Values for graph-restricted games with coalition structure

OutlinePreliminaries

Graph games with coalition structureCE G-values for games with cooperation structure

PG-valuesGeneralization to games with level structure

Sharing a river with multiple users

Values for graph-restricted gameswith 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










Aumann and Drèze (1974), Owen (1977)






Myerson (1977)






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






model of the paper






case of the coincidence of both models






model of the paper






N1 N2 Nk Nm

sharing an international river among multiple users without international firms






1 PreliminariesTU gamesGames with coalition structureGames with cooperation structure

2 Graph games with coalition structure3 CE G-values for games with cooperation structure

The Myerson valueThe position valueThe average tree solutionValues for line-graph gamesUniform approach to CE G-values

4 PG-valuesCE valuesStabilityDistribution of Harsanyi dividends

5 Generalization to games with level structure6 Sharing a river with multiple users






TU gamesGames with coalition structureGames with cooperation structure

A cooperative TU game is a pair 〈N, v〉 whereN = {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.(GN = IR2n−1 of vectors {v(S)} S⊆N

S 6=∅)

A subgame of a game v is a game v |T with a player set T ⊂ N,T 6= ∅, 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 ⊆ Nsuch that S ∩ T = ∅.







The unanimity games {uT} T⊆NT 6=∅

,

uT (S) =

{1, T ⊆ S,0, T 6⊆ S, for all S ⊆ N,

create a basis for GN , i.e., every game v ∈ GN can be uniquelypresented in the linear form

v =∑T⊆NT 6=∅

λvT uT ,

where λvT =

∑S⊆T

(−1)t−s v(S), for all T ⊆ N, T 6= ∅.

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

v(S) =∑T⊆ST 6=∅

λvT , for all S ⊆ N, S 6= ∅.







For a permutation π : N → N, let πi = {j ∈ N |π(j) ≤ π(i)} be the setof players with rank number not greater than the rank number of i ,including i itself.

The marginal contribution vector mπ(v) ∈ IRn of a game v and apermutation π is given by

mπi (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 orderingn,n − 1, . . . ,1, i.e., l(i) = n + 1− i , i ∈ N.







For any G ⊆ GN , a value on G is a mapping ξ : G → IRn,

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⊆NT3i

λvTt, for all i ∈ N,

orShi (v) =

1n!

∑π∈Π

mπi (v), for all i ∈ N.

The core of a game v

C(v) = {x ∈ IRn | 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






A coalition structure or a system of a priori unions on N is a partitionP={N1, ...,Nm}, N1 ∪ ... ∪ Nm =N, Ni ∩ Nj =∅, i 6= j .

A pair 〈v ,P〉 presents a game with coalition structure (P-game).GPN is the set of all games with coalition structure with fixed N.A P-value is an operator ξ : GPN → IRn that assigns a vector of payoffsto any game with coalition structure.

For a game with coalition structure 〈v ,P〉, following Owen we define aquotient game vP on the player set M = {1, . . . ,m}:

vP(Q) = v(⋃

k∈Q

Nk ), for all Q ⊆ M.

For any payoff x ∈ IRn we denote xP =(x(Nk )

)k∈M ∈ IRm.

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).







A cooperation structure on N is specified by an undirected graphwithout loops L, L ⊆ Lc = { {i , j} | i , j ∈ N, i 6= j}, where Lc is thecomplete 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 otherterms, a graph game (G-game).

GLN is the set of all games with cooperation structure with fixed N.A G-value is an operator ξ : GLN → IRn that assigns a vector of payoffsto 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






A payoff vector x ∈ IRn is component efficient if x(C) = v(C), forevery component C ∈ N/L.

