Noody's resolve to maintain his status as "Andy's Favourite Toy" is unravelled throughout the course of the story, until by the end he concedes that status to Buzz. At the moment of greatest crisis (right before the rocket explodes), Woody lets someone else be in charge for once, allowing himself to be literally taken under Buzz's wings. From the experience of his separation from Andy, Woody comes to believe his own words, "It doesn't matter how much we're played with-what matters is that we're here for Andy when he needs us." In the final scene we see Woody loosened up and dancing, satisfied to be part of the group rather than its leader; he's more comfortable with himself and chummier to Buzz.
Buzz Lightyear appears to make a change when he comes to see himself as Andy's Toy rather than a Space Ranger. His presence forces Woody to confront his personal issues, and that impact remains constant until Woody's own "change" resolves the inequity (unfairness) between them.
Woody needs to stop feeling entitled to sole possession of the "spot" on Andy's bed. He needs to stop being insecure, competitive, and jealous. He needs to stop measuring himself in terms of "playtime." If he would stop all these things, he could relax and accept a new state of affairs which is out of his control anyway.
Woody is a very active, take-charge kind of cowboy. He calls meetings, plots strategies, mobilizes other toys, and approaches all problems by jumping into the fray, even if it means starting an unpleasant confrontation. .
Woody doesn't think of the consequences "Hmm, if I cause Buzz to disappear behind the desk, Andy will have to pick me to go to Pizza Planet, and I'll win his favour back." He then attempts a very simple operation to use RC Car to push Buzz off the desk, but fails to see the relationships among the objects on the desk that will make his plan go wrong.