Anyone but Him: The Complexity of Precluding an Alternative

Preference aggregation in a multiagent setting is a central issue in both human and computer contexts. In this paper, we study in terms of complexity the vulnerability of preference aggregation to destructive control. That is, we study the ability of an election's chair to, through such mechanisms as voter/candidate addition/suppression/partition, ensure that a particular candidate (equivalently, alternative) does not win. And we study the extent to which election systems can make it impossible, or computationally costly (NP-complete), for the chair to execute such control. Among the systems we study--plurality, Condorcet, and approval voting--we find cases where systems immune or computationally resistant to a chair choosing the winner nonetheless are vulnerable to the chair blocking a victory. Beyond that, we see that among our studied systems no one system offers the best protection against destructive control. Rather, the choice of a preference aggregation system will depend closely on which types of control one wishes to be protected against. We also find concrete cases where the complexity of or susceptibility to control varies dramatically based on the choice among natural tie-handling rules.Comment: Preliminary version appeared in AAAI '05. Also appears as URCS-TR-2005-87

Downward Collapse from a Weaker Hypothesis

Hemaspaandra et al. proved that, for $m > 0$ and $0 < i < k - 1$: if \Sigma_i^p \BoldfaceDelta DIFF_m(\Sigma_k^p) is closed under complementation, then $DIFF_m(\Sigma_k^p) = coDIFF_m(\Sigma_k^p)$. This sharply asymmetric result fails to apply to the case in which the hypothesis is weakened by allowing the $\Sigma_i^p$ to be replaced by any class in its difference hierarchy. We so extend the result by proving that, for $s,m > 0$ and $0 < i < k - 1$: if DIFF_s(\Sigma_i^p) \BoldfaceDelta DIFF_m(\Sigma_k^p) is closed under complementation, then $DIFF_m(\Sigma_k^p) = coDIFF_m(\Sigma_k^p)$

What's Up with Downward Collapse: Using the Easy-Hard Technique to Link Boolean and Polynomial Hierarchy Collapses

During the past decade, nine papers have obtained increasingly strong consequences from the assumption that boolean or bounded-query hierarchies collapse. The final four papers of this nine-paper progression actually achieve downward collapse---that is, they show that high-level collapses induce collapses at (what beforehand were thought to be) lower complexity levels. For example, for each $k\geq 2$ it is now known that if \psigkone=\psigktwo then \ph=\sigmak. This article surveys the history, the results, and the technique---the so-called easy-hard method---of these nine papers.Comment: 37 pages. an extended abstract appeared in SIGACT News, 29, 10-22, 199

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