tldr journalctl

man journalctl

If you launch a process from CLI, you can redirect its standard output (stdout) and standard error (stderr) to files. Errors messages might be there.

foo >> /tmp/gamelog.out 2>> /tmp/gamelog.err  

Maybe the application is already saving logs or crashdumps to disk. If the game has just crashed, you can search for files modified during the last minute: find $HOME /var/log -mmin -1

That might turn up some clues on where to look.

If you have to dig deeper, look into strace (premium subscribers only).

i doubt they help, but sometimes core dump files are created when programs terminate abnormally

I think Mint uses systemd, so after rebooting, you should be able to review the system log from the previous boot with this command:

journalctl --boot=-1

The kernel might still be running even when the display is frozen. If that's the case, you can try rebooting more gently than a forced power-off would do. While holding down the Alt key, type these keys in order, with a couple seconds between each tap: SysRq R E I S U B

https://en.wikipedia.org/wiki/Magic_SysRq_key

Note: SysRq is disabled by default on some distros, and unsupported by some keyboard controllers. You can check (when the system is not frozen) by holding Alt and typing SysRq H while watching the output of sudo dmesg --follow. If it's working, you should see a help message.

For what it's worth, what you're describing sounds like it might be a GPU driver misbehaving, perhaps in response to power saving events. If you have an Nvidia card, you might try a different driver version. Otherwise, you might try an older or newer kernel version.