# Richard_Kennaway's Posts

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Underappreciated points about utility functions (of both sorts)
I already said that I think that thinking in terms of infinitary convex combinations, as you're doing, is the wrong way to go about it; but it took me a bit to put together why that's definitely the wrong way.
Specifically, it assumes probability! Fishburn, in the paper you link, assumes probability, which is why he's able to talk about why infinitary convex combinations are or are not allowed (I mean, that and the fact that he's not necessarily arbitrary actions).
Savage doesn't assume probability!

Savage doesn't assume probability or utility, but their construction is a mathematical consequence of the axioms. So although they come later in the exposition, they mathematically exist as soon as the axioms have been stated.

So if you want to disallow certain actions... how do you specify them?

I am still thinking about that, and may be some time.

As a general outline of the situation, you read P1-7 => bounded utility as modus ponens: you accept the axioms and therefore accept the conclusion. I read it as modus tollens: the conclusion seems wrong, so I believe there is a flaw in the axioms. In the same way, the axioms of Euclidean geometry seemed very plausible as a description of the physical space we find ourselves in, but conflicts emerged with phenomena of electromagnetism and gravity, and eventually they were superseded as descriptions of physical space by the geometry of differential manifolds.

It isn't possible to answer the question "which of P1-7 would I reject?" What is needed to block the proof of bounded utility is a new set of axioms, which will no doubt imply large parts of P1-7, but might not imply the whole of any one of them. If and when such a set of axioms can be found, P1-7 can be re-examined in their light.

Repossessing Degrees

The paper is not the degree. It is a certificate of having the degree. The degree is the fact of having had the degree conferred. This is an objective historical fact that cannot be repossessed, short of 1984 with its memory holes and workers keeping records updated to agree with currently decreed official truth (that is, official lies). Even if the university is obliged to rescind the conferral, that merely adds another historical fact to the record. If an employer regards the recission as a penalty for defaulting on a student loan, they are free to take that as evidence of the student's financial standing but disregard it as evidence against their academic record.

Moral uncertainty: What kind of 'should' is involved?
Instead, it's basically "Moral uncertainty is uncertainty about moral matters", which then has to be accompanied with a range of examples and counterexamples of the sort of thing we mean by that.

What need is there for a definition of "moral uncertainty"? Empirical uncertainty is uncertainty about empirical matters. Logical uncertainty is uncertainty about logical matters. Moral uncertainty is uncertainty about moral matters. These phrases mean these things in the same way that "red car" means a car that is red, and does not need a definition.

If one does not believe there are objective moral truths, then "Moral uncertainty is uncertainty about moral matters" might feel problematic. The problem lies not in "uncertainty" but in "moral matters". But that is an issue you have postponed.

How to Identify an Immoral Maze

How does this work in the military? They have a very deep hierarchy: is life in the army above private and below commander-in-chief also a maze?

How has the cost of clothing insulation changed since 1970 in the USA?

The link is 451 for me: "Unavailable due to legal reasons". The specifics:

We recognize you are attempting to access this website from a country belonging to the European Economic Area (EEA) including the EU which enforces the General Data Protection Regulation (GDPR) and therefore access cannot be granted at this time. For any issues, contact circdept@times-news.com or call (301) 722-4600.

Prima facie that looks like bullshit, but recognising that doesn't get me the web page. Time I looked into a VPN account. Any suggestions?

BTW, mousing over the same link on your web page gives me a popup saying "Too many requests", which none of the others do. What's up there?

Underappreciated points about utility functions (of both sorts)
Again, I'm simply not seeing this in the paper you linked? As I said above, I simply do not see anything like that outside of section 9, which is irrelevant. Can you point to where you're seeing this condition?

In Fishburn's "Bounded Expected Utility", page 1055, end of first paragraph (as cited previously):

However, we shall for the present take $\wp = \wp_d$ (for any $\sigma$ -algebra that contains each ${x}$ ) since this is the Blackwell-Girshick setting. Not only is $\wp_d$ an abstract convex set, but also if $\alpha_i \ge 0$ and $P_i \in \wp_d$ for $i = 1, 2, \dots$ and $\Sigma_{i=1}^\infty \alpha_i = 1$ , then $\Sigma_{i=1}^\infty \alpha_i P_i \in \wp_d$ .

That depends on some earlier definitions, e.g. $\wp_d$ is a certain set of probability distributions (the “d” stands for “discrete”) defined with reference to some particular $\sigma$ -algebra, but the important part is that last infinite sum: this is where all infinitary convex combinations are asserted to exist. Whether that is assigned to "background setup" or "axioms" does not matter. It has to be present, to allow the construction of St. Petersburg gambles.

Underappreciated points about utility functions (of both sorts)

A further short answer. In Savage's formulation, from P1-P6 he derives Theorem 4 of section 2 of chapter 5 of his book, which is linear interpolation in any interval. Clearly, linear interpolation does not work on an interval such as [17,Inf], therefore there cannot be any infinitely valuable gambles. St. Petersburg-type gambles are therefore excluded from his formulation.

Savage does not actually prove bounded utility. Fishburn did this later, as Savage footnotes in the edition I'm looking at, so Fishburn must be tackled. Theorem 14.5 of Fishburn's book derives bounded utility from Savage's P1-P7. His proof seems to construct a St. Petersburg gamble from the supposition of unbounded utility, deriving a contradiction. I shall have to examine further how his construction works, to discern what in Savage's axioms allows the construction, when P1-P6 have already excluded infinitely valuable gambles.

Underappreciated points about utility functions (of both sorts)
Or if you have some formalism where preferences can be undefined (in a way that is distinct from indifference), by all means explain it... (but what happens when you program these preferences into an FAI and it encounters this situation? It has to pick. Does it pick arbitrarily? How is that distinct from indifference?)

A short answer to this (something longer later) is that an agent need not have preferences between things that it is impossible to encounter. The standard dissolution of the St. Petersberg paradox is that nobody can offer that gamble. Even though each possible outcome is finite, the offerer must be able to cover every possible outcome, requiring that they have infinite resources.

Since the gamble cannot be offered, no preferences between that gamble and any other need exist. If your axioms require both that preference must be total and that St. Petersburg gambles exist, I would say that that is a flaw in the axioms. Fishburn (op. cit., following Blackwell and Girschick, an inaccessible source) requires that the set of gambles be closed under infinitary convex combinations. I shall take a look at Savage's axioms and see what in them is responsible for the same thing.

Looking at the argument from the other end, at what point in valuing numbers of intelligent lives does one approach an asymptote, bearing in mind the possibility of expansion to the accessible universe? What if we discover that the habitable universe is vastly larger than we currently believe? How would one discover the limits, if there are any, to one's valuing?