(v, L) = Sh(vL), CE + BLC for all G-games ⇐ ⇒ π(v, L), CE + CF for cycle-free G-games ⇐ ⇒ AT(v, L), CE + UE for line G-games ⇐ ⇒ mu(vL), CE + LE for line G-games ⇐ ⇒ ml(vL), CE + EL for line G-games ⇐ ⇒ mu(vL) + ml(vL) 2 .

Anna Khmelnitskaya Values for GR-games with CS

Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users The Myerson value The position value The average tree solution Values for line-graph games Uniform approach to CE G-values

CE + F for all G-games ⇐ ⇒ µ(v, L) = Sh(vL), CE + BLC for all G-games ⇐ ⇒ π(v, L), CE + CF for cycle-free G-games ⇐ ⇒ AT(v, L), CE + UE for line G-games ⇐ ⇒ mu(vL), CE + LE for line G-games ⇐ ⇒ ml(vL), CE + EL for line G-games ⇐ ⇒ mu(vL) + ml(vL) 2 . CE + DL for G-games with suitable graph structure ⇐ ⇒ DL(v, L), where DL is one of the axioms F , BLC, CF, LE, UE, or EL.

Anna Khmelnitskaya Values for GR-games with CS

Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users The Myerson value The position value The average tree solution Values for line-graph games Uniform approach to CE G-values

F-value = the Myerson value, F(v, L) = µ(v, L), for all G-games BLC-value = the position value, BLC(v, L) = π(v, L), for all G-games CF-value = the AT-solution, CF(v, L) = AT(v, L), for cycle-free G-games LE-value = the LE-solution, LE(v, L) = ml(vL), for line G-games UE-value = the UE-solution, UE(v, L) = mu(vL), for line G-games EL-value = EL(v, L) = LE(v, L) + UE(v, L) 2 , for any G-games

Anna Khmelnitskaya Values for GR-games with CS

Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users CE values Stability Distribution of Harsanyi dividends

A PG-value ξ is component efficient in the quotient (CEQ) if, for any PG-game v, P, LP, for each component K ∈ M/LM,

• k∈K
• i∈Nk

ξi(v, P, LP) = vPL(K). A PG-value ξ is component efficient within a priori unions (CEU) if, for any PG-game v, P, LP, for every k ∈ M, for all components C ∈ Nk/LNk , C = Nk,

• i∈C

ξi(v, P, LP) = v(C).

Anna Khmelnitskaya Values for GR-games with CS

Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users CE values Stability Distribution of Harsanyi dividends

Any PG-value is an operator ξ : GPL

N

→ I Rn that associates with a PG-game v, P, LP a vector of payoffs ξ(v, P, LP) ∈ I Rn. An operator ξ = {ξi}i∈N generates on the domain of PG-games an

• perator ξP : GPL

N

→ I Rm, ξP = {ξP

k }k∈M, with ξP k = i∈Nk ξi, k ∈ M,

and m operators ξNk : GPL

N

→ I Rnk , ξNk = {ξi}i∈Nk , k ∈ M. There exists a variety of operators ψP : GL

M → GPL N

assigning to a G-game u, L, u ∈ GM, a PG-game v, P, LP, v ∈ GN, such that vPL = u and LM = L; it is not necessary that ψP(vP, LM) = v, P, LP. ξP◦ ψP : GL

M → I

Rm is a G-value applicable to any G-game u, L∈GL

M.

∀k ∈ M, there is a variety of ψk : GL

Nk → GPL N

assigning to a G-game u, L ∈ GL

Nk , a PG-game v, P, LP ∈ GPL N , such that vk = u & Lk = L.

ξNk ◦ ψk : GL

Nk → I

Rnk is a G-value applicable to G-game u, L, u ∈ GNk .

Anna Khmelnitskaya Values for GR-games with CS

Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users CE values Stability Distribution of Harsanyi dividends

Hence, without ambiguity we may assume that a (m + 1)-tuple of deletion link axioms DLP, {DLk}k∈M with respect to PG-value ξ imposes deletion link requirements on G-values ξP ◦ ψP and ξNk ◦ ψk, k ∈ M.

