There is a theorem called Ramsey's theorem,

which is actually only something that gives us a value function,

or an ordinal utility function.

That is what he proved in 31,

and Neumann and Morgenstern in 1944 proved the variant with

the lotteries that you actually get a utility function,

which is essentially given a rational set of preferences,

i.e. those that obey these axioms we wrote down,

then you're getting a utility function from states to the reals.

That is something you can access,

assess, or you can actually observe.

You can ask people, do you like this better than that?

Or you can actually observe people or animals or whatever systems,

and get the utility function post hoc.

Utility functions are not unique.

So any linear transformation you put on

the utility function actually yields the same agent behavior.

So you can only assess the utility functions up to linear transformation,

which isn't a problem because even if we had information about K1 and K2,

it wouldn't change anything. We don't care.

So putting that together with this principle of maximizing expected utility,

that actually gives us a recipe for rational agents.

Well, we don't quite know what we do for optimizing here yet.

We're going to look into that today.

But basically, if we say,

well, we have a set of preferences,

that gives us a utility function and so on.

Then we can actually build something like that. That's the idea.

We're going to dwell on that architecture in the next time,

namely looking at the decision problems here,

scaling those decision problems up,

and then involving time and so on.

That's the plan. That's I think how far we got.

Are there any questions? Yes.

Actually, I didn't get the utility from the previous.

Where is my utility? Here? No.

So here.

Okay. This one in the middle here.

Okay. It's essentially if you get a utility function,

squiggle, squiggle, something like that,

we have state A here and state B here.

Then state B has a higher utility than state A.

Now, if you add a constant to that,

disregard my artistic skills again,

it's still A is preferred to B.

Okay? If I tilt this,

no, not this way, but if I tilt this this way,

then that's not going to change.

Now, that is only to say comparing real prizes.

The interesting bit is you