THE CORE IN FUZZY SPATIAL MODELS

A voting rule decides on a probability distribution over a set of $m$ alternatives, based on rankings of those alternatives provided by agents. We assume that agents have cardinal utility functions over the alternatives, but voting rules have access to only the rankings induced by these utilities. We evaluate how well voting rules do on measures of social welfare and of proportional fairness, computed based on the hidden utility functions. In particular, we study the distortion of voting rules, which is a worst-case measure. It is an approximation ratio comparing the utilitarian social welfare of the optimum outcome to the welfare of the outcome selected by the voting rule, in the worst case over possible input profiles and utility functions that are consistent with the input. The literature has studied distortion with unit-sum utility functions, and left a small asymptotic gap in the best possible distortion. Using tools from the theory of fair multi-winner elections, we propose the first voting rule which achieves the optimal distortion $\Theta(\sqrt{m})$ for unit-sum utilities. Our voting rule also achieves optimum $\Theta(\sqrt{m})$ distortion for unit-range and approval utilities. We then take a similar worst-case approach to a quantitative measure of the fairness of a voting rule, called proportional fairness. Informally, it measures whether the influence of cohesive groups of agents on the voting outcome is proportional to the group size. We show that there is a voting rule which, without knowledge of the utilities, can achieve an $O(\log m)$-approximation to proportional fairness, the best possible approximation. As a consequence of its proportional fairness, we show that this voting rule achieves $O(\log m)$ distortion with respect to Nash welfare, and provides an $O(\log m)$-approximation to the core, making it interesting for applications in participatory budgeting.

Lu and Boutilier proposed a novel approach based on "minimax regret" to use classical score based voting rules in the setting where preferences can be any partial (instead of complete) orders over the set of alternatives. We show here that such an approach is vulnerable to a new kind of manipulation which was not present in the classical (where preferences are complete orders) world of voting. We call this attack "manipulative elicitation." More specifically, it may be possible to (partially) elicit the preferences of the agents in a way that makes some distinguished alternative win the election who may not be a winner if we elicit every preference completely. More alarmingly, we show that the related computational task is polynomial time solvable for a large class of voting rules which includes all scoring rules, maximin, Copeland α for every α ∈ [0,1], simplified Bucklin voting rules, etc. We then show that introducing a parameter per pair of alternatives which specifies the minimum number of partial preferences where this pair of alternatives must be comparable makes the related computational task of manipulative elicitation NP-complete for all common voting rules including a class of scoring rules which includes the plurality, k-approval, k-veto, veto, and Borda voting rules, maximin, Copeland α for every α ∈ [0,1], and simplified Bucklin voting rules. Hence, in this work, we discover a fundamental vulnerability in using minimax regret based approach in partial preferential setting and propose a novel way to tackle it.

In this paper, we study a maximum likelihood estimation (MLE) approach to voting when the set of alternatives has a multi-issue structure, and the voters' preferences are represented by CP-nets.We first consider general multi-issue domains, and study whether and how issue-by-issue voting rules and sequential voting rules can be represented by MLEs. We first show that issue-by-issue voting rules in which each local rule is itself an MLE (resp. a candidate scoring rule) can be represented by MLEs with a weak (resp. strong) decomposability property. Then, we prove two theorems that state that if the noise model satisfies a very weak decomposability property, then no sequential voting rule that satisfies unanimity can be represented by an MLE, unless the number of voters is bounded.We then consider multi-issue domains in which each issue is binary; for these, we propose a general family of distance-based noise models, of which give an axiomatic characterization. We then propose a more specific family of natural distance-based noise models that are parameterized by a threshold. We identify the complexity of winner determination for the corresponding MLE voting rule in the two most important subcases of this framework.

