<p>Don't commit the cardinal mistake of equating preferences with choices.  </p>

<p>In the context of Expected Utility Theory, the fact that a risk-averse agent ($RA$) would choose $N$ over $M$ implies that </p>

<p>$$E[u_{RA}(N)] &gt; E[u_{RA}(M)]$$</p>

<p>The fact that a risk-neutral agent ($RN$) could choose $M$ over $N$ implies that</p>

<p>$$E[u_{RN}(N)] &lt; E[u_{RN}(M)] \implies E(N) &lt; E(M)$$</p>

<p>A risk-lover is a person that could choose $M$ over $N$ even if $E(N) &gt; E(M)$. So here that the opposite inequality holds, it is certain that it will also choose $M$ over $N$, <em>given</em> also the fact that there appears to be no other lotteries to choose from. </p>

<p>But never say that the risk-neutral and the risk-lover have "same preferences". They don't -it just happens, given the available lotteries, that they choose the same lottery, even though they have different preferences.</p>

CC BY-SA 3.0
2018-01-20T19:27:29.730Z
Thank you for your great help. I think similarly. But I cannot imagine why risk lover agent can also choose the preference of risk natural agent. This point is not clear for me yet. Can you expand this point in mathematical aspect? Thanks a lot !
none009
CC BY-SA 3.0
2018-01-20T18:38:52.857Z
2018-01-20T20:40:54.503Z
2018-01-20T20:40:54.503Z
Alecos Papadopoulos
<p>Another way of looking at this problem is to consider the means and variances of the lotteries. </p>

<ul>
<li>A risk averse agent (RA) likes high mean and low variance </li>
<li>A risk neutral agent (RN) likes high mean and is indifferent to changes in variance</li>
<li>A risk loving agent (RL) likes high mean and high variance </li>
</ul>

<p>From the fact that RN chooses $M$ over $N$, we known that the mean of $M$ is higher than the mean of $N$, or $E(M)&gt;E(N)$. </p>

<p>The fact that RA chooses $N$ over $M$ despite the latter having a higher mean must imply that $N$ has a much lower variance than $M$. </p>

<p>Thus, given the features of RL's preference (she likes high mean and high variance), the obvious choice is therefore $M$. </p>

CC BY-SA 3.0
2018-01-23T22:14:52.103Z
That is really good answer.  Thanks a lot :)
none009
CC BY-SA 3.0
2018-01-23T21:55:25.990Z
2018-01-23T21:55:25.990Z
Herr K.
<p>Consider two lotteries $N$ and $M$. Agent $i$ is risk-averse and prefers $N$. Agent $j$ is risk-neutral and prefers $M$. Would any risk-loving agent $k$ also prefer $M$? That is, would $j$ and $k$ have the same preferences in this scenario?</p>

<p>My attempt: </p>

<p>For example, I can easily show that a risk averse agent can behave as if it is risk natural. I can show this on indifference curves by using the equal marginal rate of substitution. </p>

<p>Then I consider and follow the same way to demonstrate a risk lover agent behave as if risk natural agent by using MRS. But I cannot a result that does make sense. </p>

<p>But I know and assume that I need to use MRS and indifference curve. After that point, I am glad if you give any help. Thanks a lot!</p>

CC BY-SA 3.0
2018-01-20T05:06:13.093Z
What is G? What do you mean by "prefer the preference"?
Kenneth Rios
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2018-01-20T05:20:29.123Z
Sorry I edited @KennethRios do you have any idea? I will be happy if you share your idea about this question.
none009
CC BY-SA 3.0
2018-01-20T05:29:36.683Z
@KennethRios thank you for editing. Hopefully you will also answer it :) how can I demonstrate / prove this idea. I think such a thing. But I cannot show it logically.
none009
<microeconomics><self-study><choice-theory><risk-aversion>
a risk lover agent preferences and the preference of risk natural agent may be the same
none009
# a risk lover agent preferences and the preference of risk natural agent may be the same

(Asked by: username_0 on 2018-01-20)

Consider two lotteries $N$ and $M$. Agent $i$ is risk-averse and prefers $N$. Agent $j$ is risk-neutral and prefers $M$. Would any risk-loving agent $k$ also prefer $M$? That is, would $j$ and $k$ have the same preferences in this scenario?

My attempt:

For example, I can easily show that a risk averse agent can behave as if it is risk natural. I can show this on indifference curves by using the equal marginal rate of substitution.

Then I consider and follow the same way to demonstrate a risk lover agent behave as if risk natural agent by using MRS. But I cannot a result that does make sense.

But I know and assume that I need to use MRS and indifference curve. After that point, I am glad if you give any help. Thanks a lot!

**Tags:** microeconomics, self-study, choice-theory, risk-aversion
> **Comments:**
> **username_1**: What is G? What do you mean by "prefer the preference"?  
> **username_0**: Sorry I edited @KennethRios do you have any idea? I will be happy if you share your idea about this question.  
> **username_0**: @KennethRios thank you for editing. Hopefully you will also answer it :) how can I demonstrate / prove this idea. I think such a thing. But I cannot show it logically.  

---

## 2 answers

### Answer by username_3 on 2018-01-23. Score: 3 | Accepted answer

Another way of looking at this problem is to consider the means and variances of the lotteries.

* A risk averse agent (RA) likes high mean and low variance
* A risk neutral agent (RN) likes high mean and is indifferent to changes in variance
* A risk loving agent (RL) likes high mean and high variance

From the fact that RN chooses $M$ over $N$, we known that the mean of $M$ is higher than the mean of $N$, or $E(M)>E(N)$.

The fact that RA chooses $N$ over $M$ despite the latter having a higher mean must imply that $N$ has a much lower variance than $M$.

Thus, given the features of RL's preference (she likes high mean and high variance), the obvious choice is therefore $M$.

> **Comments:**
> **username_0**: That is really good answer.  Thanks a lot :)  

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### Answer by username_2 on 2018-01-20. Score: 6

Don't commit the cardinal mistake of equating preferences with choices.

In the context of Expected Utility Theory, the fact that a risk-averse agent ($RA$) would choose $N$ over $M$ implies that

$$E[u\_{RA}(N)] > E[u\_{RA}(M)]$$

The fact that a risk-neutral agent ($RN$) could choose $M$ over $N$ implies that

$$E[u\_{RN}(N)] < E[u\_{RN}(M)] \implies E(N) < E(M)$$

A risk-lover is a person that could choose $M$ over $N$ even if $E(N) > E(M)$. So here that the opposite inequality holds, it is certain that it will also choose $M$ over $N$, *given* also the fact that there appears to be no other lotteries to choose from.

But never say that the risk-neutral and the risk-lover have "same preferences". They don't -it just happens, given the available lotteries, that they choose the same lottery, even though they have different preferences.

> **Comments:**
> **username_0**: Thank you for your great help. I think similarly. But I cannot imagine why risk lover agent can also choose the preference of risk natural agent. This point is not clear for me yet. Can you expand this point in mathematical aspect? Thanks a lot !  

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