Anna Khmelnitskaya Values for GR-games with CS

Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users CE values Stability Distribution of Harsanyi dividends

Hence, without ambiguity we may assume that a (m + 1)-tuple of deletion link axioms DLP, {DLk}k∈M with respect to PG-value ξ imposes deletion link requirements on G-values ξP ◦ ψP and ξNk ◦ ψk, k ∈ M. We say that PG-value ξ satisfies a (m + 1)-tuple of deletion link axioms DLP, {DLk}k∈M, if the corresponding G-value ξP ◦ ψP satisfies axiom DLP and the corresponding G-values ξNk ◦ ψk, k ∈ M, satisfy axioms DLk respectively.

Anna Khmelnitskaya Values for GR-games with CS

Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users CE values Stability Distribution of Harsanyi dividends

Hence, without ambiguity we may assume that a (m + 1)-tuple of deletion link axioms DLP, {DLk}k∈M with respect to PG-value ξ imposes deletion link requirements on G-values ξP ◦ ψP and ξNk ◦ ψk, k ∈ M. We say that PG-value ξ satisfies a (m + 1)-tuple of deletion link axioms DLP, {DLk}k∈M, if the corresponding G-value ξP ◦ ψP satisfies axiom DLP and the corresponding G-values ξNk ◦ ψk, k ∈ M, satisfy axioms DLk respectively. We say that the cooperation structure LP = LM, {Lk}k∈M in a PG-game v, P, LP, is suitable for consideration of a (m + 1)-tuple of deletion link axioms DLP, {DLk}k∈M, if the corresponding graph quotient game vPL, LM is suitable for application of the DLP-value, and for every k ∈ M, the corresponding G-game vk, Lk is suitable for application of the DLk-value.

Anna Khmelnitskaya Values for GR-games with CS

Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users CE values Stability Distribution of Harsanyi dividends

We concentrate on solution concepts for PG-games that reflect the two-stage distribution procedure when at first it is played a graph quotient game vPL, LM, and then the total payoff yk, k ∈ M, obtained by the a priori union Nk is distributed among its members by playing the graph yk-game vyk

k , Lk.

To ensure that benefits of cooperation between a priori unions can be fully distributed among single players we assume that the solutions in all graph yk-games vyk

k , Lk, k ∈ M, are efficient.

Since we focus on component efficient solutions, the requirement of efficiency (E) should not contradict to the component efficiency (CE).

Anna Khmelnitskaya Values for GR-games with CS

Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users CE values Stability Distribution of Harsanyi dividends

If the graph Lk is connected, E follows from CE automatically. Else, for every k ∈ M for which Lk is disconnected, i.e., Nk / ∈ Nk/Lk, it is necessary to require

• C∈Nk/Lk

v(C) = yk. (1) We say that in a PG-game v, P, LP, LP = LM, {Lk}k∈M, the cooperation structure within a priori unions {Lk}k∈M is compatible with the payoff vector y ∈ I Rm in the graph quotient game vPL, LM, if the equality (1) holds for every k ∈ M such that Nk / ∈ Nk/Lk.

Anna Khmelnitskaya Values for GR-games with CS

Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users CE values Stability Distribution of Harsanyi dividends

If the graph Lk is connected, E follows from CE automatically. Else, for every k ∈ M for which Lk is disconnected, i.e., Nk / ∈ Nk/Lk, it is necessary to require

• C∈Nk/Lk

v(C) = yk. (1) We say that in a PG-game v, P, LP, LP = LM, {Lk}k∈M, the cooperation structure within a priori unions {Lk}k∈M is compatible with the payoff vector y ∈ I Rm in the graph quotient game vPL, LM, if the equality (1) holds for every k ∈ M such that Nk / ∈ Nk/Lk. Remark It is worth to emphasize that if all graphs Lk, k ∈ M, are connected, then the cooperation structure within a priori unions {Lk}k∈M is always compatible with any payoff vector y ∈ I Rm in the graph quotient game vPL, LM,.

