I have seen both wordings ("game" singular or plural) in the literature sometimes with a dash between "game" and "based" and sometimes without. I was wondering why there is not a single formulation used by everyone.
From my experience, game-based learning is a bit more commonly used, at least in the US. Lack of consistency in terminology can sometimes be so frustrating. Perhaps as it becomes more common practice, we will see a chosen upon terminology.
A simple search in Google gave me more than a million of results when I looked for "game-based learning". By contrast, "game-base learning" gave me only 18,000 results and suggested to search for "game-based learning". In my opinion, the correct term is the first one, the other could be an erroneous variation... at least when we are talking scientifically.
I used to use it when teaching young learners as part of the teaching process, mainly at the end of a learning event, games helps to make sure that they can use the language taught within the topic or the lesson.
Regarding the term, I will go for the 'game-based learning'.
I had tried doing a search on google for both terms, unfortunately google doesn't care for plural so "games" also returns "game", hence a higher number of hits.
I'm doing a PhD in UK, is the British grammar different than the American one on that regard? (I am not a native speaker)
From my side, I don't think difference between American and British Englishes matters here. I believe when we title something we intend to make it single, that's only- although I am also not native.
I intend to use the word 'Englishes'- some people may not be happy about it!