I totally agree and never meant to imply otherwise. But just as any consistent system of degrees of belief can be put into correspondence with the axioms of probability, so there are certain stipulations about what can reasonably called a utility function.
I would argue that if you meet a conscious agent and your model of their utility function says that it doesn’t converge (in the appropriate L1 norm of the appropriate modeled probability space) then something’s wrong with that model of utility function… not with the assumption that utility functions should converge. There are many subtleties, I’m sure, but non-integrable utility functions seem futile to me. If something can be well-modeled by a non-integrable utility function, then I’m fine updating my position, but in years of learning and teaching probability theory, I’ve never encountered anything that would convince me of that.
Yes, good point. Is there any study of the most general objects to which integrability theory applies? Also, are you familiar with Martin Kruskal’s work on generalizing calculus to the surreal numbers? I am having difficulty locating any of his papers.
What comes to my mind are Bochner integrals and random elements. I’m not sure how much integrability theory one can develop outside of a Banach space, although you can get interesting fractal type integrals when dealing with Hausdorff measure. Integrability theory is really just an extension of measure theory, which was pinned down in painstaking detail by Lebesgue, Caratheodory, Perron, Henstock, and Kurzweil (no relation to the singularity Kurzweil). The Henstock-Kurzweil (HK) integral is the most generalized integral over the reals and complexs that preserves certain nice properties, like the fundamental theorem of calculus. The name of the game in integration theory was never an attempt to find the most abstract workable definitions of integration, but rather to see under what general assumptions you could get physically meaningful results, like mean value theorem or fundamental theorem of calculus, to hold. Complex integration theory, especially in higher dimensions shattered a lot of the preconceived notions of how functions should behave.
In looking up surreal numbers, it appears that Conway and Knuth invented them. I was surprised to learn that the hyperreal numbers (developed by Abraham Robinson) are contained in the surreals. To my knowledge, which is a bit limited because I focus more on applied math and so I am probably not as familiar with the literature on something like surreal numbers as other LWers may be, there hasn’t been much work, if any, on defining an integral over the surreals. My guess, though, is that such an integral would wind up being an unsatisfyingly trivial extension of integration over the regular reals, as is the case for hyperreals.
I’ll definitely take a look at Kruskal’s papers and see what he’s come up with.
I was surprised to learn that the hyperreal numbers (developed by Abraham Robinson) are contained in the surreals.
Every ordered field is contained within the surreals, which is why I find them promising for utility theory. The surreals themselves are not a field but a Field, since they form a proper class.
Another point worth noting is that on a set D of finite measure (which any measurable subset of a probability space is), L^{N}(D) is contained in L^{N-1}(D), and so if the first moment fails to exist (non-integrable, no defined expectation) then all higher moments fail and computation of order statistics fails. Of course nature doesn’t have to be modeled by statistics, but you’d be hard pressed to out-perform simple axiomatic formulations that just assume a topolgy, continuous preference functions, and get on with it and have access to higher order moments.
I totally agree and never meant to imply otherwise. But just as any consistent system of degrees of belief can be put into correspondence with the axioms of probability, so there are certain stipulations about what can reasonably called a utility function.
I would argue that if you meet a conscious agent and your model of their utility function says that it doesn’t converge (in the appropriate L1 norm of the appropriate modeled probability space) then something’s wrong with that model of utility function… not with the assumption that utility functions should converge. There are many subtleties, I’m sure, but non-integrable utility functions seem futile to me. If something can be well-modeled by a non-integrable utility function, then I’m fine updating my position, but in years of learning and teaching probability theory, I’ve never encountered anything that would convince me of that.
Doesn’t this all assume that utility functions are real-valued?
No, all of the integrability theory (w.r.t. probability measures) extends straightforwardly to complex valued functions. See this and this.
Yes, good point. Is there any study of the most general objects to which integrability theory applies? Also, are you familiar with Martin Kruskal’s work on generalizing calculus to the surreal numbers? I am having difficulty locating any of his papers.
What comes to my mind are Bochner integrals and random elements. I’m not sure how much integrability theory one can develop outside of a Banach space, although you can get interesting fractal type integrals when dealing with Hausdorff measure. Integrability theory is really just an extension of measure theory, which was pinned down in painstaking detail by Lebesgue, Caratheodory, Perron, Henstock, and Kurzweil (no relation to the singularity Kurzweil). The Henstock-Kurzweil (HK) integral is the most generalized integral over the reals and complexs that preserves certain nice properties, like the fundamental theorem of calculus. The name of the game in integration theory was never an attempt to find the most abstract workable definitions of integration, but rather to see under what general assumptions you could get physically meaningful results, like mean value theorem or fundamental theorem of calculus, to hold. Complex integration theory, especially in higher dimensions shattered a lot of the preconceived notions of how functions should behave.
In looking up surreal numbers, it appears that Conway and Knuth invented them. I was surprised to learn that the hyperreal numbers (developed by Abraham Robinson) are contained in the surreals. To my knowledge, which is a bit limited because I focus more on applied math and so I am probably not as familiar with the literature on something like surreal numbers as other LWers may be, there hasn’t been much work, if any, on defining an integral over the surreals. My guess, though, is that such an integral would wind up being an unsatisfyingly trivial extension of integration over the regular reals, as is the case for hyperreals.
I’ll definitely take a look at Kruskal’s papers and see what he’s come up with.
Every ordered field is contained within the surreals, which is why I find them promising for utility theory. The surreals themselves are not a field but a Field, since they form a proper class.
Another point worth noting is that on a set D of finite measure (which any measurable subset of a probability space is), L^{N}(D) is contained in L^{N-1}(D), and so if the first moment fails to exist (non-integrable, no defined expectation) then all higher moments fail and computation of order statistics fails. Of course nature doesn’t have to be modeled by statistics, but you’d be hard pressed to out-perform simple axiomatic formulations that just assume a topolgy, continuous preference functions, and get on with it and have access to higher order moments.