The ostrogorski paradox and its relation to nontransitive choice

Let Q,I,J be sets of social goods, aspects and individuals, respectively, and consider the functions v: Q×I×J ? {‐1,1} and w: Q×J ? {‐1,1}. We study two 2‐stage methods for ordering Q characterized by majority vote and, respectively, amalgamating v first over I and then over J (method IP) or vice versa (method PI). An Ostrogorski paradox occurs when IP and PI give different outcomes. Conditions for this paradox and its relations to Condorcet's paradox are investigated, particularly via a scalogram structure of w. While PI seems more like direct democracy than IP, for decisions on k ≥ 3 social goods PI is transitive whereas IP may give a cyclic outcome. Extensions to richer than binary data and to decision systems different from majority vote are explored.     

Heights of representative systems

Representative systems are hierarchical aggregation schemes that are applicable in social choice theory, multiattribute decision making, and in the study of three-valued logics. For example, many procedures for voting on issues—including simple majority voting and weighted voting—can be characterized as representative system. Such systems also include procedures in which vote outcomes of constituencies are treated as votes in a higher level of an election system. The general form of a representative system consists of a “supreme council” which aggregates vote outcomes of secondary councils, which in turn aggregate vote outcomes of tertiary councils, and so forth.An n-variable representative system maps n-tuples of 1's, 0's and ?1's into {1,0,?1} through a nested hierarchy of sign functions. The height of a representative system is the fewest number of hierarchical levels that are needed to characterize the system. The height μ(n) of all n-variable representative systems is the largest height of such systems. It was shown previously that μ(n) ? n ? 1 for all positive integers n and that μ(n) = n ? 1 for n from 1 to 4 inclusive. The present paper proves that μ(5) = μ(6) = 4 and that μ(n) ? ?2 for all n ? 6. The height function μ is known to be unbounded.     

The Condorcet criterion and committee selection

Recent studies have evaluated election procedures on their propensity to select committees that meet a Condorcet criterion. The Condorcet criterion has been defined to use majority agreement from voters' preferences to compare the selected committee to all other committees. This study uses a different definition of the Condorcet criterion as defined on committees. The focus of the new definition is on candidates. That is, we consider majority agreement on each candidate in the selected committee as compared to each candidate not in the selected committee.This new definition of the Condorcet criterion allows for the existence of majority cycles on candidates within the selected committee. However, no candidate in the non-selected group is able to defeat any candidate in the selected committee by majority rule. Of particular interest is the likelihood that a committee meeting this Condorcet criterion exists. Attention is also given to the likelihood that various simple voting procedures will select a committee meeting this Condorcet criterion when one does exist.     

Inequality in population weights,majority threshold and the inversion probability in the case of three states

Two-tier voting systems are prone to majority inversions, when the outcome of an election is not backed by a majority of popular vote. We study the inversion probability in a model with two candidates, three states and uniformly distributed fractions of supporters for each candidate. We show that the inversion probability in a two-tier voting system with three states eventually decreases with a majority threshold in the states and increases with the inequality in the size distribution of the states.     

Is politics power or policy oriented? A comparative analysis of dynamic access models in policy networks*

We will show that these base models and some intermediate ones result in fundamentally different network structures and predicted outcomes. Moreover, we will show that the policy driven models do fundamentally better than the power driven models.

In policy networks actors use access relations to influence preferences of other actors. Establishment and shifts of access relations and their consequences for outcomes of decisions are the main focal points in this paper. Unlike most policy network studies, we therefore do not take the network and its relations as given and constant. Instead we device computer simulation models to account for the dynamics in policy networks. We compare different models and investigate the resulting network structures and predicted outcomes of decisions. The choice among the alternative models is made by their correspondence with empirical network structures and actual outcomes of decisions.

In our models, we assume that all relevant actors aim at policy outcomes as close as possible to their own preferences. Policy outcomes are determined by the preferences of the final decision makers at the moment of the vote. In general, only a small fraction of the actors takes part in the final vote. Most actors have therefore to rely on access relations for directly or indirectly shaping the preferences of the final decision makers. For this purpose actors make access requests to other actors. An access relation is assumed to be established if such a request is accepted by the other actor.

Access relations require time and resources. Actors are therefore assumed to be restricted in the number of access requests they can make and the number of requests they can accept Moreover, due to incomplete information and simultaneous actions by other actors, actors have to make simplifying assumptions in the selection of their “best” requests and learn by experience.

We device two base models that correspond to two basic views on the nature of political processes. In the first view politics is seen as power driven. Corresponding to this view, actors aim at access relations with the most powerful actors in the field. They estimate their likelihood of success by comparing their own resources with those of the target actors. Power also determines the order in which actors accept requests. In the second view, policy matters and actors roughly estimate the effects access relations might have on the outcome of decisions. Actors select requests to “bolster” their own preference as much as possible.     