Anna Khmelnitskaya Values for GR-games with CS

Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users CE values Stability Distribution of Harsanyi dividends

Theorem On the class of PG-games v, P, LP with cooperation structure LP = LM, {Lk}k∈M such that (i) LP is suitable for consideration of (m+1)-tuple of deletion link axioms DLP, {DLk}k∈M, (ii) {Lk}k∈M is compatible with DLP(vPL, LM), there is a unique PG-value that satisfies component efficiency axioms CEQ, CEU and (m +1)-tuple of deletion link axioms DLP, {DLk}k∈M. Moreover, for a relevant PG-game v, P, LP it is given by ξi(v, P, LP) =    DLP

k(i)(vPL, LM),

Nk(i) = {i}, DLk(i)

i

(vDLP

k(i) , Lk(i)),

nk(i) > 1, for all i ∈ N.

Anna Khmelnitskaya Values for GR-games with CS

Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users CE values Stability Distribution of Harsanyi dividends

From now on we refer to the PG-value ξ as the DLP, {DLk}k∈M-value.

Anna Khmelnitskaya Values for GR-games with CS

Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users CE values Stability Distribution of Harsanyi dividends

From now on we refer to the PG-value ξ as the DLP, {DLk}k∈M-value. A simple algorithm to compute the DLP, {DLk}k∈M-value: (i) compute the DLP-value for the graph quotient game vPL, LM, (ii) distribute the rewards of the a priori unions DLP

k (vPL, LM),

k ∈ M, among single players within each union by applying the relevant DLk-value, k ∈ M, to the corresponding graph DLP-game vDLP

k

, Lk within Nk.

Anna Khmelnitskaya Values for GR-games with CS

Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users CE values Stability Distribution of Harsanyi dividends

Example: LE, CF, . . . , CF

• m
• value ξ of a GC-game v, P, LP with

line-graph LM and undirected trees Lk, k ∈ M. N contains 6 players, the game v is defined as follows: v({i}) = 0, for all i ∈ N; v({2, 3}) = 1, v({4, 5}) = v({4, 6}) = 2.8, v({5, 6}) = 2.9,

• therwise v({i, j}) = 0, for all i, j ∈N;

v({1, 2, 3}) = 2, v({1, 2, 3, i}) = 3, for i =4, 5, 6;

• therwise v(S) = |S|, if |S| ≥ 3;

and the coalition and cooperation structures are given by the picture N1 N2 N3 1 2 3 4 5 6

Anna Khmelnitskaya Values for GR-games with CS

Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users CE values Stability Distribution of Harsanyi dividends

N = N1 ∪ N2 ∪ N3; N1 ={1}, N2 ={2, 3}, N3 ={4, 5, 6}; L1 =∅, L2 ={{2, 3}}, L3 ={{4, 5}, {5, 6}}; M = {1, 2, 3}; LM = {(1, 2), (2, 3)}; the quotient game vPL is given by vPL({1}) = 0, vPL({2}) = 1, vPL({3}) = 3, vPL({1, 2}) = 2, vPL({2, 3}) = 5, vPL({1, 3}) = 4, vPL({1, 2, 3}) = 6; (i) compute the lower equivalent solution for the line-graph quotient game vPL, LM, i.e., LEP

k (vPL, LM), k ∈ M,

(ii) apply the average-tree solution to graph LEP-games within a priori unions Nk, k ∈ M, to distribute the rewards of the a priori unions LEP

k (vPL, LM) among single players.

Anna Khmelnitskaya Values for GR-games with CS

Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users CE values Stability Distribution of Harsanyi dividends

LE1(vPL, LM) = vLM

PL({1, 2, 3}) − vLM PL({2, 3}) = 1,

LE2(vPL, LM) = vLM

PL({2, 3}) − vLM PL({3}) = 2,

LE3(vPL, LM) = vLM

PL({3}) = 3;

N1 = {1} = ⇒ ξ1(v, P, LP) = 1; AT2(vLE2

2

, L2) = 1, AT3(vLE2

2

, L2) = 1, AT4(vLE3

3

, L3) = 1 30, AT5(vLE3

3

, L3) = 227 30, AT6(vLE3

3

, L3) = 2 30.

