Markus Brill

Bypassing combinatorial protections: polynomial-time algorithms for single-peaked electorates

For many election systems, bribery (and related) attacks have been shown NP-hard using constructions on combinatorially rich structures such as partitions and covers. It is important to learn how robust these hardness protection results are, in order to find whether they can be relied on in practice. This paper shows that for voters who follow the most central political-science model of electorates—single-peaked preferences—those protections vanish. By using single-peaked preferences to simplify combinatorial covering challenges, we show that NP-hard bribery problems—including those for Kemeny and Llull elections—fall to polynomial time. By using single-peaked preferences to simplify combinatorial partition challenges, we show that NP-hard partition-of-voters problems fall to polynomial time. We furthermore show that for single-peaked electorates, the winner problems for Dodgson and Kemeny elections, though Θp2-complete in the general case, fall to polynomial time. And we completely classify the complexity of weighted coalition manipulation for scoring protocols in single-peaked electorates.

The Computational Complexity of Random Serial Dictatorship

In social choice settings with linear preferences, random dictatorship is known to be the only social decision scheme satisfying strategyproofness and ex post efficiency. When also allowing indifferences, random serial dictatorship RSD is a well-known generalization of random dictatorship that retains both properties. RSD has been particularly successful in the special domain of random assignment where indifferences are unavoidable. While executing RSD is obviously feasible, we show that computing the resulting probabilities is #P-complete and thus intractable, both in the context of voting and assignment.

Possible and necessary winners of partial tournaments

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.

Necessary and sufficient conditions for the strategyproofness of irresolute social choice functions

While the Gibbard-Satterthwaite theorem states that every non-dictatorial and resolute, i.e., single-valued, social choice function is manipulable, it was recently shown that a number of appealing irresolute Condorcet extensions are strategyproof according to Kellys preference extension. In this paper, we study whether these results carry over to stronger preference extensions due to Fishburn and Gärdenfors. For both preference extensions, we provide sufficient conditions for strategyproofness and identify social choice functions that satisfy these conditions, answering a question by Gärdenfors [15] in the affirmative. We also show that some more discriminatory social choice functions fail to satisfy necessary conditions for strategyproofness.

On the fixed-parameter tractability of composition-consistent tournament solutions

Tournament solutions, i.e., functions that associate with each complete and asymmetric relation on a set of alternatives a non-empty subset of the alternatives, play an important role within social choice theory and the mathematical social sciences at large. Laffond et al. have shown that various tournament solutions satisfy composition-consistency, a structural invariance property based on the similarity of alternatives. We define the decomposition degree of a tournament as a parameter that reflects its decomposability and show that computing any composition-consistent tournament solution is fixed-parameter tractable with respect to the decomposition degree. Furthermore, we experimentally investigate the decomposition degree of two natural distributions of tournaments and its impact on the running time of computing the tournament equilibrium set.

On the Complexity of Iterated Weak Dominance in Constant-Sum Games

In game theory, a players action is said to be weakly dominated if there exists another action that, with respect to what the other players do, is never worse and sometimes strictly better. We investigate the computational complexity of the process of iteratively eliminating weakly dominated actions (IWD) in two-player constant-sum games, i.e., games in which the interests of both players are diametrically opposed. It turns out that deciding whether an action is eliminable via IWD is feasible in polynomial time whereas deciding whether a given subgame is reachable via IWD is NP-complete. The latter result is quite surprising as we are not aware of other natural computational problems that are intractable in constant-sum games. Furthermore, we slightly improve a result by Conitzer and Sandholm [6] by showing that typical problems associated with IWD in win-lose games with at most one winner are NP-complete.

On the structure of stable tournament solutions

A fundamental property of choice functions is stability, which, loosely speaking, prescribes that choice sets are invariant under adding and removing unchosen alternatives. We provide several structural insights that improve our understanding of stable choice functions. In particular, (1) we show that every stable choice function is generated by a unique simple choice function, which never excludes more than one alternative, (2) we completely characterize which simple choice functions give rise to stable choice functions, and (3) we prove a strong relationship between stability and a new property of tournament solutions called local reversal symmetry. Based on these findings, we provide the first concrete tournament—consisting of 24 alternatives—in which the tournament equilibrium set fails to be stable. Furthermore, we prove that there is no more discriminating stable tournament solution than the bipartisan set and that the bipartisan set is the unique most discriminating tournament solution which satisfies standard properties proposed in the literature.

Extending tournament solutions

An important subclass of social choice functions, so-called majoritarian (or C1) functions, only take into account the pairwise majority relation between alternatives. In the absence of majority ties—e.g., when there is an odd number of agents with linear preferences—the majority relation is antisymmetric and complete and can thus conveniently be represented by a tournament. Tournaments have a rich mathematical theory and many formal results for majoritarian functions assume that the majority relation constitutes a tournament. Moreover, most majoritarian functions have only been defined for tournaments and allow for a variety of generalizations to unrestricted preference profiles, none of which can be seen as the unequivocal extension of the original function. In this paper, we argue that restricting attention to tournaments is justified by the existence of a conservative extension, which inherits most of the commonly considered properties from its underlying tournament solution.

Testing Substitutability of Weak Preferences

In various models of matching markets, substitutable preferences constitute the largest domain for which stable matchings are guaranteed to exist. Recently, Hatfield et al. (2012) have proposed an efficient algorithm to test substitutability of strict preferences. In this note we show how the algorithm by Hatfield et al. can be adapted in such a way that it can test substitutability of weak preferences as well. When restricted to the domain of strict preferences, our algorithm is faster than Hatfield et al.’s original algorithm by a linear factor.

The Computational Complexity of Weak Saddles

We continue the recently initiated study of the computational aspects of weak saddles, an ordinal set-valued solution concept proposed by Shapley. Brandt et al. gave a polynomial-time algorithm for computing weak saddles in a subclass of matrix games, and showed that certain problems associated with weak saddles of bimatrix games are NP-complete. The important question of whether weak saddles can be found efficiently was left open. We answer this question in the negative by showing that finding weak saddles of bimatrix games is NP-hard, under polynomial-time Turing reductions. We moreover prove that recognizing weak saddles is coNP-complete, and that deciding whether a given action is contained in some weak saddle is hard for parallel access to NP and thus not even in NP unless the polynomial hierarchy collapses. Our hardness results are finally shown to carry over to a natural weakening of weak saddles.