Would you say it’s possible to define the line between giving more options and just adding more stuff? An extreme example, let’s say a bigger weapon loadout contributing to the same combat system versus a jetskiing minigame in a shooter.
A minigame is segmented off from the rest of the game. It cannot interact with any of the other elements, therefore it cannot multiply or exponentiate the number of game states, only add. Think about how many matchups a fighting game can have, based on the number of characters. If you have 10, then there’s 45 different matchups. If you add 2 more, then that’s 66 different matchups (12 characters). 2 more on that is 91 matchups total (14 characters). Adding even a small number of new options can vastly increase the state size of the game. And the state size doesn’t merely get bigger additively, it has a rate of increase that is similar to an exponent, because you’re increasing the number of combinations. So as you add more elements, the state size increases drastically more depending on the number of existing elements.
This is why Go is so much more complex than Chess, you’re allowed to pick many more things on any given turn. There are many many more ways the stones can be combined.
Imagine if you made a fighting game with 10 characters, then you added 4, but those 4 new characters can only fight each other. 10 characters is 45 matchups, 4 characters is 6 matchups. Together that makes 51 matchups, which is significantly less than the 91 you could have if you integrated the cast together.
This isn’t a perfect example, you could nitpick it by examining whether each of these matchups themselves is deep (which for evaluating the game’s quality would be more important than just the sheer number of matchups, because you can consider each matchup segmented off from the others in much the same way as the segmented off mini-game is). The point is to make the math behind state size a little more concrete.
Given the way the number of combinations is related to state size, we can infer that an increase in depth translates to a perceptual increase in quality across an exponential, or logarithmic scale, similar to the way decibels are measured, rather than a strictly linear scale. Of course, this is theoretical and measuring the depth of a game precisely, I don’t know if it’s really possible except for extremely simple examples like tic tac toe. Especially because state size is not the only determining factor for depth, but also redundancy, and relevancy to the playerbase (in terms of their skillset and knowledge of the game).
Because redundancy and relevancy are also a factor, things that are pure increases to the state size of a game can ironically decrease the depth of the game, because the number of relevant states might decrease because of how new elements affect the existing elements, and the new states introduced might just be rehashes of the existing states. I have my 4 criteria/rules-of-thumb for depth to prevent this.