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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..?

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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}

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