Maximal Condorcet Domains

A Condorcet domain is a subset of the set of linear orders on a finite set of candidates (alternatives to vote), such that if voters preferences are linear orders belonging to this subset, then the simple majority rule does not yield cycles. It is well-known that the set of linear orders is the Bruhat lattice. We prove that a maximal Condorcet domain is a distributive sublattice in the Bruhat lattice. An explicit lattice formula for the simple majority rule is given. We introduce the notion of a symmetric Condorcet domain and characterize symmetric Condorcet domains of maximal size.     

Collusion properties of values

Two players may enter the game with a prior proxy or association agreement in order to strengthen their positions. There exist weighted majority voting games where a proxy agreement weakens the two players' collective power: the sum of their Shapley values with the agreement is less than without the agreement. This phenomenon cannot happen in non-trivial one man-one vote majority voting games. However, an association agreement weakens the two players' collective power in one man-one vote majority voting games with a sufficiently high quorum. In contrast, the sum of the two players' Banzhaf values turns out to be always immune against manipulation via a proxy or association agreement. Each of these neutrality properties can be used as part of an axiomatic characterization of the Banzhaf value.A first draft, dealing only with collusion properties of the Shapley value, was circulated as VPI&SU, Department of Economics Working Paper E-91-01-02, Collusion Paradoxes of the Shapley Value. I am indebted to Benny Moldovanu and Eyal Winter for referring me to Ehud Lehrer's work on the Banzhaf value and to Lloyd Shapley for providing me with a copy of Shapley (1977). I am grateful to Jean Derks, Marcin Malawski, and two referees for helpful comments.     

多分类器融合系统中模糊测度的确定

有限事例集上,choquet模糊积分的计算可以转化成模糊测度的线性组合,故可以使用标准的优化技术来确定模糊测度。本文使用线性规划来确定模糊测度,数据库上的仿真实验表明,线性规划确定的模糊测度的系统融合精度比多数投票法和加权平均法的分类精度要高。     

Intra-party decision making, party formation, and moderation in multiparty systems

We establish coalitional stable party structures of a party formation game in an elected assembly. Farsighted political players can commit to form parties and to vote on policies according to the party position which is determined by intra-party majority rule. Parties may form governments and block proposals by a randomly selected member of the government. If the government recognition rule allows for the formation of multiparty governments, the median parliamentarian either realizes her ideal point or a policy lottery which she strictly prefers to the status quo. This outcome is enforced by the threat of forming a moderating centre party.     

Random majority percolation

We shall consider the discrete time synchronous random majority‐vote cellular automata on the n by n torus, in which every vertex is in one of two states and, at each time step t, every vertex goes into the state the majority of its neighbors had at time t ‐ 1 with a small chance p of error independently of all other events. We shall show that, if n is fixed and p is sufficiently small, then the process spends almost half of its time in each of two configurations. Further more, we show that the expected time for it to reach one of these configurations from the other is Θ(1/pⁿ⁺¹) despite the actual time spent in transit being O(1/p³). Unusually, it appears difficult to obtain any results for this regime by simulation. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2010     

On the practical possibility of a ‘no show paradox’ under the single transferable vote

In a recent paper Brams and Fishburn (1983) have demonstrated that the ‘single transferable vote’ system for a single member electoral district admits a ‘no show paradox’. The present work indicates that under the single transferable vote the probability of occurence of a no show paradox is very high in situations where the outcome under the single transferable vote differs from that under plurality. This considerably weakens the case for replacing the plurality system by the single transferable vote system. It is also shown that under the single transferable vote the occurrence of a ‘no show paradox’ is consistent with one-dimensional left-right voting behaviour.     

The confidence profile method: a Bayesian method for assessing health technologies.

The Confidence Profile Method is a Bayesian method for adjusting and combining pieces of evidence to estimate parameters, such as the effect of health technologies on health outcomes. The information in each piece of evidence is captured in a likelihood function that gives the likelihood of the observed results of the evidence as a function of possible values of the parameter. A posterior distribution is calculated from Bayes formula as the product of the likelihood function and a prior distribution. Multiple pieces of evidence are incorporated by successive applications of Bayes' formula. Pieces of evidence are adjusted for biases to internal or external validity by modeling the biases and deriving "adjusted" likelihood functions that incorporate the models. Likelihood functions have been derived for one-, two- and multi-arm prospective studies; 2 x 2, 2 x n and matched case-control studies, and cross-sectional studies. Biases that can be incorporated in likelihood functions include crossover in controlled trials, error in measurement outcomes, patient selection biases, differences in technologies, and differences in length of follow-up. Effect measures include differences of rates, ratios of rates, and odds ratios. The elements of the method are illustrated with an analysis of the effect of a thrombolytic agent on the difference in probability of 1-year survival after a heart attack.     

A bargaining set for monotonic simple games based on external and internal stability

A new bargaining set based on notions of both internal and external stability is developed in the context of endogenous coalition formation. It allows one to make an explicit distinction between within-group and outside-group deviation options. This type of distinction is not present in current bargaining sets. For the class of monotonic proper simple games, the outcomes in the bargaining set are characterized. Furthermore, it is shown that the bargaining set of any homogeneous weighted majority game contains an outcome for which the underlying coalition structure consists of a minimal winning coalition and its complement.     

