I try to add a general consideration: when is it useful to write in blockchain and when is it useful to solve locally on the off chain server?
The answer is simple: every time the results of your elaboration has a significant economic value that could be possibly reverted and/or nullified or damaged by someone’s action in the future, you should definitively use blockchain.
The same if they should be the starting point for further high value elaboration in the future.
On the contrary, if your transaction is closed here and now, cannot be reverted or nullified or damaged by nature, you can still evaluate to use blockchain in the case you could need to maintain some track of the process to be checked in the future for any true reason.
If not, go off chain immediately.
In the case of your game, if you summarize the effects of all the game actions in the final result and the true economic value of the single action is negligible, you can go off chain.
On the other hand if the value of the game results could be huge, it can be useful to being able to show them all the single actions which bring the game to the result in order to nullify eventual claims.
So it is something that must be evaluated for the single case, does not exist any yes/no automatic rule!
This apply to the data you could plan to compress and send to blockchain, of course.