The JB framework as standardly formulated assumes complete and consistent preferences. Of course, you can keep the same JB-style objects of preference (the propositions) and change modify the preference axioms. For incomplete preferences, there’s a nice paper by Richard Bradley, Revising Incomplete Attitudes, that looks at incomplete attitudes in a very Jeffrey-Bolker style framework (all prospects are propositions). It has a nice discussion of different things that might lead to incompleteness (one of which is “Ignorance”, related to the kind of Knightian uncertainty you asked about), and also some results and perspectives on attitude changes for imprecise Bayesian agents.
I’m less sure about inconsistent preferences—it depends what exactly you mean by that. Something related might be work on aggregating preferences, which can involve aggregating preferences that disagree and so look inconsistent. John Broome’s paper Bolker-Jeffrey Expected Utility Theory and Axiomatic Utilitarianism is excellent on this—it examines both the technical foundations of JB and its connections to social choice and utilitarianism, proving a version of the Harsanyi Utilitarian Theorem in JB.
On imprecise probabilities: the JB framework actually has a built-in form of imprecision. Without additional constraints, the representation theorem gives non-unique probabilities (this is part of Bolker’s uniqueness theorem). You can get uniqueness by adding extra conditions, like unbounded utility or primitive comparative probability judgments, but the basic framework allows for some probability imprecision. I’m not sure about deeper connections to infraprobability/Bayesianism, but given that these approaches often involve sets of probabilities, there may be interesting connections to explore.
Ah, so not like, A is strongly preferred to B and B is strongly preferred to A, but more of a violation of transitivity. Then I still think that the Broome paper is a place I’d look at, since you get that exact kind of structure in preference aggregation.
The Bradley paper assumes everything is transitive throughout, so I don’t think you get the kind of structure you want there. I’m not immediately aware of any work of that kind of inconsistency in JB that isn’t in the social choice context, but there might be some. I’ll take a look.
There are ways to think about degrees and measures of incoherence, and how that connects up to decision making. I’m thinking mainly of this paper by Schervish, Seidenfeld, and Kadane, Measures of Incoherence: How Not to Gamble if You Must. There might a JB-style version of that kind of work, and if there isn’t, I think it would be good to have one.
But to your core goal or weakening the preference axioms to more realistic standards, you can definitely do that in JB by weakening the preference axioms, but still keeping the background objects of preference be propositions in a single algebra. I think this would still preserve many of what I consider the naturalistic advantages of the JB system. For modifying the preference axioms, I would guess descriptively you might want something like prospect theory, or something else along those broad lines. Also depends on what kinds of agents we want to describe.
I want to be able to describe agents that do not have (vNM, geometric, other) rational preferences because of incompleteness or inconsistency but self-modify to become so.
Eg. In vNM utility theory there is a fairly natural weakening one can do which is ask for a vNM-style representation theorem after dropping transitivity.
[ Incidentally, there is some interesting math here having to do with conservative vs nonconservative vector fields and potentials theory all the way to hodge theory. ]
does JB support this ?
Im confused since in vNM we start with a preference order over probability distributions. But in JB irs over propositions?
The JB framework as standardly formulated assumes complete and consistent preferences. Of course, you can keep the same JB-style objects of preference (the propositions) and change modify the preference axioms. For incomplete preferences, there’s a nice paper by Richard Bradley, Revising Incomplete Attitudes, that looks at incomplete attitudes in a very Jeffrey-Bolker style framework (all prospects are propositions). It has a nice discussion of different things that might lead to incompleteness (one of which is “Ignorance”, related to the kind of Knightian uncertainty you asked about), and also some results and perspectives on attitude changes for imprecise Bayesian agents.
I’m less sure about inconsistent preferences—it depends what exactly you mean by that. Something related might be work on aggregating preferences, which can involve aggregating preferences that disagree and so look inconsistent. John Broome’s paper Bolker-Jeffrey Expected Utility Theory and Axiomatic Utilitarianism is excellent on this—it examines both the technical foundations of JB and its connections to social choice and utilitarianism, proving a version of the Harsanyi Utilitarian Theorem in JB.
On imprecise probabilities: the JB framework actually has a built-in form of imprecision. Without additional constraints, the representation theorem gives non-unique probabilities (this is part of Bolker’s uniqueness theorem). You can get uniqueness by adding extra conditions, like unbounded utility or primitive comparative probability judgments, but the basic framework allows for some probability imprecision. I’m not sure about deeper connections to infraprobability/Bayesianism, but given that these approaches often involve sets of probabilities, there may be interesting connections to explore.
Mmmmm
Inconsistent and incomplete preferences are necessary for descriptive agent foundations.
In vNM preference theory an inconsistent preference can be described as cyclic preferences that can be moneypumped.
How to see this in JB ?
Ah, so not like, A is strongly preferred to B and B is strongly preferred to A, but more of a violation of transitivity. Then I still think that the Broome paper is a place I’d look at, since you get that exact kind of structure in preference aggregation.
The Bradley paper assumes everything is transitive throughout, so I don’t think you get the kind of structure you want there. I’m not immediately aware of any work of that kind of inconsistency in JB that isn’t in the social choice context, but there might be some. I’ll take a look.
There are ways to think about degrees and measures of incoherence, and how that connects up to decision making. I’m thinking mainly of this paper by Schervish, Seidenfeld, and Kadane, Measures of Incoherence: How Not to Gamble if You Must. There might a JB-style version of that kind of work, and if there isn’t, I think it would be good to have one.
But to your core goal or weakening the preference axioms to more realistic standards, you can definitely do that in JB by weakening the preference axioms, but still keeping the background objects of preference be propositions in a single algebra. I think this would still preserve many of what I consider the naturalistic advantages of the JB system. For modifying the preference axioms, I would guess descriptively you might want something like prospect theory, or something else along those broad lines. Also depends on what kinds of agents we want to describe.
I want to be able to describe agents that do not have (vNM, geometric, other) rational preferences because of incompleteness or inconsistency but self-modify to become so.
Eg. In vNM utility theory there is a fairly natural weakening one can do which is ask for a vNM-style representation theorem after dropping transitivity.
[ Incidentally, there is some interesting math here having to do with conservative vs nonconservative vector fields and potentials theory all the way to hodge theory. ]
does JB support this ?
Im confused since in vNM we start with a preference order over probability distributions. But in JB irs over propositions?