This tutorial will give a brief introduction to Computational Social Choice and highlight some of the current research trends in the field. We will review some of the fundamental concepts from Social Choice Theory, such as preference handling, aggregation problems, voting rules, fair division procedures, and mechanism design, and point out recent work that has applied tools from Computer Science to these problem domains, such as complexity-theoretic studies, algorithms, and logical modelling.

This tutorial introduces to the field of Complexity Theory with a particular focus on problems from Computational Social Choice. For example, a famous result from Social Choice Theory (known as the Gibbard-Satterthwaite Theorem) says that essentially every nondictatorial voting system can be manipulated by strategic voting. Computational complexity, however, can provide protection against manipulation and other attempts of influencing the outcome of elections, such as bribery, control, and lobbying. Since voting is a useful mechanism for preference aggregation and collective decision-making, these results have applications not only in the context of political elections within human societies, but also for various fields of computer science, including multiagent systems (within distributed Artificial Intelligence), web-page ranking algorithms for meta-search engines, and the design of recommender systems. Other problems from Social Choice Theory and Mechanism Design that have been analyzed complexity-theoretically are related to multiagent ressource allocation, power indices, weighted voting games, etc., and we will encounter some of them while we wander through the landscape of complexity classes.

An accessible survey of this area is:
A Richer Understanding of the Complexity of Election Systems. P. Faliszewski, E. Hemaspaandra, L. Hemaspaandra, and J. Rothe. In Fundamental Problems in Computing: Essays in Honor of Professor Daniel J. Rosenkrantz, S. Ravi and S. Shukla, editors. Springer-Verlag, to appear. Available at arxiv.org/abs/cs.GT/0609112

A growing interdisciplinary literature discusses the problem of judgment aggregation: How can a group of individuals make consistent collective judgments (in the form of acceptance/rejection or true/false assignments) on some propositions based on the group members' individual judgments on these propositions? The recent interest in this problem was sparked by the observation that majority voting, perhaps the most common democratic decision method, fails to guarantee consistent collective judgments whenever there are non-trivial logical connections between propositions. For instance, if one individual accepts “p”, “if p then q”, and “q”, a second accepts “p”, but “not if p then q”, and “not q”, and a third accepts “not p”, but “if p then q” as well as “not q”, then there are majorities for “p”, for “if p then q”, and for “not q”, a logically inconsistent set of propositions, despite the consistency of individual judgments. The theory of judgment aggregation asks whether, and to what extent, problems of this kind generalize to other aggregation methods and other decision problems, and offers an axiomatic analysis of possible solutions to them. The aim of this tutorial is to give an accessible overview of some of the key concepts, ideas and results of this theory.

Here is a non-technical survey paper.

In the classical theory of social choice, a theory developed by game-theorists and theoretical economists, we consider a set of agents (voters) and a set of alternatives. Each agent ranks the alternatives, and the major aim is to find a good way to aggregate the individual preferences into a social preference. The major tool offered in this theory is the axiomatic approach: study properties (termed axioms) that characterize particular aggregation rules, and analyze whether particular desired properties can be simultaneously satisfied. In a ranking system [1] the set of voters and the set of alternatives coincide, e.g. they are both the pages in the web; in this case the links among pages are interpreted as votes: pages that page p links to are preferable by page p to pages it does not link to; the problem of preference aggregation becomes the problem of page ranking. Trust systems are personalized ranking systems [3] where the ranking is done for (and from the perspective of) each individual agent. Here the idea is to see how to rank agents from the perspective of a particular agent/user, based on the trust network generated by the votes. In a trust-based recommendation system the agents also express opinions about external topics, and a user who has not expressed an opinion should be recommended one based on the opinions of others and the trust network [6]. Hence, we get a sequence of very interesting settings, extending upon classical social choice, where the axiomatic approach can be used.

On the practical side, ranking, reputation, recommendation, and trust systems have become essential ingredients of web-based multi-agent systems (e.g. [9,13,7,14,8]). These systems aggregate agents' reviews of products and services, and of each other, into valuable information. Notable commercial examples include Amazon and E-Bay's recommendation and reputation systems (e.g. [12]), Google's page ranking system [11], and the Epinions web of trust/reputation system (e.g. [10]). Our work shows that an extremely powerful way for the study and design of such systems is the axiomatic approach, extending upon the classical theory of social choice. In this talk we discuss some representative results of our work [14,1,2,4,5,3,6].