Anna Khmelnitskaya Values for GR-games with CS

Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users CE values Stability Distribution of Harsanyi dividends

LE1(vPL, LM) = vLM

PL({1, 2, 3}) − vLM PL({2, 3}) = 1,

LE2(vPL, LM) = vLM

PL({2, 3}) − vLM PL({3}) = 2,

LE3(vPL, LM) = vLM

PL({3}) = 3;

N1 = {1} = ⇒ ξ1(v, P, LP) = 1; AT2(vLE2

2

, L2) = 1, AT3(vLE2

2

, L2) = 1, AT4(vLE3

3

, L3) = 1 30, AT5(vLE3

3

, L3) = 227 30, AT6(vLE3

3

, L3) = 2 30. Thus, ξ(v, P, LP) = (1, 1, 1, 1 30, 227 30, 2 30).

Anna Khmelnitskaya Values for GR-games with CS

Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users CE values Stability Distribution of Harsanyi dividends

Any F, {DLk}k∈N-value of a PG-game v, N, LN and any DL, F-value of a PG-game v, {N}, L{N} coincide with the Myerson value of the G-game v, LN; moreover, if the graph LN is complete, they coincide also with the Shapley value and the Owen value.

Anna Khmelnitskaya Values for GR-games with CS

Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users CE values Stability Distribution of Harsanyi dividends

Any F, {DLk}k∈N-value of a PG-game v, N, LN and any DL, F-value of a PG-game v, {N}, L{N} coincide with the Myerson value of the G-game v, LN; moreover, if the graph LN is complete, they coincide also with the Shapley value and the Owen value. A DLP, F, . . . , F

• m
• value for a PG-game v, P, LP with empty graph

LM and complete graphs LNk , k ∈ M, i.e., LNk = Lc

Nk , for all k ∈ M,

coincides with the Aumann-Drèze value of P-game v, P.

Anna Khmelnitskaya Values for GR-games with CS

Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users CE values Stability Distribution of Harsanyi dividends

Any F, {DLk}k∈N-value of a PG-game v, N, LN and any DL, F-value of a PG-game v, {N}, L{N} coincide with the Myerson value of the G-game v, LN; moreover, if the graph LN is complete, they coincide also with the Shapley value and the Owen value. A DLP, F, . . . , F

• m
• value for a PG-game v, P, LP with empty graph

LM and complete graphs LNk , k ∈ M, i.e., LNk = Lc

Nk , for all k ∈ M,

coincides with the Aumann-Drèze value of P-game v, P. However, a DLP, {DLk}k∈M-value of a PG-game v, P, LP with nontrivial coalition structure P never coincides with the Owen value (and therefore with the φ-value of Vázquez-Brage, García-Jurado, and Carreras as well) because in our model no cooperation is allowed between a proper subcoalition of any a priori union with members of

• ther a priori unions but in the case of Owen the cooperation of a

proper subcoalition of an a priori union with other entire a priori unions is permitted.

Anna Khmelnitskaya Values for GR-games with CS

Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users CE values Stability Distribution of Harsanyi dividends

Theorem Let a game v be superadditive and let all deletion link axioms under consideration be of one of the types CF, LE, UE, or EL. Then for any PG-game v, P, LP with a suitable to m + 1-tuple of deletion link axioms DLP, {DLk}k∈M cooperation structure LP = LM, {Lk}k∈M, in which {Lk}k∈M, is compatible with DLP(vPL, LM), the DLP, {DLk}k∈M-value belongs to the core. Remark If all graphs Lk, k ∈ M, determining cooperation within a priori unions are connected, then under the conditions of Theorem they are either trees or line-graphs.

Anna Khmelnitskaya Values for GR-games with CS

Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users CE values Stability Distribution of Harsanyi dividends

Example: LE, CF, . . . , CF

• m
• value ξ of a GC-game v, P, LP with

line-graph LM and undirected trees Lk, k ∈ M. N contains 6 players, the game v is defined as follows: v({i}) = 0, for all i ∈ N; v({2, 3}) = 1, v({4, 5}) = v({4, 6}) = 2.8, v({5, 6}) = 2.9,

• therwise v({i, j}) = 0, for all i, j ∈N;

v({1, 2, 3}) = 2, v({1, 2, 3, i}) = 3, for i =4, 5, 6;

• therwise v(S) = |S|, if |S| ≥ 3;

and the coalition and cooperation structures are given by the picture N1 N2 N3 1 2 3 4 5 6

Anna Khmelnitskaya Values for GR-games with CS

Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users CE values Stability Distribution of Harsanyi dividends

As we’ve seen ξ(v, P, LP) = (1, 1, 1, 1 30, 227 30, 2 30). Notice, v is superadditive and ξ(v, P, LP) ∈ C(v, P, LP).

