Optimal Scaling for Random Walk Metropolis on Spherically Constrained Target Densities

We consider the problem of optimal scaling of the proposal variance for multidimensional random walk Metropolis algorithms. It is well known, for a wide range of continuous target densities, that the optimal scaling of the proposal variance leads to an average acceptance rate of 0.234. Therefore a natural question is, do similar results hold for target densities which have discontinuities? In the current work, we answer in the affirmative for a class of spherically constrained target densities. Even though the acceptance probability is more complicated than for continuous target densities, the optimal scaling of the proposal variance again leads to an average acceptance rate of 0.234.     

On minimum sum representations for weighted voting games

A proposal in a weighted voting game is accepted if the sum of the (non-negative) weights of the ??yea?? voters is at least as large as a given quota. Several authors have considered representations of weighted voting games with minimum sum, where the weights and the quota are restricted to be integers. In Freixas and Molinero (Ann. Oper. Res. 166:243?C260, 2009) the authors have classified all weighted voting games without a unique minimum sum representation for up to 8 voters. Here we exhaustively classify all weighted voting games consisting of 9?voters which do not admit a unique minimum sum integer weight representation.     

Achieving fully proportional representation by clustering voters

Both the Chamberlin–Courant and Monroe rules are voting rules that solve the problem of fully proportional representation: given a set of candidates and a set of voters, they select committees of candidates whose members represent the voters so that the voters’ total dissatisfaction is minimized. These two rules suffer from a common disadvantage, namely being computationally intractable. As both the Chamberlin–Courant and Monroe rules, explicitly or implicitly, partition voters so that the voters in each part share the same representative, they can be seen as clustering algorithms. This suggests studying approximation algorithms for these voting rules by means of cluster analysis, which is the subject of this paper. Using ideas from cluster analysis we develop several approximation algorithms for the Chamberlin–Courant and Monroe rules and experimentally analyze their performance. We find that our algorithms are computationally efficient and, in many cases, are able to provide solutions which are very close to optimal.     

Scoring rule and majority agreements for large electorates with arbitrary preferences

This paper examines elections among three candidates when the electorate is large and voters can have any of the 26 nontrivial asymmetric binary relations on the candidates as their preference relations. Comparisons are made between rule-λ rankings based on rank-order ballots and simple majorities based on the preference relations. The rule-λ ranking is the decreasing point total order obtained when 1, λ and 0 points are assigned to the candidates ranked first, second and third on each voter's ballot, with 0 ? λ ? 1.Limit probabilities as the number of voters gets large are computed for events such as ‘the first-ranked rule-λ candidate has a majority over the second-ranked rule-λ candidate’ and ‘the rule-λ winner is the Condorcet candidate, given that there is a Condorcet candidate’. The probabilities are expressed as functions of λ and the distribution of voters over types of preference relations. In general, they are maximized at λ = 1/2 (Borda) and minimized at λ = 0 (plurality) and at λ = 1 for any fixed distribution of voters over preference types. The effects of more indifference and increased intransitivity in voter's preference relations are analyzed when λ is fixed.     

Equilibrium and effectiveness of two-parameter scoring rules

We study a cardinal model of voting with three alternatives where voters’ von Neumann Morgenstern utilities are private information. We consider voting protocols given by two-parameter scoring rules, as introduced by Myerson (2002). For these voting rules, we show that all symmetric Bayes Nash equilibria are sincere, and have a very specific form. These equilibria are unique for a wide range of model parameters, and we can therefore compare the equilibrium performance of different rules. Computational results regarding the effectiveness of different scoring rules (where effectiveness is captured by a modification of the effectiveness measure proposed in Weber, 1978) suggest that those which most effectively represent voters’ preferences allow for the expression of preference intensity, in contrast to more commonly used rules such as the plurality rule, and the Borda Count. While approval voting allows for the expression of preference intensity, it does not maximize effectiveness as it fails to unambiguously convey voters’ ordinal preference rankings.     

Logrolling and budget allocation games

In this paper we approach the concept of logrolling by examining a voting system where choices are made among sets of competing projects as a game in characteristic function form. We translate the question: “Will there be prices for votes on different projects which clear the market?” into a different, but equivalent question: “Is the formal game we have described amarket game?” We show that in general the answer is no, unless all voters have virtually the same preferences.     

Optimal bounds for the no-show paradox via SAT solving

One of the most important desirable properties in social choice theory is Condorcet-consistency, which requires that a voting rule should return an alternative that is preferred to any other alternative by some majority of voters. Another desirable property is participation, which requires that no voter should be worse off by joining an electorate. A seminal result by Moulin (1988) has shown that Condorcet-consistency and participation are incompatible whenever there are at least 4 alternatives and 25 voters. We leverage SAT solving to obtain an elegant human-readable proof of Moulin’s result that requires only 12 voters. Moreover, the SAT solver is able to construct a Condorcet-consistent voting rule that satisfies participation as well as a number of other desirable properties for up to 11 voters, proving the optimality of the above bound. We also obtain tight results for set-valued and probabilistic voting rules, which complement and significantly improve existing theorems.     