For a game with cooperation structure 〈v ,L〉, following Myerson weassume that cooperation possible only among connected players andconsider 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 ofcomponent efficient payoff vectors that are not dominated by anyconnected coalition, i.e.,

C(v ,L)={x ∈ IRn |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






An undirected graph L is cycle-free if it contains no cycles. Asequence of nodes {i1, . . . , ik+1} ⊆ N is a cycle in L if (i) k ≥ 2, (ii)ih 6= il , h, l = 1, . . . , k , h 6= 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 apredecessor of j .

j 6= 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 setof all subordinates of i in L, and S̄L(i) = SL(i) ∪ i .







A directed graph L is a rooted tree if there is one node in N, called theroot, having no predecessors in L and there is a unique sequence ofdirected links in L from this node to any other node in N.

A line-graph is a directed graph that contains links only betweensubsequent nodes. Without loss of generality we may assume that ina line-graph L nodes are ordered according to the natural order from1 to n, i.e., L ⊆ {(i , i + 1) | i = 1, . . . ,n − 1}.






A graph game with coalition structure, or shortly PG-game, is givenby a tuple 〈v ,P, 〈LM , {LNk }k∈M〉〉.

For simplicity of notation we denote graphs LNk within a priori unionsNk , k ∈ M, by Lk , a cooperation structure by 〈LM , {Lk}k∈M〉 or evensimply by LP , and for a PG-game write 〈v ,P,LP〉.

The primary is a coalition structure and a cooperation structure isintroduced above the given coalition structure. The cooperationstructure LP = 〈LM , {Lk}k∈M〉 is specified by means of graphs of twotypes – a graph LM connecting a priori unions as single elements, andgraphs Lk within a priori unions Nk , k ∈ M, connecting single players.

Denote by GPLN the set of all PG-games 〈v ,P,LP〉 with v ∈ GN .

A PG-value is an operator ξ : GPLN → IRn that associates with eachPG-game 〈v ,P,LP〉 a vector of payoffs ξ(v ,P,LP) ∈ IRn.


























RemarkA PG-game 〈v ,P,LP〉 can be considered, in particular, with the trivialcoalition 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〉,L〈N〉〉 and 〈v , {N},L{N}〉 reduces to aG-game 〈v ,LN〉.






For a given PG-game 〈v ,P,LP〉, LP = 〈LM , {Lk}k∈M〉, one canconsider graph games within a priori unions 〈vk ,Lk 〉, vk =v |Nk , k ∈M.Moreover, given a coalition structure one can consider a quotientgame. However, a quotient game relevant to a PG-game should takeinto account the limited cooperation within a priori unions, and thus itdiffers from the classical one of Owen. For a given PG-game〈v ,P,LP〉, LP = 〈LM , {Lk}k∈M〉, the related quotient game vPL wedefine 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〉.






Furthermore, let ξ be a G-value. For a PG-game 〈v ,P,LP〉 withcooperation structure LP = 〈LM , {Lk}k∈M〉 suitable for application of ξto the corresponding graph quotient game 〈vPL,LM〉, along with asubgame vk within the a priori union Nk , k ∈ M, one can alsoconsider a ξk -game vξk within the a priori union Nk , k ∈ M, defined as

vξk (S) =

{ξk (vPL,LM), S = Nk ,v(S), S 6= Nk ,

for all S ⊆ Nk ,

where ξk (vPL,LM) is the payoff to the union Nk in the graph quotientgame 〈vPL,LM〉 with respect to G-value ξ.In particular, for any x ∈ IRm, a xk -game vx

k within the a priori unionNk , k ∈ MP , is defined by

vxk (S) =

{xk , S = Nk ,v(S), S 6= Nk ,

for all S ⊆ Nk .

In this context one can consider graph games 〈vξk ,Lk 〉, k ∈ M, as well.Anna Khmelnitskaya Values for GR-games with CS





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 ∈ IRn |[[

xP(K ) = vPL(K ), ∀K ∈ M/LM]

&[xP(Q) ≥ vPL(Q), ∀Q ∈ CLM (M)

]]&[[

x(C) = v(C),∀C ∈ Nk/Lk ,C 6= Nk]

&[x(S) ≥ v(S),∀S ∈ CLk (Nk )

],∀k ∈M : nk > 1

]}.






