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36
Determining Possible and Necessary Winners under Common Voting Rules Given Partial Orders
"... Usually a voting rule or correspondence requires agents to give their preferences as linear orders. However, in some cases it is impractical for an agent to give a linear order over all the alternatives. It has been suggested to let agents submit partial orders instead. Then, given a profile of part ..."
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Cited by 48 (13 self)
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Usually a voting rule or correspondence requires agents to give their preferences as linear orders. However, in some cases it is impractical for an agent to give a linear order over all the alternatives. It has been suggested to let agents submit partial orders instead. Then, given a profile of partial orders and a candidate c, two important questions arise: first, is c guaranteed to win, and second, is it still possible for c to win? These are the necessary winner and possible winner problems, respectively. We consider the setting where the number of alternatives is unbounded and the votes are unweighted. We prove that for Copeland, maximin, Bucklin, and ranked pairs, the possible winner problem is NPcomplete; also, we give a sufficient condition on scoring rules for the possible winner problem to be NPcomplete (Borda satisfies this condition). We also prove that for Copeland and ranked pairs, the necessary winner problem is coNPcomplete. All the hardness results hold even when the number of undetermined pairs in each vote is no more than a constant. We also present polynomialtime algorithms for the necessary winner problem for scoring rules, maximin, and Bucklin.
A short introduction to computational social choice
 Proc. 33rd Conference on Current Trends in Theory and Practice of Computer Science
, 2007
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A Multivariate Complexity Analysis of Determining Possible Winners Given Incomplete Votes
"... The POSSIBLE WINNER problem asks whether some distinguished candidate may become the winner of an election when the given incomplete votes are extended into complete ones in a favorable way. POSSIBLE WINNER is NPcomplete for common voting rules such as Borda, many other positional scoring rules, Bu ..."
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Cited by 29 (10 self)
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The POSSIBLE WINNER problem asks whether some distinguished candidate may become the winner of an election when the given incomplete votes are extended into complete ones in a favorable way. POSSIBLE WINNER is NPcomplete for common voting rules such as Borda, many other positional scoring rules, Bucklin, Copeland etc. We investigate how three different parameterizations influence the computational complexity of POSSIBLE WINNER for a number of voting rules. We show fixedparameter tractability results with respect to the parameter “number of candidates ” but intractability results with respect to the parameter “number of votes”. Finally, we derive fixedparameter tractability results with respect to the parameter “total number of undetermined candidate pairs ” and identify an interesting polynomialtime solvable special case for Borda. 1
Towards a dichotomy of finding possible winners in elections based on scoring rules
 In Proc. 34th MFCS, volume 5734 of LNCS
, 2009
"... Abstract. 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. ..."
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Cited by 18 (2 self)
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Abstract. 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, if 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, kapproval, and Borda. Generalizing previous NPhardness results for some special cases and providing new manyone reductions, 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 NPcomplete 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. 1
Dealing with incomplete preferences in soft constraint problems
 CP 2007 (The 13th International Conference on Principles and Practice of Constraint Programming
, 2007
"... Abstract. We consider soft constraint problems where some of the preferences may be unspecified. This models, for example, situations with several agents providing the data, or with possible privacy issues. In this context, we study how to find an optimal solution without having to wait for all the ..."
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Cited by 15 (11 self)
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Abstract. We consider soft constraint problems where some of the preferences may be unspecified. This models, for example, situations with several agents providing the data, or with possible privacy issues. In this context, we study how to find an optimal solution without having to wait for all the preferences. In particular, we define an algorithm to find a solution which is necessarily optimal, that is, optimal no matter what the missing data will be, with the aim to ask the user to reveal as few preferences as possible. Experimental results show that in many cases a necessarily optimal solution can be found without eliciting too many preferences. 1
How to Rig Elections and Competitions
"... We investigate the extent to which it is possible to rig the agenda of an election or competition so as to favor a particular candidate in the presence of imperfect information about the preferences of the electorate. We assume that what is known about an electorate is the probability that any given ..."
