No matter the metric used, it’d be gamed by someone else through fake accounts, fake posts or whatever else. Sure, the metrics may not be published, but there isn’t that many different variables that could be used.
Additionally, as Sam said, compete with yourself. Be it a weekly, monthly, or 6month basis, try to grow faster than the previous chunk. Not only will you strive for something, but you’ll actually gain something from ‘winning’.
Finally, to state that its in people’s nature to compete may be a bit presumptuous. Its quite possible that its simply social constructs that are incredibly pervasive.