James Payor comments on Concave Utility Question

[James Payor]

[Edit: yeah nevermind I have the inequality backwards]

A5 seems too strong?

Consider lotteries $A$ and $B$, and a mixture $X=pA+(1-p)B$ in between. Applying A5 twice gives:

1. If $u\left(X\right)\ge u\left(A\right)$ then $u\left(B\right)\ge u\left(A\right)$

2. If $u\left(X\right)\ge u\left(B\right)$ then $u\left(B\right)\ge u\left(A\right)$

So if $u\left(X\right)\ge u\left(A\right)$ and $u\left(X\right)\ge u\left(B\right)$ then $u\left(A\right)=u\left(B\right)$?

Either I’m confused or A5 is a stricter condition than concavity.

[Scott Garrabrant]

You have the inequality backwards. You can’t apply A5 when the mixture is better than the endpoint, only when the mixture is worse than the endpoint.

[James Payor]

Got it, thanks!

[Scott Garrabrant]

You can also think of A5 in terms of its contrapositive: For all $A,B\in L$, if $A\succ B$, then for all $0<p\le 1$$A\succ pA+(1-p)B$

This is basically just the strict version of A4. I probably should have written it that way instead. I wanted to use $⪰$ instead of $\succ$, because it is closer to the base definition, but that is not how I was natively thinking about it, and I probably should have written it the way I think about it.

[Scott Garrabrant]

I haven’t actually thought about whether A5 implies A4 though. It is plausible that it does. (together with A1-A3, or some other simple axioms,)

When $A\succ B$, we get A4 from A5, so it suffices to replace A4 with the special case that $A\sim B$. If $A\sim B$, and $A,B\succ X$, a mixture of $A$ and $B$, then all we need to do is have any Y such that $A\succ Y\succ X$, then we can get $Y^{\prime }$ between $A$ and $X$ by A3, and then $X$ will also be a mixture of $Y^{\prime }$ and $B$, contradicting A5, since $B\succ Y^{\prime }$.

A1,A2,A3,A5 do not imply A4 directly, because you can have the function that assigns utility 0 to a fair coin flip between two options, and utility 1 to everything else. However, I suspect when we add the right axiom to imply continuity, I think that will be sufficient to also allow us to remove A4, and only have A5.

[James Payor]

The way I understand A4 is that it says “if moving by $\Delta$ is good, then moving by any fraction $\lambda \Delta$ is also good”.

And A5 says “if moving by $\Delta$ is good, then moving by any multiple $n\Delta$ is also good”, which is much stronger.

[Scott Garrabrant]

Your understanding of A4 is right. In A5, “good” should be replaced with “bad.”

[Scott Garrabrant]

(and everywhere you say “good” and “bad”, they are the non-strict versions of the words)

[James Payor]

yep!

[James Payor]

Okay, I now think A5 implies: “if moving by $\Delta$ is good, then moving by any negative multiple $-n\Delta$ is bad”. Which checks out to me re concavity.