# Decision theory: elicitation method

I'm stuck with the following question:

Let's say that C1, C2 and C3 represent the certainty equivalents and (x,p,y) the prospects.

C1 ~ (x, p, 0) C2 ~ (x, p, C1) C3 ~ (C1, p, 0)

What is C3 such that no random variables are left?

I suppose that it can be solves usiong the certainty equivalent method but im stuck. When normalizing u(x)=1, I find p^2 for u(c3) but it still doesn't give any information about c3 itself.

• Hi! What exactly are you asking here? Are you expected to express $C_3$ without the utility function and preference relation, or did @tdm answer your question? Dec 30, 2023 at 10:29
• I'm expected to express C3 without the utility function. Jan 2 at 12:15
• If you can use the preference relation, tdm's answer works. If you cannot use any such think, you can easily come up with a numerical example where $C_3$ will depend on the utility function, i.e. it will have different values if $u(c) = \sqrt{c}$ and if $u(c) = c^2$. Jan 2 at 12:37

If you receive $$C3$$, you get lottery $$C_1$$ with probability $$p$$ and $$0$$ with probability $$(1-p)$$.
$$C_1$$ is the lottery that gives $$x$$ with probability $$p$$ and $$0$$ with probability $$(1-p)$$.
So in total, you receive $$x$$ with probability $$p^2$$ and zero with probability $$(1-p) + p(1-p) = 1 - p^2$$.
As such, $$C_3 \sim (x, p^2, 0)$$.