We study the complexity of rationalizing choice behavior. We do so by analyzing two polar cases, and a number of intermediate ones. In our most structured case, that is where choice behavior is defined in universal choice domains and satisfies the weak axiom of revealed preference, finding the complete preorder rationalizing choice behavior is a simple matter. In the polar case, where no restriction whatsoever is imposed, either on choice behavior or on choice domain, finding the complete preorders that rationalize behavior turns out to be intractable. We show that the task of finding the rationalizing complete preorders is equivalent to a graph problem. This allows the search for existing algorithms in the graph theory literature, for the rationalization of choice.

This paper combines social choice theory with discrete optimization. We assume that individuals have preferences over edges of a graph that need to be aggregated. The goal is to find the socially best spanning tree. As ranking all spanning trees is becoming infeasible even for small numbers of vertices and/or edges of a graph, our interest lies in finding algorithms that determine a socially “best” spanning tree in a simple manner. This problem is closely related to the minimum (or maximum) spanning tree problem. The main result in this paper shows that for the various underlying set ranking rules discussed in this paper, using a greedy algorithm to determine the maximal spanning tree based on such a group ranking will always provide the “best” spanning tree.

Electronic communities are relying increasingly on the results of continuous polls to rate the effectiveness of their members and offerings. Sites such as eBay and Amazon solicit feedback about merchants and products, with prior feedback and results-to-date available to participants before they register their approval ratings. In such a setting, participants are understandably prone to exaggerate their approval or disapproval, so as to move the average rating in a favored direction.

We explore several protocols that solicit approval ratings and report a consensus outcome without rewarding insincerity. One such system is rationally optimal while still reporting an outcome based on the the usual notion of averaging. That system allows all participants to manipulate the outcome in turn. Although multiple equilibria exist for that system, they all report the same average approval rating as their outcome. We generalize our results to obtain a range of declared-strategy voting systems suitable for approval-rating polls.

This paper presents a quantitative solution to the problem of aggregating referee scores for documents submitted to peer-reviewed conferences or scientific journals. The proposed approach is a particular application of the Dempster-Shafer theory to a very restricted setting, from which an interesting algebraic framework results. The paper investigates the algebraic properties of this framework and shows how to apply it to the score aggregation and document ranking problems.

The computation of Kemeny scores is central to many applications in the context of rank aggregation. Unfortunately, the problem is NP-hard. Extending our previous work (AAIM08), we show that the Kemeny score of an election can be computed efficiently whenever the average pairwise distance between two input votes is not too large. In other words, Kemeny Score is fixed-parameter tractable with respect to the parameter “average pairwise Kendall-Tau distance d_a”. We describe a fixed-parameter algorithm with running time O^*(4^d_a).

A social welfare function (SWF) takes a set of votes (linear orders over a set of alternatives) as input, and produces one or more rankings (also linear orders over the alternatives) as output. Such functions have many applications, for example, aggregating the preferences of multiple agents, or merging rankings (of, say, webpages) into a single ranking. The key issue is choosing an SWF to use. One natural and previously studied approach is to assume that there is an unobserved “correct” ranking, and the votes are noisy estimates of this. Then, we can use the SWF that always chooses the maximum likelihood estimate (MLE) of the correct ranking. In this paper, we define simple ranking scoring functions (SRSFs) and show that the class of neutral SRSFs is exactly the class of neutral SWFs that are MLEs for some noise model. We also define extended ranking scoring functions (ERSFs) and show a condition under which these coincide with SRSFs. We study key properties such as consistency and continuity, and consider some example SWFs. In particular, we study Single Transferable Vote (STV), a commonly used SWF, showing that it is an ERSF but not an SRSF, thereby clarifying the extent to which it is an MLE function. This also gives a new perspective on how ties should be broken under STV. We leave some open questions.

For the problem of adjudicating conflicting claims (O'Neill, 1982; for a survey, see Thomson, 2003), we offer simple criteria to compare rules on the basis of the Lorenz order. They exploit several recently developed techniques to structure the space of rules.

