Learning to Manipulate Under Limited Information

Perhaps the most influential result in social choice theory is Arrow’s impossibility theorem, which states that a seemingly modest set of desiderata cannot be satisfied when aggregating preferences [1]. While Arrow’s theorem might appear rather negative, it can also be interpreted in a positive way by identifying what can be achieved in preference aggregation.In this talk, I present a number of variations of Arrow’s theorem–such as those due to Mas-Colell and Sonnenschein [8] and Blau and Deb [2]–in their choice-theoretic version. The critical condition in all these theorems is the assumption of a rationalizing binary relation or equivalent notions of choice-consistency. The bulk of my presentation contains three escape routes from these results. The first one is to ignore consistency with respect to a variable set of alternatives altogether and require consistency with respect to a variable electorate instead. As Smith [12] and Young [14] have famously shown, this essentially characterizes the class of scoring rules, which contains plurality and Borda’s rule. For the second escape route, we factorize choice-consistency into two parts, contraction-consistency and expansions-consistency [11]. While even the mildest dose of the former has severe consequences on the possibility of choice, varying degrees of the latter characterize a number of appealing social choice functions, namely the top cycle, the uncovered set, and the Banks set [3,9,4]. Finally, I suggest to redefine choice-consistency by making reference to the set of chosen alternatives rather than individual chosen alternatives [6]. It turns out that the resulting condition is a weakening of transitive rationalizability and can be used to characterize the minimal covering set and the bipartisan set. Based on a two decades-old conjecture due to Schwartz [10], the tournament equilibrium set can be characterized by the same condition or, alternatively, by a weak expansion-consistency condition from the second escape route. Whether Schwartz’s conjecture actually holds remains a challenging combinatorial problem as well as one of the enigmatic open problems of social choice theory.Throughout the presentation I will discuss the algorithmic aspects of all considered social choice functions. While some of the mentioned functions can be easily computed, other ones do not admit an efficient algorithm unless P equals NP [13,5,7].

Voting systems are ways of resolving conflicts, choosing candidates, selecting policy options as well as of determining winners or rankings of competitors in various contests. There are other methodological approaches to tackle these problems: conflict analysis methods (See the chapter by Kilgour and Hipel, this volume), group decision support systems (See the chapter by Ackerman and Eden, this volume), soft computing methods (See the chapter by Carlsson, this volume), multicriteria decision analysis (See the chapter by Salo and Hamalainen, this volume) and game-theoretical tools (See the chapter by Kibris, this volume). Often voting systems are used in determining a just or fair distribution of costs and benefits among a number of participants. Also here other methods can sometimes be applied (See the chapter by Klamler, this volume). The basic motivation for the study of voting systems is the fact that different systems often produce different outcomes when applied to a given set of voter opinions. In some contexts we are able to single out plausible outcomes, e.g. candidates that – given a distribution of opinions in the electorate – ought to be elected for the system to be called reasonable or democratic in some specific sense. Social choice theorists have developed various plausibility criteria for the evaluation of voting systems. After discussing the classic paradoxes of social choice, we review the main criteria as well as the most important results in social choice theory. We also present some techniques for the analysis of opinion distributions. Our main focus is on systems that lead to one candidate being elected, but we also deal with multi-winner electoral systems. We also briefly touch upon the issue of which systems could be considered best in terms of the criteria discussed.

Part I of this paper offers a novel result in social choice theory by extending Harsanyi's well-known utilitarian theorem into a “multi-profile” context. Harsanyi was contented with showing that if the individuals' utilities ui are von Neumann-Morgenstern, and if the given utility u of the social planner is VNM as well, then the Pareto indifference rule implies that u is affine in terms of the ui. We provide a related conclusion by considering u as functionally dependent on the ui, through a suitably restricted “social welfare functional” (u1,...,un)→u=f(u1,...,un). We claim that this result is more in accordance with contemporary social choice theory than Harsanyi's “single-profile” theorem is. Besides, harsanyi's initial proof of the latter was faulty. Part II of this paper offers an alternative argument which is intended to be both general and simple enough, contrary to the recent proofs published by Fishburn and others. It finally investigates the affine independence problem on the ui discussed by Fishburn as a corollary to harsanyi's theorem.

We introduce a number of logics to reason about collective propositional attitudes that are defined by means of the majority rule. It is well known that majoritarian aggregation is subject to irrationality, as the results in social choice theory and judgment aggregation show. The proposed logics for modelling collective attitudes are based on a substructural propositional logic that allows for circumventing inconsistent outcomes. Individual and collective propositional attitudes, such as beliefs, desires, obligations, are then modelled by means of minimal modalities to ensure a number of basic principles. In this way, a viable consistent modelling of collective attitudes is obtained.