Proposition

x ∈ C(v ,P,LP) ⇐⇒[xP ∈ C(vPL,LM)

]&[xNk ∈ C(vxP

k ,Lk ),∀k ∈ M : nk >1]







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}).







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}).







For arbitrary undirected graphs the position value (Messen (1988),Borm, Owen, and Tijs (1992)) attributes to each player in 〈v ,L〉 thesum of v(i) and half of the value of each link he is involved in, wherethe value of a link is defined as the Shapley payoff to this link in theassociated link game on links of L, i.e.,

πi (v ,L) = v(i) +12

∑l∈Li

Shl (L, v0L ), for all i ∈ N,

where Li = {l ∈ L|l 3 i}, v0 is the zero-normalization of v , i.e., for allS ⊆ 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 linksin L is defined as

vL(L′) = vL′(N), for all L′ ∈ 2L.







Slikker (2005) characterizes the position value via componentefficiency and balanced link contributions.

A G-value ξ satisfies balanced link contributions (BLC) if, for anygraph 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})

].







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

t ij (v ,L), for all j ∈ N,

where

t ij (v ,L) = v(S̄T (i)(j)) −

∑h∈FT (i)(j)

v(S̄T (i)(h)), for all j ∈ (N/L)i ,

T (i) is a unique rooted tree with the root i in (N/L)i .







For superadditive games the average tree solution belongs to thecore.

The AT -solution for cycle-free graph games is characterized viacomponent efficiency and component fairness.

A G-value ξ is component fair (CF) if, for any graph game 〈v ,L〉, forevery 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}

).







Values for games with cooperation structure given by line-graphsare studied in Brink, Laan, Vasil’ev (Economic Theory, 2007).

The upper equivalent solution (UE-solution)

ξUEi (v ,L) = mu

i (vL), for all i ∈ N.

The lower equivalent solution (LE-solution)

ξLEi (v ,L) = ml

i (vL), for all i ∈ N.

The equal loss solution (EL-solution)

ξELi (v ,L) =

mui (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 efficiencyand one of the three following fairness axioms.







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 anyline-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})

).







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.







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.







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






CE valuesStabilityDistribution of Harsanyi dividends

A PG-value ξ is component efficient in the quotient (CEQ) if, for anyPG-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, forany PG-game 〈v ,P,LP〉, for every k ∈ M, for all componentsC ∈ Nk/LNk , C 6= Nk , ∑

i∈C

ξi (v ,P,LP) = v(C).







Any PG-value is an operator ξ : GPLN → IRn that associates with aPG-game 〈v ,P,LP〉 a vector of payoffs ξ(v ,P,LP) ∈ IRn.An operator ξ = {ξi}i∈N generates on the domain of PG-games anoperator ξP : GPLN → IRm, ξP = {ξPk }k∈M , with ξPk =

∑i∈Nk

ξi , k ∈ M,and m operators ξNk : GPLN → IRnk , ξNk = {ξi}i∈Nk , k ∈ M.

There exists a variety of operators ψP : GLM → GPLN assigning to aG-game 〈u,L〉, u ∈ GM , a PG-game 〈v ,P,LP〉, v ∈ GN , such thatvPL = u and LM = L; it is not necessary that ψP(vP ,LM) = 〈v ,P,LP〉.

ξP◦ψP : GLM → IRm is a G-value applicable to any G-game 〈u,L〉∈GLM .

∀k ∈ M, there is a variety of ψk : GLNk→ GPLN assigning to a G-game

〈u,L〉 ∈ GLNk, a PG-game 〈v ,P,LP〉 ∈ GPLN , such that vk = u & Lk = L.

ξNk ◦ ψk : GLNk→ IRnk is a G-value applicable to G-game 〈u,L〉, u ∈ GNk .







