COMPETITIONKyou begins to have a very competitive personality, stemming directly from the fact that he is the year of the cat. For his whole life, he was put down and told he was a nuisance pretty much. All of his hate and anger towards the world built up, and he decided he was going to prove people wrong. When he learned that the rat was the main cause of his pain, he was determined to win. Akito made a bet that Kyou wouldn't be able to defeat Yuki in anything by the time high school ended. She said if she won, he would have to calmly go into the cats chambers for the rest of his life, but if Kyou defeated Yuki, he could be free and accepted into the blood-bond. Kyou wanted freedom and respect more than anything from the Sohma family, so he accepted the challenge. Kyou leaves for six months to train in the mountains with Kazuma in order to master martial arts. When he comes back, he feels determined that he can win. However, Yuki is still too strong, and he loses. Kyou gets angry because he realizes that even if he trains and trains and trains, he will still never live up Yuki's strength and power. We also see Kyou try to defeat him in other things like races at school, cards, table tennis, and other things in school. He also gets competitive when they are discussing what the school play should be, and people pick Yuki's option over his. The only thing in live that wasn't a competition was Tohru. Tohru was always going to love and pick Kyou over Yuki, and Kyou never acted out when it came to her. I don't think Kyou felt threatened by Yuki in this regard because Kyou finally found someone to cherish and someone who cherished him back.
Another example of how Kyou is competitive is with his classmate, and Tohru's friend, Uotoni, who is basically the female version of Kyou. The two are constantly challenging each other to duals in cards, respect, and other games in school. When they were playing a card game, Poor Man together, Uotoni states "Nevermind I was going to say that you're stupid and clumsy. I thought maybe I could make some easy money on you," which pumps Kyou up to defeat her as many times as it takes until she cries! I think that Uotoni challenging him is actually a good thing for his people-skills because he learns how to take his competitive anger out in a more friendly and fun manner. He doesn't actually get super angry if he loses, but it makes him willing to try again and again. Competition doesn't always have to look ugly, sometimes it can be fun and rewarding. This type of competition is healthy because the two are able to become friendly and gain respect for one another. Uotoni likes to see Kyou get worked up because it shows her that he has some spunk and fighting spirit in him. She comments that Kyoko, Tohru's mom, would have loved to meet Kyou and tease him because of this. Uotoni's opinion of Kyou is important because she is one of Tohru's best friends, and more than that, she is protective of her just like a mother.
Hatsuharu is another member of the zodiac family that Kyou gets in fights with. However, this is only when he turns into Black Haru, or the evil side of him. Kyou is often able to win fights against Hatsuharu because we see in a flash back that Kyou was the one who taught him marital arts as a child. Therefore, Kyou has the upper hand. Kyou also teases him a lot when he turns back to White Haru and says he is still going to beat him up. Haru often pleads with Kyou to take it easy on him afterwards. This type of competition seems a bit more brotherly to me. I feel as though Kyou is the older brother who puts Haru back into place when he acts up at home!
Regardless of who it's with, Kyou needs competition in his life to make him feel worthy, or respected. He tries really hard to be the best at something because he knows deep down that his life isn't pointless or a burden, and he wants to find something that is unique to him. Competition for Kyou started off pretty unhealthy, and he used it to blame others for his pain and suffering. However, as he met people like Haru and Uotoni, I believe that he was able to kind of use that competition in a healthier way, allowing him to mess around and have fun instead of just doing it to win.