The first results concern the family of “ICI rules” (Thomson, 2008, forthcoming). This family contains the constrained equal awards (Maimonides, 12th Century), constrained equal losses (Maimonides, 12th Century), Talmud (Aumann and Maschler, 1985), and minimal overlap (Ibn Ezra, 12th Centrury; O'Neill, 1982) rules. We obtain a condition relating the parameters associated with two rules in the family guaranteeing that one Lorenz dominates the other. We prove parallel results for a second family (CIC family, Thomson, 2008, forthcoming), which is obtained from the first one by exchanging, for each problem, how well agents with relatively larger claims are treated as compared to agents with relatively smaller claims. This second family also contains the constrained equal awards and constrained equal losses rules.

The next results concern the family of “consistent” rules (Young, 1987). A rule is consistent if the recommendation it makes for each problem is never contradicted by the recommendation it makes for each reduced problem obtained by imagining some claimants leaving the scene with their awards and reassessing the situation at that point. The main result here is the identification of circumstances under which the Lorenz order is “lifted by consistency” from the two-claimant case to arbitrarily many claimants. (The concept of lifting is adapted from one developed for properties of rules by Hokari and Thomson, 2008, forthcoming.) This means that if two rules are consistent and one of them Lorenz dominates the other in the two-claimant case, this domination extends to arbitrarily many claimants.

Finally, we exploit the notion of an operator on the space of rules (Thomson and Yeh, 2008, forthcoming). An operator is a mapping that associates with each rule another one. The operators we consider are the duality operator, the claims truncation operator, and the attribution of minimal right operator. We also consider the operator that associates with each rule and each list of non-negative weights for them adding up to one, their weighted average. An operator “preserves an order” if whenever a rule Lorenz dominates another one, this domination extends to the two rules obtained by subjecting them to the operator. We identify circumstances under which certain operators preserve the Lorenz order (or reverse it), and circumstances under which a rule can be Lorenz compared to the rule obtained by subjecting it to the operator.

In studies of multi-agent interaction, especially in game theory, the notion of equilibrium often plays a prominent role. A typical scenario for the belief merging problem is one in which several agents pool their beliefs together to form a consistent “group” picture of the world. The aim of this paper is to define and study new notions of equilibria in belief merging. To do so, we assume the agents arrive at consistency via the use of a social belief removal function, in which each agent, using his own individual removal function, removes some belief from his stock of beliefs. We examine several notions of equilibria in this setting, assuming a general framework for individual belief removal due to Booth et al. We look at their inter-relations as well as prove their existence or otherwise. We also show how our equilibria can be seen as a generalisation of the idea of taking maximal consistent subsets of agents.

In the logic based framework of knowledge representation and reasoning many operators have been defined in order to capture different kinds of change: revision, update, merging and many others. There are close links between revision, update, and merging. Merging operators can be considered as extensions of revision operators to multiple belief bases. And update operators can be considered as pointwise revision, looking at each model of the base, instead of taking the base as a whole. Thus, a natural question is the following one: Are there natural operators that are pointwise merging, just as update are pointwise revision? The goal of this work is to give a positive answer to this question. In order to do that, we introduce a new class of operators: the confluence operators. These new operators can be useful for modelling negotiation processes.

Control and bribery are settings in which an external agent seeks to influence the outcome of an election. Constructive control of elections refers to attempts by an agent to, via such actions as addition/deletion/partition of candidates or voters, ensure that a given candidate wins. Destructive control refers to attempts by an agent to, via the same actions, preclude a given candidate's victory. An election system in which an agent can affect the result and in which recognizing the inputs on which the agent can succeed is NP-hard (respectively, polynomial-time solvable) is said to be resistant (respectively, vulnerable) to the given type of control. Aside from election systems with an NP-hard winner problem, the only systems previously known to be resistant to all the standard control types are highly artificial election systems created by hybridization.

