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Budget-proposal Aggregation
Budget-proposal aggregation (BPA) is a problem in social choice theory. A group has to decide on how to distribute its budget among several issues. Each group-member has a different idea about what the ideal budget-distribution should be. The problem is how to aggregate the different opinions into a single budget-distribution program. BPA is a special case of Participatory budgeting rule, participatory budgeting, with the following characteristics: # The issues are ''divisible'' and ''unbounded'' – each issue can be allocated any amount, as long as the sum of allocations equals the total budget. # Agents' preferences are given by single-peaked preferences over an ''ideal budget''. It is also a special case of fractional social choice (portioning), in which agents express their preferences by stating their ideal distribution, rather than by a ranking of the issues. Another sense in which aggregation in budgeting has been studied is as follows. Suppose a manager asks his worker ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Social Choice Theory
Social choice theory is a branch of welfare economics that extends the Decision theory, theory of rational choice to collective decision-making. Social choice studies the behavior of different mathematical procedures (social welfare function, social welfare functions) used to combine individual preferences into a coherent whole.Amartya Sen (2008). "Social Choice". ''The New Palgrave Dictionary of Economics'', 2nd EditionAbstract & TOC./ref> It contrasts with political science in that it is a Normative economics, normative field that studies how a society can make good decisions, whereas political science is a Positive economics, descriptive field that observes how societies actually do make decisions. While social choice began as a branch of economics and decision theory, it has since received substantial contributions from mathematics, philosophy, political science, and game theory. Real-world examples of social choice rules include constitution, constitutions and Parliamentary ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Anonymity (social Choice)
In economics and social choice, a function satisfies anonymity, neutrality, or symmetry if the rule does not discriminate between different participants ahead of time. For example, in an election, a voter-anonymous function is one where it does not matter who casts which vote, i.e. all voters' ballots are equal ahead of time. Formally, this is defined by saying the rule returns the same outcome (whatever this may be) if the votes are "relabeled" arbitrarily, e.g. by swapping votes #1 and #2. Similarly, outcome-neutrality says the rule does not discriminate between different outcomes (e.g. candidates) ahead of time. Formally, if the labels assigned to each outcome are permuted arbitrarily, the returned result is permuted in the same way. Some authors reserve the term anonymity for agent symmetry and neutrality for outcome-symmetry, but this pattern is not perfectly consistent.{{Rp, 75 Examples Most voting rules are anonymous and neutral by design. For example, plurality voting i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
L-Infinity-Norm
In mathematical analysis, the uniform norm (or ) assigns, to real- or complex-valued bounded functions defined on a set , the non-negative number :\, f\, _\infty = \, f\, _ = \sup\left\. This norm is also called the , the , the , or, when the supremum is in fact the maximum, the . The name "uniform norm" derives from the fact that a sequence of functions converges to under the metric derived from the uniform norm if and only if converges to uniformly. If is a continuous function on a closed and bounded interval, or more generally a compact set, then it is bounded and the supremum in the above definition is attained by the Weierstrass extreme value theorem, so we can replace the supremum by the maximum. In this case, the norm is also called the . In particular, if is some vector such that x = \left(x_1, x_2, \ldots, x_n\right) in finite dimensional coordinate space, it takes the form: :\, x\, _\infty := \max \left(\left, x_1\ , \ldots , \left, x_n\\right). This is ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Strategyproof
In mechanism design, a strategyproof (SP) mechanism is a game form in which each player has a weakly-dominant strategy, so that no player can gain by "spying" over the other players to know what they are going to play. When the players have private information (e.g. their type or their value to some item), and the strategy space of each player consists of the possible information values (e.g. possible types or values), a truthful mechanism is a game in which revealing the true information is a weakly-dominant strategy for each player. An SP mechanism is also called dominant-strategy-incentive-compatible (DSIC), to distinguish it from other kinds of incentive compatibility. A SP mechanism is immune to manipulations by individual players (but not by coalitions). In contrast, in a group strategyproof mechanism, no group of people can collude to misreport their preferences in a way that makes every member better off. In a strong group strategyproof mechanism, no group of people can col ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Leontief Utilities
In economics, especially in consumer theory, a Leontief utility function is a function of the form: u(x_1,\ldots,x_m)=\min\left\ . where: * m is the number of different goods in the economy. * x_i (for i\in 1,\dots,m) is the amount of good i in the bundle. * w_i (for i\in 1,\dots,m) is the weight of good i for the consumer. This form of utility function was first conceptualized by Wassily Leontief. Examples Leontief utility functions represent complementary goods. For example: * Suppose x_1 is the number of left shoes and x_2 the number of right shoes. A consumer can only use pairs of shoes. Hence, his utility is \min(x_1,x_2). * In a cloud computing environment, there is a large server that runs many different tasks. Suppose a certain type of a task requires 2 CPUs, 3 gigabytes of memory and 4 gigabytes of disk-space to complete. The utility of the user is equal to the number of completed tasks. Hence, it can be represented by: \min(, , ). Properties A consumer with a Leont ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
