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Fractional Approval Voting
In fractional social choice, fractional approval voting refers to a class of electoral systems using approval ballots (each voter selects one or more candidate alternatives), in which the outcome is ''fractional'': for each alternative ''j'' there is a fraction ''pj'' between 0 and 1, such that the sum of ''pj'' is 1. It can be seen as a generalization of approval voting: in the latter, one candidate wins (''pj'' = 1) and the other candidates lose (''pj'' = 0). The fractions ''pj'' can be interpreted in various ways, depending on the setting. Examples are: * ''Time sharing'': each alternative ''j'' is implemented a fraction ''pj'' of the time (e.g. each candidate ''j'' serves in office a fraction ''pj'' of the term). * ''Budget'' ''distribution'': each alternative ''j'' receives a fraction ''pj'' of the total budget.A video of the EC'21 conference talk/ref> * ''Probabilities'': after the fractional results are computed, there is a lottery for selecting a single candidate, whe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Range Voting
Score voting, sometimes called range voting, is an electoral system for single-seat elections. Voters give each candidate a numerical score, and the candidate with the highest average score is elected. Score voting includes the well-known approval voting (used to calculate approval ratings), but also lets voters give partial (in-between) approval ratings to candidates. Usage Political use Historical A crude form of score voting was used in some elections in ancient Sparta, by measuring how loudly the crowd shouted for different candidates. This has a modern-day analog of using clapometers in some television shows and the judging processes of some athletic competitions. Beginning in the 13th century, the Republic of Venice elected the Doge of Venice using a multi-stage process with multiple rounds of score voting. This may have contributed to the Republic's longevity, being partly responsible for its status as the longest-lived democracy in world history. Score voting w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Participatory Budgeting
Participatory budgeting (PB) is a type of citizen sourcing in which ordinary people decide how to allocate part of a municipal or public budget through a process of democratic deliberation and decision-making. These processes typically begin with a series of neighborhood Popular assembly, popular assemblies to initiate and discuss proposals and end with Participatory budgeting ballot types, voting on the final decisions. Participatory budgeting allows citizens or residents of a locality to identify, discuss, and prioritize public spending projects, and gives them the power to make real decisions about how money is spent. Participatory budgeting processes are typically designed to involve those left out of traditional methods of public engagement, such as low-income residents, non-citizens, and youth. A comprehensive case study of eight municipalities in Brazil analyzing the successes and failures of participatory budgeting has suggested that it often results in more equitable ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Price Of Fairness
In the theory of fair division, the price of fairness (POF) is the ratio of the largest economic welfare attainable by a division to the economic welfare attained by a ''fair'' division. The POF is a quantitative measure of the loss of welfare that society has to take in order to guarantee fairness. In general, the POF is defined by the following formula:POF=\frac The exact price varies greatly based on the kind of division, the kind of fairness and the kind of social welfare we are interested in. The most well-studied type of social welfare is '' utilitarian social welfare'', defined as the sum of the (normalized) utilities of all agents. Another type is '' egalitarian social welfare'', defined as the minimum (normalized) utility per agent. Numeric example In this example we focus on the ''utilitarian price of proportionality'', or UPOP. Consider a heterogeneous land-estate that has to be divided among 100 partners, all of whom value it as 100 (or the value is normalized to 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Party-approval Voting
Multiwinner approval voting, sometimes also called approval-based committee (ABC) voting, refers to a family of multi-winner electoral systems that use approval ballots. Each voter may select ("approve") any number of candidates, and multiple candidates are elected. Multiwinner approval voting is an adaptation of approval voting to multiwinner elections. In a single-winner approval voting system, it is easy to determine the winner: it is the candidate approved by the largest number of voters. In multiwinner approval voting, there are many different ways to decide which candidates will be elected. Approval block voting In approval block voting (also called unlimited voting), each voter either approves or disapproves of each candidate, and the ''k'' candidates with the most approval votes win (where ''k'' is the predetermined committee size). It does not provide proportional representation. Proportional approval voting Proportional approval voting refers to voting methods w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Justified Representation
Justified representation (JR) is a criterion of fairness in multiwinner approval voting. It can be seen as an adaptation of the proportional representation criterion to approval voting. Background Proportional representation (PR) is an important consideration in designing electoral systems. It means that the various groups and sectors in the population should be represented in the parliament in proportion to their size. The most common system for ensuring proportional representation is the party-list system. In this system, the candidates are partitioned into parties, and each citizen votes for a single party. Each party receives a number of seats proportional to the number of citizens who voted for it. For example, for a parliament with 10 seats, if exactly 50% of the citizens vote for party A, exactly 30% vote for party B, and exactly 20% vote for party C, then proportional representation requires that the parliament contains exactly 5 candidates from party A, exactly 3 can ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
SAT Solver
