a risk lover agent behave as if risk natural.

Question

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 get a result that does make sense.

But I know and assume that I need to use MRS and indifference curve.

Answer 1

No, not necessarily

Let's suppose both lotteries have possible outcomes of $1,2,4,8$:

• lottery $M$ with probabilities $0.50,0.00,0.50,0.00$ respectively
• lottery $N$ with probabilities $0.00,0.95,0.00,0.05$ respectively

Let's suppose the utilities for an outcome $x$ are

• $\log_2(x)$ for the risk adverse player, so $0,1,2,3$ respectively
• $x$ for the risk neutral player, so $1,2,4,8$ respectively
• $2^x$ for the risk loving player, so $2,4,16,256$ respectively

Then I think the players value the lotteries with a counter-intuitive pattern:

• the risk adverse player's expected utility from $M$ is $1$ and from $N$ is $1.1$, so prefers $N$
• the risk neutral player's expected utility from $M$ is $2.5$ and from $N$ is $2.3$, so prefers $M$
• the risk loving player's expected utility from $M$ is $9$ and from $N$ is $16.6$, so prefers $N$