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.
Moved to here. Different ways of combining voting rules to make manipulation computationally hard have been investigated recently. Conitzer and Sandholm @cite_10 studied the impact on the computational complexity of manipulation of adding an initial round of the Cup rule to a voting rule. This initial round eliminates half the candidates and makes manipulation NP-hard to compute for several voting rule including plurality and Borda. Elkind and Lipmaa @cite_2 extended this idea to a general technique for combining two voting rules. The first voting rule is run for some number of rounds to eliminate some of the candidates, before the second voting rule is applied to the candidates that remain. They proved that many such combinations of voting rules are NP-hard to manipulate. Note that theirs is a sequential combinator, in which the two rules are run in sequence, whilst ours (as we will see soon) is a parallel combinator, in which the two rules are run in parallel. More recently, Walsh and Xia @cite_8 showed that using a lottery to eliminate some of the voters (instead of some of the candidates) is another mechanism to make manipulation intractable to compute.
@cite_8 @cite_10 @cite_2
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Voting is a general method for preference aggregation in multiagent settings, but seminal results have shown that all (nondictatorial) voting protocols are manipulable. One could try to avoid manipulation by using voting protocols where determining a beneficial manipulation is hard computationally. A number of recent papers study the complexity of manipulating existing protocols. This paper is the first work to take the next step of designing new protocols that are especially hard to manipulate. Rather than designing these new protocols from scratch, we instead show how to tweak existing protocols to make manipulation hard, while leaving much of the original nature of the protocol intact. The tweak studied consists of adding one elimination preround to the election. Surprisingly, this extremely simple and universal tweak makes typical protocols hard to manipulate! The protocols become NP-hard, #P-hard, or PSPACE-hard to manipulate, depending on whether the schedule of the preround is determined before the votes are collected, after the votes are collected, or the scheduling and the vote collecting are interleaved, respectively. We prove general sufficient conditions on the protocols for this tweak to introduce the hardness, and show that the most common voting protocols satisfy those conditions. These are the first results in voting settings where manipulation is in a higher complexity class than NP (presuming PSPACE @math NP). This paper addresses the problem of constructing voting protocols that are hard to manipulate. We describe a general technique for obtaining a new protocol by combining two or more base protocols, and study the resulting class of (vote-once) hybrid voting protocols, which also includes most previously known manipulation-resistant protocols. We show that for many choices of underlying base protocols, including some that are easily manipulable, their hybrids are NP-hard to manipulate, and demonstrate that this method can be used to produce manipulation-resistant protocols with unique combinations of useful features.