speaker-0: Yeah, so classically, Bayesian decision theory says you have your belief function, your utility function, you take them, you glue them together and you take the expectation over that. So you say, what is the maximum expected utility I can achieve? And I think that's fine for ⁓ decision makers who are risk neutral. If you think losses and wins are equally valuable,

speaker-1: ⁓

speaker-0: or that you're comfortable with uncertainty so long as that on average things will work out, then that's the right decision-making framework for you. And there's a kind of a robust philosophical dispute about whether risk neutrality is like uniquely best position. But whatever the answer to that question is, people in practice are risk-averse and businesses might have reasons to be risk-averse.

speaker-1: ⁓ ⁓

speaker-0: that might be kind of beyond your control. You can build in all kinds of constraints and

speaker-1: ⁓

speaker-0: ⁓ special modifications to your function. And I think this illustrates this kind of nice division of labor is that ⁓ optimal decision making is a complex process and that you need to kind of cognitively separate it from the modeling part to be able to keep track of everything. So there's a whole family of different ways of handling risks. If we took a look at what the objective function looks like, from the point of view of our uncertainty. You realize that at any given price, there's a wide range of possible sales levels you could achieve. And for those listening, visualize a nice...

speaker-1: ⁓ And

speaker-0: ⁓ inverted convex function that has a big cone of uncertainty around it. You can see that at really small prices, the uncertainty shrinks because you're kind of moving towards that more degenerate case where ⁓ you know your sales are gonna move to infinity. People are gonna be buying your product like hotcakes because it's getting closer and closer to being free. And there's like less complexity in the behavior. Whereas on the other side, as you move to higher and higher prices, there's a huge difference between if one person buys it or if two people buy it. And as a result, you get a much wider level of uncertainty in the kind of profit you might expect to achieve. So the different prices are asymmetrical with respect to uncertainty. And you might want to be on the safe side for some reason. One way to be on the safe side is to have any utility function that diminishes at higher and higher profits. like this is kind long-standing idea in social sciences that people who are really rich aren't that much more happy than people who are a little bit rich. Once you're kind of past the half million threshold, there's not that much more to be gained. ⁓ in happiness. And that's the kind of idea with these exponential utility functions. This is a very popular technique from economics to model diminishing returns in utility for money. So there's a whole family of them, but all of them have the same behaviors that they saturate as you get more and more money. instead of just taking your profit, sorry, you're taking your profit and passing it and calling that the same thing as utility. you can add an extra layer in between your profit and your utility, which is some function that modifies that. And if you pass in this exponential utility, if you insert that into the computational graph, ⁓ you can get a different utility function.

speaker-1: ⁓ And

speaker-0: and it gives you a kind of adjustable parameter for risk aversion. The different functional shapes of the ⁓ exponential utility correspond to different levels of risk aversion. And you can see that when you have no risk aversion, it still recommends a price of about $4. But as you slowly increase this function, this risk aversion, the recommended price drops. Yeah, because all of this stuff in the tail of the posterior that has really high profits but very high uncertainty, it counts less and less for you. So the function isn't attracted to anything with a big tail, tends to prioritize small tailed regions. So yeah, this is one common and powerful approach to representing risk aversion. But we also could explore a couple more if we're feeling in the mood for it.