Claims on What Remains

We study ordered claims problems in which availability of a resource changes across positions according to a stationary linear transition structure. Our main motivating example is the allocation of river water with conveyance loss. Allocations assigned to agents at different positions have different implications for their feasibility and for resource requirements. We characterise a class of geometric allocation rules in which claims are weighted by a constant multiplicative factor along the ordering and then scaled to the feasibility boundary. We apply these rules to two settings that capture opposing transition effects. In river water allocation the geometric weighting factor balances proportional sharing with efficiency considerations that would favour upstream agents. In a setting of river pollution with pollution decay, a geometric rule balances equal access to pollution rights with a reduction of pollution damage when favouring upstream agents.