Independence of the axioms of Theorem I
1. The solution fi(N, v, D) = 0 for all (N, v, D) ∈ GD and i ∈ N satisfies add, npp, imt, nsp, slb,
and sub. It does not satisfy eff.
2. The solution f (N, v, D) = v(N )ϕSL(N, uE(N,v), D) for all (N, v, D) ∈ GD, with E(N, v) being the set of all non-null players in (N, v), satisfies eff, npp, imt, nsp, slb, and sub. It does not satisfy add.
3. Let ω ∈ RΩ++ be an exogenous vector. For a given (N, v, D) ∈ GD, let
R = {i ∈ N : i is not a null player and SD(A(i)) = ∅} .
Then, consider the solution f defined for each (N, v, D) ∈ GD and each i ∈ N as follows. First, if v = uT for some T ⊆ N , fi(N, v, D) = ωi P j∈A(i)∩Rωj P j∈A(i)ϕSLi (N, v, D) if N = Ω and i ∈ R, ϕSLi (N, v, D) otherwise.
Second, for an arbitrary (N, v) ∈ G, f (N, v, D) = P
∅6=T ⊆N f (N, ∆v(T )uT, D). Then f satisfies
add, eff, npp, imt, slb, and sub. Moreover, f does not satisfy nsp.
4. The solution f (N, v, D) = v(N )ϕSL(N, uN, D) for all (N, v, D) ∈ GD satisfies add, eff, imt,
nsp, slb, and sub. It does not satisfy npp.
5. The solutionϕeSL introduced in Definition 5.2 satisfies add, eff, npp, nsp, slb, and sub. It does not satisfy imt.
6. Recall that i0denotes the root of the tree. The solution given by fi0(N, v, D) = v(N )−v(N \{i0}),
and fi(N, v, D) = ϕSL(N, v |N \{i0}, D) if i ∈ N \ {i0}, satisfies eff, add, npp, imt, nsp, and
slb. It does not satisfy sub.
7. The solution ϕSL introduced in Definition 5.1 satisfies eff, add, npp, imt, nsp, and sub. It does not satisfy slb.
Independence of the axioms of Theorem II
1. The solution fi(N, v, D) = 0 for all (N, v, D) ∈ GD and i ∈ N satisfies add, npp, imt, nsp, tlb, and tub. It does not satisfy eff.
2. The solution f (N, v, D) = v(N )ϕSL(N, uE(N,v), D) for all (N, v, D) ∈ GD, with E(N, v) being
the set of all non-null players in (N, v), satisfies eff, npp, imt, nsp, tlb, and tub. It does not satisfy add.
3. The solution f (N, v, D) = v(N )ϕSL(N, uN, D) for all (N, v, D) ∈ GD satisfies add, eff, imt,
nsp, tlb, and tub. It does not satisfy npp.
4. Let ω ∈ RΩ++ be an exogenous vector. For a given (N, v, D) ∈ GD, let
R = {i ∈ N : i is not a null player and SD(A(i)) = ∅} .
Then, consider the solution f defined for each (N, v, D) ∈ GD and each i ∈ N as follows. First, if v = uT for some T ⊆ N , fi(N, v, D) = ωi P j∈A(i)∩Rωj P j∈A(i)ϕSLi (N, v, D) if N = Ω and i ∈ R, ϕSL i (N, v, D) otherwise.
Second, for an arbitrary (N, v) ∈ G, f (N, v, D) = P
∅6=T ⊆N f (N, ∆v(T )uT, D). Then f satisfies
add, eff, npp, imt, tlb, and tub. Moreover, f does not satisfy nsp.
5. Let N∗ = {1, 2, 3, 4, 5} and D∗ = {(1, 2), (1, 3), (2, 4), (3, 5)}. Let also α ∈ [0, 1] \ {0.5}. Then, consider the solution f defined for each (N, uT, D) ∈ GD, with T ⊆ N , and i ∈ N as follows:
fi(N, u{3,4}, D) = α if (N, v, D) = (N∗, u{3,4}, D∗) and i = 3, 1 − α if (N, v, D) = (N∗, u{3,4}, D∗) and i = 4, ϕSLi (N, uT, D) otherwise.
The solution f on GD is then simply obtained as the additive extension on the whole class of games with hierarchical structure, and it satisfies add, eff, npp, nsp, tlb, and tub. Moreover, f does not satisfy imt.
6. The solution f (N, v, D) = Sh(N, v) satisfies add, eff, npp, imt, nsp, and tub. It does not satisfy tlb.
7. The solution ϕSL satisfies add, eff, npp, imt, nsp, and tlb. It does not satisfy tub. Independence of the axioms of Theorem III
1. The solution fi(N, v, D) = 0 for all (N, v, D) ∈ GD and i ∈ N satisfies add, npp, snsp, sslb,
ssub, and nop. It does not satisfy eff.
2. The solution f (N, v, D) = v(N )ϕeSL(N, uE(N,v), D) for all (N, v, D) ∈ GD, with E(N, v) being
the set of all non-null players in (N, v), satisfies eff, npp, snsp, sslb, ssub, and nop. It does not satisfy add.
3. The solution f (N, v, D) = v(N )ϕeSL(N, uN, D) for all (N, v, D) ∈ GD satisfies eff, add, snsp,
sslb, ssub, and nop. It does not satisfy npp.
4. Let N∗= {1, 2} and D∗ = {(1, 2)}, and consider the solution f defined for each (N, uT, D) ∈ GD,
with T ⊆ N , and i ∈ N as follows:
fi(N, uT, D) = i 3 if (N, v, D) = (N ∗, u {1,2}, D∗) and i ∈ {1, 2}, e ϕSLi (N, uT, D) otherwise.
The solution f on GD is then simply obtained as the additive extension on the entire class of games with hierarchical structure and it satisfies eff, add, npp, snsp, ssub, and nop. It does not satisfy sslb.
5. Let N∗= {1, 2} and D∗ = {(1, 2)}, and consider the solution f defined for each (N, uT, D) ∈ GD,
with T ⊆ N , and i ∈ N as follows:
fi(N, ut, D) = 3−i 3 if (N, v, D) = (N ∗, u {1,2}, D∗) and i ∈ {1, 2}, e ϕSLi (N, uT, D) otherwise.
The solution f on GD is then simply obtained as the additive extension on the whole class of games with hierarchical structure and it satisfies eff, add, npp, snsp, sslb, and nop. It does not satisfy ssub.
6. Let N∗ = {1, 2, 3} and D∗ = {(1, 2), (1, 3)}, and consider the solution f defined for each (N, uT, D) ∈ GD, with T ⊆ N , and i ∈ N as follows:
fi(N, v, D) = i 5 if (N, v, D) = (N ∗, u {2,3}, D∗) and i ∈ {2, 3}, e ϕSLi (N, uT, D) otherwise.
Then, solution f on GD is then simply obtained as the additive extension on the whole class of games with hierarchical structure and it satisfies eff, add, npp, ssub, sslb, and nop. It does not satisfy snsp.
7. Let N∗ = {1, 2, 3, 4, 5} and D∗ = {(1, 2), (1, 3), (2, 4), (3, 5)}, and consider the solution f defined for each (N, uT, D) ∈ GD, with T ⊆ N , and i ∈ N as follows:
fi(N, v, D) = i 9 if (N, v, D) = (N ∗, u {4,5}, D∗) and i ∈ {4, 5}, e ϕSL i (N, uT, D) otherwise.
Then, solution f on GD is then simply obtained as the additive extension on the whole class of games with hierarchical structure and it satisfies eff, add, npp, snsp, ssub, and sslb. It does not satisfy nop.