Restricted Bargaining Sets in a Club Economy

Proof of Lemma 4.1.

Let ɛ ∈ (0, λ(T)). Since T , h , ν is a counter-objection to S , g , μ , we have

We now define a vector-valued measure κ on Σ| _T as

Therefore, we have

It follows from Lyapunov’s convexity theorem that κ B : B ∈ Σ | T is a convex set. For α = ε λ T , the convexity of κ B : B ∈ Σ | T guarantees existence of a coalition R ⊆ T such that

Therefore, we have that λ R = ε and the following conditions are satisfied

1. ∫ R h a d λ + ∫ R τ ν a d λ = ∫ R e a d λ ; and

2. ∫ R ν a d λ ∈ C o n s , as C o n s is a subspace.

Thus, (R, (h, ν)) forms a counter-objection to (S, (g, μ)). This completes the proof. □

Proof of Theorem 4.3.

By abuse of notation, let us denote by B ( E ) both the local and global bargaining sets of the economy E . Likewise, the notation B ε δ E is employed to denote both local and global ɛδ-bargaining sets of the economy E . Suppose f , l ∈ B E \ B ε δ E . This means that there exists an ɛ-justified δ-objection to the feasible state (f, l). Let S , g , μ be an ɛ-justified δ-objection to f , l , where λ(S) ≤ δ. By Lemma 4.2, it follows that (S, (g, μ)) itself constitutes a justified objection to f , l . This is a contradiction as ( f , l ) ∈ B ( E ) . So B E ⊆ B ε δ E . □

Proof of Theorem 4.5.

By Theorem 4.3, we have B g ( E ) ⊆ B ε δ g ( E ) . We show that B ε δ g ( E ) ⊆ B g ( E ) . Let f , l ∈ B ε δ g E . Then, either f , l has no objection, and if there exists a δ-objection to it, there exists an ɛ-counter-objection to it. If (f, l) has no objection, then ( f , l ) ∈ C ( E ) ⊆ B g ( E ) . So let, (S, (g, μ)) be an objection to (f, l). Lyapnov’s convexity theorem guarantees the existence of a sub-coalition S ̃ of S such that λ ( S ̃ ) ≤ δ and ( S ̃ , ( g , μ ) ) constitutes a δ-feasible objection to (f, l). Since ( f , l ) ∈ B ε δ g E , there exists an ɛ-feasible counter-objection (T, (h, ν)) to ( S ̃ , ( g , μ ) ) . It can be easily observed that (T, (h, ν)) constitutes a counter-objection to (S, (g, μ)), which means that f , l ∈ B g E . This completes the proof of the theorem. □

Proof of Proposition 4.6.

Since T , h , ν is a counter-objection to ( S , g , μ ) then it must follow that

1. (h _a , ν _a ) ∈ X _a λ-a.e. on T;

2. u a h a , ν a > u a g a , μ a λ -a.e. in T;

3. ∫ _T h _a dλ + ∫ _T τ(ν _a )dλ = ∫ _T e _a dλ; and

4. ∫ T ν a d λ ∈ C o n s .

By Proposition 4.6, without loss of generality, we assume that λ(T) < α < 1. Define δ such that

By the continuity and monotonicity of preferences, we can find a function ζ : T → R + N such that u _a (ζ _a , ν _a ) > u _a (g _a , μ _a )λ-a.e. on T and

By Lemma 3.1 in Bhowmik and Kaur (2023), there exists a state (ξ, ν′) such that

1. u a ξ a , ν a ′ > u a g a , μ a , λ-a.e. on T;

2. ∫ T ξ a d λ = ∫ T δ ⋅ ζ a + 1 − δ ⋅ g a d λ ; and

3. ∫ T ν a ′ d λ = ∫ T δ ⋅ ν a + 1 − δ ⋅ μ a d λ .

Furthermore, by the Lyapunov convexity theorem, there exists a coalition R ⊆ A\T such that

1. λ R = ( 1 − δ ) λ A \ T ;

2. ∫ _R μ _a dλ = (1 − δ)∫ _A\T μ _a dλ; and

3. ∫ _R (g _a + τ(μ _a ) − e _a )dλ = (1 − δ)∫ _A\T (g _a + τ(μ _a ) − e _a )dλ.

Lastly, let us define E ≔ T ∪ R and an allocation y , κ : A → R + N × R M such that

It can be readily verified that

Since ∫ T ν a ′ ∈ C o n s and ∫ A μ a d λ ∈ C o n s , we have ∫ E κ a d λ ∈ C o n s . Therefore, we have

Using the above equality, we derive that

Thus, (E, (y, κ)) forms a global counter-objection to ( S , g , μ ) and λ(E) = α. □

Proof of Theorem 4.7.

We just need to versify that B g E ⊆ α - B g E . To this end, let f , l ∈ B g E . If f , l has no objection, then ( f , l ) ∈ C ( E ) ⊆ B g ( E ) . Thus, we assume that (f, l) has a global objection (S, (g, μ)), which is globally counter-objected by (T, (h, ν)). Then, by Proposition 4.6, we can guarantee the existence of a global counter-objection (E, (y, κ)) such that it forms a global counter-objection to (S, (g, μ)) and λ(E) = α. Hence, f , l ∈ α - B g E .