L-infinity
In mathematics, \ell^\infty, the (real or complex) vector space of bounded sequences with the supremum norm, and L^\infty = L^\infty(X,\Sigma,\mu), the vector space of essentially bounded measurable functions with the essential supremum norm, are two closely related Banach spaces. In fact the former is a special case of the latter. As a Banach space they are the continuous dual of the Banach spaces \ell_1 of absolutely summable sequences, and L^1 = L^1(X,\Sigma, \mu) of absolutely integrable measurable functions (if the measure space fulfills the conditions of being localizable and therefore semifinite). Pointwise multiplication gives them the structure of a Banach algebra, and in fact they are the standard examples of abelian Von Neumann algebras. Sequence space The vector space \ell^\infty is a sequence space whose elements are the bounded sequences. The vector space operations, addition and scalar multiplication, are applied coordinate by coordinate. With respect to the nor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Average Voting Rule
The average voting rule is a rule for group decision-making when the decision is a ''distribution'' (e.g. the allocation of a budget among different issues), and each of the voters reports his ideal distribution. This is a special case of budget-proposal aggregation. It is a simple aggregation rule, that returns the arithmetic mean of all individual ideal distributions. The average rule was first studied formally by Michael Intrilligator. This rule and its variants are commonly used in economics and sports. Characterization Intrilligator proved that the average rule is the unique rule that satisfies the following three axioms: * ''Completeness:'' for every ''n'' distributions, the rule returns a distribution. * ''Unanimity for losers'': if an issue receives 0 in all individual distributions, then it receives 0 in the collective distribution. * ''Strict and equal sensitivity to individual allocations'': if one voter increases his allocation to one issue while all other allocation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Utilitarian Approval Voting
Approval voting is a single-winner rated voting system where voters can approve of all the candidates as they like instead of choosing one. The method is designed to eliminate vote-splitting while keeping election administration simple and easy-to-count (requiring only a single score for each candidate). Approval voting has been used in both organizational and political elections to improve representativeness and voter satisfaction. Critics of approval voting have argued the simple ballot format is a disadvantage, as it forces a binary choice for each candidate (instead of the expressive grades of other rated voting rules). Effect on elections Research by social choice theorists Steven Brams and Dudley R. Herschbach found that approval voting would increase voter participation, prevent minor-party candidates from being spoilers, and reduce negative campaigning. Brams' research concluded that approval can be expected to elect majority-preferred candidates in practical el ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Convex Set
In geometry, a set of points is convex if it contains every line segment between two points in the set. For example, a solid cube (geometry), cube is a convex set, but anything that is hollow or has an indent, for example, a crescent shape, is not convex. The boundary (topology), boundary of a convex set in the plane is always a convex curve. The intersection of all the convex sets that contain a given subset of Euclidean space is called the convex hull of . It is the smallest convex set containing . A convex function is a real-valued function defined on an interval (mathematics), interval with the property that its epigraph (mathematics), epigraph (the set of points on or above the graph of a function, graph of the function) is a convex set. Convex minimization is a subfield of mathematical optimization, optimization that studies the problem of minimizing convex functions over convex sets. The branch of mathematics devoted to the study of properties of convex sets and convex f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Utilitarian Rule
In social choice theory, social choice and operations research, the utilitarian rule (also called the max-sum rule) is a Social welfare function, rule saying that, among all possible alternatives, society should pick the alternative which maximizes the ''sum of the utilities'' of all individuals in society. It is a formal mathematical representation of the Utilitarianism, utilitarian philosophy, and is often justified by reference to Harsanyi's utilitarian theorem or the Von Neumann–Morgenstern utility theorem, Von Neumann–Morgenstern theorem. 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 of the resource. 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, utilit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clemens Puppe
Clemens Dieter Puppe (*14 April 1960 in Heidelberg) is a German economist. He is known for his contributions to individual and collective decision theory (social choice theory). Clemens Puppe is Professor at Karlsruhe Institute of Technology (KIT) and co-director of the Institute of Economics (ECON). Life and academic career Clemens Puppe was born on April 14, 1960, in Heidelberg, Germany. After having abandoned the plan to study music, he began his studies in Mathematics and Philosophy at the University of Heidelberg. In the year 1983 he moved to the Free University of Berlin where he completed his M.A. studies in the year 1987 with highest distinction. From 1987 to 1991 he worked as an academic assistant at the Institute of Statistics and Mathematical Economics of the University of Karlsruhe where he finished his Ph.D. in the year 1990 with a thesis on individual decision theory under risk. From 1991 to 1993 Clemens Puppe held a scholarship of the German Research Foundation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