public choice theory to familiarize themselves with the masterworks in political theory. [6, 99-101] With this call for broader scholarship in mind, we were impressed in a recent reading of Aristotle's Politics with two aspects of his thought that, in our opinion, should be brought to the attention of modern scholars. First, we examine Aristotle's commentary on voting rules and propose a resolution of an important conflict in the literature interpreting his position on majority rule. Second, we examine Aristotle's general, but not unqualified, belief, presented mainly in analogies, that the collective wisdom of a people was superior to that of the lawgiver's. The main question that we address is whether this superior wisdom was to be embodied in the degree of restrictiveness in the voting rule or in the limits of suffrage. We show that under the former interpretation Aristotle hypothesized an important link between the restrictiveness of the voting rule and the amount of information that is brought to bear in making collective decisions. This is a link perhaps overlooked by modern writers on public choice, and we examine its effect in a model of the optimal voting rule, originally developed by J. M. Buchanan and G. Tullock, to observe its impact on the minimum cost decision rule. [7, 63-84]

Ordinary least squares econometric approaches to estimating election vote outcomes potentially ignore spatial dependence (or autocorrelation) in the data that may affect estimates of voting behavior. The presence of spatial autocorrelation in the data can yield biased or inconsistent point estimates when ordinary least squares is used inappropriately. Therefore, this paper puts forward a spatial econometric model to estimate the vote outcomes in the 2004 presidential election. We contribute to the literature in two ways. One, we extend the voting behavior literature by considering newly developed spatial specification tests to determine the proper econometric model. The results of two different spatial specification tests suggest that a spatial Durbin model provides a better fit to the data. Two, we offer a richer interpretation of the spatial effects, which differ from standard ordinary least squares estimates, of the county-level vote outcome for the 2004 presidential election.

We have observed that some recent results cast serious doubts uponthe applicability of majority rule as a device for forming a group-willout of individual wills. On the other hand, majority rule has manyproperties to recommend it. Especially, the decisiveness propertyshould please anyone interested in efficiency considerations. How-ever, when it comes to the second major criterion, consistency, major-ity rule with pairwise comparison of alternatives often leads to diffi-culties. Indeed, we have seen that there is typically no core or Con-dorcet winner when the policy space is multidimensional real space.And when there is no core, the situation looks grim indeed for major-ity rule: literally any outcome may become the majority winner. Somerelief of this very strong result is provided by the fact that no majoritycycles can go through the outcomes that belong to the M2 bargainingset. This finding, of course, solves the problem facing majority rule onlyinsofar as a method can be devised by which the M2 bargaining set out-comes can be determined in a reliable way. Here, thus, we are con-fronted with the sincerity problem. The absence of the core can lead to chaotic trajectories of votingoutcomes in multidimensional space. However, there are ways of over-coming the difficulties if a sufficient amount of flexibility is allowedin institutional design. If we consider the various nonequilibrium solu-tion concepts, and the minimax set in particular, we can always designa decision rule which (under mild restrictions) makes the vote-maximiz-ing outcome trajectory enter the solution set. The flexibility needed isgreat, indeed: the decision rule, λ-majority, must be alterable for eachvoting situation. The determination of the winner in a pair-wise comparison of alter-natives is not the main source of the difficulties surrounding majorityrule. As a matter of fact, the method of building the agenda is the mostcrucial part of the problem of cyclical majorities. In multicandidatecontests the occurrence of cyclical majorities has motivated the studyof other.decision procedures in addition to the majority comparisonof alternatives. The most recent proposal, approval voting, is morestrategyproof than its competitors in the class of single-ballot, non-ranked voting systems. At the moment the relationships betweenapproval voting and majority rule are not clear. It seems that furtherstudy of the interrelationships of these procedures is called for. Thefeatures in which approval voting is superior to majority comparisonof pairs of alternatives, however, are not, strictly speaking, procedural,but are external to the method of making collective decisions.