Anna Khmelnitskaya Values for GR-games with CS

Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users CE values Stability Distribution of Harsanyi dividends

As we’ve seen ξ(v, P, LP) = (1, 1, 1, 1 30, 227 30, 2 30). Notice, v is superadditive and ξ(v, P, LP) ∈ C(v, P, LP). Consider the F, F, F, F-value φ that is the superposition of the Myerson values, i.e., φi(v, P, LP) = µi(vµ

k(i), Lk(i)).

µ(vP, LM) = (0.5, 2, 3.5), φ(v, P, LP) = (0.5, 1, 1, 2 3, 2 7 60, 43 60). But φ(v, P, LP) / ∈ C(v, P, LP), since φN3 / ∈ C(vµ

3 , L3)

because φ4 + φ5 = 247 60 < vL3

3 ({4, 5}) = 2.8 = 248

60 (see Proposition above).

Anna Khmelnitskaya Values for GR-games with CS

Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users CE values Stability Distribution of Harsanyi dividends

Theorem Let a game v be superadditive and all deletion link axioms DL under scrutiny be of one of the types CF, LE, UE, or EL. Then for any PG-game v, P, LP with a suitable to (m + 1)-tuple of deletion link axioms DLP, {DLk}k∈M cooperation structure LP = LM, {Lk}k∈M, in which {Lk}k∈M, is compatible with DLP(vPL, LM), the DLP, {DLk}k∈M-value is the unique core selector that satisfies the axioms DLP, {DLk}k∈M. Theorem For every superadditive PG-game v, P, LP for which all graphs determining the cooperation structure LP = LM, {Lk}k∈M are either cycle-free or line-graphs and, moreover, all graphs Lk, k ∈ M, are connected, the core C(v, P, LP) is non-empty.

Anna Khmelnitskaya Values for GR-games with CS

Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users CE values Stability Distribution of Harsanyi dividends

The worth of any coalition is equal to the sum of the Harsanyi dividends of the coalition itself and all its proper subcoalitions, the Harsanyi dividend of a coalition has natural interpretation as the extra revenue of the cooperation between the players of the coalition that they did not already realize by cooperation in proper subcoalitions.

Anna Khmelnitskaya Values for GR-games with CS

Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users CE values Stability Distribution of Harsanyi dividends

The worth of any coalition is equal to the sum of the Harsanyi dividends of the coalition itself and all its proper subcoalitions, the Harsanyi dividend of a coalition has natural interpretation as the extra revenue of the cooperation between the players of the coalition that they did not already realize by cooperation in proper subcoalitions. How the value under scrutiny distributes the dividend of a coalition among the players provides important information concerning the interest of different players to create the coalition. This information is especially important in games with limited cooperation structure when it might happen that one player (or some group of players) is responsible for creation of a coalition. If in such a case the player responsible for the creation of a coalition obtains no quota from the dividend of this coalition she may simply block the creation of this coalition at all.

Anna Khmelnitskaya Values for GR-games with CS

Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users CE values Stability Distribution of Harsanyi dividends

The only coalitions allowed in graph games with coalition structure are either the coalitions of entire a priori unions or subcoalitions within a priori unions.

Anna Khmelnitskaya Values for GR-games with CS

Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users CE values Stability Distribution of Harsanyi dividends

The only coalitions allowed in graph games with coalition structure are either the coalitions of entire a priori unions or subcoalitions within a priori unions. Every DLP, {DLk}k∈M-value is a superposition of DLP-value in a quotient game and DLk-values, k ∈ M, in corresponding games within a priori unions.

