Search versus Decision for Election Manipulation Problems

Abstract

Most theoretical definitions about the complexity of manipulating elections focus on the decision problem of recognizing which instances can be successfully manipulated, rather than the search problem of finding the successful manipulative actions. Since the latter is a far more natural goal for manipulators, that definitional focus may be misguided if these two complexities can differ. Our main result is that they probably do differ: If integer factoring is hard, then for election manipulation, election bribery, and some types of election control, there are election systems for which recognizing which instances can be successfully manipulated is in polynomial time but producing the successful manipulations cannot be done in polynomial time. * Also appears as URCS-TR-2012-971. † For each fixed election system E, one can define the election winner problem as follows (see [BTT89b]).Name: E-winner, or the winner problem for E.Given: Election (C, V ) and candidate p ∈ C.Question: Is p a winner of the election (C, V ) under election system E? This is actually, in the way universally accepted in computer science (see [GJ79]), describing a set, i.e., a language. That set is the set of all triples C, V, p such that the answer to the question is "Yes." 3 We now briefly present the key definitions for the three most commonly studied types of manipulative actions: manipulation, bribery, and control. These three types were first studied, respectively, by Bartholdi, Orlin, Tovey, and Trick [BTT89a,BO91], Faliszewski, Hemaspaandra, and Hemaspaandra [FHH09], and Bartholdi, Tovey, and Trick [BTT92], for the "constructive" cases, i.e., where the goal is to make a particular candidate be a winner. 4 The "destructive" cases, where the goal is to ensure that a particular candidate is not a winner, were introduced by Conitzer, Sandholm, and Lang [CSL07] for manipulation, by Faliszewski, Hemaspaandra, and Hemaspaandra for bribery [FHH09], and by Hemaspaandra, Hemaspaandra, and Rothe for control [HHR07].The manipulation problem is defined as follows, and models whether a coalition of strategic voters can make a certain candidate win.Name: E-unweighted coalition manipulation, or the unweighted coalition manipulation problem for E; for short, the manipulation problem for E.Given: Candidate set C, nonmanipulative voter set V 1 (as a set of (name, preferences) ballots with preferences over C), manipulative voter set V 2 (as a set of names of the manipulative voters, none of whose names may belong to V 1 ), and a candidate p ∈ C.Question: Is there some choice of preferences for the manipulative voters such that p is a winner in the election in system E with candidates C and voters V 1 ∪ V 2 ?This again is a decision problem consisting of the set of all inputs yielding the answer "Yes." However, there is a very natural search problem associated with this, which we will call manipulation no one is so strong as to receive the support of 75 percent of the voters, then no one is inducted into the Baseball Hall of Fame that year. Indeed, in thr...

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Towards completing the puzzle: complexity of control by replacing, adding, and deleting candidates or voters

“…Further, there are two variants of control by partition of candidates, one with runoff (where both subgroups send their winners to the final round) and one without (where the winners of one subgroup face all candidates of the other in the final round). Hemaspaandra, Hemaspaandra, and Menton [35] showed that certain destructive variants of these problems in fact are the same. In this paper, we will not consider any cases of control by partition, though.…”