Impact statement: This work focuses on better understanding of mathematical properties of real world games and how they could be used to understand successful AI techniques that were developed in the past. Since we focus on retrospective analysis of a mathematical phenomenon, on exposing an existing structure, and deepening our understanding of the world, we do not see any direct risks it entails. Introduced notions and insights could be used to build better, more engaging AI agents for people to play with in real world games (e.g. AIs that grow with the player, matching their strengths and weaknesses). In a broader spectrum, some of the insights could be used for designing and implementing new games, that humans would fine enjoyable though challenges they pose. In particular it could be a viewed as a model for measuring how much notion of progress the game consists of. However, we acknowledge that methods enabling improved analysis of games may be used for designing products with potentially negative consequences (e.g., games that are highly addictive) rather than positive (e.g., games that are enjoyable and mentally developing).