<p>The answer depends on the "following solution concepts", which you did not specify. I guess the Nash solution and the Kalai-Smorodinsky solution will be among them.</p>

<p>The frontier functions you seem to propose are of the form <span class="math-container">$y_n(x)=\sqrt{1-\frac{100x^2}{n^2}}$</span>, where <span class="math-container">$n\in\{1,2,\ldots,100\}$</span>.</p>

<p>For the Nash solution you maximize <span class="math-container">$xy_n(x)$</span> and after substituting for <span class="math-container">$y_n(x)$</span> and setting the derivative to zero you will find <span class="math-container">$x_n=\frac{n}{\sqrt{200}}$</span> and <span class="math-container">$y_n=\frac{1}{\sqrt{2}}$</span>. For the KS solution you set <span class="math-container">$y_n(x)=\frac{10x}{n}$</span> and after substituting you will again find <span class="math-container">$x_n=\frac{n}{\sqrt{200}}$</span> and <span class="math-container">$y_n=\frac{1}{\sqrt{2}}$</span>. Graphing these solutions is easy.</p>

<p><em>Remark:</em>
That the solutions coincide is not a coincidence. The different frontier functions are constructed in such a way that they are all just horizontally "compressed" or "stretched" versions of the frontier function <span class="math-container">$y_{10}(x)=\sqrt{1-x^2}$</span>. This means that they are really all equivalent in the sense that they result from each other after some rescaling of <span class="math-container">$x$</span> (the utility of agent 1). But the frontier function <span class="math-container">$y_{10}$</span> just describes the quarter unit circle in the positive orthant, which is symmetric w.r.t. the agents. In symmetric feasible sets the Nash and the KM solutions are known to coincide.</p>

CC BY-SA 4.0
2020-03-11T11:34:49.163Z
2020-03-11T14:30:47.620Z
2020-03-11T14:30:47.620Z
VARulle
<p>I need help with the following question. </p>

<p>Take a two-agent bargaining problem. d=(0,0). Pick a strictly concave function for the bargaining frontier. Fix the intersection of the frontier with the y-axis to be at 1. Vary the intersection of the frontier with the x-axis between 0.1 and 10 with increments of 0.1. Graph the points proposed by the following solution concepts. </p>

<p>So y^2=1-x^2/0.1 can be the function. Does anyone know how can it be computed in matlab or a similar program? </p>

CC BY-SA 4.0
2020-03-11T03:43:53.417Z
Not sure if there's an existing MATLAB program for it, but it should be pretty easy to code one yourself. Have you tried this at all?
Herr K.
<game-theory><bargaining>
Computation of Different Axiomatic Barganing Solutions
sad
# Computation of Different Axiomatic Barganing Solutions

(Asked by: username_0 on 2020-03-10)

I need help with the following question.

Take a two-agent bargaining problem. d=(0,0). Pick a strictly concave function for the bargaining frontier. Fix the intersection of the frontier with the y-axis to be at 1. Vary the intersection of the frontier with the x-axis between 0.1 and 10 with increments of 0.1. Graph the points proposed by the following solution concepts.

So y^2=1-x^2/0.1 can be the function. Does anyone know how can it be computed in matlab or a similar program?

**Tags:** game-theory, bargaining

> **Comments:**
> **username_1**: Not sure if there's an existing MATLAB program for it, but it should be pretty easy to code one yourself. Have you tried this at all?  

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## 1 answers

### Answer by username_2 on 2020-03-11. Score: 2

The answer depends on the "following solution concepts", which you did not specify. I guess the Nash solution and the Kalai-Smorodinsky solution will be among them.

The frontier functions you seem to propose are of the form $y\_n(x)=\sqrt{1-\frac{100x^2}{n^2}}$, where $n\in\{1,2,\ldots,100\}$.

For the Nash solution you maximize $xy\_n(x)$ and after substituting for $y\_n(x)$ and setting the derivative to zero you will find $x\_n=\frac{n}{\sqrt{200}}$ and $y\_n=\frac{1}{\sqrt{2}}$. For the KS solution you set $y\_n(x)=\frac{10x}{n}$ and after substituting you will again find $x\_n=\frac{n}{\sqrt{200}}$ and $y\_n=\frac{1}{\sqrt{2}}$. Graphing these solutions is easy.

*Remark:*
That the solutions coincide is not a coincidence. The different frontier functions are constructed in such a way that they are all just horizontally "compressed" or "stretched" versions of the frontier function $y\_{10}(x)=\sqrt{1-x^2}$. This means that they are really all equivalent in the sense that they result from each other after some rescaling of $x$ (the utility of agent 1). But the frontier function $y\_{10}$ just describes the quarter unit circle in the positive orthant, which is symmetric w.r.t. the agents. In symmetric feasible sets the Nash and the KM solutions are known to coincide.


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