| Abstract: |
| This paper investigates a one-dimensional Mean-Field Planning (MFP) system characterized by non-local, rank-based interactions, where individual costs depend on an agent`s relative standing within the population distribution. Using a potential formulation, we reduce the coupled system to a scalar partial differential equation and establish a rigorous equivalence between classical solutions of the ranking MFP and the associated potential problem, including explicit reconstruction formulas. By identifying a monotonicity structure within the associated operator, we prove the uniqueness of classical solutions under strict convexity assumptions. Furthermore, we address the existence of solutions in low-regularity regimes by formulating a relaxed variational inequality. Using a q-Laplacian regularization and Minty`s method, we establish the existence of weak solutions in the space of functions of bounded variation (BV). These results provide a mathematical framework for deterministic first-order planning problems with cumulative distribution couplings, such as those arising in competitive models of emission regulation. |
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