Parameterized Control Complexity in Fallback Voting

Similar issues arise throughout science, in the current era of vastly expanded pools of raw information. Many of these issues can be thought of as specialized Google-style queries: for example, “which genes in the database(s) seem to be most relevant to this new information …?”

In all areas of science, government and industry, the collation of information, and prioritization of allied strategic options, has moved front-and-center both tactically and strategically.

This general situation has profoundly stimulated, and drawn upon, a mathematical subject that used to be a bit of a backwater, concerned with “ideal” voting systems and so forth. But now (and forever more) rapid and frequent, algorithmically-powered, elections about relevant information are becoming a cornerstone of civilization.

As in (almost) all things algorithmic, rich questions inevitably arise about the tractability of the desired information-election processes, and their susceptibility to manipulation (or primary data error). This paper is about this context of research.

Given the many information-resolution contexts in which voting systems are relevant, presenting different characteristics and challenges, it is an important resource that various voting systems have been proposed. These are now being vigorously investigated in regards algorithms and complexity issues of their strengths and weaknesses in various applied contexts. There are many papers regarding the complexity-theoretic aspects of the many different ways of changing the outcome of an election, like manipulation [BTT89, BO91, CSL07, HH07, FHHR09b], where a group of voters cast their votes strategically, bribery [FHH09,FHHR09a], where an external agents bribes a group of voters in order to change their votes, and control [BTT92, HHR07, FHHR09a, HHR09, ENR09, FHHR09b, EPR10], where an external agent-which is referred to as “The Chair”-changes the structure of the election (for example, by adding/deleting/partitioning either candidates or voters).

In this paper, we are concerned with control issues for the relatively recently introduced voting system of fallback voting (FV, for short) [BS09]. Fallback voting is the natural voting system that currently has the most resistances (i.e., makes the chair’s task hard) for control attacks [EPR10]. We investigate the issues in the framework parameterized complexity. Many voting systems present NPhard algorithmic challenges. Parameterized complexity is a particularly appropriate framework in many contexts of voting systems because it is concerned with exact results that exploit the structure of input distributions. It is not appropriate in political contexts, for example, to algorithmically determine a winner “approximately”.

In this section, we explicate voting systems in general, fallback voting in particular, parameterized complexity theory, and some graph theory that we will use.

An election (C,V ) consists of a finite set of candidates C and a finite collection of voters V who express their preferences over the candidates in C, and distinct voters can have the same preferences. A voting system is a set of rules determining the winners of an election. In our paper we only consider the unique-winner model, where we want to have exactly one winner at the time. Votes can be represented in different ways, depending on the voting system used. One widely-used representations of votes is via preference rankings. In this case each voter has to specify a tie-free linear ordering of all candidates. Such voting systems are for example Condorcet, Borda count, plurality or veto; see, e.g., [BF02]. Approval voting, introduced by Brams and Fishburn [BF78,BF83] is not a preference based voting system. In approval voting each voter has to vote “yes” or “no” for each candidate and the candidates with the most “yes” votes are the winners of the election. Clearly, approval voting completely ignores preference rankings.

Brams and Sanver [BS09] introduced two voting systems that combine preference-based with approval voting. One of these systems is fallback voting.

). Let (C,V ) be an election. Every voter v ∈ V has to divide the set of candidates C into two subsets S v ⊆ C indicating that v approves of all candidates in S v and disapproves of all candidates in C -S v . S v is called v’s approval strategy. In addition, each voter v ∈ V provides also a tie-free linear ordering of all candidates in S v .

Representation of votes: Let S v = {c 1 , c 2 , . . . , c k } for a voter v who ranks the candidates in S v as follo

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