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Cited by 14 (0 self)
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We investigate the extent to which it is possible to rig the agenda of an election or competition so as to favor a particular candidate in the presence of imperfect information about the preferences of the electorate. We assume that what is known about an electorate is the probability that any given candidate will beat another. As well as presenting some analytical results relating to the complexity of finding and verifying agenda, we develop heuristics for agenda rigging, and investigate the performance of these heuristics for both randomly generated data and realworld data from tennis and basketball competitions. 1
Aggregating partially ordered preferences
"... Abstract. Preferences are not always expressible via complete linear orders: sometimes it is more natural to allow for the presence of incomparable outcomes. This may hold both in the agents ’ preference ordering and in the social order. In this paper we consider this scenario and we study what prop ..."
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Cited by 14 (0 self)
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Abstract. Preferences are not always expressible via complete linear orders: sometimes it is more natural to allow for the presence of incomparable outcomes. This may hold both in the agents ’ preference ordering and in the social order. In this paper we consider this scenario and we study what properties it may have. In particular, we show that, despite the added expressivity and ability to resolve conflicts provided by incomparability, classical impossibility results (such as Arrow’s theorem, MullerSatterthwaite’s theorem, and GibbardSatterthwaite’s theorem) still hold. We also prove some possibility results, generalizing Sen’s theorem for majority voting. To prove these results, we define new notions of unanimity, monotonicity, dictator, triplewise valuerestriction, and strategyproofness, which are suitable and natural generalizations of the classical ones for complete orders. 1
Dealing with Incomplete Agents ’ Preferences and an Uncertain Agenda in Group Decision Making via Sequential Majority Voting
"... We consider multiagent systems where agents ’ preferences are aggregated via sequential majority voting: each decision is taken by performing a sequence of pairwise comparisons where each comparison is a weighted majority vote among the agents. Incompleteness in the agents ’ preferences is common i ..."
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Cited by 13 (8 self)
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We consider multiagent systems where agents ’ preferences are aggregated via sequential majority voting: each decision is taken by performing a sequence of pairwise comparisons where each comparison is a weighted majority vote among the agents. Incompleteness in the agents ’ preferences is common in many reallife settings due to privacy issues or an ongoing elicitation process. In addition, there may be uncertainty about how the preferences are aggregated. For example, the agenda (a tree whose leaves are labelled with the decisions being compared) may not yet be known or fixed. We therefore study how to determine collectively optimal decisions (also called winners) when preferences may be incomplete, and when the agenda may be uncertain. We show that it is computationally easy to determine if a candidate decision always wins, or may win, whatever the agenda. On the other hand, it is computationally hard to know whether a candidate decision wins in at least one agenda for at least one completion of the agents ’ preferences. These results hold even if the agenda must be balanced so that each candidate decision faces the same number of majority votes. Such results are useful for reasoning about preference elicitation. They help understand the complexity of tasks such as determining if a decision can be taken collectively, as well as knowing if the winner can be manipulated by appropriately ordering the agenda.
Fixing a tournament
 In Proceedings of AAAI’10
, 2010
"... We consider a very natural problem concerned with game manipulation. Let G be a directed graph where the nodes represent players of a game, and an edge from u to v means that u can beat v in the game. (If an edge (u, v) is not present, one cannot match u and v.) Given G and a “favorite ” node A, is ..."
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Cited by 12 (5 self)
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We consider a very natural problem concerned with game manipulation. Let G be a directed graph where the nodes represent players of a game, and an edge from u to v means that u can beat v in the game. (If an edge (u, v) is not present, one cannot match u and v.) Given G and a “favorite ” node A, is it possible to set up the bracket of a balanced singleelimination tournament so that A is guaranteed to win, if matches occur as predicted by G? We show that the problem is NPcomplete for general graphs. For the case when G is a tournament graph we give several interesting conditions on the desired winner A for which there exists a balanced singleelimination tournament which A wins, and it can be found in polynomial time.
Towards a Dichotomy for the Possible Winner Problem in Elections Based on Scoring Rules
, 2010
"... 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 direc ..."
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Cited by 12 (0 self)
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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, kapproval, and Borda. Generalizing previous NPhardness 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 NPcomplete 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.