Anna Khmelnitskaya Values for GR-games with CS

Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users CE values Stability Distribution of Harsanyi dividends

The only coalitions allowed in graph games with coalition structure are either the coalitions of entire a priori unions or subcoalitions within a priori unions. Every DLP, {DLk}k∈M-value is a superposition of DLP-value in a quotient game and DLk-values, k ∈ M, in corresponding games within a priori unions. Thus, any DLP, {DLk}k∈M-value distributes the dividend of a coalition containing the entire a priori unions according to the DLP-value and the dividend of any subcoalition of an a priori union according to the corresponding DLk-value.

Anna Khmelnitskaya Values for GR-games with CS

Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users

A level structure is a finite sequence of partitions L = (P1, ..., Pq) such that every Pr, is a refinement of Pr+1, that is, if P ∈ Pr, then P ⊂ Q for some Q ∈ Pr+1; elements of each coalition structure Pr, 1≤r ≤q, are given by Nkr , kr ∈ MPr , i.e., Pr =

• Nkr
• kr ∈MPr , for all 1 ≤ r ≤ q.

PG-games, can be naturally extended to graph games with level

• structure. It is assumed that at each level r = 1, . . . , q cooperation

possible only among a priori unions Nkr , Nlr ∈ Pr, kr, lr ∈ MPr , that simultaneously belong to the same a priori union of the coalition structure at the next level, i.e., Nkr , Nlr ⊂ Nkr+1 ∈ Pr+1, and no cooperation is allowed between elements from different levels. The cooperation structure in this situation can be specified by LL = LMq, {{Lkr }kr ∈Mr }q

r=1.

The combination of a TU game with level structure and with limited cooperation possibilities represented by means of level structure dependent graph structure results in a so-called graph game with level structure (LG-game), given by a tuple v, L, LL.

Anna Khmelnitskaya Values for GR-games with CS

Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users

2-level level structure

Anna Khmelnitskaya Values for GR-games with CS

Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users

A GL-value ξ is component efficient with respect to level structure (CEL) if, for any LG-game v, L, LL, LL = LMq, {{Lkr }kr ∈Mr }q

r=1,

(i) for all k1 ∈ M1, for any component C ∈ Nk1/Lk1, C = Nk1,

• i∈C

ξi(v, L, LL) = v(C), (ii) for every level r = 2, . . . , q, for all kr ∈ Mr, for any component C ∈ Nkr /Lkr , C = Nkr ,

• kr ∈C
• i∈Nkr

ξi(v, L, LL) = vPr−1L(C). (iii) for any component C ∈ Mq/LMq,

• kq∈C
• i∈Nkq

ξi(v, L, LL) = vPqL(C).

Anna Khmelnitskaya Values for GR-games with CS

Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users

To ensure that benefits of cooperation obtained by a priori unions at the upper level q can be fully distributed among elements of coalition structures at levels below, similar to PG-games it is assumed that the cooperation structures within a priori unions {Lkr }kr ∈Mr , r = 1, . . . , q, are compatible with the payoffs in the graph quotient games at the levels above.

Anna Khmelnitskaya Values for GR-games with CS

Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users

N =

k∈M Nk, Nk ∩ Nl = ∅, k, l ∈ M, is a set players (users of water)

composed of the communities of players Nk, k ∈ M, located along the river and numbered successively from upstream to downstream.

Anna Khmelnitskaya Values for GR-games with CS

Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users

N =

k∈M Nk, Nk ∩ Nl = ∅, k, l ∈ M, is a set players (users of water)

composed of the communities of players Nk, k ∈ M, located along the river and numbered successively from upstream to downstream. ek ≥ 0 is the inflow of water entering the river between communities k − 1 and k, k ∈ M, with e1 the inflow in front of N1.

Anna Khmelnitskaya Values for GR-games with CS

Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users

N =

k∈M Nk, Nk ∩ Nl = ∅, k, l ∈ M, is a set players (users of water)

composed of the communities of players Nk, k ∈ M, located along the river and numbered successively from upstream to downstream. ek ≥ 0 is the inflow of water entering the river between communities k − 1 and k, k ∈ M, with e1 the inflow in front of N1. Each Nk is equipped by a connected pipe system connecting all its

• members. Without loss of generality we assume that all graphs Lk,

k ∈ M, presenting the pipe systems in Nk, k ∈ M, are cycle free.

