There could be a mode that users opt into where any time you pass someone you're not doing an activity with, you are awarded a Greyhound badge. The person that gets passed would receive a Rabbit demerit. If the Rabbit doesn't achieve a new PR on a segment in the next few days, the demerit stays on the account.
This could be a fun way to encourage a friendly rivalry with strangers. It could be a fun motivator to see if you can catch up to the next person you see, or encourage you to keep up the pace if someone is coming up behind you. You never know who else might be on Strava. Even if you do get passed, the motivation is there to do your best on your next activity. If user names are shared when someone passes you, or you pass someone else, it could be an introduction to people you share routes with.