Proof of Proposition 5.2.

Let (g, μ) be a robustly efficient state of E . Suppose, by way of contradiction, we assume that (S, (g, μ)) is not a justified objection. By Proposition 4.6, there exists a coalition T with λ(T) < λ(A) and a state (h, ν) such that

1. u _a (h _a , ν _a ) > u _a (g _a , μ _a ), λ-a.e. on T;

2. ∫ _T g _a dλ + ∫ _T τ(ν _a )dλ = ∫ _T e _a dλ; and

3. ∫ T ν a d λ ∈ C o n s .

The rest of the argument follows analogously as the proof of Theorem 3.4 of Bhowmik and Kaur (2023). □

Proof of Theorem 5.5.

Assume that (S, (g, μ)) is a justified objection to the state (f, l). Thus, ( g , μ ) ∈ C ( E ) and by Theorem 5.1 of Ellickson et al. (1999), we can claim that (g, μ) is a club equilibrium state of the economy E . Let (p, q) be an equilibrium price. Suppose, by way of contradiction, that (g, μ) is not an ɛ-sequentially robustly efficient state for some ɛ > 0. This implies that there exists a sequence { E ( S n , B n , g , μ , α n ) : n ≥ 1 } of economies and a sequence {(h ⁿ , ν ⁿ ): n ≥ 1} of states such that (g, μ) is dominated by (h ⁿ , ν ⁿ ) in E ( S n , B n , g , μ , α n ) , which means

1. u a h a n , ν a n > u a g a , μ a , λ -a.e. on A;

2. ∫ A h a n d λ + ∫ A τ ν a n d λ = ∫ A e S n , g , α n d λ + ∫ A τ β a ( B n , μ ) d λ ; and

3. ∫ A ν a n d λ , ∫ A β a ( B n , μ ) d λ ∈ C o n s .

   In addition, the following conditions are satisfied:

4. there is a coalition R such that u a ψ a n , ν a n > u a g a , μ a for all ψ a n ∈ h a n + B 0 , ε with a ∈ R and n ≥ 1; and

5. I B n μ = I S n μ and λ B n i ≥ α n λ S n i for all n ≥ 1 and i ∈ I S n ; and

6. α n , λ B n : n ≥ 1 converges to (0, 0).

For each n ≥ 1, there is a sub-coalition C _n of B _n such that λ C n i = α n λ S n i for all i ∈ I S . Thus, we have

Since { λ B n : n ≥ 1 } converges to 0, we have { q ⋅ ∫ B n \ C n μ a d λ : n ≥ 1 } converges to 0. Let n ₀ ≥ 1 be an integer such that

Letting

we note that δ < ε 2 N . It follows that z 0 ≔ δ , … , δ ∈ B 0 , ε . Thus we consider h ̃ : A → R + N such that

As a consequence, we have

It follows from (iv) that

λ-a.e. on A. Thus,

Consequently,

λ-a.e. on S n 0 , and hence

This further implies that

which immediately yields that^[16]

This is equivalent to

Thus, we have that

Now it can be observed that

Now from (iii) we have that ∫ A ( ν a n 0 − β a ( B n 0 , μ ) ) d λ ∈ C o n s and thus there exists a real number δ(π, γ) such that

The above equation along with the fact that ∑ ω ∈ Ω π ( ω ) q ( ω , π , γ ) = p ⋅ i n p ( π , γ ) ^[17] implies

A simple algebraic manipulation yields

We further observe that

which is equivalent to

This further yields that

which means

equivalently,

It can be observed from the definition of the perturbed economy that

Thus, it follows from (ii) that

This contradicts (6.1). □

Proof of Theorem 5.8.

Let (S, (g, μ)) be a globally justified objection to the state (f, l) in the economy E . Thus, (g, μ) belongs to the core of the economy E . Hence, from Theorem 5.1 of Ellickson et al. (1999), we can claim that (g, μ) is a club equilibrium state of the economy E . Suppose, by way of contradiction, let us assume that (g, μ) is not approximately robustly efficient. Then one can argue along the lines of Theorem 3.12 of Bhowmik and Kaur (2023) to show that (g, μ) is approximately robustly efficient.

Conversely, let (S, (g, μ)) be a global objection to the state (f, l) such that (g, μ) is an approximately robustly efficient state. Suppose, by way of contradiction, that (S, (g, μ)) is not globally justified. Hence, by Proposition 4.6, there exists a coalition T with λ(T) < λ(A) and a state (h, ν) such that

1. u _a (h _a , ν _a ) > u _a (g _a , μ _a ), λ-a.e. on T;

2. ∫ _T h _a dλ + ∫ _T τ(ν _a )dλ = ∫ _T e _a dλ; and

3. ∫ T ν a d λ ∈ C o n s .

We can proceed analogously as the proof of Theorem 3.12 of Bhowmik and Kaur (2023) to show that (g, μ) fails to be approximately robustly efficient which is a contradiction. Thus, (S, (g, μ)) is a global justified objection. This completes the proof. □