Positional scoring-based allocation of indivisible goods

Abstract

International audienceWe define a family of rules for dividing $m$ indivisible goods among agents, parameterized by a scoring vector and a social welfare aggregation function. We assume that agents' preferences over sets of goods are additive, but that the input is ordinal: each agent reports her preferences simply by ranking single goods. Similarly to positional scoring rules in voting, a scoring vector s = (s1, ... , sm) consists of m nonincreasing, nonnegative weights, where si is the score of a good assigned to an agent who ranks it in position i. The global score of an allocation for an agent is the sum of the scores of the goods assigned to her. The social welfare of an allocation is the aggregation of the scores of all agents, for some aggregation function * such as, typically, + or min. The rule associated with s and * maps a profile to (one of) the allocation(s) maximizing social welfare. After defining this family of rules, and focusing on some key examples, we investigate some of the social-choice-theoretic properties of this family of rules, such as various kinds of monotonicity, and separability. Finally, we focus on the computation of winning allocations, and on their approximation: we show that for commonly used scoring vectors and aggregation functions this problem is NP-hard and we exhibit some tractable particular case

Fair and Efficient Resource Allocation with Partial Information

“…Some of the work does not assume any underlying cardinal preferences; instead, it aims to obtain guarantees defined directly in terms of the ordinal preferences. For example, Baumeister et al [2017] and use the socalled scoring vectors to convert agents' ordinal preferences into numerical proxies for their utility and then consider maximizing various notions of social welfare or guaranteeing various fairness properties in terms of such utilities.…”

Using the Borda rule for ranking sets of objects

“…Despite its problem discussed in the previous section, the Borda-sum ranking is actually used in various occasions and, in particular, in the literature on fair division (see, e.g., Brams and Taylor 2000;Brams et al 2003;Brams and King 2005;Bouveret and Lang 2011;Baumeister et al 2017;Brams et al 2017 or Kilgour andVetschera 2018). However, many other potential applications do exist where a comparison of sets with fixed cardinality may be relevant.…”

“…Recently, various papers explicitly use the sum of some kind of positional scores 1 with respect to the ranking of the single objects and, in particular, Borda scores 2 of its respective elements (and their sum of points) to compare the different sets (see, e.g., Brams and Taylor 2000;Brams et al 2003;Brams and King 2005;Bouveret and Lang 2011;Baumeister et al 2017;Brams et al 2017 or Kilgour and Vetschera 2018). In that literature the Borda-sum ranking is used to deal with issues such as efficiency, proportionality or envy-freeness, because these are hard to tackle with purely ordinal and non-additive information.…”

Borda Count in Collective Decision Making: A Summary of Recent Results

“…In voting, such a shift of scores would not matter (Hemaspaandra. For the allocation problem, however,Baumeister et al (2017) show that such a shift actually does matter.…”