The Adjusted Winner (AW) method is a fundamental procedure for the fair division of indivisible resources between two agents. However, its reliance on splitting resources can lead to practical complications. To address this limitation, we propose an extension of AW that allows the sale of selected resources under a budget constraint, with the proceeds subsequently redistributed, thereby aiming for allocations that remain as equitable as possible. Alongside developing this extended framework, we provide an axiomatic analysis that examines how equitability and envy-freeness are modified in our setting. We then formally define the resulting combinatorial problems, establish their computational complexity, and design a fully polynomial-time approximation scheme (FPTAS) to mitigate their inherent intractability. Finally, we complement our theoretical results with computer-based simulations.

翻译：调整赢家（AW）方法是两个主体间不可分割资源公平分配的基本流程。然而，该方法对资源分割的依赖可能导致实际应用中的复杂性。为应对这一局限，我们提出一种AW的扩展方法，允许在预算约束下出售选定资源，并将所得收益重新分配，从而力求实现尽可能公平的分配方案。在构建这一扩展框架的同时，我们提供了公理化分析，以探究公平性与无嫉妒性在本研究设定下的变化。随后，我们正式定义了由此产生的组合优化问题，确立了其计算复杂度，并设计了完全多项式时间近似方案（FPTAS）以缓解其固有的计算困难。最后，我们通过计算机模拟对理论结果进行了补充验证。