Adjusted Winner Under Restricted Divisibility

We study two-person fair division when multiple items are indivisible and exactly one item, interpreted as money, is divisible. This setting imposes a natural restriction relative to the classical Adjusted Winner (AW) procedure, in which any item may become the split item. We introduce Adjusted Winner with Money (AWm), an ordered-allocation procedure tailored to this mixed environment. We identify a sufficient condition, Condition C, under which AWm produces an allocation that is both envy-free and equitable without splitting any indivisible item. When Condition C fails, AWm need not achieve full envy-freeness, but it still guarantees envy-freeness for mixed items. We also clarify the relation between AWm and Pareto-optimality. When classical AWwould split the money item, AWm coincides with AW; more generally, however, AWm need not be Pareto-optimal among all allocations that divide only money. Simulations based on Dirichletdistributed valuations indicate that Condition C is frequently satisfied, especially when the divisible item is relatively valuable. To illustrate the procedure, we apply it to the division of property in a real-life divorce case.