We study the control complexity of fallback voting. Like manipulation and bribery, electoral control describes ways of changing the outcome of an election; unlike manipulation or bribery attempts, control actions---such as adding/deleting/partitioning either candidates or voters---modify the participative structure of an election. Via such actions one can try to either make a favorite candidate win ("constructive control") or prevent a despised candidate from winning ("destructive control"). Computational complexity can be used to protect elections from control attempts, i.e., proving an election system resistant to some type of control shows that the success of the corresponding control action, though not impossible, is computationally prohibitive. We show that fallback voting, an election system combining approval with majority voting, is resistant to each of the common types of candidate control and to each common type of constructive control. Among natural election systems with a polynomial-time winner problem, only plurality and sincere-strategy preference-based approval voting (SP-AV) were previously known to be fully resistant to candidate control, and only Copeland voting and SP-AV were previously known to be fully resistant to constructive control. However, plurality has fewer resistances to voter control, Copeland voting has fewer resistances to destructive control, and SP-AV (which like fallback voting has 19 out of 22 proven control resistances) is arguably less natural a system than fallback voting.
Deep Dive into Control Complexity in Fallback Voting.
We study the control complexity of fallback voting. Like manipulation and bribery, electoral control describes ways of changing the outcome of an election; unlike manipulation or bribery attempts, control actions—such as adding/deleting/partitioning either candidates or voters—modify the participative structure of an election. Via such actions one can try to either make a favorite candidate win (“constructive control”) or prevent a despised candidate from winning (“destructive control”). Computational complexity can be used to protect elections from control attempts, i.e., proving an election system resistant to some type of control shows that the success of the corresponding control action, though not impossible, is computationally prohibitive. We show that fallback voting, an election system combining approval with majority voting, is resistant to each of the common types of candidate control and to each common type of constructive control. Among natural election systems with a p
Voting is a way of aggregating individual preferences (or votes) to achieve a societal consensus on which among several alternatives (or candidates) to choose. This is a central method of decision-making not only in human societies but also in, e.g., multiagent systems where autonomous software agents may have differing individual preferences on a given number of alternatives. Voting has been studied intensely in areas as diverse as social choice theory and political science, economics, operations research, artificial intelligence, and other fields of computer science. Voting applications in computer science include the web-page ranking problem [DKNS01], similarity search [FKS03], planning [ER93], and recommender systems [GMHS99]. For such applications, it is important to understand the computational properties of election systems.
Various ways of tampering with the outcome of elections have been studied from a complexitytheoretic perspective, in particular the complexity of changing an election’s outcome by manipulation [BTT89, BO91, CSL07, HH07, FHHR09c], control [BTT92, HHR07, FHHR09a, HHR09, ENR09, FHHR09c], and bribery [FHH09,FHHR09a], see also the surveys by Faliszewski et al. [FHHR09b] and Baumeister et al. [BEH + 09].
In control scenarios, an external actor-commonly referred to as the “chair”-seeks to either make a favorite candidate win (constructive control) or block a despised candidate’s victory (destructive control) via actions that change the participative structure of the election. Such actions include adding, deleting, and partitioning either candidates or voters; the 22 commonly studied control actions, and their corresponding control problems, are described formally in Section 2.
We study the control complexity of fallback voting, an election system introduced by Brams and Sanver [BS09] as a way of combining approval and preference-based voting. We prove that fallback voting is resistant (i.e., the corresponding control problem is NP-hard) to each of the 14 common types of candidate control. In addition, we show that fallback voting is resistant to five types of voter control. In particular, fallback voting is resistant to each of the 11 common types of constructive control. Among natural election systems with a polynomial-time winner determination procedure, only plurality and sincere-strategy preference-based approval voting (SP-AV) were previously known to be fully resistant to candidate control [BTT92, HHR07, ENR09], and only Copeland voting and SP-AV were previously known to be fully resistant to constructive control [FHHR09a,ENR09]. However, SP-AV (as modified by [ENR09]) is arguably less natural a system than fallback voting, 1 and plurality has fewer resistances to voter control and Copeland voting has fewer resistances to destructive control than fallback voting.
This paper is organized as follows. In Section 2, we recall some notions from voting theory, define the commonly studied types of control, and explain Brams and Sanver’s fallback voting procedure [BS06] in detail. Our results on the control complexity of fallback voting are presented in Section 3. Finally, Section 4 provides some conclusions and open questions. 1 SP-AV is (a variant of) another hybrid system combining approval and preference-based voting that was also proposed by Brams and Sanver (see [BS06]). The reason we said SP-AV is less natural than fallback voting is that, in order to preserve “admissibility” of votes (as required by Brams and Sanver [BS06] to preclude trivial approval strategies), SP-AV (as modified by Erdélyi et al. [ENR09]) employs an additional rule to (re-)coerce admissibility (in particular, if in the course of a control action an originally admissible vote becomes inadmissible). This point has been discussed in detail by Baumeister et al. [BEH + 09]. In a nutshell, this rule, if applied, changes the approval strategies of the votes originally cast by the voters. The effect of this rule is that SP-AV can be seen as a hybrid between approval and plurality voting, and it indeed possesses each resistance either of these two systems has (and many of these resistance proofs are based on slightly modified constructions from the resistance proofs for either plurality or approval due to Hemaspaandra et al. [HHR07]). In contrast, here we study the original variant of fallback voting, as proposed by Brams and Sanver [BS09], in which votes, once cast, do not change.
An election is a pair (C,V ), where C is a finite set of candidates and V is a collection of votes over C. How the votes are represented depends on the election system used. Many systems (such as majority, Condorcet, and Copeland voting as well as the class of scoring protocols including plurality, veto, and Borda voting; see, e.g., [BF02]) represent the voters’ preferences as strict, linear orders over the candidates. In approval voting [BF78,BF83,Bra80], however, a vote over C is a socalled approval strategy, a subset of C containing
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