In their AAMAS 2020 paper, Szufa et al. presented a "map of elections" that visualizes a set of 800 elections generated from various statistical cultures. While similar elections are grouped together on this map, there is no obvious interpretation of the elections' positions. We provide such an interpretation by introducing four canonical “extreme” elections, acting as a compass on the map. We use them to analyze both a dataset provided by Szufa et al. and a number of real-life elections. In effect, we find a new parameterization of the Mallows model, based on measuring the expected swap distance from the central preference order, and show that it is useful for capturing real-life scenarios.

We introduce the ELECTION ISOMORPHISM problem and a family of its approximate variants, which we refer to as dISOMORPHISM DISTANCE (d-ID) problems (where d is a metric between preference orders). We show that ELECTION ISOMORPHISM is polynomial-time solvable, and that the d-ISOMORPHISM DISTANCE problems generalize various classic rank-aggregation methods (e.g., those of Kemeny and Litvak). We establish the complexity of our problems (including their inapproximability) and provide initial experiments regarding the ability to solve them in practice.

We extend the map-of-elections framework to the case of approval elections. While doing so, we study a number of statistical cultures, including some new ones, and we analyze their properties. We find that approval elections can be understood in terms of the average number of approvals in the votes, and the extent to which the votes are chaotic.

Motivated by putting empirical work based on (synthetic) election data on a more solid mathematical basis, we analyze six distances among elections, including, e.g., the challenging-to-compute but very precise swap distance and the distance used to form the so-called map of elections. Among the six, the latter seems to strike the best balance between its computational complexity and expressiveness.

We describe the PArticipatory BUdgeting LIBrary website (in short, Pabulib), which can be accessed via http://pabulib.org/, and which is a library of participatory budgeting data. In particular, we describe the file format (.pb) that is used for instances of participatory budgeting.