Abstract: The possibility of `tacit collusion', in which interactions across  market making algorithms lead to an outcome similar to collusion among market makers, has increasingly received regulatory scrutiny.  We model the interaction of  market makers in a dealer market as a stochastic differential game of intensity control with partial information and study the resulting dynamics of bid-ask spreads. Competition among dealers is modeled as a Nash equilibrium, which we characterise  in terms of a system of coupled Hamilton-Jacobi-Bellman (HJB) equations, while   Pareto optima correspond  to collusion.  Using a decentralized multi-agent deep reinforcement learning algorithm to model how competing market makers   learn  to adjust their  quotes, we show  how the interaction of market making algorithms may lead to  tacit collusion  with spread levels strictly above the competitive equilibrium level, without any explicit sharing of information. Joint work with Wei Xiong (Oxford)