<p>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 $(\overline x_1,\overline x_2)$. </p>

<p>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}</p>

CC BY-SA 3.0
2018-03-07T22:44:36.963Z
2018-03-07T23:21:12.637Z
2018-03-07T23:21:12.637Z
Herr K.
<p>Under standard assumptions on the domain of bargaining problems like <em>Pareto Optimality, Symmetric, Scale Invariance and Independent of Irrelevant Alternatives / Individual Monotonicty</em>, in which conditions (on feasible set S, or disagreemnt point d) that characterize <strong>the domain in which Kalai-Smorodinsky and Nash solutions coincide</strong>?</p>

<p>My idea is that when feasible set is symmetric, both solutions should coincide but it should be other cases..?</p>

CC BY-SA 3.0
2018-03-07T22:45:35.123Z
Please do not [cross post on different SE platforms](https://math.stackexchange.com/q/2681030/70119).
Herr K.
<game-theory><bargaining>
Axiomatic Bargaining: Kalai-Smorodinsky and Nash solutions coincide
John Stern
# Axiomatic Bargaining: Kalai-Smorodinsky and Nash solutions coincide

(Asked by: username_0 on 2018-03-07)

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..?

**Tags:** game-theory, bargaining
> **Comments:**
> **username_1**: Please do not [cross post on different SE platforms](https://math.stackexchange.com/q/2681030/70119).  

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## 1 answers

### Answer by username_1 on 2018-03-07. Score: 5 | Accepted answer

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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