Nash Bargaining Solution Axioms

Question

For each of the four axioms characterizing the Nash bargaining solution, construct a bargaining solution that violates that axiom but satisfies the remaining three axioms.

EDIT: Here are the axioms I am referring to:

1. (Pareto Efficiency) A bargaining solution $f$ is Pareto efficient if for any bargaining problem $(U, d)$, there does not exist $(u_1, u_2) \in U$ such that $u_1\ge f_1(U, d)$ and $u_2\ge f_2(U,d)$, with at least one strict inequality.

2. (Symmetry) A bargaining solution $f$ is symmetric if for any symmetric bargaining problem $(U,d)$ ($(u1,u2) \in U$ if and only if $(u2,u1) \in U$ and $d_1 = d_2$), we have $f_1(U, d) = f_2(U, d)$.

3. Invariance to Linear Transformations

4. (Independence of Irrelevant Alternatives) A bargaining solution $f$ is independent if for any two bargaining problems $(U,d)$ and $(U',d)$ with $U' \subset U$ and $f(U,d) \in U'$, we have $f(U',d) = f(U,d)$.

Answer 1

Here are four counterexamples, one for each axiom.

1) Reverse the direction of preference and consider the anti-Nash solution $$f(U,d) = \arg \min (u_1-d_1)(u_2-d_2)$$

2) The asymmetric Nash solution is $$f(U,d) = \arg \max (u_1-d_1)^k(u_2-d_2)^{1-k}$$ for some $k \ne 1/2$ in $[0,1]$.

3) The egalitarian solution is $$f(U,d) = \arg \max \min \{ (u_1-d_1), (u_2-d_2)\}$$

4) The Kalai-Smorodinski solution is characterised by assuming monotonicity instead of IIA.