Game Theory: Isomorphism and Theme Integration

This is the first post in my series of Game Theory topics. Today, I share my thoughts on isomorphism in game design. What is that? Read on to find out.

Recently, TC Petty III wrote a post about a new game he was developing about isolating the element Xenon.
The game looks like it is off to an interesting start, but there was something in the post that stuck with me. The post talks about the author’s dislike for games which add a rather bland theme to complement what is essentially a bland mechanic. Then I read this sentence:

“I’ve been challenging myself to design games that absolutely cannot be transposed into other themes without drastically altering the gameplay.”

This got me thinking about isomorphism. Isomorphism, in the mathematical sense, is the ability to map relationships between sets. Game design is typically interested in mapping the relationships between the elements of the theme to the relationships between game mechanics (excluding abstract games with no theme).

A mapping can be bad in three ways. The first way is when relationships between thematic elements are not preserved when applied to the mechanics. Most theme versions of Monopoly are very close to this, by simply renaming pieces without changing how the game works. The second way is when relationships between mechanics are only trivially related. Many role playing games fall into this category, by relying heavily on card text and imagery to drive the game and using a generic playing field and dice to add random elements.
The third bad mapping is more complex, in which some relationships are maintained, but some thematic elements seem forced to meet the mechanics, or some mechanics seem arbitrary to meet the theme. Games in the third category are more rare because this type of bad mapping typically makes something “seem” wrong.

There are different degrees of mapping that can occur in a game. The more the theme and mechanics reinforce each other, the more isomorphic they are, and the more “natural” the game feels. Agricola is often cited as a game that is very natural, because the farming mechanics (mostly) make sense with farming in real life. A game like Settlers of Catan contains much less mapping, because although wood comes from a forest and ore comes from a mountain, the elements are somewhat arbitrary, as shown by the existence of Star Trek: Catan.
Here is where the challenge of isomorphism comes in. Not only can ideas be mapped onto mechanics, but ideas can be mapped onto other ideas, and mechanics can be mapped onto other mechanics. (For instance, 36 Cards used in place of 2 six sided dice.) The downside of isomorphism is that for any given mechanic, there is more than one theme that can be applied, and vice versa. But, there is also a side of isomorphism that works for the designer. The more the theme and mechanic are isomorphic, the more the game begins to act like a model of the world. The more isomorphism there is between the two, the more detailed the model becomes, and the harder it is to separate them.

As a designer, any theme will help suggest mechanics. A theme that is isomorphic to the mechanic will make this step easier. Likewise, new mechanics will be easier to fit into the theme. It becomes easy to see how different elements will fit into that model. Due to the isomorphism, both the real world and the model will be affected very similarly. This is how I do a lot of my own design, creating mechanics and finding thematic elements and matching them up. All of this is not to say that games need a high degree of isomorphism between theme and mechanic, or any theme at all. Several highly rated and popular games are only lightly thematic, but are good games on the strength of their mechanics.

The goal of designing a game that cannot be separated from its theme is really just the goal of finding the perfect theme for the mechanics. In the end, isomorphism is just another element that should be considered when designing a game. When done right, it can make both designing and playing the game easier. It can make a game that is powerful in its flexibility or a game with a strong identity.


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  1. #1 by Thomas on August 21, 2016 - 11:41 pm

    Either something is isomorphic or it is not. There are no degrees of isomorphism. Mappings also cannot be mapped well or poorly, either elements from sets are mapped to each other or they are not.

    That being said, I think I mostly understood the point you were making, though your analogy with mathematical isomorphism was pretty bad.

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