Dynamics in Human-AI interaction should lead to more satisfying and engaging collaboration. Key open questions are how to design such interactions and the role personal goals and expectations play. We developed three AI partners of varying initiative (leader, follower, shifting) in a collaborative game called Geometry Friends. We conducted a within-subjects experiment with 60 participants to assess personal AI partner preference and performance satisfaction as well as perceived warmth and competence of AI partners. Results show that AI partners following human initiative are perceived as warmer and more collaborative. However, some participants preferred AI leaders for their independence and speed, despite being seen as less friendly. This suggests that assigning a leadership role to the AI partner may be suitable for time-sensitive scenarios. We identify design factors for developing collaborative AI agents with varying levels of initiative to create more effective human-AI teams that consider context and individual preference.