This paper analyzes a class of MFGs with singular controls motivated from the partially reversible problem. It establishes the existence of the solution when controls are of bounded velocity, solves explicitly the game when controls are of finite variation, and presents sensitivity analysis to compare the single-player game with the MFG. Our analysis shows that MFGs, when appropriately formulated, can demonstrate genuine game effects even without heterogeneity among players and additional common noise.
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