The construction and characterization of egalitarian solutions for multi-choice NTU games

Abstract

In this article we construct a procedure to define the egalitarian solutions in the context of multi-choice non-transferable utility (NTU) games. Also, we show that in the presence of other weak axioms the egalitarian solutions are the only monotonic ones.

Keywords:

• egalitarian solutions
• monotonic
• multi-choice NTU games

AMS Subject Classification:

• 91A12

Notes

1. Kalai and Samet 14 named the additivity of endowments.

2. For precise definitions of axioms, see 14, pp. 311–312].

3. An n-person multi-choice NTU game is a generalization of an NTU game, according to the terminology in 14.

4. V(1, 0) means V((1, 0)), S(1, 0) means S((1,0)) and so on.

5. If β ∧ α = 0 _N , where β ≠ 0 _N , then V(β) ≠ V(β ∧ α). So, it is necessary to add in the formula

6. In Example 2, (N, m, V) is the game (N, m, â_α), where and α = (1, 1).

7. The game (N, m, â_α) is the notion of a minimal effort game. Intuitively, it seems that the solution of game (N, m, â_α) is a _S(α). That is, if ψ is a solution then for all action vectors α ∈ M ^N and for all vectors a ∈ ℝ ^N , ψ(N, m, â_α) = a _S(α). Here we do not impose the above assumption. We only require that for an action vector α ∈ M ^N and a vector a ∈ ℝ ^N , if ψ(N, m, â_α) = a _S(α) then we call a is acceptable to S(α) under α and ψ.

8. An alternative formula of TOE is as follows: for each (N, m, V) ∈ Γ and for each a ∈ ℝ ^N , if ψ(N, m, â_α) = a _S(α) then ψ(N, m, V + â_α) = ψ(N, m, V) + a _S(α).

9. A similar procedure also arises in the definition of the Harsanyi value.

10. This is inspired from 14, p. 313].

11. For example, let N = {1, 2} and m = (1, 1). Let (N, m, V) be a game in Γ defined by V(1, 0) = {(x ₁, x ₂) ∈ ℝ^{1}|x ₁ ≤ 0}, V(0, 1) = {(x ₁, x ₂) ∈ ℝ^{2}|x ₂ ≤ 1}, V(1, 1) = {(x ₁, x ₂) ∈ ℝ^{1,2}| x ₁ ≤ 0, x ₂ ≤ 1}. Clearly, by the definition of ψ¹, . This implies that ψ¹ violates IRMG.

12. For example, let N = {1, 2}, m = (1, 1). Take λ = (2, 1). Let a = (1, 1) ∈ ℝ ^N , by the definition of ψ³, it is easy to see that a is acceptable to N under m and ψ³. That is, ψ³ (N, m, â _m ) = a = (1, 1). Similarly, let b = (0, 0) ∈ ℝ ^N , by the definition of ψ³, it is easy to see that b is ‘not’ acceptable to N under m and ψ³. And, . Clearly, . Hence we derive that

This implies that ψ³ violates TOE.

13. We use cch A to denote the convex and comprehensive hull of A. Let N = {1, 2} and m = (1, 1). Let (N, m, V) and (N, m, W) be two games in Γ defined by V(1, 0) = W(1, 0) = {(x ₁, x ₂) ∈ ℝ^{1}|x ₁ ≤ 0}, V(0, 1) = W(0, 1) = {(x ₁, x ₂) ∈ ℝ^{2}|x ₂ ≤ 0}, and . Clearly, V(1, 1) ⊆ W(1, 1). It is easy to derive that and . This implies that ψ⁴ violates MON.

14. Let N = {1, 2}, m = (1, 1) and let λ = (0, 1). Let be a sequence of games converging to (N, m, V), where for all t, V(1, 0) = V _t (1, 0) = {(x ₁, x ₂) ∈ ℝ^{1}|x ₁ ≤ 0}, V(0, 1) = V _t (0, 1) = {(x ₁, x ₂) ∈ ℝ^{2}|x ₂ ≤ 0}, V(1, 1) = cch{(0, 1)} and . It is easy to derive that ψ⁵ (N, m, V) = (0, 1) and ψ⁵(N, m, V _t ) = (0, 0) for all t. This implies that ψ⁵ violates CONT.

15. See 16, Chap. 1, p. 8].

16. For these precise proofs, see 25.

17. The game in Example 2 is a bargaining game.