Well, you can’t have some states as “avoid at all costs” and others as “achieve at all costs”, because having them in the same lottery leads to nonsense, no matter what averaging you use. And allowing only one of the two seems arbitrary. So it seems cleanest to disallow both.
Fine. But the purpose of exploring different averaging methods is to see whether it expands the richness of the kind of behaviour we want to describe. The point is that using arithmetic averaging is a choice which limits the kind of behaviour we can get. Maybe we want to describe behaviours which can’t be described under expected utility. Having an ‘avoid at all costs state’ is one such behaviour which finds natural description using a non-arithmetic averaging which can’t be described in more typical VNM terms.
If your position is ‘I would never want to describe normative ethics using anything other than expected utility’ then that’s fine, but some people (like me) are interested in looking at what alternatives to expected utility might be. That’s why I wrote this post. As it stands, I didn’t find geometric averaging very satisfactory (as I wrote in the post), but I think things like this are worth exploring.
But geometric averaging wouldn’t let you do that either, or am I missing something?
You are right. Geometric averaging on its own doesn’t give allow violations of independence. But some other protocol for deciding over lotteries does. It’s described more in the Garrabrant post linked above.
Fine. But the purpose of exploring different averaging methods is to see whether it expands the richness of the kind of behaviour we want to describe. The point is that using arithmetic averaging is a choice which limits the kind of behaviour we can get. Maybe we want to describe behaviours which can’t be described under expected utility. Having an ‘avoid at all costs state’ is one such behaviour which finds natural description using a non-arithmetic averaging which can’t be described in more typical VNM terms.
If your position is ‘I would never want to describe normative ethics using anything other than expected utility’ then that’s fine, but some people (like me) are interested in looking at what alternatives to expected utility might be. That’s why I wrote this post. As it stands, I didn’t find geometric averaging very satisfactory (as I wrote in the post), but I think things like this are worth exploring.
You are right. Geometric averaging on its own doesn’t give allow violations of independence. But some other protocol for deciding over lotteries does. It’s described more in the Garrabrant post linked above.