Winner Problem Research Articles

The distribution of argmaximum or a winner problem

We consider a limit theorem for the distribution of a random variable (r.v.) An=argmaxi:1…n{Xi}, where Xi’s are independent continuous non-negative random r.v.’s. The Xi,i=1.…,n, may be interpreted as the gains of n players in a game, and the r.v. An itself as the number of a “winner”. The paper contains some limit theorems for the distribution of An as n→∞.

The possible winner with uncertain weights problem

The original possible winner problem consists of an unweighted election with partial preferences and a distinguished candidate c and asks whether the preferences can be extended to total ones such that c wins the given election. We introduce a novel variant of this problem: possible winner with uncertain weights. In this variant, for a given weighted election, not some of the preferences but some of the preferences' weights are uncertain. We introduce a general framework to study this problem for nonnegative integer and rational weights as well as for four different variations of the problem itself: with and without given upper bounds on the total weight and with and without given ranges to choose weights from. We study the complexity of these problems for important voting systems such as scoring protocols, (simplified) Bucklin and fallback voting, plurality with runoff and veto with runoff, Copelandα, ranked pairs, and Borda.

On the exact amount of missing information that makes finding possible winners hard

In the possible winner problem, we need to compute if a set of partial votes can be completed such that a given candidate wins the election under some specific voting rule. In this paper, we determine the smallest number of undetermined pairs per partial vote for which the Possible Winner problem is NP-complete. In particular, we find the exact values of t for which the Possible Winner problem transitions to being NP-complete from being in P, where t is the maximum number of undetermined pairs in every vote. We demonstrate tight results for a broad class of scoring rules, Copelandα for every α∈[0,1], maximin, and Bucklin voting rules. A somewhat surprising aspect of our results is that for many of these rules, the Possible Winner problem turns out to be hard even if every vote has at most one undetermined pair of candidates.

Control in the presence of manipulators: cooperative and competitive cases

Control and manipulation are two of the most studied types of attacks on elections. In this paper, we study the complexity of control attacks on elections in which there are manipulators. We study both the case where the “chair” who is seeking to control the election is allied with the manipulators, and the case where the manipulators seek to thwart the chair. In the latter case, we see that the order of play substantially influences the complexity. We prove upper bounds, holding over every election system with a polynomial-time winner problem, for all standard control cases, and some of these bounds are at the second or third level of the polynomial hierarchy, and we provide matching lower bounds to prove these tight. Nonetheless, for important natural systems the complexity can be much lower. We prove that for approval and plurality elections, the complexity of even competitive clashes between a controller and manipulators falls far below those high bounds, even as low as polynomial time. Yet for a Borda-voting case we show that such clashes raise the complexity unless NP = coNP.

Very Hard Electoral Control Problems

It is important to understand how the outcome of an election can be modified by an agent with control over the structure of the election. Electoral control has been studied for many election systems, but for all these systems the winner problem is in P, and so control is in NP. There are election systems, such as Kemeny, that have many desirable properties, but whose winner problems are not in NP. Thus for such systems control is not in NP, and in fact we show that it is typically complete for ∑p2 (i.e., NPNP, the second level of the polynomial hierarchy). This is a very high level of complexity. Approaches that perform quite well for solving NP problems do not necessarily work for ∑p2-complete problems. However, answer set programming is suited to express problems in ∑p2, and we present an encoding for Kemeny control.

Cycles and Intractability in a Large Class of Aggregation Rules

We introduce the (j,k)-Kemeny rule -- a generalization of Kemeny's voting rule that aggregates j-chotomous weak orders into a k-chotomous weak order. Special cases of (j,k)-Kemeny include approval voting, the mean rule and Borda mean rule, as well as the Borda count and plurality voting. Why, then, is the winner problem computationally tractable for each of these other rules, but intractable for Kemeny? We show that intractability of winner determination for the (j,k)-Kemeny rule first appears at the j=3, k=3 level. The proof rests on a reduction of max cut to a related problem on weighted tournaments, and reveals that computational complexity arises from the cyclic part in the fundamental decomposition of a weighted tournament into cyclic and cocyclic components. Thus the existence of majority cycles -- the engine driving both Arrow's impossibility theorem and the Gibbard-Satterthwaite theorem -- also serves as a source of computational complexity in social choice.

