GNS theory is, or was, a popular theoretical tool for the examination of tabletop RPG game design, though it was mostly fixed upon the player, rather than the game itself. Ron Edwards, and the forum that the theory came from, have a bad reputation in some circles, but regardless of these issues I have found the theory structure useful in the past while working on games and judging the design in terms of something to aim for.
That isn’t to say that GNS doesn’t also have some issues and I’ve been rooting around for some time to improve it to aid with my own game design structure and I think I have hit on the missing element for a structure to examine and design games of most types. The idea of something being a ‘Toy’.
There would, then, be four elements to any game.
The Game factor is the element of skill to the activity. This can take many forms from honing reflexive skills to learning how to build characters with maximum damage output. The Game factor is the amount to which you can ‘git gud’ and improve your personal skill and interaction with the game. Game elements tend to conflict with Toy and Narrative elements.
Examples of high ‘game’, games would include Dark Souls, 3.5 Edition D&D (character builds), or Chess.
Narrative is the extent to which story constitutes and controls the activity. Narrative factors tend to conflict with Toy and Game elements. Narrative elements can include story heaviness (often limiting free exploration) and tends to create a more passive, less player oriented experience.
Examples of high ‘narrative’ games would include Choose Your Own Adventures, visual novels or Once Upon a Time.
Simulation is the extent to which an activity tries to faithfully replicate something. This does not necessarily mean realism, activities can attempt to replicate the tropes and conventions of unrealistic genres (superheroic comics or action films for example). Simulation tends to conflict with Narrative and can conflict with Game elements.
Examples of high ‘simulation’ games would include Basic Roleplaying, Microsoft Flight Simulator and SSI Wargames.
A Toy factor determines how much the activity is a ‘plaything’ and how much it is self directed. This is another way of expressing the degree of ‘sandbox’ and latitude a game might have. Toy elements tend to conflict with Narrative and Game elements as a sandbox must allow for latitude and escape from the directed narrative and the constriction of needing to ‘git gud’ gets in the way of freedom.
Examples of high ‘toy’ games would include Lego, Minecraft and Hexcrawl tabletop settings.
You could then plot a game’s position by measuring the tension between these various factors to measure it against other games.