An important aspect in systems of multiple autonomous agents is the exploitation of synergies via coalition formation. In this paper, we solve various open problems concerning the computational complexity of stable partitions in additively separable hedonic games. First, we propose a polynomial-time algorithm to compute a contractually individually stable partition. This contrasts with previous results such as the NP-hardness of computing individually stable or Nash stable partitions. Secondly, we prove that checking whether the core or the strict core exists is NP-hard in the strong sense even if the preferences of the players are symmetric. Finally, it is shown that verifying whether a partition consisting of the grand coalition is contractually strict core stable or Pareto optimal is coNP-complete.

Two fundamental axioms in social choice theory are consistency with respect to a variable electorate and consistency with respect to components of similar alternatives. In the context of traditional non-probabilistic social choice, these axioms are incompatible with each other. We show that in the context of probabilistic social choice, these axioms uniquely characterize a function proposed by Fishburn (Rev. Econ. Stud., 51(4), [683][684][685][686][687][688][689][690][691][692] 1984). Fishburn's function returns so-called maximal lotteries, i.e., lotteries that correspond to optimal mixed strategies in the symmetric zero-sum game induced by the pairwise majority margins. Maximal lotteries are guaranteed to exist due to von Neumann's Minimax Theorem, are almost always unique, and can be efficiently computed using linear programming.

Tournament solutions constitute an important class of social choice functions that only depend on the pairwise majority comparisons between alternatives. Recent analytical results have shown that several concepts with appealing axiomatic properties such as the Banks set or the minimal covering set tend to not discriminate at all when the tournaments are chosen from the uniform distribution. This is in sharp contrast to empirical studies which have found that real-world preference profiles often exhibit Condorcet winners, i.e., alternatives that all tournament solutions select as the unique winner. In this work, we aim to fill the gap between these extremes by examining the distribution of the number of alternatives returned by common tournament solutions for empirical data as well as data generated according to stochastic preference models such as impartial culture, impartial anonymous culture, Mallows mixtures, spatial models, and Pólya-Eggenberger urn models.

We study the problem of computing possible and necessary winners for partially specified weighted and unweighted tournaments. This problem arises naturally in elections with incompletely specified votes, partially completed sports competitions, and more generally in any scenario where the outcome of some pairwise comparisons is not yet fully known. We specifically consider a number of well-known solution concepts---including the uncovered set, Borda, ranked pairs, and maximin---and show that for most of them, possible and necessary winners can be identified in polynomial time. These positive algorithmic results stand in sharp contrast to earlier results concerning possible and necessary winners given partially specified preference profiles.

Many hardness results in computational social choice make use of the fact that every directed graph may be induced as the pairwise majority relation of some preference profile. However, this fact requires a number of voters that is almost linear in the number of alternatives. It is therefore unclear whether these results remain intact when the number of voters is bounded, as is, for example, typically the case in search engine aggregation settings. In this paper, we provide a systematic study of majority digraphs for a constant number of voters resulting in analytical, experimental, and complexity-theoretic insights. First, we characterize the set of digraphs that can be induced by two and three voters, respectively, and give sufficient conditions for larger numbers of voters. Second, we present a surprisingly efficient implementation via SAT solving for computing the minimal number of voters that is required to induce a given digraph and experimentally evaluate how many voters are required to induce the majority digraphs of real-world and generated preference profiles. Finally, we leverage our sufficient conditions to show that the winner determination problem of various well-known voting rules remains hard even when there is only a small constant number of voters. In particular, we show that Kemeny's rule is hard to evaluate for 7 voters, while previous methods could only establish such a result for constant even numbers of voters.