Compositional game theory is a new mathematical formulation of game theory. It offers a new perspective on things we already know a lot about. So far, it has not produced new theorems on the foundations of game theory or the like.
Thus, from the view point of an econ theorist, it (currently) does not solve any problem. As a consequence, if your aim is to do publishable research based on game theory, you mostly likely will not find it interesting.
Why bother then?
When mapping a formal model into a concrete application, we need to provide details. Moreover, the correct model does not exist. There is uncertainty how to best capture a situation. We need a collage of different models and different specifications in order to get a robust reading of a situation.
Compositional game theory offers a novel way how to accomplish this map from theory to concrete application. It provides a programming language for representing and analyzing strategic interactions. You can think about the software implementation as a canvas in which you can instantiate models.
Turning modelling into programming has two consequences for the modelling process.
First, modularity: you can assemble components into larger models. Secondly, reuse: once you have models implemented you can reuse them in different ways (analogously to programming modules).
These features help to iterate through a range of scenarios quickly. And as for other software, it is easy to do that in teams. Thus, the benefit of the software behind compositional game theory is that it reduces development time. Which means modelling time.
To be clear: this is still mostly an hypothesis. We have seen some evidence for the speedup advantage in some use cases (non-standard auction formats and protocols in the blockchain sphere).
But these are just a handful of examples. We have more projects coming up and will report. If you are interested, drop me a mail!