a risk lover agent preferences and the preference of risk natural agent may be the same 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!
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 

$$E[u_{RA}(N)] &gt; E[u_{RA}(M)]$$

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

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

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 $N$, 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.