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Register a new userControl Complexity in Bucklin and Fallback Voting
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Added by
Joerg Rothe
Publication date
2011
fields
Informatics Engineering
and research's language is
English
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Electoral control models ways of changing the outcome of an election via such actions as adding/deleting/partitioning either candidates or voters. To protect elections from such control attempts, computational complexity has been investigated and the corresponding NP-hardness results are termed resistance. It has been a long-running project of research in this area to classify the major voting systems in terms of their resistance properties. We show that fallback voting, an election system proposed by Brams and Sanver (2009) to combine Bucklin with approval voting, is resistant to each of the common types of control except to destructive control by either adding or deleting voters. Thus fallback voting displays the broadest control resistance currently known to hold among natural election systems with a polynomial-time winner problem. We also study the control complexity of Bucklin voting and show that it performs at least almost as well as fallback voting in terms of control resistance. As Bucklin voting is a special case of fallback voting, each resistance shown for Bucklin voting strengthens the corresponding resistance for fallback voting. Such worst-case complexity analysis is at best an indication of security against control attempts, rather than a proof. In practice, the difficulty of control will depend on the structure of typical instances. We investigate the parameterized control complexity of Bucklin and fallback voting, according to several parameters that are often likely to be small for typical instances. Our results, though still in the worst-case complexity model, can be interpreted as significant strengthenings of the resistance demonstrations based on NP-hardness.
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Walsh [Wal10, Wal09], Davies et al. [DKNW10, DKNW11], and Narodytska et al. [NWX11] studied various voting systems empirically and showed that they can often be manipulated effectively, despite their manipulation problems being NP-hard. Such an experimental approach is sorely missing for NP-hard control problems, where control refers to attempts to tamper with the outcome of elections by adding/deleting/partitioning either voters or candidates. We experimentally tackle NP-hard control problems for Bucklin and fallback voting. Among natural voting systems with efficient winner determination, fallback voting is currently known to display the broadest resistance to control in terms of NP-hardness, and Bucklin voting has been shown to behave almost as well in terms of control resistance [ER10, EPR11, EFPR11]. We also investigate control resistance experimentally for plurality voting, one of the first voting systems analyzed with respect to electoral control [BTT92, HHR07]. Our findings indicate that NP-hard control problems can often be solved effectively in practice. Moreover, our experiments allow a more fine-grained analysis and comparison-across various control scenarios, vote distribution models, and voting systems-than merely stating NP-hardness for all these control problems.
A central theme in computational social choice is to study the extent to which voting systems computationally resist manipulative attacks seeking to influence the outcome of elections, such as manipulation (i.e., strategic voting), control, and bribery. Bucklin and fallback voting are among the voting systems with the broadest resistance (i.e., NP-hardness) to control attacks. However, only little is known about their behavior regarding manipulation and bribery attacks. We comprehensively investigate the computational resistance of Bucklin and fallback voting for many of the common manipulation and bribery scenarios; we also complement our discussion by considering several campaign management problems for Bucklin and fallback.
In the context of computational social choice, we study voting methods that assign a set of winners to each profile of voter preferences. A voting method satisfies the property of positive involvement (PI) if for any election in which a candidate x would be among the winners, adding another voter to the election who ranks x first does not cause x to lose. Surprisingly, a number of standard voting methods violate this natural property. In this paper, we investigate different ways of measuring the extent to which a voting method violates PI, using computer simulations. We consider the probability (under different probability models for preferences) of PI violations in randomly drawn profiles vs. profile-coalition pairs (involving coalitions of different sizes). We argue that in order to choose between a voting method that satisfies PI and one that does not, we should consider the probability of PI violation conditional on the voting methods choosing different winners. We should also relativize the probability of PI violation to what we call voter potency, the probability that a voter causes a candidate to lose. Although absolute frequencies of PI violations may be low, after this conditioning and relativization, we see that under certain voting methods that violate PI, much of a voters potency is turned against them - in particular, against their desire to see their favorite candidate elected.
We introduce a new algebraic proof system, which has tight connections to (algebraic) circuit complexity. In particular, we show that any super-polynomial lower bound on any Boolean tautology in our proof system implies that the permanent does not have polynomial-size algebraic circuits (VNP is not equal to VP). As a corollary to the proof, we also show that super-polynomial lower bounds on the number of lines in Polynomial Calculus proofs (as opposed to the usual measure of number of monomials) imply the Permanent versus Determinant Conjecture. Note that, prior to our work, there was no proof system for which lower bounds on an arbitrary tautology implied any computational lower bound. Our proof system helps clarify the relationships between previous algebraic proof systems, and begins to shed light on why proof complexity lower bounds for various proof systems have been so much harder than lower bounds on the corresponding circuit classes. In doing so, we highlight the importance of polynomial identity testing (PIT) for understanding proof complexity. More specifically, we introduce certain propositional axioms satisfied by any Boolean circuit computing PIT. We use these PIT axioms to shed light on AC^0[p]-Frege lower bounds, which have been open for nearly 30 years, with no satisfactory explanation as to their apparent difficulty. We show that either: a) Proving super-polynomial lower bounds on AC^0[p]-Frege implies VNP does not have polynomial-size circuits of depth d - a notoriously open question for d at least 4 - thus explaining the difficulty of lower bounds on AC^0[p]-Frege, or b) AC^0[p]-Frege cannot efficiently prove the depth d PIT axioms, and hence we have a lower bound on AC^0[p]-Frege. Using the algebraic structure of our proof system, we propose a novel way to extend techniques from algebraic circuit complexity to prove lower bounds in proof complexity.
Computers are known to solve a wide spectrum of problems, however not all problems are computationally solvable. Further, the solvable problems themselves vary on the amount of computational resources they require for being solved. The rigorous analysis of problems and assigning them to complexity classes what makes up the immense field of complexity theory. Do protein folding and sudoku have something in common? It might not seem so but complexity theory tells us that if we had an algorithm that could solve sudoku efficiently then we could adapt it to predict for protein folding. This same property is held by classic platformer games such as Super Mario Bros, which was proven to be NP-complete by Erik Demaine et. al. This article attempts to review the analysis of classical platformer games. Here, we explore the field of complexity theory through a broad survey of literature and then use it to prove that that solving a generalized level in the game Celeste is NP-complete. Later, we also show how a small change in it makes the game presumably harder to compute. Various abstractions and formalisms related to modelling of games in general (namely game theory and constraint logic) and 2D platformer video games, including the generalized meta-theorems originally formulated by Giovanni Viglietta are also presented.
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