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

We say that PG-value ξ satisfies a (m + 1)-tuple of deletion linkaxioms 〈DLP , {DLk}k∈M〉, if the corresponding G-value ξP ◦ ψPsatisfies 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 aPG-game 〈v ,P,LP〉, is suitable for consideration of a (m + 1)-tuple ofdeletion link axioms 〈DLP , {DLk}k∈M〉, if the corresponding graphquotient game 〈vPL,LM〉 is suitable for application of the DLP -value,and for every k ∈ M, the corresponding G-game 〈vk ,Lk 〉 is suitablefor application of the DLk -value.

























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

k ,Lk 〉.

To ensure that benefits of cooperation between a priori unions can befully distributed among single players we assume that the solutions inall graph yk -games 〈vyk

k ,Lk 〉, k ∈ M, are efficient.

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







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 , itis 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〉, thecooperation structure within a priori unions {Lk}k∈M is compatiblewith the payoff vector y ∈ IRm in the graph quotient game 〈vPL,LM〉, ifthe equality (1) holds for every k ∈ M such that Nk /∈ Nk/Lk .

RemarkIt is worth to emphasize that if all graphs Lk , k ∈ M, are connected,then the cooperation structure within a priori unions {Lk}k∈M isalways compatible with any payoff vector y ∈ IRm in the graphquotient game 〈vPL,LM〉,.







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 , itis 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〉, thecooperation structure within a priori unions {Lk}k∈M is compatiblewith the payoff vector y ∈ IRm in the graph quotient game 〈vPL,LM〉, ifthe equality (1) holds for every k ∈ M such that Nk /∈ Nk/Lk .

RemarkIt is worth to emphasize that if all graphs Lk , k ∈ M, are connected,then the cooperation structure within a priori unions {Lk}k∈M isalways compatible with any payoff vector y ∈ IRm in the graphquotient game 〈vPL,LM〉,.







Theorem

On the class of PG-games 〈v ,P,LP〉 with cooperation structureLP = 〈LM , {Lk}k∈M〉 such that

(i) LP is suitable for consideration of (m+1)-tuple of deletion linkaxioms 〈DLP, {DLk}k∈M〉,

(ii) {Lk}k∈M is compatible with DLP(vPL,LM),

there is a unique PG-value that satisfies component efficiency axiomsCEQ, 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) =

DLPk(i)(vPL,LM), Nk(i) = {i},

DLk(i)i (vDLP

k(i) ,Lk(i)), nk(i) > 1,for all i ∈ N.







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 DLPk (vPL,LM),k ∈ M, among single players within each union by applying therelevant DLk -value, k ∈ M, to the corresponding graphDLP -game 〈vDLP

k ,Lk 〉 within Nk .







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 DLPk (vPL,LM),k ∈ M, among single players within each union by applying therelevant DLk -value, k ∈ M, to the corresponding graphDLP -game 〈vDLP

k ,Lk 〉 within Nk .







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,otherwise 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;

otherwise v(S) = |S|, if |S| ≥ 3;

and the coalition and cooperation structures are given by the picture

N1 N2 N3

12

3

45

6







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 quotientgame 〈vPL,LM〉, i.e., LEPk (vPL,LM), k ∈ M,

(ii) apply the average-tree solution to graph LEP -games within apriori unions Nk , k ∈ M, to distribute the rewards of the a prioriunions LEPk (vPL,LM) among single players.







LE1(vPL,LM) = vLMPL({1,2,3})− vLM

PL({2,3}) = 1,

LE2(vPL,LM) = vLMPL({2,3})− vLM

PL({3}) = 2,

LE3(vPL,LM) = vLMPL({3}) = 3;

N1 = {1} =⇒ ξ1(v ,P,LP) = 1;

AT2(vLE22 ,L2) = 1, AT3(vLE2

2 ,L2) = 1,

AT4(vLE33 ,L3) =

130, AT5(vLE3

3 ,L3) = 22730, AT6(vLE3

3 ,L3) =230.