We study a parameterized version of Copeland voting, denoted by Copeland^α, where the parameter α is a rational number between 0 and 1 that specifies how ties are valued in the pairwise comparisons of candidates. In every previously studied constructive or destructive control scenario, we determine which of resistance or vulnerability holds for Copeland^α for each rational α, 0 ≤ α ≤ 1. In particular, we prove that Copeland^0.5, the system commonly referred to as “Copeland voting”, provides full resistance to constructive control. Among the systems with a polynomial-time winner problem, this is the first natural election system proven to have full resistance to constructive control. In addition, we prove that both Copeland⁰ and Copeland¹ (interestingly, the latter is an election system developed by the thirteenth-century mystic Ramon Llull) are resistant to all the standard types of constructive control other than one variant of addition of candidates. Moreover, we show that for each rational α, 0 ≤ α ≤ 1, Copeland^α voting is fully resistant to bribery attacks, and we establish fixed-parameter tractability of bounded-case control for Copeland^α.

We also study Copeland^α elections under more flexible models such as microbribery and extended control, we integrate the potential irrationality of voter preferences into many of our results, and we prove our results in both the unique-winner and the nonunique-winner model. Our vulnerability results for microbribery are proven via a technique involving min-cost network flow.

We study sincere-strategy preference-based approval voting (SP-AV), a system proposed by Brams and Sanver [Electoral Studies, 25(2):287-305, 2006], with respect to procedural control. In such control scenarios, an external agent seeks to change the outcome of an election via actions such as adding/deleting/partitioning either candidates or voters. SP-AV combines the voters' preference rankings with their approvals of candidates, and we adapt it here so as to keep its useful features with respect to approval strategies even in the presence of control actions. We prove that this system is computationally resistant (i.e., the corresponding control problems are NP-hard) to 19 out of 22 types of constructive and destructive control. Thus, SP-AV has more resistances to control, by three, than is currently known for any other natural voting system with a polynomial-time winner problem. In particular, SP-AV is (after Copeland voting, see Faliszewski et al. [AAIM-2008, Springer LNCS 5034, pp. 165-176, 2008]) the second natural voting system with an easy winner-determination procedure that is known to have full resistance to constructive control, and unlike Copeland voting it in addition displays broad resistance to destructive control.

In this paper, we set up a framework to study approximation of manipulation, control, and bribery in elections. We show existence of approximation algorithms (even fully polynomial time approximation schemes) as well as obtain inapproximability results. In particular, we show that a large subclass of scoring protocols admits fully-polynomial time approximation schemes for the coalitional weighted manipulation problem and that if certain families of scoring protocols (e.g., veto) admitted such approximation schemes then P = NP. We also show that bribery for Borda count is NP-complete and that there is no approximation algorithm that achieves even a polynomial approximation ratio for bribery in Borda count for the case where voters have prices.

In this paper, we study the computational complexity of the unweighted coalitional manipulation (UCM) problem under some common voting rules. We show that the UCM problem under maximin is NP-complete. We also show that the UCM problem under ranked pairs is NP-complete, even if there is only one manipulator. Finally, we present a polynomial-time algorithm for the UCM problem under Bucklin.

A recent trend (especially in electronic commerce) is higher levels of expressiveness in the mechanisms that mediate interactions such as auctions, exchanges, catalog offers, voting systems, matching of peers, and so on. Participants can express their preferences in drastically greater detail than ever before. In many cases this trend is fueled by modern algorithms for winner determination that can handle the richer inputs.

But is more expressiveness always a good thing? What forms of expressiveness should be offered?

In this talk I will first report on our experience from over $40 billion of combinatorial multi-attribute sourcing auctions. Then, I will present recent theory that ties the expressiveness of a mechanism to an upper bound on efficiency in a domain-independent way in private-information settings. Time permitting, I will also discuss theory and experiments on applying expressiveness to ad auctions, such as sponsored search and real-time banner ad auctions with temporal span and complex preferences.

The negative conclusions of the Gibbard-Satterthwaite theorem—that only dictatorial social choice functions are non-manipulable—can be overcome by restricting the class of admissible preference profiles. A common approach is to assume that the preferences of the agents can be represented by quasilinear utility functions. This restriction allows for the positive results of the Vickrey auction and the Vickrey-Clarke-Groves mechanism. Quasilinear preferences, however, involve the controversial assumption that there is some commonly desired commodity or numeraire—money, shells, beads, etcetera—the utility of which is commensurable with the utility of the other alternatives in question. We propose a generalization of the Vickrey auction, which does not assume the agents' preferences being quasilinear but still has some of its desirable properties. In this auction a bid can be any alternative, rather than just a monetary bid. Such an auction is also applicable to situations where no numeraire is available, when there is a fixed budget, or when money is no issue. In the presence of quasilinear preferences, however, the traditional Vickrey turns out to be a special case. In order to sidestep the Gibbard-Satterthwaite theorem, we restrict the preferences of the bidders. We show that this qualitative Vickrey auctions always has a dominant strategy equilibrium, which moreover invariably yields a weakly Pareto efficient outcome.

