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General Possibility Theorem
Arrow's impossibility theorem, the general possibility theorem or Arrow's paradox is an impossibility theorem in social choice theory that states that when voters have three or more distinct alternatives (options), no ranked voting electoral system can convert the ranked preferences of individuals into a community-wide (complete and transitive) ranking while also meeting the specified set of criteria: '' unrestricted domain'', '' non-dictatorship'', ''Pareto efficiency'', and ''independence of irrelevant alternatives''. The theorem is often cited in discussions of voting theory as it is further interpreted by the Gibbard–Satterthwaite theorem. The theorem is named after economist and Nobel laureate Kenneth Arrow, who demonstrated the theorem in his doctoral thesis and popularized it in his 1951 book ''Social Choice and Individual Values''. The original paper was titled "A Difficulty in the Concept of Social Welfare". In short, the theorem states that no rank-order electoral syste ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proof Of Impossibility
In mathematics, a proof of impossibility is a proof that demonstrates that a particular problem cannot be solved as described in the claim, or that a particular set of problems cannot be solved in general. Such a case is also known as a negative proof, proof of an impossibility theorem, or negative result. Because they show that something cannot be done, proofs of impossibility can be the resolutions to decades or centuries of work attempting to find a solution. Proving that something is impossible is usually much harder than the opposite task, as it is often necessary to develop a proof that works in general, rather than to just show a particular example. Impossibility theorems are usually expressible as negative existential propositions or universal propositions in logic. The irrationality of the square root of 2 is one of the oldest proofs of impossibility. It shows that it is impossible to express the square root of 2 as a ratio of two integers. Another consequential proof of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Welfare Economics
Welfare economics is a branch of economics that uses microeconomic techniques to evaluate well-being (welfare) at the aggregate (economy-wide) level. Attempting to apply the principles of welfare economics gives rise to the field of public economics, the study of how government might intervene to improve social welfare. Welfare economics also provides the theoretical foundations for particular instruments of public economics, including cost–benefit analysis, while the combination of welfare economics and insights from behavioral economics has led to the creation of a new subfield, behavioral welfare economics. The field of welfare economics is associated with two fundamental theorems. The first states that given certain assumptions, competitive markets produce ( Pareto) efficient outcomes; it captures the logic of Adam Smith's invisible hand. The second states that given further restrictions, any Pareto efficient outcome can be supported as a competitive market equilibrium. Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
A Priori Knowledge
("from the earlier") and ("from the later") are Latin phrases used in philosophy to distinguish types of knowledge, justification, or argument by their reliance on empirical evidence or experience. knowledge is independent from current experience (e.g., as part of a new study). Examples include mathematics,Some associationist philosophers have contended that mathematics comes from experience and is not a form of any a priori knowledge () tautologies, and deduction from pure reason.Galen Strawson has stated that an argument is one in which "you can see that it is true just lying on your couch. You don't have to get up off your couch and go outside and examine the way things are in the physical world. You don't have to do any science." () knowledge depends on empirical evidence. Examples include most fields of science and aspects of personal knowledge. The terms originate from the analytic methods found in ''Organon'', a collection of works by Aristotle. Prior analytics ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diagram For Part One Of Arrow's Impossibility Theorem
A diagram is a symbolic representation of information using visualization techniques. Diagrams have been used since prehistoric times on walls of caves, but became more prevalent during the Enlightenment. Sometimes, the technique uses a three-dimensional visualization which is then projected onto a two-dimensional surface. The word ''graph'' is sometimes used as a synonym for diagram. Overview The term "diagram" in its commonly used sense can have a general or specific meaning: * ''visual information device'' : Like the term "illustration", "diagram" is used as a collective term standing for the whole class of technical genres, including graphs, technical drawings and tables. * ''specific kind of visual display'' : This is the genre that shows qualitative data with shapes that are connected by lines, arrows, or other visual links. In science the term is used in both ways. For example, Anderson (1997) stated more generally: "diagrams are pictorial, yet abstract, representat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Economic Theory (journal)
''Economic Theory'' is a peer-reviewed academic journal that focuses on theoretical economics, particularly social choice, general equilibrium theory, and game theory. Mathematically rigorous articles are also published in the fields of experimental economics, public economics, international economics, development economics, and industrial organisation. The journal is the official journal of the Society for the Advancement of Economic Theory. Both the society and the journal were founded by Charalambos D. Aliprantis, David Cass, Douglas Gale, Mukul Majumdar, Edward C. Prescott Edward Christian Prescott (December 26, 1940 – November 6, 2022) was an American economist. He received the Nobel Memorial Prize in Economics in 2004, sharing the award with Finn E. Kydland, "for their contributions to dynamic macroeconomics: ..., Nicholas C. Yannelis, and Yves Younes. External links * {{DEFAULTSORT:Economic Theory (Journal) Economics journals Publications established in 1991 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tuple
