Hare和Droop配额下的多项式时间完全正当代表权
Full Justified Representation under Hare and Droop Quotas in Polynomial Time
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中文总结 AI 辅助
该研究针对认可多赢家选举，提出降预算算法，分别在Hare和Droop配额下多项式时间内得到满足对应完全正当代表权的委员会，算法确定性且复杂度为O(kmn)。
中文摘要 AI 辅助
我研究了基于认可的多赢家选举中，在Hare配额和Droop配额两种规则下的完全正当代表权（FJR）。我提出了一种降预算算法，其中选民将剩余预算分配到当前代表缺口，当产生的报价覆盖共同价格时就购买候选人。当候选人价格为λ_H=n/k时，该算法返回满足Hare-FJR的委员会；当候选人价格为λ_D=n/(k+1)时，返回满足Casey和Elkind提出的要求更严格的Droop-FJR公理的委员会。两种保证共享历史支付不变性和终端行列核算论证，而Droop的证明需要在所有k个付费席位填满时使用新的剩余预算论证。一旦选民和候选人的顺序固定，两种变体都是确定性的，使用O(kmn)次有理运算。
英文摘要
I study Full Justified Representation (FJR) in approval-based multiwinner elections under both the Hare and Droop quota conventions. I introduce a descending-budget algorithm in which voters distribute their remaining budgets across their current representation gaps and candidates are purchased whenever the resulting offers cover a common price. With candidate price $λ_H=n/k$, the algorithm returns a Hare-FJR committee; with candidate price $λ_D=n/(k+1)$, it returns a committee satisfying the more demanding Droop-FJR axiom of Casey and Elkind. The two guarantees share a historical-payment invariant and a terminal row--column accounting argument, while the Droop proof requires a new residual-budget argument when all $k$ paid seats are filled. Both variants are deterministic once the voter and candidate orders are fixed and use $O(kmn)$ rational operations.
