I think that the core of the problem this post points out actually has very little to do with utility function. The core problem in using an extremely confusing term “possible world” for an element of a sample space.
Now, I don’t mind when people use weird terminology for historical reasons. If everybody understood that “possible world” is simply a synonym for “mutually exclusive outcome of a probability experiment”, there won’t be an issue.
But at this point:
The sample spaceΩ of a rational agent’s beliefs is, more or less, the set of possible ways the world could be—which is to say, the set of possible physical configurations of the universe. Hence, each world ω∈Ω is one such configuration.
We absolutely have a problem.
Let’s make it clear: ω is not a whole physical configurations of the universe. It’s one of a state of probability experiment, which itself is an approximation of our knowledge about some physical process in our world.
When I toss a coin it doesn’t actually cause the universe to split in two where it’s Tails in one world and Heads in the other. Neither I need to imagine as if it does it. I don’t need to conceptualize two coherent universes from the ground up where in one the coin is Tails and in the other it’s Heads to be able to reason about probability of a coin to come Heads. That would’ve been insane!
All I need to understand is that some coin tosses result in Heads and some in Tails and I have no idea which is the one I’m dealing with. I approximate “This particular coin toss with all its properties” to “Some coin toss about which I know as much as I do about this one”. While there is a very specific territory, I make a more abstract map of it.
This absolutely is not a “view from nowhere”. The notion of Probability Experiment already captures my knowledge state of the process I’m trying to reason about.
When this is cleared, the situation becomes much less confusing.
You don’t need to come up with some new axiomatics for probability theory. You can keep use Kolmogorovs axioms, just don’t interpret them in a ridiculous way.
Of course we do not need to assume that there are some “possible worlds” at all. We just need to know what are the mutually exclusive outcomes of the experiment we are reasoning about so that we could formally construct the Event Space from sets of individual outcomes.
Of course utility is not a function of the world. As a matter of fact it’s not a function of an outcome, either. Just like with probability function, its domain is not the Sample Space, but the Event Space. It doesn’t mean that you need to get rid of simple and intuitive formula that allows you to calculate expected utility of an event from its utility and probability, though.
And so on and so forth.
Now, you may still want to get rid of the notion of utility function for some reason, but frankly, I don’t see what it gets you after we’ve cleared the whole “possible worlds” confusion.
I think that the core of the problem this post points out actually has very little to do with utility function. The core problem in using an extremely confusing term “possible world” for an element of a sample space.
Now, I don’t mind when people use weird terminology for historical reasons. If everybody understood that “possible world” is simply a synonym for “mutually exclusive outcome of a probability experiment”, there won’t be an issue.
But at this point:
We absolutely have a problem.
Let’s make it clear: ω is not a whole physical configurations of the universe. It’s one of a state of probability experiment, which itself is an approximation of our knowledge about some physical process in our world.
When I toss a coin it doesn’t actually cause the universe to split in two where it’s Tails in one world and Heads in the other. Neither I need to imagine as if it does it. I don’t need to conceptualize two coherent universes from the ground up where in one the coin is Tails and in the other it’s Heads to be able to reason about probability of a coin to come Heads. That would’ve been insane!
All I need to understand is that some coin tosses result in Heads and some in Tails and I have no idea which is the one I’m dealing with. I approximate “This particular coin toss with all its properties” to “Some coin toss about which I know as much as I do about this one”. While there is a very specific territory, I make a more abstract map of it.
This absolutely is not a “view from nowhere”. The notion of Probability Experiment already captures my knowledge state of the process I’m trying to reason about.
When this is cleared, the situation becomes much less confusing.
You don’t need to come up with some new axiomatics for probability theory. You can keep use Kolmogorovs axioms, just don’t interpret them in a ridiculous way.
Of course we do not need to assume that there are some “possible worlds” at all. We just need to know what are the mutually exclusive outcomes of the experiment we are reasoning about so that we could formally construct the Event Space from sets of individual outcomes.
Of course utility is not a function of the world. As a matter of fact it’s not a function of an outcome, either. Just like with probability function, its domain is not the Sample Space, but the Event Space. It doesn’t mean that you need to get rid of simple and intuitive formula that allows you to calculate expected utility of an event from its utility and probability, though.
And so on and so forth.
Now, you may still want to get rid of the notion of utility function for some reason, but frankly, I don’t see what it gets you after we’ve cleared the whole “possible worlds” confusion.