And if we assume that the nodes are whole worlds, rather than pieces of worlds.
For example, if I’m also ordering a soda, and prefer Pepsi to Coke, then the relative magnitudes become important. (There’s an implicit assumption here that the utility of the whole is the sum of the utilities of the parts.) However, if the node includes the entire meal, so that there are six nodes (chicken, pepsi), (chicken, coke), (pork, pepsi), (pork, coke), (steak, pepsi), (steak, coke), then the magnitude doesn’t matter. Are utility functions generally assumed to be whole-world like this?
Yes, although I would word it as “the nodes include everything relevant to our implied preferences”, rather than “whole worlds”, just to be clear what we’re talking about. Certainly the entire notion of adding together two utilities is something which requires additional structure.
However, if the node includes the entire meal, so that there are six nodes (chicken, pepsi), (chicken, coke), (pork, pepsi), (pork, coke), (steak, pepsi), (steak, coke), then the magnitude doesn’t matter.
I don’t think this is right; you still want to be able to decide between actions which might have probabilistic “outcomes” (given that your action is necessarily being taken under incomplete information about its exact results).
You could define a continuous DAG over probability distributions, but that structure is actually too general; you do want to be able to rely on linear additivity if you’re using utilitarianism (rather than some other consequentialism that cares about the whole distribution in some nonlinear way).
Of course, once you have your function from worlds to utilities, you can construct the ordering between nodes of {100% to be X | outcomes X}, but that transformation is lossy (and you don’t need the full generality of a DAG, since you’re just going to end up with a linear ordering.
(For modeling incomplete preferences, DAGs are great! Not so great for utility functions.)
And if we assume that the nodes are whole worlds, rather than pieces of worlds.
For example, if I’m also ordering a soda, and prefer Pepsi to Coke, then the relative magnitudes become important. (There’s an implicit assumption here that the utility of the whole is the sum of the utilities of the parts.) However, if the node includes the entire meal, so that there are six nodes (chicken, pepsi), (chicken, coke), (pork, pepsi), (pork, coke), (steak, pepsi), (steak, coke), then the magnitude doesn’t matter. Are utility functions generally assumed to be whole-world like this?
Yes, although I would word it as “the nodes include everything relevant to our implied preferences”, rather than “whole worlds”, just to be clear what we’re talking about. Certainly the entire notion of adding together two utilities is something which requires additional structure.
I don’t think this is right; you still want to be able to decide between actions which might have probabilistic “outcomes” (given that your action is necessarily being taken under incomplete information about its exact results).
You could define a continuous DAG over probability distributions, but that structure is actually too general; you do want to be able to rely on linear additivity if you’re using utilitarianism (rather than some other consequentialism that cares about the whole distribution in some nonlinear way).
Of course, once you have your function from worlds to utilities, you can construct the ordering between nodes of {100% to be X | outcomes X}, but that transformation is lossy (and you don’t need the full generality of a DAG, since you’re just going to end up with a linear ordering.
(For modeling incomplete preferences, DAGs are great! Not so great for utility functions.)