The standard assumption underlying most of the negative results of the social choice theory is that the individuals have complete and transitive preference relations over the candidates. As an alternative to this assumption we consider the possibility that individuals can be characterized as possessing preference tournaments (i.e. asymmetric and complete relations) over the candidate set. We discuss the implications of the latter assumption to the negative results of social choice theory. Finally some solution concepts applicable in the individual preference tournament framework are outlined.

This paper offers a short introduction to some of the fundamental results of social choice theory. Topics include Nash implementability, monotonic social choice correspondences, the Muller-Satterthwaite impossibility theorem, anonymous and neutral social choice correspondences, sophisticated solutions of binary agendas, the top cycle of a tournament, the bipartisan set for two-party competition, and median voter theorems. The paper begins with a simple example to illustrate the importance of multiple equilibria in game-theoretic models of political institutions.

Previous articleNext article No AccessArrow's Theorem, Indeterminacy, and Multiplicity Reconsidered*Mathias RisseMathias Risse Search for more articles by this author PDFPDF PLUS Add to favoritesDownload CitationTrack CitationsPermissionsReprints Share onFacebookTwitterLinkedInRedditEmail SectionsMoreDetailsFiguresReferencesCited by Ethics Volume 111, Number 4July 2001 Article DOIhttps://doi.org/10.1086/233570 Views: 53Total views on this site Citations: 18Citations are reported from Crossref © 2001 by The University of Chicago. All rights reserved.PDF download Crossref reports the following articles citing this article:Henrik D. Kugelberg Social choice problems with public reason proceduralism, Economics and Philosophy 38, no.11 (Feb 2021): 51–70.https://doi.org/10.1017/S0266267120000498Wesley H. Holliday, Eric Pacuit Axioms for defeat in democratic elections, Journal of Theoretical Politics 33, no.44 (Nov 2021): 475–524.https://doi.org/10.1177/09516298211043236Hannu Nurmi The Incidence of Some Voting Paradoxes Under Domain Restrictions, Group Decision and Negotiation 29, no.66 (Aug 2020): 1107–1120.https://doi.org/10.1007/s10726-020-09697-9Adiel Teixeira de Almeida, Danielle Costa Morais, Hannu Nurmi More Than Two Alternatives, (Oct 2019): 21–30.https://doi.org/10.1007/978-3-030-30955-8_4Eerik Lagerspetz Interpretations, (Jan 2016): 247–340.https://doi.org/10.1007/978-3-319-23261-4_5MARK A. BURGMAN, HELEN M. REGAN, LYNN A. MAGUIRE, MARK COLYVAN, JAMES JUSTUS, TARA G. MARTIN, KRIS ROTHLEY Voting Systems for Environmental Decisions, Conservation Biology 28, no.22 (Jan 2014): 322–332.https://doi.org/10.1111/cobi.12209Rosa Camps, Xavier Mora, Laia Saumell A continuous rating method for preferential voting. The incomplete case, Social Choice and Welfare 40, no.44 (Apr 2012): 1111–1142.https://doi.org/10.1007/s00355-012-0663-5Aki Lehtinen A welfarist critique of social choice theory, Journal of Theoretical Politics 23, no.33 (Jul 2011): 359–381.https://doi.org/10.1177/0951629811411753Mathias Risse On the Philosophy of Group Decision Methods I: The Nonobviousness of Majority Rule, Philosophy Compass 4, no.55 (Sep 2009): 793–802.https://doi.org/10.1111/j.1747-9991.2009.00224.xMathias Risse On the Philosophy of Group Decision Methods II: Alternatives to Majority Rule, Philosophy Compass 4, no.55 (Sep 2009): 803–812.https://doi.org/10.1111/j.1747-9991.2009.00225.xArnold Shepperson An economy of impossibility: a preliminary study for an ordinal approach to research methods in cultural studies, Critical Arts 22, no.22 (Nov 2008): 333–357.https://doi.org/10.1080/02560040802472492Joel Parker Peter Emerson, ed., Designing an All-Inclusive Democracy: Consensual Voting Procedures For Use in Parliaments, Councils and Committees, Public Choice 135, no.3-43-4 (Nov 2007): 493–496.https://doi.org/10.1007/s11127-007-9261-yMathias Risse On the Philosophy of Group Decision Methods I: The Non-Obviousness of Majority Rule, SSRN Electronic Journal (Jan 2008).https://doi.org/10.2139/ssrn.1296638MARC FLEURBAEY SOCIAL CHOICE AND JUST INSTITUTIONS: NEW PERSPECTIVES, Economics and Philosophy 23, no.11 (Mar 2007): 15–43.https://doi.org/10.1017/S0266267107001204Donald G. Saari Which is better: the Condorcet or Borda winner?, Social Choice and Welfare 26, no.11 (Jan 2006): 107–129.https://doi.org/10.1007/s00355-005-0046-2Mathias Risse Why the count de Borda cannot beat the Marquis de Condorcet, Social Choice and Welfare 25, no.11 (Nov 2005): 95–113.https://doi.org/10.1007/s00355-005-0045-3Gerry Mackie Democracy Defended, 20 (Sep 2009).https://doi.org/10.1017/CBO9780511490293 by Donald G. Saari Capturing the “Will of the People” Saari, Ethics 113, no.22 (Jul 2015): 333–349.https://doi.org/10.1086/342857

