In the Odyssey, Athena says to Telemachus:

It’s true few men are like their fathers. Most of them are worse. Only very few of them are better.

Athena was pretty sharp! Of course, “better” and “worse” are rather vague terms, but we can talk more precisely if we consider reproductive fitness, defined as the number of offspring one has. Then we can restate Athena’s theorem as:

The expected fitness of a random organism is ≤ the expected fitness of its parent.

Intuitively, one can reason as follows: the expected fitness of a random organism is the average fitness of the population. But the expected fitness of its parent is above average, because we already know that the parent has at least one child.

A rigorous proof is left as an exercise to the reader.

Zhan Shen, a fellow student, mentioned this really cool game last week in his practice thesis defense on insurance theory. Zhan talked a bit about applications of an asymmetric version of the game (with different army sizes). His defense is today.

I’m working on a paper on this topic, but I figured I would post about it now and clear up some confusion.

Let’s say we have an agent interacting with an environment. The agent sends the environment some number (an action), and the environment sends the agent some number in response (a perception).

When we have an infinite set of hypotheses, the expected utility of some action is determined by an infinite series. It’s important for that series to converge, or else we won’t (in general) be able to compare the expected utilities of two different actions.

Given the following assumptions:

- We have some finite set of data about the environment.
- We assign a nonzero probability to all computable hypotheses consistent with our data.
- Either this probability distribution is computable, or it is bounded below by some positive computable function.
- Our utility function is determined by our perceptions, and is
*unbounded*.
- Either this utility function is computable, or it is bounded below in absolute value by some unbounded computable function.

…then it follows that our expected utilities *do not converge.*

This seems like a pretty strong reason to stick with bounded utility functions. It certainly seems much more plausible to me now than it did before that my utility function is bounded (to the extent that I can be said to have a utility function).

But we shouldn’t jump to conclusions. Here are some reasons why the above might not apply:

- You might have an uncomputable utility function.
- You might have a utility function which is not determined solely by your perceptions.
- You might assign a probability of zero to some hypotheses which are consistent with your data.