发表机构
Software College, Northeastern University（东北大学软件学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究解决了4主体SD-EF矩阵必存在Dec-EF分解的公开问题，拓展了多主体、多偏好下的相关结论，证明1/2为最优嫉妒概率阈值，同时指出任意随机分配的Dec-EF分解判定是强NP完全问题。
AI 中文摘要
将n个不可分割物品随机分配给n个主体的分配方式由分配矩阵定义，并通过从Birkhoff-von Neumann（伯克霍夫-冯·诺依曼）分解中抽取确定性分配来实现。Kawase等人发现，分解方式的选择会影响公平性：一个满足随机占优无嫉妒（SD-EF）的矩阵，其分解结果可能出现某个主体以接近1的概率嫉妒另一主体的情况。他们将每个主体对其他任一主体的嫉妒概率不超过1/2的分解定义为Dec-EF分解，并证明了当n≤3或最多存在两种不同偏好时，每个SD-EF矩阵都存在Dec-EF分解，而一般情形的结论仍未明确。\n我们解决了首个未决情形：4个主体的每个SD-EF矩阵都存在Dec-EF分解。某偏好剖面下SD-EF多面体的最坏情况出现在顶点处，我们借助计算机辅助证明，通过精确算术枚举了考虑对称性后的762个偏好剖面对应的全部26927个顶点，并通过有理分解逐一验证其满足条件。同一方法还解决了最多4种不同偏好的5主体情形，以及所有5主体偏好剖面下的概率串行规则情形；针对最多7个主体的对抗性搜索也未找到反例。\n对于一般n值，嫉妒预算恒等式表明1/2是最优阈值。我们证明了最多包含两行不同行的SD-EF矩阵都存在Dec-EF分解；且当除两个主体外其余主体偏好均相同时，最大熵分解是Dec-EF的，后者的证明基于一个新的加权最小二乘排序单调性引理。一般情况下，常见分解方法均不满足要求：贪心Birkhoff-von Neumann分解的嫉妒概率可任意接近(n-1)/n，而当n=4且所有偏好均不同时，最大熵分解不满足Dec-EF。判断任意随机分配（不一定是SD-EF）是否存在Dec-EF分解是强NP完全问题。
英文摘要
A random assignment of n indivisible objects to n agents is specified by its assignment matrix and implemented by drawing a deterministic assignment from a Birkhoff-von Neumann decomposition. Kawase et al. observed that the choice of decomposition matters for fairness: a matrix that is envy-free in the sense of stochastic dominance (SD-EF) can be decomposed so that some agent envies another with probability close to 1. They call a decomposition Dec-EF if every agent envies every other agent with probability at most 1/2, proved that every SD-EF matrix admits a Dec-EF decomposition when n <= 3 or when there are at most two distinct preferences, and left the general case open. We settle the first open case: every SD-EF matrix with four agents admits a Dec-EF decomposition. The worst case over the SD-EF polytope of a profile is attained at a vertex, and our computer-aided proof enumerates all 26,927 vertices for the 762 profiles up to symmetry in exact arithmetic and certifies each by a rational decomposition. The same method settles five agents with at most four distinct preferences and the probabilistic serial rule for all five-agent profiles, and adversarial search up to seven agents finds no counterexample. For general n, an envy-budget identity shows that 1/2 is the best possible threshold. We prove that every SD-EF matrix with at most two distinct rows admits a Dec-EF decomposition, and that the maximum-entropy decomposition is Dec-EF whenever all agents but two share a preference; the latter proof rests on a new monotonicity lemma for weighted least-squares rankings. In general, natural decompositions fail: greedy Birkhoff-von Neumann can come arbitrarily close to envy probability (n-1)/n, and maximum entropy fails at n = 4 when all preferences differ. Deciding whether an arbitrary random assignment, not necessarily SD-EF, admits a Dec-EF decomposition is strongly NP-complete.
Comments11 pages; code and computational results in ancillary files
