You really, really, really don’t want to be touching continuity without knowing exactly what you’re doing. See the hyperreals for an example of the sort of thing that happens in this case. Also look at non-measurable functions to see the fun in store.
But most of the time, when people deny continuity, it’s not on theoretical grounds but because they have a particular non-continuous preference theory in mind. That’s perfectly fine. But generally, the non-continous theory can be approximated arbitrarily well by a continuous version that looks exactly the same in virtually all circumstances.
This has been helpful. I’m much more familiar with the mathematics than the economics. Presently, I’m more worried about the mathematical chicanery involved in approximating a consistent continuous utility function out of things.
If we’re using the Independence II as an axiom, you should be a little more precise, when you introduced it above, you referred to the base four axioms, including continuity.
Now, I only noticed consistency needed to convert between the two Independence formulations, which would make your statement correct. But on the face of things, it looks like you are trying to show a money pump theorem under discontinuous preferences by calling upon the continuity axiom.
For all A, B, C, D and p, if A ≤ B and C ≤ D, then pA + (1-p)C ≤ pB + (1-p)D.
If we now take C and D to be the same lottery, we get independence, as long as C ≤ C. Now, given completeness, C ≤ C is always true (because at least one of C=C, CC must be true, and thus we can always get C ≤ C, -- switching C with C if needed!).
So we don’t need consistency, we need a weak form of completeness, in which every lottery can be at least compared with itself.
You really, really, really don’t want to be touching continuity without knowing exactly what you’re doing. See the hyperreals for an example of the sort of thing that happens in this case. Also look at non-measurable functions to see the fun in store.
But most of the time, when people deny continuity, it’s not on theoretical grounds but because they have a particular non-continuous preference theory in mind. That’s perfectly fine. But generally, the non-continous theory can be approximated arbitrarily well by a continuous version that looks exactly the same in virtually all circumstances.
This has been helpful. I’m much more familiar with the mathematics than the economics. Presently, I’m more worried about the mathematical chicanery involved in approximating a consistent continuous utility function out of things.
Continuity is no longer needed for these results...
If we’re using the Independence II as an axiom, you should be a little more precise, when you introduced it above, you referred to the base four axioms, including continuity.
Now, I only noticed consistency needed to convert between the two Independence formulations, which would make your statement correct. But on the face of things, it looks like you are trying to show a money pump theorem under discontinuous preferences by calling upon the continuity axiom.
Mathematically:
Independence + other 3 axioms ⇒ Independence II
Independence II ⇒ Independence
Hence: ~Independence ⇒ ~Independence II
My theorem implies: ~Independence II ⇒ You can be money pumped
Hence: ~Independence ⇒ You can be money pumped
Note, Independence II does not imply Independence, without using at least the consistency axiom.
The contrapositive of independence II is:
For all A, B, C, D and p, if A ≤ B and C ≤ D, then pA + (1-p)C ≤ pB + (1-p)D.
If we now take C and D to be the same lottery, we get independence, as long as C ≤ C. Now, given completeness, C ≤ C is always true (because at least one of C=C, CC must be true, and thus we can always get C ≤ C, -- switching C with C if needed!).
So we don’t need consistency, we need a weak form of completeness, in which every lottery can be at least compared with itself.
Transitivity and Continuity are unnecessary, however.
That is my reading of it too. I know Stuart is putting forward analytic results here, I was concerned that this one was not correctly represented.