A mechanism is manipulable if it is in agents� best interest to misrepresent their private information (lie) to the center. We provide the first formal treatment of the windfall of manipulability, the seemingly paradoxical quality by which the failure of any agent to play their best manipulation yields a strictly better result than an optimal truthful mechanism. We dub such mechanisms manipulation optimal. We prove that any manipulation-optimal mechanism can have at most one manipulable type per agent. We show the existence of manipulation-optimal multiagent mechanisms with the goal of social welfare maximization, but not in dominant strategies when agents are anonymous and the mechanism is symmetric, the most common setting. For this setting, we show the existence of manipulation-optimal mechanisms when the goal is affine welfare maximization.

Knockout tournaments are very common in practice for various settings such as sport events and sequential pairwise elimination elections. In this paper, we investigate the computational aspect of tournament agenda control, i.e., finding the agenda that maximizes the chances of a target player winning the tournament. We consider several modelings of the problem based on different constraints that can be placed on the structure of the tournament or the model of the players. In each setting, we analyze the complexity of finding the solution and show how it varies depending on the modelings of the problem. In general, constraints on the tournament structure make the problem become harder.

We investigate the extent to which it is possible to rig the agenda of an election or competition so as to favor a particular candidate in the presence of imperfect information about the preferences of the electorate. We assume that what is known about an electorate is the probability that any given candidate will beat another. As well as presenting some analytical results relating to the complexity of finding and verifying agenda, we develop heuristics for agenda rigging, and investigate the performance of these heuristics for both randomly generated data and real-world data from tennis and basketball competitions.

Voting trees describe an iterative procedure for selecting a single vertex from a tournament. It has long been known that there is no voting tree that always singles out a vertex with maximum degree. In this paper, we study the power of voting trees in approximating the maximum degree. We give upper and lower bounds on the worst-case ratio between the degree of the vertex chosen by a tree and the maximum degree, both for the deterministic model concerned with a single fixed tree, and for randomizations over arbitrary sets of trees. Our main positive result is a randomization over surjective trees of polynomial size that provides an approximation ratio of at least 1/2. The proof is based on a connection between a randomization over caterpillar trees and a rapidly mixing Markov chain.

A recurring theme in the mathematical social sciences is how to select the "most desirable" elements given a binary dominance relation on a set of alternatives. Schwartz's tournament equilibrium set (TEQ) ranks among the most intriguing, but also among the most enigmatic, tournament solutions proposed so far in this context. Due to its unwieldy recursive definition, little is known about TEQ. In particular, its monotonicity remains an open problem to date. Yet, if TEQ were to satisfy monotonicity, it would be a very attractive solution concept refining both the Banks set and Dutta's minimal covering set. We show that the problem of deciding whether a given alternative is contained in TEQ is NP-hard. Furthermore, we propose a heuristic that significantly outperforms the naive algorithm for computing TEQ. Early experimental results support the conjecture that TEQ is indeed monotonic.

All nontrivial voting rules are manipulable at some preference profiles when defined on a universal domain. This manipulation is possible even by single individuals, and “a fortiori”, by coalitions of voters. This negative result may be reversed if the functions are defined in appropriately restricted domains of preferences, and only preferences in that same domain can be used to manipulate.

In the first part of the talk I will exemplify the type of results regarding non-manipulability by individuals (or strategy-proofness) that one can obtain under domain restrictions. I will pay special attention to the domain of separable preferences on grids. I will present a characterization of all strategy proof rules in that context, consider the complications that arise in the presence of feasibility constraints and discuss the relevance of that paradigmatic case. I will also briefly touch upon the case of voting rules whose outcomes are lotteries.