In mathematics, a tuple is a finite ordered list (sequence) of elements. An -tuple is a sequence (or ordered list) of elements, where is a non-negative integer. There is only one 0-tuple, referred to as ''the empty tuple''. An -tuple is defined inductively using the construction of an ordered pair. Mathematicians usually write tuples by listing the elements within parentheses "" and separated by a comma and a space; for example, denotes a 5-tuple. Sometimes other symbols are used to surround the elements, such as square brackets "nbsp; or angle brackets "⟨ ⟩". Braces "" are used to specify arrays in some programming languages but not in mathematical expressions, as they are the standard notation for sets. The term ''tuple'' can often occur when discussing other mathematical objects, such as vectors. In computer science, tuples come in many forms. Most typed functional programming languages implement tuples directly as product types, tightly associated with algebr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Total Order
In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( reflexive). # If a \leq b and b \leq c then a \leq c ( transitive). # If a \leq b and b \leq a then a = b ( antisymmetric). # a \leq b or b \leq a (strongly connected, formerly called total). Total orders are sometimes also called simple, connex, or full orders. A set equipped with a total order is a totally ordered set; the terms simply ordered set, linearly ordered set, and loset are also used. The term ''chain'' is sometimes defined as a synonym of ''totally ordered set'', but refers generally to some sort of totally ordered subsets of a given partially ordered set. An extension of a given partial order to a total order is called a linear extension of that partial order. Strict and non-strict total orders A on a set X is a strict partial ord ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Surjective Function
In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of its domain. It is not required that be unique; the function may map one or more elements of to the same element of . The term ''surjective'' and the related terms ''injective'' and ''bijective'' were introduced by Nicolas Bourbaki, a group of mainly French 20th-century mathematicians who, under this pseudonym, wrote a series of books presenting an exposition of modern advanced mathematics, beginning in 1935. The French word '' sur'' means ''over'' or ''above'', and relates to the fact that the image of the domain of a surjective function completely covers the function's codomain. Any function induces a surjection by restricting its codomain to the image of its domain. Every surjective function has a right inverse assuming the axiom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monotonicity Criterion
The monotonicity criterion is a voting system criterion used to evaluate both single and multiple winner ranked voting systems. A ranked voting system is monotonic if it is neither possible to prevent the election of a candidate by ranking them higher on some of the ballots, nor possible to elect an otherwise unelected candidate by ranking them lower on some of the ballots (while nothing else is altered on any ballot).D R Woodall"Monotonicity and Single-Seat Election Rules" ''Voting matters'', Issue 6, 1996 That is to say, in single winner elections no winner is harmed by up-ranking and no loser is helped by down-ranking. Douglas Woodall called the criterion mono-raise. Raising a candidate on some ballots ''while changing'' the orders of other candidates does ''not'' constitute a failure of monotonicity. E.g., harming candidate by changing some ballots from to would violate the monotonicity criterion, while harming candidate by changing some ballots from to would not. The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Learning To Rank
Learning to rank. Slides from Tie-Yan Liu's talk at WWW 2009 conference aravailable online or machine-learned ranking (MLR) is the application of machine learning, typically supervised, semi-supervised or reinforcement learning, in the construction of ranking models for information retrieval systems. Training data consists of lists of items with some partial order specified between items in each list. This order is typically induced by giving a numerical or ordinal score or a binary judgment (e.g. "relevant" or "not relevant") for each item. The goal of constructing the ranking model is to rank new, unseen lists in a similar way to rankings in the training data. Applications In information retrieval Ranking is a central part of many information retrieval problems, such as document retrieval, collaborative filtering, sentiment analysis, and online advertising. A possible architecture of a machine-learned search engine is shown in the accompanying figure. Training data con ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Independence Of Irrelevant Alternatives
The independence of irrelevant alternatives (IIA), also known as binary independence or the independence axiom, is an axiom of decision theory and various social sciences. The term is used in different connotation in several contexts. Although it always attempts to provide an account of rational individual behavior or aggregation of individual preferences, the exact formulation differs widely in both language and exact content. Perhaps the easiest way to understand the axiom is how it pertains to casting a ballot. There the axiom says that if Charlie (the irrelevant alternative) enters a race between Alice and Bob, with Alice (leader) liked better than Bob (runner-up), then the individual voter who likes Charlie less than Alice will not switch his vote from Alice to Bob. Because of this, a violation of IIA is commonly referred to as the "spoiler effect": support for Charlie "spoils" the election for Alice, while it "logically" should not have. After all, Alice ''was'' liked better t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Deterministic System (mathematics)
In mathematics, computer science and physics, a deterministic system is a system in which no randomness is involved in the development of future states of the system. A deterministic model will thus always produce the same output from a given starting condition or initial state. at In physics Physical laws that are described by represent deterministic systems, even though the state of the system at a given point in time may be difficult to describe explicitly. In |