Possible winner problems on partial tournaments: a parameterized study

We study possible winner problems related to the uncovered set and the Banks set on partial tournaments from the viewpoint of parameterized complexity. We first study a problem where given a partial tournament D and a subset X of vertices, we are asked to add some arcs to D such that all vertices in X are included in the uncovered set. We focus on two parameterizations: parameterized by |X| and parameterized by the number of arcs to be added. In addition, we study a parameterized variant of the problem which is to determine whether all vertices of X can be included in the uncovered set by reversing at most k arcs. Finally, we study some parameterizations of a possible winner problem on partial tournaments, where we are given a partial tournament D and a distinguished vertex p, and asked whether D has a maximal transitive subtournament with p being the 0-indegree vertex. These parameterized problems are related to the Banks set. We achieve $$\mathcal {XP}$$XP results, $$\mathcal {W}$$W-hardness results as well as $$\mathcal {FPT}$$FPT results along with a kernelization lower bound for the problems studied in this paper.

Kernelization complexity of possible winner and coalitional manipulation problems in voting

In the Possible winner problem in computational social choice theory, we are given a set of partial preferences and the question is whether a distinguished candidate could be made winner by extending the partial preferences to linear preferences. Previous work has provided, for many common voting rules, fixed parameter tractable algorithms for the Possible winner problem, with number of candidates as the parameter. However, the corresponding kernelization question is still open and in fact, has been mentioned as a key research challenge [10]. In this paper, we settle this open question for many common voting rules.We show that the Possible winner problem for maximin, Copeland, Bucklin, ranked pairs, and a class of scoring rules that includes the Borda voting rule does not admit a polynomial kernel with the number of candidates as the parameter. We show however that the Coalitional manipulation problem which is an important special case of the Possible winner problem does admit a polynomial kernel for maximin, Copeland, ranked pairs, and a class of scoring rules that includes the Borda voting rule, when the number of manipulators is polynomial in the number of candidates. A significant conclusion of our work is that the Possible winner problem is harder than the Coalitional manipulation problem since the Coalitional manipulation problem admits a polynomial kernel whereas the Possible winner problem does not admit a polynomial kernel.

Control complexity in Bucklin and fallback voting: A theoretical analysis

Electoral control models ways of changing the outcome of an election via such actions as adding, deleting, or partitioning either candidates or voters. To protect elections from such control attempts, computational complexity has been used to establish so-called resistance results. We show that fallback voting, an election system proposed by Brams and Sanver [12] to combine Bucklin with approval voting, displays the broadest control resistance currently known to hold among natural election systems with a polynomial-time winner problem. We also study the control complexity of Bucklin voting and show that it performs almost as well as fallback voting in terms of control resistance. Furthermore, we investigate the parameterized control complexity of Bucklin and fallback voting, according to several parameters that are often likely to be small for typical instances. In a companion paper [28], we challenge our worst-case complexity results from an experimental point of view.

Taking the final step to a full dichotomy of the possible winner problem in pure scoring rules

The Possible Winner problem asks, given an election where the votersʼ preferences over the candidates are specified only partially, whether a designated candidate can become a winner by suitably extending all the votes. Betzler and Dorn (2010) [2] proved a result that is only one step away from a full dichotomy of this problem for the important class of pure scoring rules in the case of unweighted votes and an unbounded number of candidates: Possible Winner is NP-complete for all pure scoring rules except plurality, veto, and the scoring rule with vector ( 2 , 1 , … , 1 , 0 ) , but is solvable in polynomial time for plurality and veto. We take the final step to a full dichotomy by showing that Possible Winner is NP-complete also for the scoring rule with vector ( 2 , 1 , … , 1 , 0 ) .

Possible Winners when New Candidates Are Added: The Case of Scoring Rules

In some voting situations, some new candidates may show up in the course of the process. In this case, we may want to determine which of the initial candidates are possible winners, given that a fixed number k of new candidates will be added. Focusing on scoring rules, we give complexity results for the above possible winner problem.

Towards a dichotomy for the Possible Winner problem in elections based on scoring rules

To make a joint decision, agents (or voters) are often required to provide their preferences as linear orders. To determine a winner, the given linear orders can be aggregated according to a voting protocol. However, in realistic settings, the voters may often only provide partial orders. This directly leads to the Possible Winner problem that asks, given a set of partial votes, whether a distinguished candidate can still become a winner. In this work, we consider the computational complexity of Possible Winner for the broad class of voting protocols defined by scoring rules. A scoring rule provides a score value for every position which a candidate can have in a linear order. Prominent examples include plurality, k-approval, and Borda. Generalizing previous NP-hardness results for some special cases, we settle the computational complexity for all but one scoring rule. More precisely, for an unbounded number of candidates and unweighted voters, we show that Possible Winner is NP-complete for all pure scoring rules except plurality, veto, and the scoring rule defined by the scoring vector ( 2 , 1 , … , 1 , 0 ) , while it is solvable in polynomial time for plurality and veto.