In the second part of the talk I will consider the added difficulties that arise if one wants to guarantee that a rule is non-manipulable by coalitions of voters, in addition to being individually strategy-proof. I will also consider intermediate cases, where only “large enough” groups may manipulate. I will also show that, in spite of these difficulties, some domains of preferences have the nice property that all strategy proof rules that one can define on them are also necessarily group strategy-proof (or at least strategy proof in front of coalitions that are not “too large”). I will provide a characterization of those nice domains of preferences for which both types of requirements become equivalent.

We extend the Gibbard-Satterthwaite theorem in the following way. We prove that an onto, non-dictatorial social choice rule which is employed to choose one of at least three alternatives is safely manipulable by a single voter. This means that on occasion a voter will have an incentive to make a strategic vote and know that he will not be worse off regardless of how other voters with similar preferences would vote, sincerely or not.

The problem of the manipulability of known social choice rules in the case of multiple choice is considered. Several concepts of expanded preferences (preferences over the sets of alternatives) are elaborated. As a result of this analysis ordinal and nonordinal methods of preferences expanding are defined. The notions of the degree of manipulability are extended to the case under study. Using the results of theoretical investigation, 22 known social choice rules are studied via computational experiments to reveal their degree of manipulability.

In judgment aggregation, unlike preference aggregation, not much is known about domain restrictions that guarantee consistent majority outcomes. We introduce several conditions on individual judgments sufficient for consistent majority judgments. Some are based on global orders of propositions or individuals, others on local orders, still others not on orders at all. Some generalize classic social-choice-theoretic domain conditions, others have no counterpart. Our most general condition generalizes Seni's triplewise value-restriction, itself the most general classic condition. We also give a new characterization theorem: for a large class of domains, if there exists any aggregation function satisfying some democratic conditions, then majority voting is the unique such function. Taken together, our results provide new support for the robustness of majority rule.

Don Saari developed a geometric approach to the analysis of voting paradoxes such as the Condorcet paradox or Arrow's general possibility theorem. In this paper we want to extend this approach to judgment aggregation. In particular we will use Saari's representation cubes to provide a geometric presentation of profiles and majority rule outcomes. We will then show how profile decompositions can be used to derive restrictions on the set of profiles that guarantee consistent majority outcomes. Finally, the geometric approach is used to determine the likelihood of inconsistencies.

The paper studies the interrelationships of the Preference Aggregation and Judgment Aggregation problems from the point of view of logical semantics. The result of the paper is twofold. On the one hand, the Preference Aggregation problem is viewed as a special case of the Judgment Aggregation one. On the other hand, the Judgment Aggregation problem is viewed as a special case of the Preference Aggregation one. As an example, it is shown how to import an impossibility result from each framework to the other. This shows how the aggregation of preferences and judgments can be viewed as two faces of a same coin.

In the paper we restate the discursive dilemma and relate it to another well-known problem regarding the rational acceptance of logically connected propositions — the lottery paradox. We recall how the latter has been tackled by I. Levi in the decision-theoretic framework and consider the possibility of applying a similar approach to the judgment aggregation problem. Finally, we propose a method of aggregation built on the concepts of epistemic and social utility of accepting a collective judgment, which accounts for such parameters — arguably of importance for aggregation tasks — as the factual truth of the propositions, reliability of agents, information content (completeness) of possible collective judgments and the level of agreement between the agents. We claim that the expected utility of accepting a collective judgment depends on the degree to which all those objectives are satisfied and that groups of rational agents aim at maximizing it, while solving a judgment aggregation problems.

Applications of Epistemic Logic have by now become a major industry and an area once dominated by philosophers has now attracted large followings among both AI people and economists. We will discuss some of our own work in this area, including applications to various social issues, like consensus, common knowledge, elections, the sorts of things which candidates running for office are apt to say, and why.

The paper discusses the interaction properties between preference and choice of coalitions in a strategic interaction. A language is presented to talk about the conflict between coalitionally optimal and socially optimal choices. Norms are seen as social constructions that enable to enforce socially desirable outcomes.

