The rapidly growing field of computational social choice, at the intersection of computer science and economics, deals with the computational aspects of collective decision making. This handbook, written by thirty-six prominent members of the computational social choice community, covers the field comprehensively. Chapters devoted to each of the field's major themes offer detailed introductions. Topics include voting theory (such as the computational complexity of winner determination and manipulation in elections), fair allocation (such as algorithms for dividing divisible and indivisible goods), coalition formation (such as matching and hedonic games), and many more. Graduate students, researchers, and professionals in computer science, economics, mathematics, political science, and philosophy will benefit from this accessible and self-contained book.

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Handbook of Computational Social Choice

False-name manipulation refers to the question of whether a player in a weighted voting game can increase her power by splitting into several players and distributing her weight among these false identities. Relatedly, the beneficial merging problem asks whether a coalition of players can increase their power in a weighted voting game by merging their weights. For the problems of whether merging or splitting players in weighted voting games is beneficial in terms of the Shapley-Shubik and the normalized Banzhaf index, merely NP-hardness lower bounds are known, leaving the question about their exact complexity open. For the Shapley-Shubik and the probabilistic Banzhaf index, we raise these lower bounds to hardness for PP, "probabilistic polynomial time," a class considered to be by far a larger class than NP. For both power indices, we provide matching upper bounds for beneficial merging and, whenever the new players' weights are given, also for beneficial splitting, thus resolving previous conjectures in the affirmative. Relatedly, we consider the beneficial annexation problem, asking whether a single player can increase her power by taking over other players' weights. It is known that annexation is never disadvantageous for the Shapley-Shubik index, and that beneficial annexation is NP-hard for the normalized Banzhaf index. We show that annexation is never disadvantageous for the probabilistic Banzhaf index either, and for both the Shapley-Shubik index and the probabilistic Banzhaf index we show that it is NP-complete to decide whether annexing another player is advantageous. Moreover, we propose a general framework for merging and splitting that can be applied to different classes and representations of games.

False-name manipulation in weighted voting games is hard for probabilistic polynomial time

We investigate the fairness of Bayesian estimators (BEs) by viewing them as (irresolute) voting rules and evaluating them by satisfaction of desirable social choice axioms. We characterize the class of BEs that satisfy neutrality by the class of BEs with neutral structures. We prove that a BE with a neutral structure is a minimax rule if it further satisfies parameter connectivity. We prove that no BE satisfies strict Condorcet criterion. We also propose three new BEs of natural frameworks and investigate their computational complexity and satisfaction of monotonicity and Condorcet criterion.

/pdf/bayesian-estimators-as-voting-rules-56xbqafsra.pdf

Bayesian estimators as voting rules

Path-disruption games, recently introduced by Bachrach and Porat, are coalitional games played on graphs where one or multiple adversaries each seeks to reach a given target vertex from a given source vertex, while a coalition of agents seeks to prevent that from happening by blocking every path from the source to the target for each adversary. These coalitional games model, for instance, security issues in computer networks. Inspired by bribery in voting, we introduce the notion of bribery for path-disruption games. We analyze the question of how hard it is to decide whether the adversaries can bribe some of the agents such that no coalition will form that blocks all paths for them. We show that this problem is NP-complete for a single adversary and complete for Σ2P=NPNP${\Sigma }^{\mathrm {P}}_{2} = \text {NP}^{\text {NP}}$, the second level of the polynomial hierarchy, for the case of multiple adversaries. We also expand the model by allowing uncertainty about the targets: In probabilistic path-disruption games, we assign to each vertex the probability that an adversary wants to reach it, and we study the complexity of problems related to common solution concepts (such as the core and the ź-core) and other properties of such games.

Path-Disruption Games: Bribery and a Probabilistic Model

Jiři Matousek (1963–2015) had many breakthrough contributions in mathematics and algorithm design. His milestone results are not only profound but also elegant. By going beyond the original objects—such as Euclidean spaces or linear programs—Jirka found the essence of the challenging mathematical/algorithmic problems as well as beautiful solutions that were natural to him, but were surprising discoveries to the field.

