<p>In your last inequality, point <span class="math-container">$A$</span> is where <span class="math-container">$p_1=1$</span> and point <span class="math-container">$C$</span> is where <span class="math-container">$p_1=0$</span>. So the condition <span class="math-container">$p_1&lt;\frac34$</span> is captured by the line segment <span class="math-container">$\overline{CD}$</span>.</p>

CC BY-SA 4.0
2021-11-14T00:58:24.120Z
2021-11-14T00:58:24.120Z
Herr K.
<p>You can find point D on line AC with the following explanation -</p>
<p>Key here is here is to find a point where expected utility from a lottery on line AC equals the utility derived from $9 . Proceeding with this idea , equate 2p1 + (1-p1).6 = 3 and you will find p1= 3/4 (point D).</p>

CC BY-SA 4.0
2024-04-21T12:10:11.937Z
2024-04-21T12:10:11.937Z
Dhruv Sardana
<p>The question asks how to find the point D.</p>
<p>For the strategy below we find the vector AC, scale it down to AD then set the vector's tail at A so that the tip is at D. Finally, adding the vector to A gives us point D.</p>
<p>The vector from point A to point C is found as point C minus point A so is (-1,0,1). You want to travel a quarter of the way down this vector to get to point D .</p>
<p>Thus point D is at the tip of the vector (-1/4, 0, 1/4) when the vector starts at A.</p>
<p>so D = (1, 0, 0) + (-1/4, 0, 1/4) = the point at (3/4,0, 1/4)</p>

CC BY-SA 4.0
2024-02-21T19:01:38.860Z
2024-02-28T17:29:53.420Z
2024-02-28T17:29:53.420Z
Goose
<p>There are 3 possible payoffs - \$4, \$9 and \$36. The utility function for these payoffs is <span class="math-container">$\sqrt x $</span>.</p>
<p>I have to find all the lotteries preferable over getting $9 with probability 1 in a probability simplex. The answer is the shaded area in the probability simplex below.</p>
<p><a href="https://i.sstatic.net/Dv7mt.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/Dv7mt.png" alt="Probability simplex" /></a></p>
<p>I am not able to find the 3D equation of line required for it. But I have approached the problem as follows:</p>
<p>Any point on line BC gives expected utility of <span class="math-container">$ 6p_{3} + 3(1-p_{3}) = 3p_{3} + 3 &gt; 3 $</span>, so all points on BC are to be included.</p>
<p>All points on AB are eliminated as expected utility on them is less than 3.</p>
<p>Only those points on AC are included which give the expected utility greater than 3 i.e.</p>
<p><span class="math-container">$$ 2p_{1} + 6(1-p_{1}) = 6 - 4p_{1} &gt;3 \implies p_{1} &lt; \dfrac{3}{4} $$</span></p>
<p>From here how do we arrive at point D ?</p>

<microeconomics><mathematical-economics><uncertainty>
Drawing a Probability simplex
Kritika
# Drawing a Probability simplex

(Asked by: username_0 on 2021-11-13)

There are 3 possible payoffs - \$4, \$9 and \$36. The utility function for these payoffs is $\sqrt x $.

I have to find all the lotteries preferable over getting $9 with probability 1 in a probability simplex. The answer is the shaded area in the probability simplex below.

[![Probability simplex](https://i.sstatic.net/Dv7mt.png)](https://i.sstatic.net/Dv7mt.png)

I am not able to find the 3D equation of line required for it. But I have approached the problem as follows:

Any point on line BC gives expected utility of $ 6p\_{3} + 3(1-p\_{3}) = 3p\_{3} + 3 > 3 $, so all points on BC are to be included.

All points on AB are eliminated as expected utility on them is less than 3.

Only those points on AC are included which give the expected utility greater than 3 i.e.

$$ 2p\_{1} + 6(1-p\_{1}) = 6 - 4p\_{1} >3 \implies p\_{1} < \dfrac{3}{4} $$

From here how do we arrive at point D ?

**Tags:** microeconomics, mathematical-economics, uncertainty


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## 3 answers

### Answer by username_1 on 2021-11-14. Score: 2

In your last inequality, point $A$ is where $p\_1=1$ and point $C$ is where $p\_1=0$. So the condition $p\_1<\frac34$ is captured by the line segment $\overline{CD}$.


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### Answer by username_2 on 2024-04-21. Score: 0

You can find point D on line AC with the following explanation -

Key here is here is to find a point where expected utility from a lottery on line AC equals the utility derived from $9 . Proceeding with this idea , equate 2p1 + (1-p1).6 = 3 and you will find p1= 3/4 (point D).


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### Answer by username_3 on 2024-02-21. Score: 0

The question asks how to find the point D.

For the strategy below we find the vector AC, scale it down to AD then set the vector's tail at A so that the tip is at D. Finally, adding the vector to A gives us point D.

The vector from point A to point C is found as point C minus point A so is (-1,0,1). You want to travel a quarter of the way down this vector to get to point D .

Thus point D is at the tip of the vector (-1/4, 0, 1/4) when the vector starts at A.

so D = (1, 0, 0) + (-1/4, 0, 1/4) = the point at (3/4,0, 1/4)


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