Since its introduction in the mid-nineties, Dung's theory of abstract argumentation frameworks has been influential in artificial intelligence. Dung viewed arguments as abstract entities with a binary defeat relation among them. This enabled extensive analysis of different (semantic) argument acceptance criteria. However, little attention was given to comparing such criteria in relation to the preferences of self-interested agents who may have conflicting preferences over the final status of arguments. In this paper, we define a number of agent preference relations over argumentation outcomes. We then analyse different argument evaluation rules taking into account the preferences of individual agents. Our framework and results inform the mediator (e.g. judge) to decide which argument evaluation rule (i.e. semantics) to use given the type of agent population involved.

With the Dodgson rule, cloning the electorate can change the winner, which Young (1977) considers an “absurdity”. We show that removing this absurdity results in a new rule (DC) for which we can compute the winner in Polynomial time, unlike the traditional Dodgson rule. We introduce two related rules (DR and D&). Dodgson did not explicitly propose the “Dodgson rule” (Tideman, 1987); we argue that DC and DR are better realizations of the principle behind the Dodgson rule than the traditional Dodgson rule. These rules, especially D&, are also effective approximations to the traditional Dodgson's rule. We show that, unlike the rules we have considered previously, the DC, DR and D& scores differ from the Dodgson score by no more than a fixed amount, and thus these new rules converge to Dodgson under any reasonable assumption on voter behaviour.

The needed amount of information to make a social choice is the cost of information processing, and it is a practically important feature of social choice rules. We introduce informational aspects into the analysis of social choice rules and prove that (i) if an anonymous, neutral, and monotonic social choice rule operates on minimal informational requirements, then it is a supercorrespondence of either the plurality rule or the antiplurality rule, and (ii) if the social choice rule is furthermore Pareto efficient, then it is a supercorrespondence of the plurality rule.

In many practical situations where agents vote for making a common decision, the votes do not come all together at the same time (for instance, when voting about a date for a meeting, it often happens that one or two participants express their preferences later than others). In such situations, we might want to preprocess the information given by the subelectorate (consisting of those voters who have expressed their votes) so as to “compile” the known votes for the time when the latecomers will have expressed their votes. We study the amount of space necessary to such a compilation, in function of the voting rule used, the number of candidates, the number of voters who have already expressed their votes and the number of remaining voters. We position our results with respect to existing work, noticeably on vote elicitation and communication complexity.

The central question in mechanism design is how to implement a given social choice function. One of the most studied concepts is that of truthful implementations in which truth-telling is always the best response of the players. The Revelation Principle says that one can focus on truthful implementations without loss of generality (if there is no truthful implementation then there is no implementation at all). Green and Laffont (1986) showed that, in the scenario in which players' responses can be partially verified, the revelation principle holds only in some particular cases.

When the Revelation Principle does not hold, non-truthful implementations become interesting since they might be the only way to implement a social choice function of interest. In this work we show that, although non-truthful implementations may exist, they are hard to find. Namely, it is NP-hard to decide if a given social choice function can be implemented in a non-truthful manner, or even if it can be implemented at all. This is in contrast to the fact that truthful implementability can be recognized efficiently, even when partial verification of the agents is allowed. Our results also show that there is no “simple” characterization of those social choice functions for which it is worth looking for non-truthful implementations.

Coalitional voting games appear in different forms in multi-agent systems, social choice and threshold logic. In this paper, the complexity of comparison of influence between players in coalitional voting games is characterized. The possible representations of simple games considered are simple games represented by winning coalitions, minimal winning coalitions, weighted voting game or a multiple weighted voting game. The influence of players is gauged from the viewpoint of basic player types, desirability relations and classical power indices such as Shapley-Shubik index, Banzhaf index, Holler index, Deegan-Packel index and Chow parameters. Among other results, it is shown that for a simple game represented by minimal winning coalitions, although it is easy to verify whether a player has zero or one voting power, computing the Banzhaf value of the player is #P-complete. Moreover, it is proved that multiple weighted voting games are the only representations for which it is NP-hard to verify whether the game is linear or not. For a simple game with a set W^m of minimal winning coalitions and n players, a O(n.|W^m|+n²log(n)) algorithm is presented which returns no if the game is non-linear and returns the strict desirability ordering otherwise. The complexity of transforming simple games into compact representations is also examined.

