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).