I've promised you utilities and given you preference structures.

It'll turn out that from preference structures we can make utilities.

So that's really what Ramsey's theorem is, independently discovered by John von Neumann

and Morgenstern relatively early.

The von Neumann and so on, they were thinking about decision theory in a situation where

of course decision went horribly wrong for a while.

So they were quite interested in this.

But Ramsey's theorem says, given a preference that satisfies the axioms from before, I

can translate this preference structure into a utility function such that if I prefer A

to B, then the utility of A is greater or equal to the utility of B.

And what's more, the utility of A, the utility of a multiple auction, right?

We've only seen binary auctions here.

But of course, you can kind of iterate those and get multi-ary auctions.

It's relatively easy to do that.

Just chain them together.

So basically, the utility of an auction that has n outcomes with n probabilities is just

the sum of the probabilities of the year.

It's actually an additive function, which is what you need to interpret auctions correctly.

Yes?

Whether you formulate all of these things in a strict ordering relation or a non-strict

one doesn't really matter.

You can always factor out equality.

But I did say it wrong, yes.

But I'm bad with moths.

I should probably...

You could just say, if I prefer A to B, then the utility is greater.

So if you look at the preference operators, there they are, one of them is really redundant.

If you basically, in the language in which you talk, have a not, which we're using here,

then that's what you get.

You don't really need all three.

But they're convenient.

I have a feeling I'm not answering your question.

I think the utility is better when of course I should prefer it.

No.

No.

Preference is the strict relation.

Okay, good.

Yes.

Which is not preferring with switch arguments.

Think about less or equal and less.

Not less than is greater or equal.

Excellent.

Okay?

So Ramsey's theorem says I've been giving you utilities after all.

There's one little sticky point here.

This doesn't give us a unique utility.

Right?

I'm not getting the utility is 22.

It turns out that there's two things I can do and not destroy the thing of being a utility

function.