We characterize single-crossing preference profiles in terms of two forbidden substructures, one of which contains three voters and six (not necessarily distinct) alternatives, and one of which contains four voters and four (not necessarily distinct) alternatives. We also provide an efficient way to decide whether a preference profile is single-crossing.JEL Classification: D71, C78.
In their AAMAS 2020 paper, Szufa et al. presented a "map of elections" that visualizes a set of 800 elections generated from various statistical cultures. While similar elections are grouped together on this map, there is no obvious interpretation of the elections' positions. We provide such an interpretation by introducing four canonical “extreme” elections, acting as a compass on the map. We use them to analyze both a dataset provided by Szufa et al. and a number of real-life elections. In effect, we find a new parameterization of the Mallows model, based on measuring the expected swap distance from the central preference order, and show that it is useful for capturing real-life scenarios.
We investigate the problem of deciding whether a given preference profile is close to having a certain nice structure, as for instance single-peaked, single-caved, singlecrossing, value-restricted, best-restricted, worst-restricted, medium-restricted, or groupseparable profiles. We measure this distance by the number of voters or alternatives that have to be deleted to make the profile a nicely structured one. Our results classify the problem variants with respect to their computational complexity, and draw a clear line between computationally tractable (polynomial-time solvable) and computationally intractable (NP-hard) questions. * A preliminary short version of this work has been presented at Single-peakedness implies a number of nice properties, as for instance strategy-proofness of a family of voting rules [42] and transitivity of the majority relation [36]. Furthermore, Arrow's impossibility result collapses for single-peaked profiles. In a similar spirit (but in the algorithmic branch), Walsh [51], Brandt et al. [11], and Faliszewski et al. [31] show that many electoral bribery, control and manipulation problems that are NP-hard in the general case become tractable under single-peaked profiles. Besides the single-peaked domain, the literature contains many other restricted domains of nicely structured preference profiles (see Section 2 for precise mathematical definitions).• Sen [47] and Sen and Pattanaik [46] introduced the domain of value-restricted preference profiles which satisfy the following: for every triple of alternatives, one alternative is not preferred most by any individual (best-restricted profile), or one is not preferred least by any individual (worst-restricted profile), or one is not considered as the intermediate alternative by any individual (medium-restricted profile).• Inada [35,36] considered the domain of group-separable preference profiles which satisfy the following: the alternatives can be split into two groups such that every voter prefers every alternative in the first group to those in the second group, or prefers every alternative in the second group to those in the first group. Every group-separable profile is also medium-restricted.• Single-caved preference profiles are derived from single-peaked profiles by reversing the preferences of every voter. Sometimes single-caved profiles are also called singledipped [37].• Single-crossing preference profiles go back to the seminal papers of Mirrlees [40] and Roberts [45] on income taxation. A preference profile is single-crossing if there exists a linear order of the voters such that each pair of alternatives separates this order into two sub-orders where in each sub-order, all voters agree on the relative order of this pair. Similar to the single-peaked property, single-crossing profiles can also be recognized in polynomial time [12,22,26].Unfortunately, real-world elections are almost never single-peaked, value-restricted, groupseparable, single-caved or single-crossing. Usually there are maverick voters whose preferences are ...
In the SHIFT BRIBERY problem, we are given an election (based on preference orders), a preferred candidate p, and a budget. The goal is to ensure that p wins by shifting p higher in some voters' preference orders. However, each such shift request comes at a price (depending on the voter and on the extent of the shift) and we must not exceed the given budget. We study the parameterized computational complexity of SHIFT BRIBERY with respect to a number of parameters (pertaining to the nature of the solution sought and the size of the election) and several classes of price functions. When we parameterize SHIFT BRIBERY by the number of affected voters, then for each of our voting rules (Borda, Maximin, Copeland) the problem is W[2]-hard. If, instead, we parameterize by the number of positions by which p is shifted in total, then the problem is fixed-parameter tractable for Borda and Maximin, and is W[1]-hard for Copeland. If we parameterize by the budget for the cost of shifting, then the results depend on the price function class. We also show that SHIFT BRIBERY tends to be tractable when parameterized by the number of voters, but that the results for the number of candidates are more enigmatic.
We consider opinion diffusion in binary influence networks, where at each step one or more agents update their opinions so as to be in agreement with the majority of their neighbors. We consider several ways of manipulating the majority opinion in a stable outcome, such as bribing agents, adding/deleting links, and changing the order of updates, and investigate the computational complexity of the associated problems, identifying tractable and intractable cases.
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