Multi-objective optimization problems usually do not have a single unique optimal solution, for either discrete or continuous domains. Furthermore, there are usually many possible available algorithms for solving these problems, and one typically does not know in advance which of these will be the most effective for solving a particular problem instance. Hyper-heuristics (HHs) are often used as a means to make this choice. In particular, the underlying idea of HHs is to run several algorithms or heuristics and dynamically decide, based on different criteria, which problem or part of the problem should be solved by which algorithm or heuristic. On the other hand, the domain of social choice theory studies how to design collective decision processes by aggregating individual inputs into collective ones. In this paper, we explore the use of social choice theory in creating HHs. By using HHs based on different voting methods, like Borda, Copeland and Kemeny–Young, we show how we can solve both continuous and discrete engineering multi-objective optimization problems and discuss the results obtained by each of these methods. Our obtained results show that our strategy has found solutions that are at least equals to the ones generated by the best algorithm among the studied ones, and sometimes even overcomes these results.

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

There is a long tradition of fruitful interaction between logic and social choice theory. In recent years, much of this interaction has focused on computer-aided methods such as SAT solving and interactive theorem proving. In this paper, we report on the development of a framework for formalizing voting theory in the Lean theorem prover, which we have applied to verify properties of a recently studied voting method. While previous applications of interactive theorem proving to social choice (using Isabelle/HOL and Mizar) have focused on the verification of impossibility theorems, we aim to cover a variety of results ranging from impossibility theorems to the verification of properties of specific voting methods (e.g., Condorcet consistency, independence of clones, etc.). In order to formalize voting theoretic axioms concerning adding or removing candidates and voters, we work in a variable-election setting whose formalization makes use of dependent types in Lean.

Instant runoff voting (IRV) is an electoral system that produces better elections and more representative outcomes in races with three or more candidates. Used by over 50 American jurisdictions, IRV addresses the challenges of plurality voting in crowded candidate fields. It upholds majority rule, promotes voter engagement, discourages personal attacks, and minimizes strategic voting. IRV is preferable to runoff systems and other proposed single-winner methods. In comparison to runoffs, IRV produces higher, more equitable turnout and reduces costs and demands on voters. In contrast to methods based on rating candidates, voters in IRV elections have incentives to indicate their honest preferences. IRV demonstrates greater political viability and more intuitively fair outcomes than Condorcet methods. For legislative elections, IRV complements the single transferable vote, which is the best way to address the U.S.’s governance and polarization crises, end gerrymandering, and secure fairer representation for all.

In an election which uses preference ballots, there exist many different voting methods for choosing a winner of the election given the voters’ preference data. We analyse the extent to which the choice of voting method matters in American ranked-choice elections. To this end, we compiled a database of over 200 American elections which used preference ballots and, for each election, calculated the winner under the methods of plurality, instant runoff voting, plurality runoff, Condorcet, Bucklin voting, and two versions of Borda count. We find that the choice of voting method matters less frequently than what is suggested by much of the theoretical literature.

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.

Australia is the birthplace of the two main forms of preferential electoral voting - the Alternative Vote (AV) and the Single Transferable Vote (STV) methods. Accident and force of circumstances largely explain their introduction. The Hughes Government introduced the AV system in 1918 in an attempt to prevent the right-of-centre parties from splitting their support to the benefit of the ALP. In 1948 the Chifley Government moved to replace the Senate's preferential block system with the STV method in an attempt to electorally contain Menzies. Yet credit also needs be given to the deliberate efforts and clear intentions of their designers. The electoral systems proposed in 1902 were ahead of their time. Indeed, 1902 marked an important step in the design of Australia's electoral system. Once the debate had crystallized around the relative merits of preferential methods, it was only a matter of time before the Commonwealth would see their adoption.

Utilitarian Collective Choice and Voting