To borrow James Carse’s terms and combine with the OKIC metaphor, “finite game” refers to the navigation of particular gameplay.
I would further discern that orientation to particular gameplay, while generally infinite in the sense of repeated iterations, can also be timeless.
With chess a child playing for the first time can know nothing actual of navigating the nexus of play surrounding the game. The child learns the board and pieces, and meaningful play commences. This points to chess being a game that also exists in a timeless, explanatory-free and real, integrated context.
From the Krivov paper cited, Krivov, Sergei V. "Optimal dimensionality reduction of complex dynamics: the chess game as diffusion on a free-energy landscape." Physical Review E 84.1 (2011): 011135. https://arxiv.org/abs/1103.3681v1
The [optimal dimensionality] approach suggests a possible way of obtaining a simple yet accurate description of many important complex phenomena. The analysis of the chess game shows that the approach can quantitatively describe the dynamics of processes where human decision-making plays a central role, e.g., financial and social dynamics.
Applied more broadly to any game, I would characterize “optimal dimensionality” as the process of navigating while also accurately orientated.
Further, timelessness in games is an additional orientation feature of referring, as it were, to spatio-temporally generalizable phenomena, such that continuity is maintained to both finite and infinite (referring to a particular model, game, or agent) phenomena.