Thresholds of the Divisor Methods

We give an algorithm which permits calculating the maximum and minimum vote shares that allow a party to obtain h seats, that is, the threshold of exclusion and the threshold of representation. These have already been studied for some methods (such as d'Hondt or Sainte-Laguë), and are here generalized to any divisor method, and to any number of seats. The thresholds depend on the size of the constituency, the number of parties running in the constituency, and the divisor method used. Finally, we give some consequences, including a characterization of the d'Hondt method.     

Google矩阵和它的性质

网页等级(PageRank)是一个反映网页重要性的数值.当一个网页A连向另一个网页B的时候,A就等于给网页B投了有效的一票.一个网页接受的票越多,这个网页就越重要.同时,给网页B投票的网页本身的等级也决定了该选票的重要性.Google通过每张选票本身重要性和得票多少来计算一个网页的级别(重要性).Google的核心就是计算每一个网页的等级(即PageRank).本文主要介绍Google矩阵的定义和产生,解释PageRank的一些相关概念,证明Google矩阵及其第二特征值具有的一些性质,并简要介绍这些性质的应用.     

A characterization of a family of consensus rules for committee elections

A committee election rule is proposed that is based on a measure of vote concentration. Relationships are established between properties of the committee selection function, and properties of the function used to measure vote concentration.     

Posterior mode estimation for the generalized linear model

Posterior mode estimators are proposed, which arise from simply expressed prior opinion about expected outcomes, roughly as follows: a conjugate family of prior distributions is determined by a given variance function. Using a conjugate prior, a posterior mode estimator and its estimated (co-)variances are obtained through conventional maximum likelihood computations, by means of small alterations to the observed outcomes and/or to the modelled variance function. Within the conjugate family, for purposes of inference about the regression vector, a reference prior is proposed for a given choice of linear design of the canonical link. The resulting approximate reference inferences approximate the Bayesian inferences which arise from a minimally informative reference prior. A set of subjective prior upper and lower percentage points for the expected outcomes can be used to determine a conjugate family member. Alternatively, a set of subjective prior means and standard deviations determines a member. The subfamily of priors determinable by percentage points either includes or approximates the proposed reference prior.The research of the first-named author was funded in part by a Natural Sciences and Engineering Research Council grant.The second named author gratefully acknowledges the support of the National Science Foundation, grant #DMS-8901494 and of the Kansas Geological Survey where he visited during the term of the majority of this research.     

Strict majority functions on graphs

An opinion function on a graph G = (V, E) is a function f: V → {−1, +1}. The vote of a vertex v is the sum of these function values over the closed neighborhood of v. A strict majority function on a graph G is an opinion function for which more than half of the vertices have a positive vote. The strict majority number of G is the minimum sum of the values in a strict majority function of G. We prove the conjecture of Cockayne and Mynhardt (Ars. Combin. 43 (1996), 235–245) that every tree has strict majority number at most 2. We also prove that every graph has strict majority number at most 4. Both bounds are sharp. © 1998 John Wiley & Sons, Inc. J Graph Theory 28: 49–56, 1998     

An experimental study of voting rules and polls in three-candidate elections

We report the results of elections conducted in a laboratory setting, modelled on a threecandidate example due to Borda. By paying subjects conditionally on election outcomes, we create electorates with (publicly) known preferences. We compare the results of experiments with and without non-binding pre-election polls under plurality rule, approval voting, and Borda rule. We also refer to a theory of voting “equilibria,” which makes sharp predictions concerning individual voter behavior and election outcomes. We find that Condorcet losers occasionally win regardless of the voting rule or presence of polls. Duverger's law (which asserts the predominance of two candidates) appears to hold under plurality rule, but close three-way races often arise under approval voting and Borda rule. Voters appear to poll and vote strategically. In elections, voters usually cast votes that are consistent with some strategic equilibrium. By the end of an election series, most votes are consistent with a single equilibrium, although that equilibrium varies by experimental group and voting rule.     

Likelihood robust optimization for data-driven problems

We consider optimal decision-making problems in an uncertain environment. In particular, we consider the case in which the distribution of the input is unknown, yet there is some historical data drawn from the distribution. In this paper, we propose a new type of distributionally robust optimization model called the likelihood robust optimization (LRO) model for this class of problems. In contrast to previous work on distributionally robust optimization that focuses on certain parameters (e.g., mean, variance, etc.) of the input distribution, we exploit the historical data and define the accessible distribution set to contain only those distributions that make the observed data achieve a certain level of likelihood. Then we formulate the targeting problem as one of optimizing the expected value of the objective function under the worst-case distribution in that set. Our model avoids the over-conservativeness of some prior robust approaches by ruling out unrealistic distributions while maintaining robustness of the solution for any statistically likely outcomes. We present statistical analyses of our model using Bayesian statistics and empirical likelihood theory. Specifically, we prove the asymptotic behavior of our distribution set and establish the relationship between our model and other distributionally robust models. To test the performance of our model, we apply it to the newsvendor problem and the portfolio selection problem. The test results show that the solutions of our model indeed have desirable performance.