Social Software for Coalition Formation

An interdisciplinary approach to coalition formation

Suppose a "democratic" society must decide between several options, but the members of society do not unanimously agree on the desirability of the different options. Each member has his or her own ranking, and the goal is to transform this set of preference orders into a single preference order for society as a whole. The top-ranked outcomes according to that social order is what one calls a social choice rule. Ideally a social choice rule should have various desirable features, such as being nondictatorial (the social choice rule does not simply select one person's ranking and ignore the others) and monotonic (if one individual promotes an option on his or her ranking, the societal ranking of this option should not go down). In 1972 Kenneth Arrow won the Nobel Prize in economics for his work on social choice theory, in particular his "impossibility theorem." This theorem states that if society consists of at least two people and there are at least three options to rank, then it is impossible to design a social choice rule that simultaneously satisfies a particular set of desirable features including those mentioned above. In spite of this, choices must be made and a variety of social choice rules have been designed for combining individual preferences into a collective agreement, each having certain limitations. But even once we decide on a particular rule, implementing it in the real world can be a challenge. Consider a multipass voting system---for example, the American system of primary elections followed by the general election between winners of the primary. Just this spring the state of Washington overhauled its primary system to require voters to choose one party's ballot before they can vote, rather than putting candidates of all political parties on the same ballot and letting voters pick and choose among them for each office. The fear with the old system was that voters would misrepresent their own desires in the primary election, in hopes of getting a more easily defeated opposition candidate on the general electionballot. On the other hand, independent-minded voters (of which there are many in Washington state) criticize the new system for not allowing them to express their true preferences for each office. Even when members of society do not intentionally misrepresent their own preferences, decision makers in the real world must always work with incomplete information about the state of the world and people's priorities. The goal of implementation theory is to study the extent to which it is possible to implement a desired social choice rule under various assumptions on the preferences and behavior of members of society. In the following paper, Roberto Serrano, of the Economics Department at Brown University, presents a survey of implementation theory and mathematical results obtained using game theory. The theory is illustrated with a number of interesting examples covering everything from voting and public decision making to King Solomon's dilemma. For more on the subject of voting, please note that this issue contains a related paper in the Problems and Techniques section, "Single Transferable Votes with Tax Cuts," by Eivind Stensholt.

This paper establishes a clear connection between equilibrium theory, game theory and social choice theory by showing that, for a well defined social choice problem, a condition which is necessary and sufficient to solve this problem — limited arbitrage — is the same as the condition which is necessary and sufficient to establish the existence of an equilibrium and the core. The connection is strengthened by establishing that a market allocation, which is in the core, can always be realized as a social allocation, i.e. an allocation which is optimal according to an ordering chosen by a social choice rule. Limited arbitrage characterizes those economies without Condorcet triples, and those for which Arrow’s paradox can be resolved on choices of large utility values.KeywordsSocial ChoiceSocial PreferenceCompetitive EquilibriumPareto FrontierSocial Choice RuleThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Chapter 1 Impossibility theorems in the arrovian framework

In this paper, we provide axiomatic foundations for social choice rules on a domain of convex and comprehensive social choice problems when agents have cardinal utility functions. We translate the axioms of three well known approaches in bargaining theory (Nash 1950; Kalai and Smorodinsky 1975; Kalai 1977) to the domain of social choice problems and provide an impossibility result for each. We then introduce the concept of a reference function which, for each social choice set, selects a point from which relative gains are measured. By restricting the invariance and comparison axioms so that they only apply to sets with the same reference point, we obtain characterizations of social choice rules that are natural analogues of the bargaining theory solutions.

Abelian symmetry groups in social choice

Impossibility theorems are ubiquitous in social choice theory. By now, the reader has encountered several of these, among them the famous impossibility results by Arrow, Sen, and Gibbard and Satterthwaite. We think that it is high time to study various possibility results. Of course, this has a price: at least one of Arrow ‘s conditions must be weakened and the reader has to ask himself or herself whether in each individual case, the ensuing possibility is worth its price.

In this paper, we study the metric distortion of deterministic social choice rules that choose a winning candidate from a set of candidates based on voter preferences. Voters and candidates are located in an underlying metric space. A voter has cost equal to her distance to the winning candidate. Ordinal social choice rules only have access to the ordinal preferences of the voters that are assumed to be consistent with the metric distances. Our goal is to design an ordinal social choice rule with minimum distortion, which is the worst-case ratio, over all consistent metrics, between the social cost of the rule and that of the optimal omniscient rule with knowledge of the underlying metric space. The distortion of the best deterministic social choice rule was known to be between $3$ and $5$. It had been conjectured that any rule that only looks at the weighted tournament graph on the candidates cannot have distortion better than $5$. In our paper, we disprove it by presenting a weighted tournament rule with distortion of $4.236$. We design this rule by generalizing the classic notion of uncovered sets, and further show that this class of rules cannot have distortion better than $4.236$. We then propose a new voting rule, via an alternative generalization of uncovered sets. We show that if a candidate satisfying the criterion of this voting rule exists, then choosing such a candidate yields a distortion bound of $3$, matching the lower bound. We present a combinatorial conjecture that implies distortion of $3$, and verify it for small numbers of candidates and voters by computer experiments. Using our framework, we also show that selecting any candidate guarantees distortion of at most $3$ when the weighted tournament graph is cyclically symmetric.