Network Essence: PageRank Completion and Centrality-Conforming Markov Chains

Inspired by social choice theory in voting and other contexts, we provide the first axiomatic approach to community identification in a social network. We start from an abstract framework, called preference networks, which, for each member, gives their ranking of all the other members of the network. This complete-information preference model enables us to focus on the fundamental conceptual question: What constitutes a community in a social network?Within this framework, we axiomatically study the formation and structures of communities in two different ways. First, we apply social choice theory and define communities indirectly by postulating that they are fixed points of a preference aggregation function obeying certain desirable axioms. Second, we directly postulate six desirable axioms for communities to satisfy, without reference to preference aggregation. For the second approach, we prove a taxonomy theorem that provides a beautiful structural characterization of the family of axiom-conforming community rules as a lattice. We complement this structural theorem with a complexity result, showing that, while for some rules in the lattice, community characterization is straightforward, it is coNP-complete to characterize subsets according to others. Our study also sheds light on the limitations of defining community rules solely based on preference aggregation, namely that many aggregation functions lead to communities which violate at least one of our community axioms. These include any aggregation function satisfying Arrow's "independence of irrelevant alternatives" axiom, as well as commonly used aggregation schemes like the Borda count or generalizations thereof. Finally, we give a polynomial-time rule consistent with five axioms and weakly satisfying the sixth axiom.

/pdf/an-axiomatic-approach-to-community-detection-3mijug9nx5.pdf

An Axiomatic Approach to Community Detection

Path-disruption games, a class of cooperative games introduced by Bachrach and Porat [1], model situations where the players, sitting on the vertices of a given graph, try to prevent - by blocking all possible paths - their adversaries from traveling from a set of source vertices to a set of target vertices. Rey and Rothe[3] studied bribery in these games and showed that when costs are assigned to the vertices, the corresponding problem is NP-complete in the single-adversary case, and is in Sigma_2^p = NP^NP, the second level of the polynomial hierarchy, in the multiple-adversary case. They left open whether the latter problem is Sigma_2^p-complete. In this note, we solve this open question in the affirmative.

/pdf/bribery-in-multiple-adversary-path-disruption-games-is-hard-1bxhao1tq4.pdf

Bribery in multiple-adversary path-disruption games is hard for the second level of the polynomial hierarchy

We provide the first social choice theory approach to the question of what constitutes a community in a social network. Inspired by the classic preferences models in social choice theory, we start from an abstract social network framework, called preference networks; these consist of a finite set of members where each member has a total-ranking preference of all members in the set. 
Within this framework, we develop two complementary approaches to axiomatically study the formation and structures of communities. (1) We apply social choice theory and define communities indirectly by postulating that they are fixed points of a preference aggregation function obeying certain desirable axioms. (2) We directly postulate desirable axioms for communities without reference to preference aggregation, leading to eight natural community axioms. 
These approaches allow us to formulate and analyze community rules. We prove a taxonomy theorem that provides a structural characterization of the family of community rules that satisfies all eight axioms. The structure is actually quite beautiful: these community rules form a bounded lattice under the natural intersection and union operations. Our structural theorem is complemented with a complexity result: while identifying a community by the most selective rule of the lattice is in P, deciding if a subset is a community by the most comprehensive rule of the lattice is coNP-complete. Our studies also shed light on the limitations of defining community rules solely based on preference aggregation: any aggregation function satisfying Arrow's IIA axiom, or based on commonly used aggregation schemes like the Borda count or generalizations thereof, lead to communities which violate at least one of our community axioms. Finally, we give a polynomial-time rule consistent with seven axioms and weakly satisfying the eighth axiom.

Fixed-Points of Social Choice: An Axiomatic Approach to Network Communities

We generalize traditional equilibrium concepts for finite games in extensive form with behavioral strategies so that they apply to all games, including games of imperfect recall. Adapting and augmenting previous de finitions (in particular, by Piccione and Rubinstein, and by Battigalli), we de fine four notions: Distributed Agent Equilibrium (DAE), Distributed Nash Equilibrium (DNE), Distributed Sequential Equilibrium (DSE), and Distributed Perfect Equilibrium (DPE). We show that, in a precise sense, these extend the classical equilibrium notions: (a) they form a strict inclusion hierarchy (e.g., every DNE is a DAE but not necessarily vice versa, and so on up the hierarchy), (b) every game has a DPE (and therefore also a DSE, DNE and DAE), and (c) in the subclass of games of perfect recall, DAE, DNE, DSE and DPE collapse, respectively, to agent equilibrium, Nash equilibrium, sequential equilibrium, and perfect equilibrium. In service of these results we introduce several novel notions - including the distributed agent form and phantom strategies - which may be interesting in their own right.

https://papers.ssrn.com/sol3/Delivery.cfm/SSRN_ID2188646_code1945926.pdf?abstractid=2188646&mirid=1&type=2

Equilibria in Finite Games with Imperfect Recall

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