An entry-level characterization of some types of guy in decision theory, and in real life, interspersed with short stories about them

A concave function bends down. A convex function bends up. A linear function does neither.

Decisionmaking agents have utility functions. A utility function is just a mapping from conditions to numbers that say how good different outcomes are, according to the preferences of the agent.

Usually, a utility function assigns scores to outcomes or histories, but in this article we'll define a sort of utility function that takes the quantity of resources that the agent has control over, and the utility function says how good an outcome the agent could attain using that quantity of resources.

In that sense, a concave agent values resources less the more that it has, eventually barely wanting more resources at all, while a convex agent wants more resources the more it has. But that's a rough and incomplete understanding, and I'm not sure this turns out to be a meaningful claim without talking about expected values, so let's continue.

Humans generally have mostly concave utility functions in this sense. Money is more important to someone who has less of it.

Concavity manifests as a reduced appetite for variance in payouts, which is to say, concavity is risk-aversion. This is not just a fact about concave and convex agents, it's a definition of the distinction between them:

*Any* agent that always raises its expected utility by increasing how wildly its fates can vary will be convex, and vice versa.

Humans' concavity is probably the reason we have a fondness for policies that support more even distributions of wealth. If humans instead had convex utility functions, we would prefer policies that actively encourage the concentration of wealth for its own sake. We would play strange, grim games where we gather together, put all of our money into a pot, and select a random person among ourselves who shall alone receive all of everyone's money. Oh, we do something like that sometimes, it's called a lottery, but from what I can gather, we spend ten times more on welfare (redistribution) than we do on lottery tickets (concentration). But, huh, only ten times as much?!^[1] And you could go on to argue that Society is lottery-shaped in general, but I think that's an incidental result of wealth inevitably being applicable to getting more wealth, rather ...

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