Given a finite set of alternatives and choices between all pairs of alternatives, how to choose from the entire set in a way that is faithful to the pairwise comparisons? This simple, yet captivating, problem is studied in the literature on tournament solutions. A tournament solution thus seeks to identify the “best” elements according to some binary dominance relation, which is usually assumed to be asymmetric and complete. As the ordinary notion of maximality may return no elements due to cyclical dominations, numerous alternative solution concepts have been devised and axiomatized.Many problems in multiagent decision making can be addressed using tournament solutions. For instance, tournament solutions play an important role in collective decision-making (social choice theory), where the binary relation is typically defined via pairwise majority voting. Other application areas include adversarial decision-making (theory of zero-sum games) and coalitional decision-making (cooperative game theory) as well as multi-criteria decision analysis and argumentation theory.In this talk, I will present an overview of some of the most common tournament solutions such as the uncovered set, the minimal covering set, and the bipartisan set and analyze them from an algorithmic point of view.

If is a set of social alternatives, a social choice rule (SCR) assigns a subset of A to each potential profile of individuals' preferences over A, where the subset is interpreted as the set of 'welfare optima.' A game form (or 'mechanism') implements the social choice rule if, for any potential profile of preferences, (1) any welfare optimum can arise as a Nash equilibrium of the game form (implying, in particular, that a Nash equilibrium exists) and, (2) all Nash equilibria are welfare optimal. The main result of this paper establishes that any SCR that satisfies two properties - monotonicity and no veto power--can be implemented by a game form if there are three or more individuals. The proof is constructive.

Sybil attacks, in which fake or duplicate identities (a.k.a., Sybils) infiltrate an online community, pose a serious threat to such communities, as they might tilt community-wide decisions in their favor. While the extensive research on sybil identification may help keep the fraction of sybils in such communities low, it cannot however ensure their complete eradication. Thus, our goal here is to enhance social choice theory with effective group decision mechanisms for communities with bounded sybil penetration. Inspired by Reality-Aware Social Choice, we use the status quo as the anchor of Sybil Resilience, characterized by Sybil Safety -- the inability of sybils to change the status quo against the will of the genuine agents, and Sybil Liveness -- the ability of the genuine agents to change the status quo against the will of the sybils. We consider the social choice settings of deciding on a single proposal, on multiple proposals, and on updating a parameter. For each, we present social choice rules that are sybil-safe and, under certain conditions, satisfy sybil-liveness.

The problem ofjustifying democracy arises when a society sees the need for cooperative, collective action. Collective action may be needed to solve coordination problems, public goods problems, Prisoner's Dilemmas, and other structural problems of human interaction or perhaps to realize common ideals of justice in concrete political institutions. Once the need for collective action is established, one must ask how the goals of a cooperative endeavor are to be secured. Cooperation requires that individuals, at least over a specified range of activity, pursue a joint strategy. To assure voluntary compliance in a joint venture, cooperation must be to each person's advantage. However, each person may have an incentive to induce others to cooperate and to defect from the joint strategy in the hope of enjoying the fruits of cooperation without incurring the opportunity costs of compliance. This is the essence of the Prisoner's Dilemma, and considerations of this sort suggest that if collective action is to succeed, policies or strategies formed collectively must be coercively enforceable. Coercion, however, requires justification. Because solving the problem of rational noncompliance requires that collective decisions be coercively enforceable, the rules by which collective decisions are reached require justification. Douglas Rae neatly puts the problem of justification that emerges when political solutions are coercively enforceable as follows: "Once a political community has decided which of its members are to participate directly in the making of collective policy, an important question remains: 'How many of them must agree before a policy is imposed on the community?"" This is essentially the question to which the principle of democratic rule provides an answer: by what process are collective decisions to be made? Answering Rae's question requires a normative framework. We could say that a procedure for making collective decisions is justified if and

Social Aggregation Rules and Continuity

This is an advance summary of a forthcoming article in the Oxford Research Encyclopedia of Politics. Please check back later for the full article.Narrowly understood, social choice theory is a specialized branch of applied logic and mathematics that analyzes abstract objects called preference aggregation functions, social welfare functions, and social choice functions. But more broadly, social choice theory identifies, analyzes, and evaluates rules that may be used to make collective decisions. So understood, social choice is a subfield of the social sciences that examines what may be called “voting rules” of various sorts. While social choice theory typically assumes a finite set of alternatives over which voter preferences are unrestricted, the spatial model of social choice assumes that policy alternatives can be represented by points in a space of one or more dimensions, and that voters have preferences that are plausibly shaped by this spatial structure.Social choice theory has considerable relevance for the study of legislative (as well as electoral) institutions. The concepts and tools of social choice theory make possible formal descriptions of legislative institutions such as bicameralism, parliamentary voting procedures, effects of decision rules (e.g., supramajority vs. simple majority rule and executive veto rules), sincere vs. strategic voting by legislators, agenda control, and other parliamentary maneuvers. Spatial models of social choice further enrich this analysis and raise additional questions regarding policy stability and change. Spatial models are used increasingly to guide empirical research on legislative institutions and processes.

The aim of the present paper is to analyze the problem of assuring the feasibility of a mechanism (game form), implementing in Nash equilibrium a given social choice rule abbreviated as (SCR) when the mechanism is constrained as to the way in which it is permitted to depend on endowments. A social choice rule is a correspondence specifying outcomes considered to be desirable in a given economy (environment). A mechanism is defined by (a) an outcome function and (b) a strategy domain prescribed for each player. Our outcome functions are not permitted to depend at all on the initial endowments. As to strategy domains, the ith agent’s strategy domain Si is only permitted to depend on that agent’s endowment, but not on the endowments, other agents. (For earlier results concerning endoment manipulation, see Postlewaite (1979) and Sertel (1990).)