Weighted voting games are a popular model of collaboration in multiagent systems. In such games, each agent has a weight (intuitively corresponding to resources he can contribute), and a coalition of agents wins if its total weight meets or exceeds a given threshold. Even though coalitional stability in such games is important, existing research has nonetheless only considered the stability of the grand coalition. In this paper, we introduce a model for weighted voting games with coalition structures. This is a natural extension in the context of multiagent systems, as several groups of agents may be simultaneously at work, each serving a different task. We then proceed to study stability in this context. First, we define the CS-core, a notion of the core for such settings, discuss its non-emptiness, and relate it to the traditional notion of the core in weighted voting games. We then investigate its computational properties. We show that, in contrast with the traditional setting, it is computationally hard to decide whether a game has a non-empty CS-core, or whether a given outcome is in the CS-core. However, we then provide an efficient algorithm that verifies whether an outcome is in the CS-core if all weights are small (polynomially bounded). Finally, we also suggest heuristic algorithms for checking the non-emptiness of the CS-core.

Weighted voting is a well-known model of cooperation among agents in decision-making domains. In such games, each of the players has a weight, and a coalition of players wins the game if its total weight meets or exceeds a given quota. In such games, the player's ability to influence the outcome of the game is related to its weight, but not always directly proportional to it. Usually, the agents' power is measured using a power index, such as e.g., Shapley-Shubik index. In this paper, we study how the agent can manipulate its voting power (as measured by Shapley-Shubik power index) by distributing his weight among several false identities. We call this behavior a false-name manipulation. We show that such manipulations can indeed increase an agent's power and provide upper and lower bounds on the effects of such manipulations. We then study this issue from the computational perspective, and show that checking whether a beneficial split exists is NP-hard. We also discuss efficient algorithms for restricted cases of this problem, as well as randomized algorithms for the general case.

Weighted voting games (WVG) are coalitional games in which an agent's contribution to a coalition is given by his weight, and a coalition wins if its total weight meets or exceeds a given quota. These games model decision-making in political bodies as well as collaboration and surplus division in multiagent domains. The computational complexity of various solution concepts for weighted voting games received a lot of attention in recent years. In particular, Elkind et al. [5] studied the complexity of stability-related solution concepts in WVGs, namely, of the core, the least core, and the nucleolus. While they have completely characterized the algorithmic complexity of the core and the least core, for the nucleolus they have only provided an NP-hardness result. In this paper, we solve an open problem posed in [5] by showing that the nucleolus can be computed in pseudopolynomial time, i.e., there exists an algorithm that correctly computes the nucleolus and runs in time polynomial in the number of players n and the maximum weight W.

We consider the case of self-interested agents that are willing to form coalitions for increasing their individual rewards. We assume that each agent gets an individual payoff which depends on the coalition structure (CS) formed. We consider a CS to be stable if no agent has an incentive to change coalition from this CS. Stability is a desirable property of a CS: if agents form a stable CS, they do not spend further time and effort in selecting or changing CSs. When no stable CSs exist, rational agents will be changing coalitions forever unless some agents accept suboptimal results. When stable CSs exist, they may not be unique, and choosing one over the other will give an unfair advantage to some agents. In addition, it may not be possible to reach a stable CS from any CS using a sequence of myopic rational actions. We provide a payoff distribution scheme that is based on the expected utility of a rational myopic agent (an agent that changes coalitions to maximize immediate reward) given a probability distribution over the initial CS. Agents share the utility from a social welfare maximizing CS proportionally to the expected utility of the agents, which guarantees that agents receives at least as much as their expected utility from myopic behavior. This ensures sufficient incentives for the agents to use our protocol.

Knuth (1976) asked whether the stable matching problem can be generalised to three dimensions i. e., for families containing a man, a woman and a dog. Subsequently, several authors considered the three-sided stable matching problem with cyclic preferences, where men care only about women, women only about dogs, and dogs only about men. In this paper we prove that if the preference lists may be incomplete, then the problem of deciding whether a stable matching exists, given an instance of three-sided stable matching problem with cyclic preferences is NP-complete. Considering an alternative stability criterion, strong stability, we show that the problem is NP-complete even for complete lists. These problems can be regarded as special types of stable exchange problems, therefore these results have relevance in some real applications, such as kidney exchange programs.