We develop new voting mechanisms for the case when voters and candidates are located in an arbitrary unknown metric space, and the goal is to choose a candidate minimizing social cost: the total distance from the voters to this candidate. Previous work has often assumed that only ordinal preferences of the voters are known (instead of their true costs), and focused on minimizing distortion: the quality of the chosen candidate as compared with the best possible candidate. In this paper, we instead assume that a (very small) amount of information is known about the voter preference strengths, not just about their ordinal preferences. We provide mechanisms with much better distortion when this extra information is known as compared to mechanisms which use only ordinal information. We quantify tradeoffs between the amount of information known about preference strengths and the achievable distortion. We further provide advice about which type of information about preference strengths seems to be the most useful. Finally, we conclude by quantifying the ideal candidate distortion, which compares the quality of the chosen outcome with the best possible candidate that could ever exist, instead of only the best candidate that is actually in the running.
Attempts to exploit preference strength information have led to various approaches for modeling, eliciting, measuring, and aggregating people's preference intensities in a variety of fields, including Likert scales, semantic differential scales, sliders, constant sum paired comparisons, graded pair comparisons, response times, willingness to pay, vote buying, and many others (see @cite_26 @cite_21 @cite_12 for summaries). In our work we specifically consider only a small amount of coarse information about preference strengths, since obtaining detailed information is extremely difficult. Intuitively, any rule used to aggregate preference strengths must ask under what circumstances an apathetic majority' should win over a more passionate minority @cite_10 , and we provide a partial answer to this question when the objective is to minimize distortion.
@cite_26 @cite_10 @cite_21 @cite_12
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Voting is a way to aggregate individual voters' preferences. Traditionally a voter's preference is represented by a total order on the set of candidates. However, sometimes one may not have complete information about a voter's preference, and in this case, can only model a voter's preference as a partial order. Given this framework, there has been work on computing the possible and necessary winners of a (partial) profile. In this paper, we take a step further, look at sets of questions to ask in order to determine the outcome of such a partial profile. Specifically, we call a set of questions a deciding set for a candidate if the outcome of the vote for the candidate is determined no matter how the questions are answered by the voters, and a possible winning (losing) set if there is a way to answer these questions to make the candidate a winner (loser) of the vote. We discuss some interesting properties about these sets of queries, prove some complexity results about them under some well-known voting rules such as plurality and Borda, and consider their application in vote elicitation. Voting is a general method for aggregating the preferences of multiple agents. Each agent ranks all the possible alternatives, and based on this, an aggregate ranking of the alternatives (or at least a winning alternative) is produced. However, when there are many alternatives, it is impractical to simply ask agents to report their complete preferences. Rather, the agents' preferences, or at least the relevant parts thereof, need to be elicited. This is done by asking the agents a (hopefully small) number of simple queries about their preferences, such as comparison queries, which ask an agent to compare two of the alternatives. Prior work on preference elicitation in voting has focused on the case of unrestricted preferences. It has been shown that in this setting, it is sometimes necessary to ask each agent (almost) as many queries as would be required to determine an arbitrary ranking of the alternatives. In contrast, in this paper, we focus on single-peaked preferences. We show that such preferences can be elicited using only a linear number of comparison queries, if either the order with respect to which preferences are single-peaked is known, or at least one other agent's complete preferences are known. We show that using a sublinear number of queries does not suffice. We also consider the case of cardinally single-peaked preferences. For this case, we show that if the alternatives' cardinal positions are known, then an agent's preferences can be elicited using only a logarithmic number of queries; however, we also show that if the cardinal positions are not known, then a sublinear number of queries does not suffice. We present experimental results for all elicitation algorithms. We also consider the problem of only eliciting enough information to determine the aggregate ranking, and show that even for this more modest objective, a sublinear number of queries per agent does not suffice for known ordinal or unknown cardinal positions. Finally, we discuss whether and how these techniques can be applied when preferences are almost single-peaked. Preferences can be aggregated using voting rules. We consider here the family of rules which perform a sequence of pairwise majority comparisons between two candidates. The winner thus depends on the chosen sequence of comparisons, which can be represented by a binary tree. We address the difficulty of computing candidates that win for some trees, and then introduce and study the notion of fair winner, i.e. candidates who win in a balanced tree. We then consider the situation where we lack complete informations about preferences, and determine the computational complexity of computing winners in this case. The Internet gives individuals more choice in political news and information sources and more tools to filter out disagreeable information. Citing the preference described by selective exposure theory — people prefer information that supports their beliefs and avoid counter-attitudinal information — observers warn that people may use these tools to access only agreeable information and thus live in ideological echo chambers. We report on a field deployment of a browser extension that showed users feedback about the political lean of their weekly and all time reading behaviors. Compared to a control group, showing feedback led to a modest move toward balanced exposure, corresponding to 1-2 visits per week to ideologically opposing sites or 5-10 additional visits per week to centrist sites.