The next "How Not to be Stupid" may be a bit delayed for a couple of reasons.
First, there appears to be a certain unstated continuity assumption in the material I've been working from that would probably be relevant for the next posting. As I said in the intro post, I'm working from Stephen Omhunduro's vulnerability argument, but filling in what I viewed as missing bits, generalizing one or two things, and so on. Anyways, the short of it is I thought that I was able to how to derive the relevant continuity condition, but turns out I need to think about that a bit more carefully.
If I remain stumped on that bit, I'll just post and explicitly state the assumption, pointing it out as a potential problem area that needs to be dealt with one way or another. ie, either solved somehow, or demonstrate that it actually is invalid (thus causing some issues for decision theory...)
Also, I'm going to be at Penguicon the next few days.
Actually, I think I'll right now state the continuity condition I need, let others play with it too:
Basically, I need it to be that if there's a preference ranking A > C > B, there must exist some p such that the p*A + (1-p)*B lottery ranks equal to C. (That is, that the mixing lotteries correspond to a smooth spectrum of preferences between, well, the things they are a mixing of rather than having any discontinuous jumps.)
Anyways, I hope we can put this little trouble bit behind us and resume climbing the shining path of awakening to nonstupidity. :)
The continuity of p -> preference space is an additional property you must assert. Like the total ordering, you need to specify that this function is in fact of this form, since it isn't determined by the premises
A monotonically increasing function from p -> preference space will preserve the total order just fine and meet the end point criteria, even if discontinuous. And since those are the two properties you currently require you need to add another property to the theory if you want to eliminate discontinuity.
Intuitively, you probably want to do this anyway, since you haven't said anything about how decisions are made except by listing basic decision pathologies and ruling them out. First was cycles (although a total order is arguably over-kill for this one). The second was incorporating probability (which necessitated an end-point condition - A >= pA + (1-p)B >= B). Now it's time to add a new condition of continuity of the p mapping, based on the intuition that immeasurably small changes in probability should not cause measurable changes in decisions. But nothing you've laid out so far excludes such an agent.
(Good point. My error there. But do note, though, while b implies c, c definitely does not imply b.)
Edited to add: The intuition about measureable changes caused by immeasurable probability shifts removes all but point-wise discontinuitites. Those you can remove via adding something like c, i.e. probabilities are real numbers or the like.