Exploiting Polyhedral Symmetries in Social Choice

A large amount of literature in social choice theory deals with quantifying the probability of certain election outcomes. One way of computing the probability of a specific voting situation under the Impartial Anonymous Culture assumption is via counting integral points in polyhedra. Here, Ehrhart theory can help, but unfortunately the dimension and complexity of the involved polyhedra grows rapidly with the number of candidates. However, if we exploit available polyhedral symmetries, some computations become possible that previously were infeasible. We show this in three well known examples: Condorcet's paradox, Condorcet efficiency of plurality voting and in Plurality voting vs Plurality Runoff.Comment: 14 pages; with minor improvements; to be published in Social Choice and Welfar

X THEN X: Manipulation of Same-System Runoff Elections

Do runoff elections, using the same voting rule as the initial election but just on the winning candidates, increase or decrease the complexity of manipulation? Does allowing revoting in the runoff increase or decrease the complexity relative to just having a runoff without revoting? For both weighted and unweighted voting, we show that even for election systems with simple winner problems the complexity of manipulation, manipulation with runoffs, and manipulation with revoting runoffs are independent, in the abstract. On the other hand, for some important, well-known election systems we determine what holds for each of these cases. For no such systems do we find runoffs lowering complexity, and for some we find that runoffs raise complexity. Ours is the first paper to show that for natural, unweighted election systems, runoffs can increase the manipulation complexity

Fair Knapsack

We study the following multiagent variant of the knapsack problem. We are given a set of items, a set of voters, and a value of the budget; each item is endowed with a cost and each voter assigns to each item a certain value. The goal is to select a subset of items with the total cost not exceeding the budget, in a way that is consistent with the voters' preferences. Since the preferences of the voters over the items can vary significantly, we need a way of aggregating these preferences, in order to select the socially best valid knapsack. We study three approaches to aggregating voters' preferences, which are motivated by the literature on multiwinner elections and fair allocation. This way we introduce the concepts of individually best, diverse, and fair knapsack. We study the computational complexity (including parameterized complexity, and complexity under restricted domains) of the aforementioned multiagent variants of knapsack.Comment: Extended abstract will appear in Proc. of 33rd AAAI 201