Restricted Manipulation in Iterative Voting: Convergence and Condorcet Efficiency

In collective decision making, where a voting rule is used to take a collective decision among a group of agents, manipulation by one or more agents is usually considered negative behavior to be avoided, or at least to be made computationally difficult for the agents to perform. However, there are scenarios in which a restricted form of manipulation can instead be beneficial. In this paper we consider the iterative version of several voting rules, where at each step one agent is allowed to manipulate by modifying his ballot according to a set of restricted manipulation moves which are computationally easy and require little information to be performed. We prove convergence of iterative voting rules when restricted manipulation is allowed, and we present experiments showing that most iterative voting rules have a higher Condorcet efficiency than their non-iterative version.

Possible and Necessary Winner Problem in Social Polls

Social networks are increasingly being used to conduct polls. We introduce a simple model of such social polling. We suppose agents vote sequentially, but the order in which agents choose to vote is not necessarily fixed. We also suppose that an agent's vote is influenced by the votes of their friends who have already voted. Despite its simplicity, this model provides useful insights into a number of areas including social polling, sequential voting, and manipulation. We prove that the number of candidates and the network structure affect the computational complexity of computing which candidate necessarily or possibly can win in such a social poll. For social networks with bounded treewidth and a bounded number of candidates, we provide polynomial algorithms for both problems. In other cases, we prove that computing which candidates necessarily or possibly win are computationally intractable.

The SeqBin Constraint Revisited

We revisit the SeqBin constraint. This meta-constraint subsumes a number of important global constraints like Change, Smooth and IncreasingNValue. We show that the previously proposed filtering algorithm for SeqBin has two drawbacks even under strong restrictions: it does not detect bounds disentailment and it is not idempotent. We identify the cause for these problems, and propose a new propagator that overcomes both issues. Our algorithm is based on a connection to the problem of finding a path of a given cost in a restricted $n$-partite graph. Our propagator enforces domain consistency in O(nd^2) and, for special cases of SeqBin that include Change, Smooth and IncreasingNValue, in O(nd) time.

Eliminating the Weakest Link: Making Manipulation Intractable?

Successive elimination of candidates is often a route to making manipulation intractable to compute. We prove that eliminating candidates does not necessarily increase the computational complexity of manipulation. However, for many voting rules used in practice, the computational complexity increases. For example, it is already known that it is NP-hard to compute how a single voter can manipulate the result of single transferable voting (the elimination version of plurality voting). We show here that it is NP-hard to compute how a single voter can manipulate the result of the elimination version of veto voting, of the closely related Coombs' rule, and of the elimination versions of a general class of scoring rules.

Symmetry Breaking Constraints: Recent Results

Symmetry is an important problem in many combinatorial problems. One way of dealing with symmetry is to add constraints that eliminate symmetric solutions. We survey recent results in this area, focusing especially on two common and useful cases: symmetry breaking constraints for row and column symmetry, and symmetry breaking constraints for eliminating value symmetry

Combining Voting Rules Together

We propose a simple method for combining together voting rules that performs a run-off between the different winners of each voting rule. We prove that this combinator has several good properties. For instance, even if just one of the base voting rules has a desirable property like Condorcet consistency, the combination inherits this property. In addition, we prove that combining voting rules together in this way can make finding a manipulation more computationally difficult. Finally, we study the impact of this combinator on approximation methods that find close to optimal manipulations.

The RegularGcc Matrix Constraint

We study propagation of the RegularGcc global constraint. This ensures that each row of a matrix of decision variables satisfies a Regular constraint, and each column satisfies a Gcc constraint. On the negative side, we prove that propagation is NP-hard even under some strong restrictions (e.g. just 3 values, just 4 states in the automaton, or just 5 columns to the matrix). On the positive side, we identify two cases where propagation is fixed parameter tractable. In addition, we show how to improve propagation over a simple decomposition into separate Regular and Gcc constraints by identifying some necessary but insufficient conditions for a solution. We enforce these conditions with some additional weighted row automata. Experimental results demonstrate the potential of these methods on some standard benchmark problems.

A Comparison of Lex Bounds for Multiset Variables in Constraint Programming

Set and multiset variables in constraint programming have typically been represented using subset bounds. However, this is a weak representation that neglects potentially useful information about a set such as its cardinality. For set variables, the length-lex (LL) representation successfully provides information about the length (cardinality) and position in the lexicographic ordering. For multiset variables, where elements can be repeated, we consider richer representations that take into account additional information. We study eight different representations in which we maintain bounds according to one of the eight different orderings: length-(co)lex (LL/LC), variety-(co)lex (VL/VC), length-variety-(co)lex (LVL/LVC), and variety-length-(co)lex (VLL/VLC) orderings. These representations integrate together information about the cardinality, variety (number of distinct elements in the multiset), and position in some total ordering. Theoretical and empirical comparisons of expressiveness and compactness of the eight representations suggest that length-variety-(co)lex (LVL/LVC) and variety-length-(co)lex (VLL/VLC) usually give tighter bounds after constraint propagation. We implement the eight representations and evaluate them against the subset bounds representation with cardinality and variety reasoning. Results demonstrate that they offer significantly better pruning and runtime.

Dominating Manipulations in Voting with Partial Information

We consider manipulation problems when the manipulator only has partial information about the votes of the nonmanipulators. Such partial information is described by an information set, which is the set of profiles of the nonmanipulators that are indistinguishable to the manipulator. Given such an information set, a dominating manipulation is a non-truthful vote that the manipulator can cast which makes the winner at least as preferable (and sometimes more preferable) as the winner when the manipulator votes truthfully. When the manipulator has full information, computing whether or not there exists a dominating manipulation is in P for many common voting rules (by known results). We show that when the manipulator has no information, there is no dominating manipulation for many common voting rules. When the manipulator's information is represented by partial orders and only a small portion of the preferences are unknown, computing a dominating manipulation is NP-hard for many common voting rules. Our results thus throw light on whether we can prevent strategic behavior by limiting information about the votes of other voters.

Online Cake Cutting (published version)

We propose an online form of the cake cutting problem. This models situations where agents arrive and depart during the process of dividing a resource. We show that well known fair division procedures like cut-and-choose and the Dubins-Spanier moving knife procedure can be adapted to apply to such online problems. We propose some fairness properties that online cake cutting procedures can possess like online forms of proportionality and envy-freeness. We also consider the impact of collusion between agents. Finally, we study theoretically and empirically the competitive ratio of these online cake cutting procedures. Based on its resistance to collusion, and its good performance in practice, our results favour the online version of the cut-and-choose procedure over the online version of the moving knife procedure.