So, a bit fascinating to see how many people are technically still actively participating

at least in the Tuesday quizzes compared to how many are actually here.

Not that I blame them.

I mean, the thing is recorded anyway, so why show up technically?

Other than doing me a favor because talking to an empty room would be very boring.

Okay, so this is where we were at.

We were talking about, if I turn this on, it will actually work.

Yes.

About utility functions and how to arrive at utility functions.

The usual way we try to determine utilities at least among human beings is by asking them

about their preferences and comparing them.

So we have a, we've talked about lotteries, which is just the economist's way of representing

random variables in such a way that we can like ask people about their preferences with

respect to various states and rank them based on probabilities in lotteries and so on and

so forth.

We've talked about the axioms of rationality for preferences.

So we call a preference, a set of preferences rational if they observe all of these axioms.

If we violate those axioms, then in theory, if we have a person that adheres to those

preferences, we can trick them out of money if they are willing to actually bet money

on their preferences.

And we've seen via Ramsey's theorem that once we have a rational set of preferences, we

can derive a utility function from that.

Notably, the utility function we obtain by applying that theorem is not unique, which

makes sense because if we have an agent that acts in such a manner that expected utility

is maximized, then it turns out that the behavior of such an agent is invariant under

utility functions that are scaled by positive linear transformations, i.e. in that sense,

the precise nature of the utility function doesn't even matter much.

The only thing that matters is the ranking of the various choices and the scale by which

they are ranked basically.

So how much do we prefer some outcome over some other outcome in relation, which means

we can just scale things up by positive linear transformations and we obtain the exact same

behavior with respect to an agent that adheres to such a utility function.

If we don't care about the scales at all, if we really just care about the ordering,

then we call a utility function a value function.

Not that that is technically a property of the function, it's more a property of what

we want to get out of that function or what we want to use it for.

So which numbers do we want to use?

Well we have a couple of choices.

We can construct a utility function from a set of rational preferences by basically constructing

a normalized utility function with values from 0 to 1, which we do by picking the lottery

based on a best possible price, i.e. basically a maximally utilitarian state and a maximally

bad utilitarian state and then scaling up probabilities until we find a indifferent

state for any possible, sorry, the other way around.

So we scale lotteries until we find the probability at which any given state happens to be indifferent

to the lottery where we get either the best possible price or the worst possible price

that gives us some probability and we can just use that probability as a utility value

that gives us a normalized utility from 0 to 1.

That is one way we can do things.

We can also use money, which as we talked about is not necessarily a good idea.

Other popular units are micromorts where a micromort is a chance of one in a million