In this post, I give a application of geometric rationality to a toy version of a real problem.

A Conflicted Agent

Let's say you are an agent with two partially conflicting goals. Part of you wants to play a video game, and part of you wants to save the world and tile the multiverse with computronium shaped exactly the way you like it (not paperclips). How should these conflicting interests figure out what to do? (Assume you have not yet had the idea of starting a video game company to save the world.)

We will assume that $2 / 3$ of you wants to save the world, and $1 / 3$ of you wants to play video games.

At first your world-saving-self has the bright idea that 2/3 is bigger that 1/3, and you should therefore devote all your time to saving the world. However, this proposal doesn't stick. Eventually, you end up Nash bargaining with your time, and devoting 2/3 of your time to saving the world and 1/3 of your time to playing video games. This works well for a while, but then your world-saving-self has a new bright idea:

"Let's look around at the world, and see how much ability we expect to have to save it. If it feels like we are in the top 60 percentile of worlds ordered by how much control we have, then we will try to save the world. If we are in the bottom 40 percentile, we will play video games!"

(The 60 percentile is arbitrarily rounding down from 2/3, so that you can both play more video games and save more worlds.)

A Nash Bargaining Model

Let's model this more carefully. Let's say there are five different types of worlds: 1, 2, 3, 4, and 5. In each world, you have two buttons in front of you. The video game button, and the save the world button. Each time step, pressing the video game button lets you play some video games, and pressing the save the world button saves the world with probability $ε \cdot i$ in the world of type $i$ .

We have five degrees of freedom. For each $i \in {1, 2, 3, 4, 5}$ , we have $p_{i}$ , which represents the proportion of our time in world i that we spend pressing the save the world button. The rest of the time is spent playing video games.

The part of you that wants to save the world has power $\frac{2}{3}$ , and utility equal to $10^{10^{100}} \cdot ε \cdot (p_{1} + 2 p_{2} + 3 p_{3} + 4 p_{4} + 5 p_{5})$ . However, since we are Nash bargaining, the coefficient out in front does not matter.

The part of you that wants to play video games has power $\frac{1}{3}$ and util...