Thanks. Though the continuity assumption is itself the thing I felt to be the problem area. Unless I’m misunderstanding your argument, you’re assuming the very continuity property I want to derive.
(Incidentally, I may be wrong on this, but I think the closure property you’re referring to would follow directly from the continuity. (alternately, one might need to show that closure property to show the continuity. Point being, I don’t think those two properties are separable, except at A and B))
Continuity is merely preserving order: if A > B and p > q, then pA+(1-p)B > qA+(1-q)B. It is a not-being-stupid assumption. or an interpretation of probability.
Mendel seems to be working in an extremely abstract version of probability where p cannot be described as a size. But once you insist on p being a number, there are many possibilities. You might allow p only to take rational values, so that they can be finitely represented. Or you might allow p to take all real values, in which case some p exists solving the problem.
The issue of closure is about where the p’s live, that is, what kinds of lotteries you can build. It isn’t about preferences or states of the world (except in that lotteries are states of the world).
ETA: Actually...the axiom of independence gives you order preservation. The axiom of continuity does only one thing: it rules out lexicographic preferences. It says that if you lexicographically care more about X than about Y, you aren’t allowed to use Y as a tie breaker, but must simply not care about about Y at all.
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.
Thanks. Though the continuity assumption is itself the thing I felt to be the problem area. Unless I’m misunderstanding your argument, you’re assuming the very continuity property I want to derive.
(Incidentally, I may be wrong on this, but I think the closure property you’re referring to would follow directly from the continuity. (alternately, one might need to show that closure property to show the continuity. Point being, I don’t think those two properties are separable, except at A and B))
Continuity is merely preserving order: if A > B and p > q, then pA+(1-p)B > qA+(1-q)B. It is a not-being-stupid assumption. or an interpretation of probability.
Mendel seems to be working in an extremely abstract version of probability where p cannot be described as a size. But once you insist on p being a number, there are many possibilities. You might allow p only to take rational values, so that they can be finitely represented. Or you might allow p to take all real values, in which case some p exists solving the problem.
The issue of closure is about where the p’s live, that is, what kinds of lotteries you can build. It isn’t about preferences or states of the world (except in that lotteries are states of the world).
ETA: Actually...the axiom of independence gives you order preservation. The axiom of continuity does only one thing: it rules out lexicographic preferences. It says that if you lexicographically care more about X than about Y, you aren’t allowed to use Y as a tie breaker, but must simply not care about about Y at all.
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.