r/simulationtheory https://www.reddit.com/r/SimulationTheory/comments/1tuoifz/duality_theory_concurrent_running_lives_based_on/
A reward function is the basis for building a reinforcement-learning-based agent. One way to optimize the reward function is through constrained optimization. In this setting, the agent is living in the "primal" space, and let's say we want to maximize its rewards, or it (the agent) wants to go for the best reward. The duality principle says that there is a "dual" space where one can search. By minimizing the reward in the dual space, you can effectively achieve the same goal. Sometimes, it is even easier to go for minimization in the dual space.

Label Primal as "+" and Dual as "-". Pure labeling, no meaning.

Simply put, there is Life 1+ and Life 1-, connected by 2+, 2-, and so on. Rewards can be positive or negative in either space/Life.

Standalone Example:

Let's say we are at Life i. A positive reward r posted in Life i+ will be posted as a negative reward -r in Life i-. The posting mechanism has no delay. You can imagine that if you suddenly get promoted in Life i+, you are getting terminated in Life i-. One may ask: I got a raise of only $10000, so why was I terminated (a once-and-for-all action instead of a demotion) in the dual space? It is because reward is calculated across all the horizons, as an expected value over the whole life. A promotion action of $10000, if added up over the whole life with interest, is equivalent to the negative of losing a job instantaneously.

Multi-agent example: In Life i+, you are interacting with another individual. If they forcibly steal x amount of goods from you in the plus world, you are stealing, or taking an equivalent action involving, -x amount of goods from them. You may encounter something odd every day; for example, someone may just come up to me and punch me in the face, causing a huge negative reward. This seems random and out of place. But by duality theory, this mysterious action might be driven by the fact that I was punching them in the dual space.

Compared to multiverse theory, there are two differences: 1. There are only two worlds, not many of them. 2. One is the exact opposite of the other in terms of rewards, not actions (as illustrated in the standlone example. So this duality theory is a significant weaker version of multivere theory.

TLDR: There is a concurrent life you are running on. If you suck in one of them, you rock in another one.

Duality Theory: Concurrent Running Lives from the Lagrangian Principle