The number of Stigmas you earn per antagonist match is determined by your score. If you lose, you don't receive stigmas. If you're above 20 stigmas, you start losing stigmas if you fail.
The number of stigmas you possess determines your antagonist rank. The rank requirements change based on the leaderboards at the time. Winning a game nets you around 1-5 stigmas depending on how well you did. And to reach rank 10 requires about 10 stigmas.