O-Sensei felt that competition was incompatible with Aikido, but that does not mean that everyone agrees. Two aikido styles seem to have embraced the theory of competitions --- Tomiki and Shin-shin Toitsu.
Kenji Tomiki proposed "rationalizing" Aikido training using kata and competition during O-sensei's life. This effectively split the Tomiki and mainstream aikido schools. Somewhat ironically, the competition is not considered a fundamental part of the style; a few offshoots of Tomiki-ryu have abandoned the idea of competition.
Koichi Tohei (Shin-shin Toitsu) has started an International Taigi competition. Judging is based on fudoshin "mind-body unification", kankyu ritsudo "rhythm", and kiryoku jujitsu "ki extension".