I didn’t say anything about risk aversion. This is about utility functions that depend on multiple different “values” in some non-convex way. You can observe that, in my original example, if you have no water, then utility (days survived) is linear with respect to food.
Oh, I see. The problem is that if the importance of a value changes depending on how well you achieve a different value, a Pareto improvement in the expected value of each value function is not necessarily an improvement overall, even if your utility with respect to each value function is linear given any fixed values for the other value functions (e.g. U = v1*v2). That’s a good point, and I now agree; Pareto optimality with respect to the expected value of each value function is not an obviously desirable criterion. (apologies for the possibly confusing use of “value” to mean two different things)
Edit: I’m going to backtrack on that somewhat. I think it makes sense if the values are independent of one another (not the case for food and water, which are both subgoals of survival). The assumption needed for the theorem is that for all i, the utility function is linear with respect to v_i given fixed expected values of the other value functions, and does not depend on the distribution of possible values of the other value functions.
I didn’t say anything about risk aversion. This is about utility functions that depend on multiple different “values” in some non-convex way. You can observe that, in my original example, if you have no water, then utility (days survived) is linear with respect to food.
Oh, I see. The problem is that if the importance of a value changes depending on how well you achieve a different value, a Pareto improvement in the expected value of each value function is not necessarily an improvement overall, even if your utility with respect to each value function is linear given any fixed values for the other value functions (e.g. U = v1*v2). That’s a good point, and I now agree; Pareto optimality with respect to the expected value of each value function is not an obviously desirable criterion. (apologies for the possibly confusing use of “value” to mean two different things)
Edit: I’m going to backtrack on that somewhat. I think it makes sense if the values are independent of one another (not the case for food and water, which are both subgoals of survival). The assumption needed for the theorem is that for all i, the utility function is linear with respect to v_i given fixed expected values of the other value functions, and does not depend on the distribution of possible values of the other value functions.