Thus,

ξ(v ,P,LP) = (1, 1, 1,130, 2

2730,

230

).







LE1(vPL,LM) = vLMPL({1,2,3})− vLM

PL({2,3}) = 1,

LE2(vPL,LM) = vLMPL({2,3})− vLM

PL({3}) = 2,

LE3(vPL,LM) = vLMPL({3}) = 3;

N1 = {1} =⇒ ξ1(v ,P,LP) = 1;

AT2(vLE22 ,L2) = 1, AT3(vLE2

2 ,L2) = 1,

AT4(vLE33 ,L3) =

130, AT5(vLE3

3 ,L3) = 22730, AT6(vLE3

3 ,L3) =230.

Thus,

ξ(v ,P,LP) = (1, 1, 1,130, 2

2730,

230

).







Any 〈F , {DLk}k∈N〉-value of a PG-game 〈v , 〈N〉,L〈N〉〉 and any〈DL,F 〉-value of a PG-game 〈v , {N},L{N}〉 coincide with the Myersonvalue 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 = LcNk

, 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〉 withnontrivial 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 allowedbetween a proper subcoalition of any a priori union with members ofother a priori unions but in the case of Owen the cooperation of aproper subcoalition of an a priori union with other entire a prioriunions is permitted.








A 〈DLP ,F , . . . ,F︸︷︷︸m



, 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〉 withnontrivial 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 allowedbetween a proper subcoalition of any a priori union with members ofother a priori unions but in the case of Owen the cooperation of aproper subcoalition of an a priori union with other entire a prioriunions is permitted.








A 〈DLP ,F , . . . ,F︸︷︷︸m



, 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〉 withnontrivial 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 allowedbetween a proper subcoalition of any a priori union with members ofother a priori unions but in the case of Owen the cooperation of aproper subcoalition of an a priori union with other entire a prioriunions is permitted.







Theorem

Let a game v be superadditive and let all deletion link axioms underconsideration be of one of the types CF, LE, UE, or EL. Then for anyPG-game 〈v ,P,LP〉 with a suitable to m + 1-tuple of deletion linkaxioms 〈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 unionsare connected, then under the conditions of Theorem they are eithertrees or line-graphs.







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,otherwise 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;

otherwise v(S) = |S|, if |S| ≥ 3;

and the coalition and cooperation structures are given by the picture

N1 N2 N3

12

3

45

6







As we’ve seen

ξ(v ,P,LP) = (1, 1, 1,130, 2

2730,

230

).

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 theMyerson values, i.e., φi (v ,P,LP) = µi (v

µk(i),Lk(i)).

µ(vP ,LM) = (0.5, 2, 3.5),

φ(v ,P,LP) = (0.5, 1, 1,23, 2

760,

4360

).

But φ(v ,P,LP) /∈ C(v ,P,LP), since φN3 /∈ C(vµ3 ,L3) because

φ4 + φ5 = 24760

< vL33 ({4,5}) = 2.8 = 2

4860

(see Proposition above).







As we’ve seen

ξ(v ,P,LP) = (1, 1, 1,130, 2

2730,

230

).

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 theMyerson values, i.e., φi (v ,P,LP) = µi (v

µk(i),Lk(i)).

µ(vP ,LM) = (0.5, 2, 3.5),

φ(v ,P,LP) = (0.5, 1, 1,23, 2

760,

4360

).

But φ(v ,P,LP) /∈ C(v ,P,LP), since φN3 /∈ C(vµ3 ,L3) because

φ4 + φ5 = 24760

< vL33 ({4,5}) = 2.8 = 2

4860

(see Proposition above).







Theorem

Let a game v be superadditive and all deletion link axioms DL underscrutiny be of one of the types CF, LE, UE, or EL. Then for anyPG-game 〈v ,P,LP〉 with a suitable to (m + 1)-tuple of deletion linkaxioms 〈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 theaxioms 〈DLP , {DLk}k∈M〉.

