Collaboration is an integral part of human dialogue. Typical task-oriented dialogue games assign asymmetric roles to the participants, which limits their ability to elicit naturalistic role-taking in collaboration and its negotiation. We present a novel and simple online setup that favors balanced collaboration: a two-player 2D object placement game in which the players must negotiate the goal state themselves. We show empirically that human players exhibit a variety of role distributions, and that balanced collaboration improves task performance. We also present an LLM-based baseline agent which demonstrates that automatic playing of our game is an interesting challenge for artificial systems.