From Election Fraud to Finding the Dream Team: A Study of the Computational Complexity in Voting Problems and Stability in Hedonic Games

In this thesis we study computational aspects of different voting problems and cooperative games with hedonic preferences. For two well-studied voting systems, namely Bucklin and fallback voting, we present a detailed study of the computational complexity of common manipulative attacks on elections. We fully describe the classical worst-case complexity of manipulation, bribery, and swap bribery in both voting systems and furthermore study extension bribery tailored to fallback elections. We extend existing studies of the complexity of electoral control in Bucklin and fallback elections by investigating control by adding/deleting candidates or voters parameterized by the number of added/deleted candidates or voters, respectively. We complement these results with the first experimental evaluation of electoral control based on randomly generated elections, which also provides results for plurality elections. Furthermore, we study the complexity of the margin of victory problem and its relation to destructive unweighted bribery. We show that for the cup rule, both destructive unweighted bribery and the margin of victory problem are NP-complete. Beyond that, we introduce and study two new variants of this problem: exact margin of victory and swap margin of victory. The exact variant can be shown to be DP-complete for the Schulze and the cup rule, as well as for the family of Copelandα systems. The swap margin of victory problem, on the other hand, can be shown to be solvable in deterministic polynomial time for k-approval and certain positional scoring rules, while for the cup rule this problem is NP-complete. We furthermore show the close connection of the swap margin of victory problem and destructive swap bribery with unit costs. Moreover, we define a new notion of the possible winner problem for weighted elections in which the uncertainty lies in the voters’ weights and the complete preferences of the voters are given. We study this problem in detail for nonnegative rational weights and show that for positional scoring rules and Bucklin and fallback voting this problem can be solved in deterministic polynomial time. In the context of hedonic games we study the computational complexity of wonderful stability existence and strict core stability existence in enemy-based hedonic games. We improve the best known lower bounds for these problems by establishing DP-hardness results. We furthermore prove that coDPhardness of these problems directly implies hardness forΘ 2 (parallel access to NP), which in turn would resolve the question of the exact complexity of the former problem, as it is known to be contained in this complexity class. Beyond that, we introduce a new class of hedonic games in which each player divides her co-players into friends, enemies, and other (neutral) players and furthermore provides a weak ranking of her friends and a weak ranking of her enemies. These preferences over players are extended to preferences over coalitions using a generalized Bossong-Schweigert extension principle. The thus defined FENhedonic games may have incomplete preferences, meaning that there can be pairs of coalitions that are incomparable with respect to this preference extension. We suggest to break these incomparabilities by defining cardinal comparability functions based on Borda-like scoring vectors leading to additively separable preferences. We show that this class of Borda-induced FEN-hedonic games is a strict subclass of additively separable hedonic games and provide a detailed study on the computational complexity of the existence and verification problems of commonly studied stability concepts such as Nash stability and (strict) core stability.