Theorem

For every superadditive PG-game 〈v ,P,LP〉 for which all graphsdetermining the cooperation structure LP = 〈LM , {Lk}k∈M〉 are eithercycle-free or line-graphs and, moreover, all graphs Lk , k ∈ M, areconnected, the core C(v ,P,LP) is non-empty.







The worth of any coalition is equal to the sum of the Harsanyidividends of the coalition itself and all its proper subcoalitions, theHarsanyi dividend of a coalition has natural interpretation as the extrarevenue of the cooperation between the players of the coalition thatthey did not already realize by cooperation in proper subcoalitions.

How the value under scrutiny distributes the dividend of a coalitionamong the players provides important information concerning theinterest of different players to create the coalition.

This information is especially important in games with limitedcooperation structure when it might happen that one player (or somegroup of players) is responsible for creation of a coalition. If in such acase the player responsible for the creation of a coalition obtains noquota from the dividend of this coalition she may simply block thecreation of this coalition at all.







The worth of any coalition is equal to the sum of the Harsanyidividends of the coalition itself and all its proper subcoalitions, theHarsanyi dividend of a coalition has natural interpretation as the extrarevenue of the cooperation between the players of the coalition thatthey did not already realize by cooperation in proper subcoalitions.

How the value under scrutiny distributes the dividend of a coalitionamong the players provides important information concerning theinterest of different players to create the coalition.

This information is especially important in games with limitedcooperation structure when it might happen that one player (or somegroup of players) is responsible for creation of a coalition. If in such acase the player responsible for the creation of a coalition obtains noquota from the dividend of this coalition she may simply block thecreation of this coalition at all.







The only coalitions allowed in graph games with coalition structureare either the coalitions of entire a priori unions or subcoalitionswithin a priori unions.

Every 〈DLP , {DLk}k∈M〉-value is a superposition of DLP -value in aquotient game and DLk -values, k ∈ M, in corresponding gameswithin a priori unions.

Thus, any 〈DLP , {DLk}k∈M〉-value distributes the dividend of acoalition containing the entire a priori unions according to theDLP -value and the dividend of any subcoalition of an a priori unionaccording to the corresponding DLk -value.
























A level structure is a finite sequence of partitions L = (P1, ...,Pq) suchthat every Pr , is a refinement of Pr+1, that is, if P ∈ Pr , then P ⊂ Qfor 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 levelstructure. It is assumed that at each level r = 1, . . . ,q cooperationpossible only among a priori unions Nkr ,Nlr ∈ Pr , kr , lr ∈ MPr , thatsimultaneously belong to the same a priori union of the coalitionstructure at the next level, i.e., Nkr ,Nlr ⊂ Nkr+1 ∈ Pr+1, and nocooperation is allowed between elements from different levels. Thecooperation structure in this situation can be specified byLL = 〈LMq , {{Lkr }kr∈Mr }

qr=1〉.

The combination of a TU game with level structure and with limitedcooperation possibilities represented by means of level structuredependent graph structure results in a so-called graph game withlevel structure (LG-game), given by a tuple 〈v ,L,LL〉.






2-level level structure






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 }

qr=1〉,

(i) for all k1 ∈ M1, for any component C ∈ Nk1/Lk1 , C 6= Nk1 ,∑i∈C

ξi (v ,L,LL) = v(C),

(ii) for every level r = 2, . . . ,q, for all kr ∈ Mr , for any componentC ∈ Nkr /Lkr , C 6= 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).






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






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 theriver and numbered successively from upstream to downstream.

ek ≥ 0 is the inflow of water entering the river between communitiesk − 1 and k , k ∈ M, with e1 the inflow in front of N1.

