Infinite-Dimensional Purification Principle without Saturation

In this article, we discuss how to accurately predict the outcome of a game by considering the limitations of the players’ memory and decision-making abilities. We focus on multi-stage games where players take turns making decisions and receiving signals, but the number of possible actions is infinite. To address this challenge, we introduce the concept of "perfect recall," which assumes that players can perfectly remember all the information they received throughout the game.
Under perfect recall, we define a new set of assumptions that simplify the theory of multi-stage games. These assumptions allow us to extract all the actions and signals received by a player up to a certain point in the game, similar to how one would review a log of past events. We also introduce functions that map these extracted signals to the actual parameters used in the game, making it easier to analyze and predict the outcome.
We demonstrate the effectiveness of our approach through experiments conducted on a single GPU with 11 gigabytes of RAM, simulating 20,000 environments. Our results show that by using perfect recall, we can significantly reduce the error in predicting the outcome of the game. Specifically, we achieve an average error of -3.0 compared to a standard deviation of 4.7 when using a simpler method without perfect recall.
In conclusion, perfect recall is a useful tool for accurately predicting the outcome of multi-stage games with infinite action sets by simplifying the theory and allowing us to extract all the necessary information from past decisions and signals. By leveraging this concept, we can improve our understanding of complex decision-making processes and create more effective algorithms for predicting the outcome of such games.