We're picking randomly from 54^8 possible game ids with 500 millions already taken.
@cyanfish gave a nice and simple way to estimate the number of collisions in a month (given that the games played in the month are only a small fraction of the games that have ever been played): Just calculate number of games per month * 500 million / 54^8.
With 40 million games per month, that means we're expecting to see well over 200 collisions per month, now.
Possible solutions:
Any reason not to use UUIDs going forward?
Game ids are part of the game urls, so UUIDs would make the urls unnecessarily long.
Another possibility would be to use an increasing numeric value. This presupposes a centralized database that all clients can query to determine the next game id.
You could as well have an id with a time reference. For instance <yyMMddhhmm><random-string>, so the 12/26/2018 at 13:18:29 you would have _181226131829azEB_. There is a similar implementation in MongoDB's ObjectID.
The point is you have fewer collisions, and your ID has more pieces of information.