We present an approach to ensure safe and deadlock-free navigation for decentralized multi-robot systems operating in constrained environments, including doorways and intersections. Although many solutions have been proposed that ensure safety and resolve deadlocks, optimally preventing deadlocks in a minimally invasive and decentralized fashion remains an open problem. We first formalize the objective as a non-cooperative, non-communicative, partially observable multi-robot navigation problem in constrained spaces with multiple conflicting agents, which we term as social mini-games. Formally, we solve a discrete-time optimal receding horizon control problem leveraging control barrier functions for safe long-horizon planning. Our approach to ensuring liveness rests on the insight that \textit{there exists barrier certificates that allow each robot to preemptively perturb their state in a minimally-invasive fashion onto liveness sets i.e. states where robots are deadlock-free}. We evaluate our approach in simulation as well on physical robots using F$1/10$ robots, a Clearpath Jackal, as well as a Boston Dynamics Spot in a doorway, hallway, and corridor intersection scenario. Compared to both fully decentralized and centralized approaches with and without deadlock resolution capabilities, we demonstrate that our approach results in safer, more efficient, and smoother navigation, based on a comprehensive set of metrics including success rate, collision rate, stop time, change in velocity, path deviation, time-to-goal, and flow rate.

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