Hybrid Elections Broaden Complexity‐Theoretic Resistance to Control

Abstract

Electoral control refers to attempts by an election's organizer ("the chair") to influence the outcome by adding/deleting/partitioning voters or candidates. The groundbreaking work of Bartholdi, Tovey, and Trick [2] on (constructive) control proposes computational complexity as a means of resisting control attempts: Look for election systems where the chair's task in seeking control is itself computationally infeasible.We introduce and study a method of combining two or more candidate-anonymous election schemes in such a way that the combined scheme possesses all the resistances to control (i.e., all the NP-hardnesses of control) possessed by any of its constituents: It combines their strengths. From this and new resistance constructions, we prove for the first time that there exists an election scheme that is resistant to all twenty standard types of electoral control. * A preliminary version was presented at the COMSOC-06 workshop and a preliminary, seven-page version of this paper appeared in IJCAI-07 [1]. †

A Richer Understanding of the Complexity of Election Systems

“…Winners in these three systems are not necessarily unique and these three winner problems ask whether a given candidate is a winner. However, as mentioned earlier, Hemaspaandra, Hemaspaandra, and Rothe [HHR06] have shown that the unique winner problems for Dodgson, Young, and Kemeny elections are Θ p 2 -complete as well. Θ p 2 -completeness suggests that the relevant problem is far from being efficiently solvable, and there are many ways in which completeness for this higher level of the polynomial hierarchy speaks more powerfully than would completeness for its kid brother, NP [HHR97b].…”

“…In fact, at the time the table's work on control was completed (early 2005), no system was proven to be immune-or-resistant to all twenty types of control (or even to the ten constructive types, or to the ten destructive types). However, recently, work of Hemaspaandra, Hemaspaandra, and Rothe [HHR06] has shown how to "hybridize" collections of elections in a way such that the hybrid election has a polynomial-time winner problem if all its constituent systems have polynomial-time winner problems, yet the hybrid system is resistant to every one of the twenty types of control to which one or more of its constituent systems is resistant. Simply put, the hybridization scheme combines strengths without adding weaknesses.…”

Copeland Voting Fully Resists Constructive Control

“…Since by Theorem 3.1, Copeland 0.5 provides resistance for all ten basic constructive control types and for CCAC u , and also for the four basic types of destructive voter control, the hybrid (in the sense of [9]) of plurality with Copeland 0.5 is resistant to each basic type of constructive and destructive control and in addition to constructive and destructive AC u control. This result follows via Theorem 3.1 and the results of Hemaspaandra et al [9]. And, unlike the hybrid system constructed by Hemaspaandra et al [9], this hybrid uses only natural systems as its constituents.…”

“…One key issue here is that there might be attempts to influence the outcome of elections. Settings in which such influence on elections can be implemented include manipulation [3,13], electoral control [2,7,8,9,13], and bribery [6,7]. Although reasonable election systems typically are susceptible to these kinds of influence (for manipulation this is universally true, via the Gibbard-Satterthwaite and Duggan-Schwartz Theorems), computational complexity can be used to provide some protection in each such setting.…”

“…In their seminal paper they introduced a number of fundamental control scenarios involving (what is now called) constructive control, i.e., where the chair's goal is to make some designated candidate win. Other papers studying control include [8,13,9,7], which in addition to constructive control also consider destructive control, where the chair tries to preclude some designated candidate from winning. The notion of bribery in elections was introduced by Faliszewski et al [6] and was also studied in [7].…”