Standard Setting Organizations and Standard Essential Patents: Voting and Markets

Fast Bargaining in Bankruptcy

In this paper, we attempt to explain the underlying strategic incentives confronting individuals when they must make a collective decision over a set of alternatives and each has information that is decision-relevant for others. A significant literature has emerged in formal political theory over the past several years that focuses on such problems, paying particular attention, first, to the extent to which voting can be expected to aggregate committee members' information and, second, to the role of communication among committee members prior to voting. Inter alia, this literature reveals a surprisingly subtle interaction between the voting rules used to make decisions and the incentives for committee members to share information prior to voting.

Geographical AnalysisVolume 15, Issue 3 p. 205-211 Free Access Majority Voting and the Location of Salutary Public Facilities Fred M. Shelley, Fred M. Shelley Fred M. Shelley is assistant professor of geography, University of Oklahoma.Search for more papers by this authorMichael F. Goodchild, Michael F. Goodchild Michael F. Goodchild is professor of geography, University of Western Ontario. The helpful comments of Daniel Fesenmaier, Diane Whalley, Frank Stetzer, Sally McLafferty, Avijit Ghosh, and two anonymous referees on earlier drafts of this paper are gratefully acknowledged.Search for more papers by this author Fred M. Shelley, Fred M. Shelley Fred M. Shelley is assistant professor of geography, University of Oklahoma.Search for more papers by this authorMichael F. Goodchild, Michael F. Goodchild Michael F. Goodchild is professor of geography, University of Western Ontario. The helpful comments of Daniel Fesenmaier, Diane Whalley, Frank Stetzer, Sally McLafferty, Avijit Ghosh, and two anonymous referees on earlier drafts of this paper are gratefully acknowledged.Search for more papers by this author First published: July 1983 https://doi.org/10.1111/j.1538-4632.1983.tb00782.xCitations: 2 Michael F. Goodchild is professor of geography, University of Western Ontario. The helpful comments of Daniel Fesenmaier, Diane Whalley, Frank Stetzer, Sally McLafferty, Avijit Ghosh, and two anonymous referees on earlier drafts of this paper are gratefully acknowledged. AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat LITERATURE CITED Black, D. (1948). “On the Rationale of Group Decision Making. Journal of Political Economy, 56, 23– 34. Hansen, P., and J.-E Thisse (1981). “Outcomes of Voting and Planning: Condorcet, Weber, and Rawls Locations. Journal of Public Economics, 16, 1– 15. Kramer, G. (1973). “On a Class of Equilibrium Conditions for Majority Rule. Econometrica, 41, 255– 97. Rushton, G., S. L. McLafferty, and A. Ghosh (1981). “Optimum Locations for Public Services: Individual Preferences and Social Choice. Geographical Analysis, 13, 196– 202. Sen, A. (1977). “Social Choice Theory: A Re-Examination. Econometrica, 45, 53– 89. Tellier, L.-N. (1972). “The Weber Problem: Solution and Interpretation. Geographical Analysis, 4, 215– 33. Wendell, R. E., and R. D. McKelvey (1981). “New Perspectives in Competitive Location Theory. European Journal of Operational Research, 6, 174– 82. Citing Literature Volume15, Issue3July 1983Pages 205-211 ReferencesRelatedInformation

This study uses GIS and spatial modelling to relate voting outcomes at the 2001 federal election for polling booths across Australia with the socio‐economic characteristics of polling booth catchment areas. The data and analysis used are more detailed and comprehensive than previous studies. It is conducted at a fine level of spatial disaggregation across the whole nation to examine voting outcomes for both major and minor political parties. Because the aim of the paper is to distinguish voting outcomes between political parties rather than to predict voting outcomes for particular political parties, a discriminant analysis is used rather than regression analysis. The statistical discriminant analysis identifies two main socio‐economic dimensions that are able to predict polling booth outcomes with a relatively high degree of accuracy. That analysis shows how, at the 2001 federal election, the middle ground, in terms of socio‐economic characteristics, was being claimed by the Liberal Party, Country Liberal Party, The Greens, and, to a lesser extent, by the Australian Labor Party. However, the Australian Democrats, National Party and One Nation had more distinctive constituencies, with the National Party and One Nation Party competing for areas with similar socio‐economic characteristics. Using GIS mapping tools, examples of actual and predicted polling booth voting outcomes are given, along with selected socio‐economic characteristics of booth catchments.

