We consider a voting scenario in which the resource to be voted upon may consist of both indivisible and divisible goods. This setting generalizes both the well-studied model of multiwinner voting and the recently introduced model of cake sharing. Under approval votes, we propose two variants of the extended justified representation (EJR) notion from multiwinner voting, a stronger one called EJR for mixed goods (EJR-M) and a weaker one called EJR up to 1 (EJR-1). We extend three multiwinner voting rules to our setting -- GreedyEJR, the method of equal shares (MES), and proportional approval voting (PAV) -- and show that while all three generalizations satisfy EJR-1, only the first one provides EJR-M. In addition, we derive tight bounds on the proportionality degree implied by EJR-M and EJR-1, and investigate the proportionality degree of our proposed rules.
翻译:我们考虑一种投票设想,即要表决的资源可以包括不可分割的和可分割的物品。这种设定将经过认真研究的多赢投票模式和最近推出的蛋糕分享模式普遍化。在批准票中,我们提议从多赢投票中扩大合理的代表权概念的两个变体,一个更强大的混合物品EJR(EJR-M),一个较弱的混合物品EJR(EJR-1),我们把三个多赢的投票规则扩大到我们的设置 -- -- 贪婪的EJR、平等份额方法和按比例核准投票 -- -- 并表明,虽然所有三种普遍办法都满足EJR-1,但只有第一个办法提供EJR-M。 此外,我们从EJR-M和EJR-1所隐含的相称程度上得出了严格的界限,并调查我们拟议规则的相称性程度。