Each Nk is equipped by a connected pipe system connecting all itsmembers. 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






N =⋃




N1 N2 Nk Nm

e1 e2 ek em






N =⋃




N1 N2 Nk Nm

e1 e2 ek em






N =⋃




N1 N2 Nk Nm

e1 e2 ek em






Following Ambec and Sprumont (2002), we assume that for each Nk ,the cumulated utility of all players is given by a quasi-linear utilityfunction uk (xk , tk ) = bk (xk ) + tk , where tk is a monetarycompensation to community Nk , xk is the amount of water allocatedto Nk , and bk : IR+ → IR is a continuous nondecreasing functionproviding 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 interms of a TU game vk that allocates the water optimally within thecommunity.






Following Ambec and Sprumont (2002), we assume that for each Nk ,the cumulated utility of all players is given by a quasi-linear utilityfunction uk (xk , tk ) = bk (xk ) + tk , where tk is a monetarycompensation to community Nk , xk is the amount of water allocatedto Nk , and bk : IR+ → IR is a continuous nondecreasing functionproviding 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 interms of a TU game vk that allocates the water optimally within thecommunity.






Since no cooperation is allowed among single users from differentlevels along the course of the river, the problem of optimal waterallocation fits the framework of the introduced above PG-game forwhich the optimal solution is provided by a suitable PG-value that is asuperposition of the solutions for a line-graph superadditive gameamong communities located along the river and cycle-free graphgames within each community.

In accordance with results obtained by Ambec and Sprumont (J EconTheory, 2002) and Brink, Laan, and Vasil’ev (Econ Theory, 2007) theoptimal water distribution among communities Nk , k ∈ M, can bemodeled as a line-graph river game 〈M, v ,L〉 withL = Lc = {(k , k + 1) | k = 1, . . . ,m − 1} and superadditive game v ;moreover, if all functions bk are differentiable with derivatives going toinfinity as xk tends to zero, strictly increasing and strictly concave, thegame v is convex.






Since no cooperation is allowed among single users from differentlevels along the course of the river, the problem of optimal waterallocation fits the framework of the introduced above PG-game forwhich the optimal solution is provided by a suitable PG-value that is asuperposition of the solutions for a line-graph superadditive gameamong communities located along the river and cycle-free graphgames within each community.

In accordance with results obtained by Ambec and Sprumont (J EconTheory, 2002) and Brink, Laan, and Vasil’ev (Econ Theory, 2007) theoptimal water distribution among communities Nk , k ∈ M, can bemodeled as a line-graph river game 〈M, v ,L〉 withL = Lc = {(k , k + 1) | k = 1, . . . ,m − 1} and superadditive game v ;moreover, if all functions bk are differentiable with derivatives going toinfinity as xk tends to zero, strictly increasing and strictly concave, thegame v is convex.






However, the Ambec and Sprumont solution for the river game withsingleton users that coincides with the UE solution, though being acore selector, is very contradictious from a perspective of thedistribution of the Harsanyi dividends, because the UE solution givesthe total dividend of a coalition to the most downstream player whilethe creation of a coalition is fully up to the most upstream one.

With respect to reasonable distribution of the Harsanyi dividends theLE solution, the EL solution, the AT solution, and the Myerson valueseem to be more attractive.

Hence, for the solution of the river game with multiple users it isreasonable to apply one of the 〈LEP , {CF k}k∈M〉-, 〈ELP , {CF k}k∈M〉-,〈CFP , {CF k}k∈M〉-, or 〈FP , {CF k}k∈M〉-values, i.e. to distribute wateramong communities located along the river applying the LE solution,the EL solution, the AT solution, or the Myerson value and then for thedistribution of shares obtained by each community among itsmembers to apply the AT solution.






Thank You!!!






Bibliography

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Bibliography

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Shapley, L.S. (1953), A value for n-person games, in: Tucker AW, Kuhn HW (eds.) Contributions to thetheory of games II, Princeton University Press, Princeton, NJ, pp. 307–317.

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Documents

Values for graph-restricted games with coalition structure