Anna Khmelnitskaya Values for GR-games with CS

Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users

N =

k∈M Nk, Nk ∩ Nl = ∅, k, l ∈ M, is a set players (users of water)

composed of the communities of players Nk, k ∈ M, located along the river and numbered successively from upstream to downstream. ek ≥ 0 is the inflow of water entering the river between communities k − 1 and k, k ∈ M, with e1 the inflow in front of N1. Each Nk is equipped by a connected pipe system connecting all its

• members. Without loss of generality we assume that all graphs Lk,

k ∈ M, presenting the pipe systems in Nk, k ∈ M, are cycle free. N1 N2 Nk Nm e1 e2 ek em

Anna Khmelnitskaya Values for GR-games with CS

Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users

Following Ambec and Sprumont (2002), we assume that for each Nk, the cumulated utility of all players is given by a quasi-linear utility function uk(xk, tk) = bk(xk) + tk, where tk is a monetary compensation to community Nk, xk is the amount of water allocated to Nk, and bk : I R+ → I R is a continuous nondecreasing function providing benefit bk(xk) through the consumption of water.

Anna Khmelnitskaya Values for GR-games with CS

Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users

Following Ambec and Sprumont (2002), we assume that for each Nk, the cumulated utility of all players is given by a quasi-linear utility function uk(xk, tk) = bk(xk) + tk, where tk is a monetary compensation to community Nk, xk is the amount of water allocated to Nk, and bk : I R+ → I R is a continuous nondecreasing function providing benefit bk(xk) through the consumption of water. Moreover, we assume that if the total shares of water for all Nk, k ∈ M, are fixed, then for each Nk there is a mechanism presented in terms of a TU game vk that allocates the water optimally within the community.

Anna Khmelnitskaya Values for GR-games with CS

Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users

Since no cooperation is allowed among single users from different levels along the course of the river, the problem of optimal water allocation fits the framework of the introduced above PG-game for which the optimal solution is provided by a suitable PG-value that is a superposition of the solutions for a line-graph superadditive game among communities located along the river and cycle-free graph games within each community.

Anna Khmelnitskaya Values for GR-games with CS

Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users

Since no cooperation is allowed among single users from different levels along the course of the river, the problem of optimal water allocation fits the framework of the introduced above PG-game for which the optimal solution is provided by a suitable PG-value that is a superposition of the solutions for a line-graph superadditive game among communities located along the river and cycle-free graph games within each community. In accordance with results obtained by Ambec and Sprumont (J Econ Theory, 2002) and Brink, Laan, and Vasil’ev (Econ Theory, 2007) the

• ptimal water distribution among communities Nk, k ∈ M, can be

modeled as a line-graph river game M, v, L with L = Lc = {(k, k + 1) | k = 1, . . . , m − 1} and superadditive game v; moreover, if all functions bk are differentiable with derivatives going to infinity as xk tends to zero, strictly increasing and strictly concave, the game v is convex.

Anna Khmelnitskaya Values for GR-games with CS

Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users

However, the Ambec and Sprumont solution for the river game with singleton users that coincides with the UE solution, though being a core selector, is very contradictious from a perspective of the distribution of the Harsanyi dividends, because the UE solution gives the total dividend of a coalition to the most downstream player while the creation of a coalition is fully up to the most upstream one. With respect to reasonable distribution of the Harsanyi dividends the LE solution, the EL solution, the AT solution, and the Myerson value seem to be more attractive. Hence, for the solution of the river game with multiple users it is reasonable to apply one of the LEP, {CF k}k∈M-, ELP, {CF k}k∈M-, CF P, {CF k}k∈M-, or F P, {CF k}k∈M-values, i.e. to distribute water among communities located along the river applying the LE solution, the EL solution, the AT solution, or the Myerson value and then for the distribution of shares obtained by each community among its members to apply the AT solution.

Anna Khmelnitskaya Values for GR-games with CS

Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users

Thank You!!!

Anna Khmelnitskaya Values for GR-games with CS

Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users

Bibliography

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Outline Preliminaries Graph games with coalition structure CE G-values for games with cooperation structure PG-values Generalization to games with level structure Sharing a river with multiple users

Bibliography

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