The present paper proposes a collaborative control strategy for CAV movement in a lane-free environment, where CAVs have different priority values. The priority values are designed to introduce fairness, relying on the interaction history of every CAV. These priority values are the output of the Stationary Nash Equilibrium (SNE) of a modified Karma game. The modification is proposed to tackle a proportional resource allocation problem. The outcome of the modified Karma game is then used to determine the priority of CAVs in their interactions with the other vehicles in their neighborhood, which may constitute potential threats, so-called Threatening Vehicle Cluster (TVC). An MPC is designed at an upper level to enforce an overall fair collaboration among CAVs so as to avoid collisions. Finally, a simulation case study analyzes the proposed control approach regarding performance, efficiency, and fairness.