# Dealing with incomplete agents' preferences and an uncertain agenda in group decision making via sequential majority voting

## TL;DR

This paper investigates the computational complexity of determining collective winners in multi-agent systems using sequential majority voting, considering incomplete preferences and uncertain agendas, with implications for preference elicitation and manipulation.

## Contribution

It provides complexity results for identifying always-winning or possibly-winning decisions under incomplete preferences and uncertain agendas in sequential majority voting.

## Key findings

- Determining if a candidate always wins is computationally easy.
- Deciding if a candidate may win in some agenda is computationally easy.
- Checking if a candidate wins in at least one agenda with some preference completion is computationally hard.

## Abstract

We consider multi-agent systems where agents' preferences are aggregated via sequential majority voting: each decision is taken by performing a sequence of pairwise comparisons where each comparison is a weighted majority vote among the agents. Incompleteness in the agents' preferences is common in many real-life settings due to privacy issues or an ongoing elicitation process. In addition, there may be uncertainty about how the preferences are aggregated. For example, the agenda (a tree whose leaves are labelled with the decisions being compared) may not yet be known or fixed. We therefore study how to determine collectively optimal decisions (also called winners) when preferences may be incomplete, and when the agenda may be uncertain. We show that it is computationally easy to determine if a candidate decision always wins, or may win, whatever the agenda. On the other hand, it is computationally hard to know wheth er a candidate decision wins in at least one agenda for at least one completion of the agents' preferences. These results hold even if the agenda must be balanced so that each candidate decision faces the same number of majority votes. Such results are useful for reasoning about preference elicitation. They help understand the complexity of tasks such as determining if a decision can be taken collectively, as well as knowing if the winner can be manipulated by appropriately ordering the agenda.