Under standard assumptions on the domain of bargaining problems like Pareto Optimality, Symmetric, Scale Invariance and Independent of Irrelevant Alternatives / Individual Monotonicty, in which conditions (on feasible set S, or disagreemnt point d) that characterize the domain in which Kalai-Smorodinsky and Nash solutions coincide?
My idea is that when feasible set is symmetric, both solutions should coincide but it should be other cases..?
Another sufficient condition for the two solutions to coincide, which is not necessarily the same as symmetry, is that the feasible set $S$ be rectangular. That is, \begin{equation} S=\text{convex hull}\{(d_1,d_2),(d_1,\overline x_2),(\overline x_1,d_2),(\overline x_1,\overline x_2)\}, \end{equation} where $d_i$ is $i$'s reservation utility and $\overline x_i$ is $i$'s best possible utility from the bargaining. Both Nash and KS solutions are at $(\overline x_1,\overline x_2)$.
Using the property that the KS solution can be obtained at the intersection of a ray from the disagreement point $d$ and the (northeast) boundary of the feasible set $S$, we can derive the following necessary condition for KS and Nash solutions to match: let $(x_1^N,x_2^N)$ denote the Nash solution to bargaining problem $(S,d)$; the KS solution $(x_1^{KS},x_2^{KS})=(x_1^N,x_2^N)$ only if \begin{equation} \frac{x_2^N-d_2}{x_1^N-d_1}=\frac{\overline x_2-d_2}{\overline x_1-d_1}. \end{equation}
