A. Initial test:
A1. Use tracert to your game server to obtain list of hops from your computer to game server.
A2. Based on the list, constant ping the nearest computer to see whether the connection has intermittent outages and evaluate the result.
A3. If there were no outages, repeat step A2 with next server (one hop further).
A4. If there were these outages you described in the question, then focus to Athat connection between last server without problems and this server. Based on their role, you can understand what to diagnose (or to whom report your problems).
B. Follow up:
B1. Try from another computer using the same connection (e.g. in your home). This will show whether the problem is directly in your computer or if it is in your home connection.
B2. If the problem appears to be in your home connection, try connecting without your router.
B3. Test your computer with different internet providers (e.g. use mobile connection or take the computer to friend's home who has different internet provider). Write down the tracert output made there so you can compare it to tracert output you obtained in step A1. This will show you whether the problem is at your home internet provider or not.
Edit:
As you found outages in A-steps at 4th hop, please check with owner of 3rd server why there is problem pinging 4th server and give them your findings. (Including evidence that 3rd server has no outages, but 4th server has them.) If you don't know who is the owner of 3rd server, you can check using online Whois services (there are more of them). I think it is quite likely your internet provider. Then raise a technical support request with them.
Of course, if there is minimum of ping outages between 3rd and 4th hop, but there is much more for example between 5th and 6th hop, then the main problem you want to address is there. The investigation based on these statistics is on you. Now you should understand the approach so you can determine the weakest point by yourself and contact the support about it.
If you compare tracert output made when using your mobile connection with tracert made at A1, you should be able to identify common servers at the bottom of the list which never fail. Because you have no failures over mobile connection.
Edit 2:
Bing pinged without problem but Google has problems? Check the tracert. Are there any servers in the route which are present when you are pinging Google, but which are not in the route when you are pinging Bing? This is unlikely, but if you can find the evidence, submit it to your ISP. You need to rule out going hop-by-hop which servers on the route work rock-solid and which have issues. Then follow up addressing that part of the infrastructure. Double check there are no other factors involved like weather, which can do outages on radio networks.