Unidimensional Continuity of Preference ≈ Assumption of “Resources”?
tl;dr, the unidimensional continuity of preference assumption in the money pumping argument used to justify the VNM axioms correspond to the assumption that there exists some unidimensional “resource” that the agent cares about, and this language is provided by the notion of “souring / sweetening” a lottery.
Various coherence theorems—or more specifically, various money pumping arguments generally have the following form:
If you violate this principle, then [you are rationally required] / [it is rationally permissible for you] to follow this trade that results in you throwing away resources. Thus, for you to avoid behaving pareto-suboptimally by throwing away resources, it is justifiable to call this principle a ‘principle of rationality,’ which you must follow.
… where “resources” (the usual example is money) are something that, apparently, these theorems assume exist. They do, but this fact is often stated in a very implicit way. Let me explain.
In the process of justifying the VNM axioms using money pumping arguments, one of the three main mathematical primitives are: (1) lotteries (probability distribution over outcomes), (2) preference relation (general binary relation), and (3) a notion of Souring/Sweetening of a lottery. Let me explain what (3) means.
Souring of A is denoted A−, and a sweetening of A is denoted A+.
A− is to be interpreted as “basically identical with A but strictly inferior in a single dimension that the agent cares about.” Based on this interpretation, we assume A>A−. Sweetening is the opposite, defined in the obvious way.
Formally, souring could be thought of as introducing a new preference relation A>uniB, which is to be interpreted as “lottery B is basically identical to lottery A, but strictly inferior in a single dimension that the agent cares about”.
On the syntactic level, such B is denoted as A−.
On the semantic level, based on the above interpretation, >uni is related to > via the following: A>uniB⟹A>B
This is where the language to talk about resources come from. “Something you can independently vary alongside a lottery A such that more of it makes you prefer that option compared to A alone” sounds like what we’d intuitively call a resource[1].
Now that we have the language, notice that so far we haven’t assumed sourings or sweetenings exist. The following assumption does it:
Unidimensional Continuity of Preference: If X>Y, then there exists a prospect X− such that 1) X− is a souring of X and 2) X>X−>Y.
Which gives a more operational characterization of souring as something that lets us interpolate between the preference margins of two lotteries—intuitively satisfied by e.g., money due to its infinite divisibility.
So the above assumption is where the assumption of resources come into play. I’m not aware of any money pump arguments for this assumption, or more generally, for the existence of a “resource.” Plausibly instrumental convergence.
I don’t actually think this + the assumption below fully capture what we intuitively mean by “resources”, enough to justify this terminology. I stuck with “resources” anyways because others around here used that term to (I think?) refer to what I’m describing here.
Thinking about for some time my feeling has been that resources are about fungibility implicitly embedded in a context of trade, multiple agents (very broadly construed. E.g. an agent in time can be thought of as multiple agents cooperating intertemporally perhaps).
A resource over time has the property that I can spend it now or I can spend it later. Glibly, one could say the operational meaning of the resource arises from the intertemporal bargaining of the agent.
Perhaps it’s useful to distinguish several levels of resources and resource-like quantities.
Discrete vs continuous, tradeable / meaningful to different agents, ?? Fungibility, ?? Temporal and spatial locatedness, ?? Additivity?, submodularity ?
Addendum: another thing to consider is that the input of the vNM theorem is in some sense more complicated than the output.
The output is just a utility function u: X → R, while your input is a preference order on the very infinite set of lotteries (= probability distributions ) L(X).
Thinking operationally about a preference ordering on a space of distribution is a little wacky.
It means you are willing to trade off uncertain options against one another. For this to be a meaningful choice would seem to necessitate some sort of (probabilistic) world model.
Unidimensional Continuity of Preference ≈ Assumption of “Resources”?
tl;dr, the unidimensional continuity of preference assumption in the money pumping argument used to justify the VNM axioms correspond to the assumption that there exists some unidimensional “resource” that the agent cares about, and this language is provided by the notion of “souring / sweetening” a lottery.
Various coherence theorems—or more specifically, various money pumping arguments generally have the following form:
… where “resources” (the usual example is money) are something that, apparently, these theorems assume exist. They do, but this fact is often stated in a very implicit way. Let me explain.
In the process of justifying the VNM axioms using money pumping arguments, one of the three main mathematical primitives are: (1) lotteries (probability distribution over outcomes), (2) preference relation (general binary relation), and (3) a notion of Souring/Sweetening of a lottery. Let me explain what (3) means.
Souring of A is denoted A−, and a sweetening of A is denoted A+.
A− is to be interpreted as “basically identical with A but strictly inferior in a single dimension that the agent cares about.” Based on this interpretation, we assume A>A−. Sweetening is the opposite, defined in the obvious way.
Formally, souring could be thought of as introducing a new preference relation A>uniB, which is to be interpreted as “lottery B is basically identical to lottery A, but strictly inferior in a single dimension that the agent cares about”.
On the syntactic level, such B is denoted as A−.
On the semantic level, based on the above interpretation, >uni is related to > via the following: A>uniB⟹A>B
This is where the language to talk about resources come from. “Something you can independently vary alongside a lottery A such that more of it makes you prefer that option compared to A alone” sounds like what we’d intuitively call a resource[1].
Now that we have the language, notice that so far we haven’t assumed sourings or sweetenings exist. The following assumption does it:
Which gives a more operational characterization of souring as something that lets us interpolate between the preference margins of two lotteries—intuitively satisfied by e.g., money due to its infinite divisibility.
So the above assumption is where the assumption of resources come into play. I’m not aware of any money pump arguments for this assumption, or more generally, for the existence of a “resource.” Plausibly instrumental convergence.
I don’t actually think this + the assumption below fully capture what we intuitively mean by “resources”, enough to justify this terminology. I stuck with “resources” anyways because others around here used that term to (I think?) refer to what I’m describing here.
Thinking about for some time my feeling has been that resources are about fungibility implicitly embedded in a context of trade, multiple agents (very broadly construed. E.g. an agent in time can be thought of as multiple agents cooperating intertemporally perhaps).
A resource over time has the property that I can spend it now or I can spend it later. Glibly, one could say the operational meaning of the resource arises from the intertemporal bargaining of the agent.
Perhaps it’s useful to distinguish several levels of resources and resource-like quantities.
Discrete vs continuous, tradeable / meaningful to different agents, ?? Fungibility, ?? Temporal and spatial locatedness, ?? Additivity?, submodularity ?
Addendum: another thing to consider is that the input of the vNM theorem is in some sense more complicated than the output. The output is just a utility function u: X → R, while your input is a preference order on the very infinite set of lotteries (= probability distributions ) L(X).
Thinking operationally about a preference ordering on a space of distribution is a little wacky. It means you are willing to trade off uncertain options against one another. For this to be a meaningful choice would seem to necessitate some sort of (probabilistic) world model.