Sincere‐Strategy Preference‐Based Approval Voting Fully Resists Constructive Control and Broadly Resists Destructive Control

AbstractWe study sincere‐strategy preference‐based approval voting (SP‐AV), a system proposed by Brams and Sanver [1] and here adjusted so as to coerce admissibility of the votes (rather than excluding inadmissible votes a priori), with respect to procedural control. In such control scenarios, an external agent seeks to change the outcome of an election via actions such as adding/deleting/partitioning either candidates or voters. SP‐AV combines the voters' preference rankings with their approvals of candidates, where in elections with at least two candidates the voters' approval strategies are adjusted – if needed – to approve of their most‐preferred candidate and to disapprove of their least‐preferred candidate. This rule coerces admissibility of the votes even in the presence of control actions, and hybridizes, in effect, approval with pluralitiy voting. We prove that this system is computationally resistant (i.e., the corresponding control problems are NP‐hard) to 19 out of 22 types of constructive and destructive control. Thus, SP‐AV has more resistances to control than is currently known for any other natural voting system with a polynomial‐time winner problem. In particular, SP‐AV is (after Copeland voting, see Faliszewski et al. [2, 3]) the second natural voting system with an easy winner‐determination procedure that is known to have full resistance to constructive control, and unlike Copeland voting it in addition displays broad resistance to destructive control (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Hybrid Elections Broaden Complexity‐Theoretic Resistance to Control

AbstractElectoral control refers to attempts by an election's organizer (“the chair”) to influence the outcome by adding/deleting/partitioning voters or candidates. The important paper of Bartholdi, Tovey, and Trick [1] that introduces (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 a neutral, anonymous election scheme (whose winner problem is computable in polynomial time) that is resistant to all twenty standard types of electoral control (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Llull and Copeland Voting Computationally Resist Bribery and Constructive Control

Control and bribery are settings in which an external agent seeks to influence the outcome of an election. Constructive control of elections refers to attempts by an agent to, via such actions as addition/deletion/partition of candidates or voters, ensure that a given candidate wins. Destructive control refers to attempts by an agent to, via the same actions, preclude a given candidate's victory. An election system in which an agent can sometimes affect the result and it can be determined in polynomial time on which inputs the agent can succeed is said to be vulnerable to the given type of control. An election system in which an agent can sometimes affect the result, yet in which it is NP-hard to recognize the inputs on which the agent can succeed, is said to be resistant to the given type of control. Aside from election systems with an NP-hard winner problem, the only systems previously known to be resistant to all the standard control types were highly artificial election systems created by hybridization. This paper studies a parameterized version of Copeland voting, denoted by Copeland^\alpha, where the parameter \alpha is a rational number between 0 and 1 that specifies how ties are valued in the pairwise comparisons of candidates. In every previously studied constructive or destructive control scenario, we determine which of resistance or vulnerability holds for Copeland^\alpha for each rational \alpha, 0 \leq \alpha \leq 1. In particular, we prove that Copeland^{0.5}, the system commonly referred to as ``Copeland voting,'' provides full resistance to constructive control, and we prove the same for Copeland^\alpha, for all rational \alpha, 0 < \alpha < 1. Among systems with a polynomial-time winner problem, Copeland voting is the first natural election system proven to have full resistance to constructive control. In addition, we prove that both Copeland^0 and Copeland^1 (interestingly, Copeland^1 is an election system developed by the thirteenth-century mystic Llull) are resistant to all standard types of constructive control other than one variant of addition of candidates. Moreover, we show that for each rational \alpha, 0 \leq \alpha \leq 1, Copeland^\alpha voting is fully resistant to bribery attacks, and we establish fixed-parameter tractability of bounded-case control for Copeland^\alpha. We also study Copeland^\alpha elections under more flexible models such as microbribery and extended control, we integrate the potential irrationality of voter preferences into many of our results, and we prove our results in both the unique-winner model and the nonunique-winner model. Our vulnerability results for microbribery are proven via a novel technique involving min-cost network flow.

A note on the query complexity of the Condorcet winner problem

Given an unknown tournament over { 1 , … , n } , we show that the query complexity of the question “Is there a vertex with outdegree n − 1 ?” (known as a Condorcet winner in social choice theory) is exactly 2 n − ⌊ log ( n ) ⌋ − 2 . This stands in stark contrast to the evasiveness of this property in general digraphs.

Exact Complexity of the Winner Problem for Young Elections

In 1977 Young proposed a voting scheme that extends the Condorcet Principle based on the fewest possible number of voters whose removal yields a Condorcet winner. We prove that both the winner and the ranking problem for Young elections is complete for \p || NP , the class of problems solvable in polynomial time by parallel access to NP. Analogous results for Lewis Carroll's 1876 voting scheme were recently established by Hemaspaandra et al. In contrast, we prove that the winner and ranking problems in Fishburn's homogeneous variant of Carroll's voting scheme can be solved efficiently by linear programming.