# 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!

• What is G? What do you mean by "prefer the preference"? – Kenneth Rios Jan 20 '18 at 5:06
• Sorry I edited @KennethRios do you have any idea? I will be happy if you share your idea about this question. – none009 Jan 20 '18 at 5:20
• @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 Jan 20 '18 at 5:29

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 $M$.

• That is really good answer. Thanks a lot :) – none009 Jan 23 '18 at 22:14

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.

• 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 Jan 20 '18 at 19:27