Properties of Open Procedure of Sequential Veto-Voting

Game-theoretic properties of joint decision making are considered. Procedures based on sequential open voting by veto are investigated. The paper is aimed at the question how to make voters’ behavior intuitively rational when they choose their optimal strategies. The review of the existing results is also presented and the connection between them is established. Further research is discussed as well.     

Properties of voting with veto power

The article investigates a game-theoretical model with veto power in application to the election of the chairman of the board in a corporation. Alternative voting rules are considered, the most interesting being open voting in a specified order. Conditions are determined when voter 1 can ensure the election of his candidate. A complete solution is obtained for three voters, both in the case of strict preferences and in the case when the preferences of voters 2 and 3 are incompletely defined (contain an uncertainty). The latter case is particularly relevant because it arises for several decision-making rules and is an inseparable part of the real-life election process, when the voters represent several interest groups.     

学英语

There are 7 questions in total,presenting various different question types.While you attempt to resolve the problems,remember to be creative.During accomplishing these flexible mathematical exercises,you can inspire your mathematical thinking.1.If a-5=0,what is the value of a+5?     

学英语（英文）

There are 9questions in total,presenting various different question types.While you attempt to resolve the problems,remember to be creative.During accomplishing these flexible mathematical exercises,you can inspire your mathematical thinking.1.What is the perimeter of a right triangle if the lengths of its two smallest sides are 15and 36?2.If a∶b∶c=6∶7∶11,what is the value of c-a?     

On the inverse power index problem

Weighted voting games are frequently used in decision making. Each voter has a weight and a proposal is accepted if the weight sum of the supporting voters exceeds a quota. One line of research is the efficient computation of so-called power indices measuring the influence of a voter. We treat the inverse problem: Given an influence vector and a power index, determine a weighted voting game such that the distribution of influence among the voters is as close as possible to the given target value. We present exact algorithms and computational results for the Shapley–Shubik and the (normalized) Banzhaf power index.     

A Power Index in Weighted Voting Systems

We examine the properties of a new method for constructing the power index in voting systems where voters have unequal influence. A system of equations derived in this study produces the sought index by optimal approximation of the original voting system with another system that allows for the weights of the voters. The new index is applied to analyze the known paradoxes of committee voting theory.     

Understanding kindergarten teachers’ perspectives of teaching basic geometric shapes: a phenomenographic research

Geometry is one of the disciplines children involve within early years of their lives. However, there is not much information about geometry education in Turkish kindergarten classes. The current study aims to examine teachers’ perspectives on teaching geometry in kindergarten classes. The researchers inquired about teachers’ in-class experiences in geometry and asked a series of questions such as “what are the benchmarks in your kindergarten class?”; “what kind of tools and materials you use to teach geometry in your class?”; “what shape do you teach first in your kindergarten class?”; “what do you expect to hear when you asked your students ‘what is square’?”; “how do you teach rectangular?”. The study utilized one of the qualitative research methods, namely phenomenography, to collect the data and analyze the data. The study involved with eight kindergarten teachers who work in different schools in central Kutahya, Turkey. The researchers collected data by conducting face-to-face half-structured interviews. The findings of this phenomenographic research showed that kindergarten teachers have some difficulties in teaching geometry and have lack of knowledge and skills in teaching geometry in kindergarten classes.     

学英语(英文)

<正>There are 7 questions in total,presenting various different question types.While you attempt to resolve the problems,remember to be creative.During accomplishing these flexible mathematical exercises,you can inspire your mathematical thinking.1.If a car travels at x miles per hour,in terms of xand y,how long does it take to travel y miles?     

学英语（英文）

<正>There are 8questions in total,presenting various different question types.While you attempt to resolve the problems,remember to be creative.During accomplishing these flexible mathematical exercises,you can inspire your mathematical thinking.1.If the ratio of A to B is 3∶4and the ratio of B to C is 5∶6,what is the ratio of A∶C?You can express this in a fraction.     

Comparing influence theories in voting games under locally generated measures of dissatisfaction

The Isbell desirability relation (I), the Shapley?CShubik index (SS) and the Banzhaf?CColeman index (BC) are power theories that grasp the notion of individual influence in a yes?Cno voting rule. Also, a yes?Cno voting rule is often used as a tool for aggregating individual preferences over any given finite set of alternatives into a collective preference. In this second context, Diffo Lambo and Moulen (DM) have introduced a power relation which ranks the voters with respect to how ably they influence the collective preference. However, DM relies on the metric d that measures closeness between preference relations. Our concern in this work is: do I, SS, BC and DM agree when the same yes?Cno voting rule is the basis for collective decision making? We provide a concrete and intuitive class of metrics called locally generated (LG). We give a characterization of the LG metrics d for which I, SS, BC and DM agree on ranking the voters.