Now, if we want to go back to more exact methods, then of course what we need is if we have

an n attribute utility, remember that all of those utilities are actually random variables,

which all have a range of size d, which means the utility function is actually a big n dimensional

matrix with b to the n values. Completely unrealistic. Luckily, very often those things

have more structure. Very often we can find kind of a function f that has much more structure,

that is kind of a simple thing, where f is something like addition or multiplication

or something like this. And where we have these kind of functions f1 to fn that kind

of do something for the various dimensions. Say we have a utility function over constant

cost and noise. Then you might be able to say, oh, noise is three times as important

as cost. Then you would just get u as what is it, three times x plus y. And the only

thing I need to know is the pattern f here and two coordinate functions. One is the identity

and the other one is three times. And that might make it much easier because this function

here I can do something with essentially two numbers. Yes?

Yes. Then you're in trouble. Or then you have to work harder.

Then you go to piecewise monotonous functions. And just basically have areas where you use

monotony. Yes? Well, but there might be things that you have to spend at least this amount

of money. A friend of mine who was a lawyer on this first work day, he was actually started

as a lawyer for the company Boss. The first day he was given 50,000 Deutschmarks at the

time and said you have to spend that on business suits until tomorrow evening. He wasn't the

suit type before that so he had a hard time. Imagine 50,000 Euros, probably more by now.

But yes, there is a sometimes funny utility function there. And then you of course have

to do something interesting. Yes. Dominance only works for...

Well, this function can be whatever it wants to be. But for the dominance stuff we want

to have it at least... We want to have these things monotonous in the dimensions. And I

think that gives you monotony all over. I could jector that. I don't know. But yes,

very often in real life we find these kind of things where we have very simple Fs. And

in your case we wouldn't. But still we would probably have it piecewise or locally and

so on. But yes, the world is bad. That's how the world is. And sometimes the world is a

little bit better like this. Or we can assume it to be better. Because we can't... All of

this thing we're doing here is like the person who's looking for his contact lenses at night

under a street light. And then he gets asked, why are you searching... Where did you lose

your contact lenses? Oh yes, somewhere over there. But why are you searching under the

street light? Well, I can't find them there anyway. So I'm going to look here. So that's

kind of always the trade off we have here. And of course if we look at this, that looks

quite a lot like Bayesian networks. And that's something that people noticed. So if somebody

notices something like this, then you go and try and define kind of independence results.

Independent results actually drive down the number of values here. And instead of looking

at probabilistically independent, which is really what we're getting here with addition.

What you can look at then is preferential independence. Because we want to model a different

distribution. So we can actually do those kind of... Make a very simple, similar definition.

We have two sets being preferentially independent from a third one. If kind of the preference

between the first two doesn't depend on the third one. So if we have noise cost and safety,

the preference for noise and what was the thing? Noise and costs are independent of

the deaths. You can actually get values independently. So if you have neutral preferential independence,

meaning everything is independent, then you have a theorem that says you have an additive

value function. Or in other words, something like this must always hold. Or in other words,

times of world is easy. We can actually just write down single attribute functions. This

is exactly for F being addition here. That's really what we have. So we have to write down

N single function. So we have something of the form N times D many values. Which is much

nicer than this. And of course, we don't always have independence. We sometimes have things