The Complexity of Manipulating kk-Approval Elections

An important problem in computational social choice theory is the complexity of undesirable behavior among agents, such as control, manipulation, and bribery in election systems. These kinds of voting strategies are often tempting at the individual level but disastrous for the agents as a whole. Creating election systems where the determination of such strategies is difficult is thus an important goal. An interesting set of elections is that of scoring protocols. Previous work in this area has demonstrated the complexity of misuse in cases involving a fixed number of candidates, and of specific election systems on unbounded number of candidates such as Borda. In contrast, we take the first step in generalizing the results of computational complexity of election misuse to cases of infinitely many scoring protocols on an unbounded number of candidates. Interesting families of systems include $k$-approval and $k$-veto elections, in which voters distinguish $k$ candidates from the candidate set. Our main result is to partition the problems of these families based on their complexity. We do so by showing they are polynomial-time computable, NP-hard, or polynomial-time equivalent to another problem of interest. We also demonstrate a surprising connection between manipulation in election systems and some graph theory problems

Computational Aspects of Multi-Winner Approval Voting

We study computational aspects of three prominent voting rules that use approval ballots to elect multiple winners. These rules are satisfaction approval voting, proportional approval voting, and reweighted approval voting. We first show that computing the winner for proportional approval voting is NP-hard, closing a long standing open problem. As none of the rules are strategyproof, even for dichotomous preferences, we study various strategic aspects of the rules. In particular, we examine the computational complexity of computing a best response for both a single agent and a group of agents. In many settings, we show that it is NP-hard for an agent or agents to compute how best to vote given a fixed set of approval ballots from the other agents

Approximation and elections

Any culture that requires that a decision be made within a group necessarily creates methods for aggregating each individual’s preferences. For instance, we see such a need in political elections, committees, and businesses. With the Internet and the increasing use of multiagent software systems, the general need for means of aggregating differing preferences has increased dramatically. Voting is one way to come to a single option (or small group of options) out of a larger pool of candidate options. Many voting systems exist, and criteria exist (within the field of social choice theory) for deciding which most fairly and accurately take into account the preferences of each voter. Since there is generally much to be gained from influencing such a vote through manipulation or bribery, one desirable criterion of fairness would be whether such activities are impossible in the system. However, it has been shown that a reasonable system that disallows manipulation does not exist [18, 43], so the next-best solution would be a system in which deciding how to bribe or otherwise influence the vote is so computationally difficult as to render it impossible or highly unlikely. While the debate over which voting systems are most fair and effective is on record of existing over the past few centuries (and likely goes further back to ancient Greece), there may exist the seeds of a renewal of this debate in the current boom in voting due to new technologies. For one, in artificial intelligence agents may vote to determine the best course of action to take given the individual’s preferences. In addition, algorithms in search engines and metasearch engines do order results in a manner that assumes a ranking was somehow approached. Voting is not only on the rise in software, of course, as most any user of the Internet could demonstrate. Internet users routinely vote most any user of the Internet could demonstrate. Internet users routinely vote online in situations ranging from the inane (e.g., rating a video on YouTube) to the potentially crucial (e.g., voting on whether a story is newsworthy or not on any of a plethora of such sites, including Digg, Reddit, and Newsvine). These newer uses of voting systems are interesting. They are used in environments where there are potentially far more candidates and voters than are conventionally seen in, say, political elections. Also, in these new environments, voting and manipulation can be automated to some degree, thus making the possibility of manipulation and control even more real than it has been in the past. Faliszewski, Hemaspaandra, and Hemaspaandra have proved for a number of voting systems that the bribery problem is too complex to be feasible (i.e., NP-complete) [15], and much research has been put forth determining the complexity of other problems related to voting. But it is still possible in the optimization cases of these problems that there exist approximation algorithms that can find a good solution with a reasonable amount of computation. That is, while a voting system may seem “resistant” to a particular form of manipulation as described by previous research, it may be that the problem is not as difficult if we allow a constant amount of error. Or, it may be that the problem is still difficult when error is allowed, thus making the voting system even more resilient with respect to some forms of manipulation. This thesis will examine the possibility of such approximations for some problems in elections