Voting in Combinatorial Domains

Abstract

Voting in Combinatorial Domains Jérôme Lang a and Lirong Xia b 10.1 Motivations and classes of problems This chapter addresses preference aggregation and voting on domains which are the Cartesian product (or sometimes, a subset of the Cartesian product) of finite domain values, each corresponding to an issue, a variable, or an attribute. As seen in other chapters of this handbook, voting rules map a profile (usually, a collection of rankings, see Chapter 1 (Zwicker, 2015)) into an alternative or a set of alternatives. A key question has to do with the structure of the set of alternatives. Sometimes, this set has a simple structure and a small cardinality (e.g., in a presidential election). But in many contexts, it has a complex combinatorial structure. We give here three typical examples:

Barriers to Manipulation in Voting

“…23 A similar negative result is given by in a different context, where multiple related binary decisions must be made and these issues are voted on in sequence (but with all the voters voting at the same time on each issue). For more on voting in such combinatorial domains, please see Chapter 9 (Lang and Xia, 2015).…”

Judgment Aggregation

“…14 Now suppose three judges truthfully report their judgment sets: Preferences induced by the Hamming distance are also discussed in Chapter 9 on voting in combinatorial domains (Lang and Xia, 2016) Under the premise-based rule, all three premises will get rejected, which means that also the two conclusions get rejected. For judge 3 the Hamming distance from this outcome to her own judgment set is 4.…”

“…One possible refinement is to choose a maximally consistent subset of the union (Baral et al, 1992), but this and similar approaches do not track which individual provided which formulas in the knowledge base, i.e., it is not possible to give equal weight to different individuals. The latter is possible using distance-based procedures (Konieczny and Pino Pérez, 2002), which are also briefly discussed in Chapter 9 (Lang and Xia, 2016).…”

Complexity results for preference aggregation over (m)CP-nets: Pareto and majority voting

“…We provide various exact complexity results showing tight lower bounds and matching upper bounds for problems that (up to date) did not have any explicit non-obvious lower bound.(a) A CP-net modeling dinner preferences.(b) The CP-net's extended preference graph. Figure 1: A CP-net and its preference graph.has a combinatorial structure [18,45,48]. By combinatorial structure, we mean that the set of candidates (or outcomes) is the Cartesian product of finite value domains for each of a set of features (also called variables, or issues, or attributes).…”

“…The problem of aggregating agents' preferences over combinatorial domains (or multi-issue domains) is called a combinatorial vote [44,45].Interestingly, voting over combinatorial domains is rather common. For example, in 2012, on the day of the US presidential election, voters in California had to vote also for eleven referenda [48]. As another example, it may be the case that the inhabitants of a town have to make a joint decision about different related issues regarding their community, which could be whether and where to build new public facilities (such as a swimming pool or a library), or whether to levy new taxes.…”