Congestion is an important management problem at mass tourist sites. This essay focuses on the social carrying capacity (SCC) of a tourist site as indicator of residents’ and visitors’ perception of crowding, intended as the maximum number of visitors (MNV) tolerated. In case of conflict between the residents’ MNV tolerated and the visitors’ MNV tolerated, the policy-maker has to mediate. We consider the case in which the residents’ SCC is lower than the visitors’ SCC, and the site SCC is the result of a compromise between these two aspects of the SCC. This can be measured by making reference to two criteria of choice: the utility maximisation criterion and the voting rule. The use of one method rather than the other depends on the data available about the individual preferences on crowding. Assuming that individual preferences are known, a maximisation model for the computation of the site SCC is conceived. It represents the case in which the residents’ SCC is the limiting factor. The site SCC is intended as the number of visitors which maximises the social welfare function. Because a local policy-maker maximises the welfare of residents, in this model visitors are represented by those residents whose welfare wholly depends on the tourism sector, while the social costs due to crowding are borne by those residents who are partially or totally independent from tourism. Nevertheless, in practice, the individual preferences about crowding are not always known. In this case, the MNV tolerated can be computed by applying the majority voting rule. It is shown that, under certain conditions, the optimum number of visitors, obtained through a maximisation model, is equal to the MNV tolerated by the majority of voters.

The Idealized Electoral College Voting Mechanism and Shareholder Power

We study dynamic committee bargaining over an infinite horizon with discounting. In each period, a committee proposal is generated by a random recognition rule, the committee chooses between the proposal and a status quo by majority rule, and the voting outcome in period t becomes the status quo in period t + 1. We study symmetric Markov equilibria of the resulting game and conduct an experiment to test hypotheses generated by the theory for pure distributional (divide-the-dollar) environments. In particular, we investigate the effects of concavity in the utility functions, the existence of a Condorcet winning alternative, and the discount factor (committee “impatience”). We report several new findings. Voting behavior is selfish and myopic. Status quo outcomes have great inertia. There are strong treatment effects that are in the direction predicted by the Markov equilibrium. We find significant evidence of concave utility functions.

Objective: Overlapping anatomy in projection mammography can degrade the detection of mammographic lesions and further create mass-like features that can confound computer-aided detection (CADe) systems and degrade their performance. Tri-plane correlation imaging (TCI) aims to reduce this impact by incorporating three projection images into CADe using a geometric correlation scheme. This study aimed to assess the efficacy of TCI for breast cancer detection. Materials and Methods: An image set of 198 human subjects was used from a breast tomosynthesis database. Each case included 25 projections acquired within a 45 degree angular span at an approximate total glandular dose equal to that of two-view mammography. Triplet images were derived from each case (the central projection along with two symmetrical images) and analyzed by two independent CADe programs. The CADe results from the three projections from each program were combined using two TCI correlation rules based on unanimous vote and majority vote. The findings were analyzed in terms of true positives (TP), sensitivity, false positives per breast volume (FP), and a positive predictive index combining the two figures (i.e., TP/(TP+FP)). Results: For the first CADe program, the TCI scheme using the majority voting rule improved sensitivity by 40% while maintaining specificity, leading to a 40% improvement in the PPI performance. For the second, a higher-sensitivity/lower-specificity CADe program, the TCI using the majority voting rule improved sensitivity by 10% while increasing false positives per breast volume by 3%, leading to an improvement of 8% in the PPI performance. The unanimous voting rule led to notably lower performance for both programs. For both CADe programs, an angular separation of six degrees (± 3 degrees) proved optimal. Conclusions: TCI was able to improve sensitivity over single projection imaging while maintaining specificity for both CADe programs, which suggests its potential as a supplement to standard mammography and as a complementary module to existing CADe algorithms.