A Unified General Formula for Arbitrary Liquidity Operations in Weighted AMMs: Potential Applications to Intelligent Transportation Systems
Intelligent transportation systems increasingly rely on decentralized mechanisms to allocate limited resources such as freight capacity, warehouse availability, charging infrastructure, and network bandwidth. Efficient allocation requires pricing mechanisms that adapt dynamically to demand while preserving system stability. This paper investigates weighted constant-function market makers as a decentralized resource allocation mechanism for intelligent transportation systems, adapting the weighted invariant from Balancer-type automated market makers to model a generalized formulation over multiple tokenized resources. The standard literature documents exactly four resource allocation operations: proportional contribution, proportional withdrawal, single-resource contribution, and single-resource withdrawal, each obtained via separate derivations. This paper presents a single closed-form formula that unifies all four cases and extends them to two previously undocumented operations: partial-proportional contributions and fully non-proportional operations. The unified formula reveals that the conservation invariant and the allocation formula are structurally identical; the invariant itself is the general allocation formula. We prove two swap-decomposition theorems showing that, in a fee-less environment, any non-proportional operation is equivalent to an internal rebalancing swap combined with a proportional operation. Both theorems generalize previous propositions from single-resource to arbitrary multi-resource operations. The proposed framework provides a mathematically grounded mechanism for decentralized market-based coordination in transportation networks.