In computer science and formal methods, a SAT solver is a computer program which aims to solve the Boolean satisfiability problem (SAT). On input a formula over Boolean data type, Boolean variables, such as "(''x'' or ''y'') and (''x'' or not ''y'')", a SAT solver outputs whether the formula is satisfiability, satisfiable, meaning that there are possible values of ''x'' and ''y'' which make the formula true, or unsatisfiable, meaning that there are no such values of ''x'' and ''y''. In this case, the formula is satisfiable when ''x'' is true, so the solver should return "satisfiable". Since the introduction of algorithms for SAT in the 1960s, modern SAT solvers have grown into complex software, software artifacts involving a large number of heuristics and program optimizations to work efficiently. By a result known as the CookâLevin theorem, Boolean satisfiability is an NP-completeness, NP-complete problem in general. As a result, only algorithms with exponential worst-case comple ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Single-peaked Preferences
Single-peaked preferences are a class of preference relations. A group has single-peaked preferences over a set of outcomes if the outcomes can be ordered along a line such that: # Each agent has a "best outcome" in the set, and # For each agent, outcomes that are further from his or her best outcome are preferred less. Single-peaked preferences are typical of one-dimensional domains. A typical example is when several consumers have to decide on the amount of public good to purchase. The amount is a one-dimensional variable. Usually, each consumer decides on a certain quantity which is best for him or her, and if the actual quantity is more/less than that ideal quantity, the agent is then less satisfied. With single-peaked preferences, there is a simple truthful mechanism for selecting an outcome, which is to select the median quantity; this results in the median voter theorem. It is truthful because the median function satisfies the strong monotonicity property. The notion wa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Random Permutation
A random permutation is a sequence where any order of its items is equally likely at random, that is, it is a permutation-valued random variable of a set of objects. The use of random permutations is common in games of chance and in randomized algorithms in coding theory, cryptography, and simulation. A good example of a random permutation is the fair shuffling of a standard deck of cards: this is ideally a random permutation of the 52 cards. Computation of random permutations Entry-by-entry methods One algorithm for generating a random permutation of a set of size ''n'' uniformly at random, i.e., such that each of the ''n''! permutations is equally likely to appear, is to generate a sequence by uniformly randomly selecting an integer between 1 and ''n'' (inclusive), sequentially and without replacement ''n'' times, and then to interpret this sequence (''x''1, ..., ''x''''n'') as the permutation : \begin 1 & 2 & 3 & \cdots & n \\ x_1 & x_2 & x_3 & \cdots & x_n \\ \end, shown h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Permutation
In mathematics, a permutation of a set can mean one of two different things: * an arrangement of its members in a sequence or linear order, or * the act or process of changing the linear order of an ordered set. An example of the first meaning is the six permutations (orderings) of the set : written as tuples, they are (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), and (3, 2, 1). Anagrams of a word whose letters are all different are also permutations: the letters are already ordered in the original word, and the anagram reorders them. The study of permutations of finite sets is an important topic in combinatorics and group theory. Permutations are used in almost every branch of mathematics and in many other fields of science. In computer science, they are used for analyzing sorting algorithms; in quantum physics, for describing states of particles; and in biology, for describing RNA sequences. The number of permutations of distinct objects is factorial, us ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Dictatorship Mechanism
In social choice theory, a dictatorship mechanism is a degenerate voting rule or mechanism where the result depends on one person's. A serial dictatorship is similar, but also designates a series of "backup dictators", who break ties in the original dictator's choices when the dictator is indifferent. Formal definition Non-dictatorship is one of the necessary conditions in Arrow's impossibility theorem.''Game Theory'' Second Edition Guillermo Owen Ch 6 pp124-5 Axiom 5 Academic Press, 1982 In ''Social Choice and Individual Values'', Kenneth Arrow defines non-dictatorship as: :There is no voter i in such that, for every set of orderings in the domain of the constitution, and every pair of social states ''x'' and ''y'', ''x \succeq_i y'' implies x \succeq y. Unsurprisingly, a dictatorship is a rule that does not satisfy non-dictatorship. Anonymous voting rules automatically satisfy non-dictatorship (so long as there is more than one voter). Serial dictatorship When the dictator ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Concave Function
In mathematics, a concave function is one for which the function value at any convex combination of elements in the domain is greater than or equal to that convex combination of those domain elements. Equivalently, a concave function is any function for which the hypograph is convex. The class of concave functions is in a sense the opposite of the class of convex functions. A concave function is also synonymously called concave downwards, concave down, convex upwards, convex cap, or upper convex. Definition A real-valued function f on an interval (or, more generally, a convex set in vector space) is said to be ''concave'' if, for any x and y in the interval and for any \alpha \in ,1/math>, :f((1-\alpha )x+\alpha y)\geq (1-\alpha ) f(x)+\alpha f(y) A function is called ''strictly concave'' if :f((1-\alpha )x+\alpha y) > (1-\alpha ) f(x)+\alpha f(y) for any \alpha \in (0,1) and x \neq y. For a function f: \mathbb \to \mathbb, this second definition merely states that for ev ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Egalitarian Rule
In social choice and operations research, the egalitarian rule (also called the max-min rule or the Rawlsian rule) is a rule saying that, among all possible alternatives, society should pick the alternative which maximizes the ''minimum utility'' of all individuals in society. It is a formal mathematical representation of the egalitarian philosophy. It also corresponds to John Rawls' principle of maximizing the welfare of the worst-off individual. Definition Let X be a set of possible `states of the world' or `alternatives'. Society wishes to choose a single state from X. For example, in a single-winner election, X may represent the set of candidates; in a resource allocation setting, X may represent all possible allocations. Let I be a finite set, representing a collection of individuals. For each i \in I, let u_i:X\longrightarrow\mathbb be a ''utility function'', describing the amount of happiness an individual ''i'' derives from each possible state. A '' social choice rule ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
