As far as I can tell, this is the entire point. I don’t see this 2D vector space actually being used in modeling agents, and I don’t think Abram does either.
I largely agree. In retrospect, a large part of the point of this post for me is that it’s practical to think of decision-theoretic agents as having expected value estimates for everything without having a utility function anywhere, which the expected values are “expectations of”.
A utility function is a gadget for turning probability distributions into expected values. This object makes sense in a context like VNM, where you are asking agents to judge between arbitrary gambles. In the jeffrey-bolker setting, you instead only ask agents to choose between events, not gambles. This allows us to directly derive coherence constraints on expectations without introducing a function they’re expectations “of”.
For me, this fits better with the way humans seem to think; it’s relatively easy to compare events to each other, but nigh impossible to take entire world-descriptions and compare them (which is what a utility function does).
The rotation comes into play because looking at preferences this way is much more ‘situated’: you are only required to have preferences relating to your current beliefs, rather than relating to arbitrary probability distributions (as in VNM). We can intuit from our experience that there is some wiggle room between probability vs preference when representing situations in the real world. VNM doesn’t model this, because probabilities are simply given to us in the VNM setting, and we’re to take them as gospel truth.
So jeffrey-bolker seems to do a better job of representing the subjective nature of probability, and the vector rotations illustrate this.
On the other hand, I think there is a real advantage to the 2d vector representation of a preference structure. For agents with identical beliefs (the “common prior assumption”), Harsanyi showed that cooperative preference structures can be represented by simple linear mixtures (Harsanyi’s utilitarian theorem). However, Critch showed that combining preferences in general is not so simple. You can’t separately average two agent’s beliefs and their utility function; you have to dynamically change the weights of the utility-function averaging based on how bayesian updates shift the weights of the probability mixture.
Averaging the vector-valued measures together works fine, though, I believe. (I haven’t worked it out in detail.) If true, this makes vector-valued measures an easier way to think about coalitions of cooperating agents who merge preferences in order to select a pareto-optimal joint policy.
I largely agree. In retrospect, a large part of the point of this post for me is that it’s practical to think of decision-theoretic agents as having expected value estimates for everything without having a utility function anywhere, which the expected values are “expectations of”.
A utility function is a gadget for turning probability distributions into expected values. This object makes sense in a context like VNM, where you are asking agents to judge between arbitrary gambles. In the jeffrey-bolker setting, you instead only ask agents to choose between events, not gambles. This allows us to directly derive coherence constraints on expectations without introducing a function they’re expectations “of”.
For me, this fits better with the way humans seem to think; it’s relatively easy to compare events to each other, but nigh impossible to take entire world-descriptions and compare them (which is what a utility function does).
The rotation comes into play because looking at preferences this way is much more ‘situated’: you are only required to have preferences relating to your current beliefs, rather than relating to arbitrary probability distributions (as in VNM). We can intuit from our experience that there is some wiggle room between probability vs preference when representing situations in the real world. VNM doesn’t model this, because probabilities are simply given to us in the VNM setting, and we’re to take them as gospel truth.
So jeffrey-bolker seems to do a better job of representing the subjective nature of probability, and the vector rotations illustrate this.
On the other hand, I think there is a real advantage to the 2d vector representation of a preference structure. For agents with identical beliefs (the “common prior assumption”), Harsanyi showed that cooperative preference structures can be represented by simple linear mixtures (Harsanyi’s utilitarian theorem). However, Critch showed that combining preferences in general is not so simple. You can’t separately average two agent’s beliefs and their utility function; you have to dynamically change the weights of the utility-function averaging based on how bayesian updates shift the weights of the probability mixture.
Averaging the vector-valued measures together works fine, though, I believe. (I haven’t worked it out in detail.) If true, this makes vector-valued measures an easier way to think about coalitions of cooperating agents who merge preferences in order to select a pareto-optimal joint policy.