The Complexity of Online Manipulation of Sequential Elections

Most work on manipulation assumes that all preferences are known to the manipulators. However, in many settings elections are open and sequential, and manipulators may know the already cast votes but may not know the future votes. We introduce a framework, in which manipulators can see the past votes but not the future ones, to model online coalitional manipulation of sequential elections, and we show that in this setting manipulation can be extremely complex even for election systems with simple winner problems. Yet we also show that for some of the most important election systems such manipulation is simple in certain settings. This suggests that when using sequential voting, one should pay great attention to the details of the setting in choosing one's voting rule. Among the highlights of our classifications are: We show that, depending on the size of the manipulative coalition, the online manipulation problem can be complete for each level of the polynomial hierarchy or even for PSPACE. We obtain the most dramatic contrast to date between the nonunique-winner and unique-winner models: Online weighted manipulation for plurality is in P in the nonunique-winner model, yet is coNP-hard (constructive case) and NP-hard (destructive case) in the unique-winner model. And we obtain what to the best of our knowledge are the first P^NP[1]-completeness and P^NP-completeness results in the field of computational social choice, in particular proving such completeness for, respectively, the complexity of 3-candidate and 4-candidate (and unlimited-candidate) online weighted coalition manipulation of veto elections.Comment: 24 page

A Sound and Complete Axiomatization of Majority-n Logic

Manipulating logic functions via majority operators recently drew the attention of researchers in computer science. For example, circuit optimization based on majority operators enables superior results as compared to traditional logic systems. Also, the Boolean satisfiability problem finds new solving approaches when described in terms of majority decisions. To support computer logic applications based on majority a sound and complete set of axioms is required. Most of the recent advances in majority logic deal only with ternary majority (MAJ- 3) operators because the axiomatization with solely MAJ-3 and complementation operators is well understood. However, it is of interest extending such axiomatization to n-ary majority operators (MAJ-n) from both the theoretical and practical perspective. In this work, we address this issue by introducing a sound and complete axiomatization of MAJ-n logic. Our axiomatization naturally includes existing majority logic systems. Based on this general set of axioms, computer applications can now fully exploit the expressive power of majority logic.Comment: Accepted by the IEEE Transactions on Computer

School algebra and the computer

How are we to use the computer in the teaching and learning of algebra? In the longterm the new technology is introducing new possibilities that may radically change the algebra curriculum. However, in the short-term we already have the National Curriculum placing its template on the development of algebra in school. The recent regrouping of topics into five attainment targets has integrated number pattern with the development of algebraic symbolism. It seems natural to build from expressing patterns in words to expressing them in a shorthand algebraic notation, but, although this proves a sound tactic for the more able, there are subtle difficulties for the majority of children. Instead I shall advocate introducing algebraic symbolism by using it as a language of communication with the computer, through programming in a suitable computer language. This has two distinct benefits – it develops a meaningful algebraic language which can be used to describe number patterns, and it gives a foundation for traditional algebra and its manipulation

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

Computer-Aided Conceptual Design Through TRIZ-based Manipulation of Topological Optimizations

Organised by: Cranfield UniversityIn a recent project the authors proposed the adoption of Optimization Systems [1] as a bridging element between Computer-Aided Innovation (CAI) and PLM to identify geometrical contradictions [2], a particular case of the TRIZ physical contradiction [3]. A further development of the research has revealed that the solutions obtained from several topological optimizations can be considered as elementary customized modeling features for a specific design task. The topology overcoming the arising geometrical contradiction can be obtained through a manipulation of the density distributions constituting the conflicting pair. Already two strategies of density combination have been identified as capable to solve geometrical contradictions.Mori Seiki – The Machine Tool Compan