Piotr Faliszewski

How hard is bribery in elections

We study the complexity of influencing elections through bribery: How computationally complex is it for an external actor to determine whether by paying certain voters to change their preferences a specified candidate can be made the elections winner? We study this problem for election systems as varied as scoring protocols and Dodgson voting, and in a variety of settings regarding homogeneous-vs.-nonhomogeneous electorate bribability, bounded-size-vs.-arbitrary-sized candidate sets, weighted-vs.-unweighted voters, and succinct-vs.-nonsuccinct input specification. We obtain both polynomial-time bribery algorithms and proofs of the intractability of bribery, and indeed our results show that the complexity of bribery is extremely sensitive to the setting. For example, we find settings in which bribery is NP-complete but manipulation (by voters) is in P, and we find settings in which bribing weighted voters is NP-complete but bribing voters with individual bribe thresholds is in P. For the broad class of elections (including plurality, Borda, k- approval,and veto) known as scoring protocols, we prove a dichotomy result for bribery of weighted voters: We find a simple-to-evaluate condition that classifies every case as either NP-complete or in P.

Using complexity to protect elections

Computational complexity may truly be the shield against election manipulation.

AI’s War on Manipulation: Are We Winning?

We provide an overview of more than two decades of work, mostly in AI, that studies computational complexity as a barrier against manipulation in elections.

A Richer Understanding of the Complexity of Election Systems

We provide an overview of some recent progress on the complexity of election systems. The issues studied include the complexity of the winner, manipulation, bribery, and control problems.

Swap Bribery

In voting theory, bribery is a form of manipulative behavior in which an external actor (the briber) offers to pay the voters to change their votes in order to get her preferred candidate elected. We investigate a model of bribery where the price of each vote depends on the amount of change that the voter is asked to implement. Specifically, in our model the briber can change a voters preference list by paying for a sequence of swaps of consecutive candidates. Each swap may have a different price; the price of a bribery is the sum of the prices of all swaps that it involves. We prove complexity results for this model, which we call swap bribery , for a broad class of voting rules, including variants of approval and k -approval, Borda, Copeland, and maximin.

Properties of multiwinner voting rules

A committee selection rule (or, multiwinner voting rule) is a mapping that takes a collection of strict preference rankings and a positive integer k as input, and outputs one or more subsets of candidates of size k. In this paper we consider committee selection rules that can be viewed as generalizations of single-winner scoring rules, including SNTV, Bloc, k-Borda, STV, as well as several variants of the Chamberlin–Courant rule and the Monroe rule and their approximations. We identify two natural broad classes of committee selection rules, and show that many of the existing rules belong to one or both of these classes. We then formulate a number of desirable properties of committee selection rules, and evaluate the rules we consider with respect to these properties.

Clone structures in voters' preferences

In elections, a set of candidates ranked consecutively (though possibly in different order) by all voters is called a clone set, and its members are called clones. A clone structure is the family of all clone sets of a given election. In this paper we study properties of clone structures. In particular, we give an axiomatic characterization of clone structures, show that they are organized hierarchically, and analyze clone structures in single-peaked and single-crossing elections. We describe a polynomial-time algorithm that finds a minimal collection of clones that need to be collapsed for an election to become single-peaked, and we show that this problem is NP-hard for single-crossing elections.

The complexity of fully proportional representation for single-crossing electorates

We study the complexity of winner determination in single-crossing elections under two classic fully proportional representation rules-Chamberlin-Courants rule and Monroes rule. Winner determination for these rules is known to be NP-hard for unrestricted preferences. We show that for single-crossing preferences this problem admits a polynomial-time algorithm for Chamberlin-Courants rule, but remains NP-hard for Monroes rule. Our algorithm for Chamberlin-Courants rule can be modified to work for elections with bounded single-crossing width. We then consider elections that are both single-peaked and single-crossing, and develop an efficient algorithm for the egalitarian variant of Monroes rule for such elections. While Betzler et al. [3] have recently presented a polynomial-time algorithm for this rule under single-peaked preferences, our algorithm has considerably better worst-case running time than that of Betzler et al.

Approximation algorithms for campaign management

We study electoral campaign management scenarios in which an external party can buy votes, i.e., pay the voters to promote its preferred candidate in their preference rankings. The external partys goal is to make its preferred candidate a winner while paying as little as possible. We describe a 2-approximation algorithm for this problem for a large class of electoral systems known as scoring rules. Our result holds even for weighted voters, and has applications for campaign management in commercial settings. We also give approximation algorithms for our problem for two Condorcet-consistent rules, namely, the Copeland rule and maximin.

Campaign Management Under Approval-Driven Voting Rules

Approval-like voting rules, such as sincere-strategy preference-based approval voting (SP-AV), the Bucklin rule (an adaptive variant of k-approval voting), and the Fallback rule (a hybrid of the Bucklin rule and SP-AV) have many desirable properties: for example, they are easy to understand and encourage the candidates to choose electoral platforms that have a broad appeal. In this paper, we investigate both classic and parameterized computational complexity of electoral campaign management under such rules. We focus on two methods that can be used to promote a given candidate: asking voters to move this candidate upwards in their preference order or asking them to change the number of candidates they approve of. We show that finding an optimal campaign management strategy of the first type is easy for both Bucklin and Fallback. In contrast, the second method is computationally hard even if the degree to which we need to affect the votes is small. Nevertheless, we identify a large class of scenarios that admit fixed-parameter tractable algorithms.