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Stochastic Transitivity
Stochastic transitivity models are stochastic versions of the transitivity property of binary relations studied in mathematics. Several models of stochastic transitivity exist and have been used to describe the probabilities involved in experiments of paired comparisons, specifically in scenarios where transitivity is expected, however, empirical observations of the binary relation is probabilistic. For example, players' skills in a sport might be expected to be transitive, i.e. "if player A is better than B and B is better than C, then player A must be better than C"; however, in any given match, a weaker player might still end up winning with a positive probability. Tightly matched players might have a higher chance of observing this inversion while players with large differences in their skills might only see these inversions happen seldom. Stochastic transitivity models formalize such relations between the probabilities (e.g. of an outcome of a match) and the underlying transiti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stochastic
Stochastic (; ) is the property of being well-described by a random probability distribution. ''Stochasticity'' and ''randomness'' are technically distinct concepts: the former refers to a modeling approach, while the latter describes phenomena; in everyday conversation, however, these terms are often used interchangeably. In probability theory, the formal concept of a '' stochastic process'' is also referred to as a ''random process''. Stochasticity is used in many different fields, including image processing, signal processing, computer science, information theory, telecommunications, chemistry, ecology, neuroscience, physics, and cryptography. It is also used in finance (e.g., stochastic oscillator), due to seemingly random changes in the different markets within the financial sector and in medicine, linguistics, music, media, colour theory, botany, manufacturing and geomorphology. Etymology The word ''stochastic'' in English was originally used as an adjective with the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rational Choice Theory
Rational choice modeling refers to the use of decision theory (the theory of rational choice) as a set of guidelines to help understand economic and social behavior. The theory tries to approximate, predict, or mathematically model human behavior by analyzing the behavior of a rational actor facing the same costs and benefits.Gary Browning, Abigail Halcli, Frank Webster (2000). ''Understanding Contemporary Society: Theories of the Present'', London: Sage Publications. Rational choice models are most closely associated with economics, where mathematical analysis of behavior is standard. However, they are widely used throughout the social sciences, and are commonly applied to cognitive science, criminology, political science, and sociology. Overview The basic premise of rational choice theory is that the decisions made by individual actors will collectively produce aggregate social behaviour. The theory also assumes that individuals have preferences out of available choice ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Decision Theory
Decision theory or the theory of rational choice is a branch of probability theory, probability, economics, and analytic philosophy that uses expected utility and probabilities, probability to model how individuals would behave Rationality, rationally under uncertainty. It differs from the Cognitive science, cognitive and Behavioural sciences, behavioral sciences in that it is mainly Prescriptive economics, prescriptive and concerned with identifying optimal decision, optimal decisions for a rational agent, rather than Descriptive economics, describing how people actually make decisions. Despite this, the field is important to the study of real human behavior by Social science, social scientists, as it lays the foundations to Mathematical model, mathematically model and analyze individuals in fields such as sociology, economics, criminology, cognitive science, moral philosophy and political science. History The roots of decision theory lie in probability theory, developed by Blai ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nontransitive Game
An intransitive or non-transitive game is a zero-sum game in which pairwise competitions between the strategies contain a cycle. If strategy A beats strategy B, B beats C, and C beats A, then the binary relation "to beat" is intransitive, since transitivity would require that A beat C. The terms "transitive game" or "intransitive game" are not used in game theory. A prototypical example of an intransitive game is the game rock, paper, scissors. In probabilistic games like Penney's game, the violation of transitivity results in a more subtle way, and is often presented as a probability paradox. Examples * Rock, paper, scissors * Penney's game * Intransitive dice * ''Fire Emblem'', the video game franchise that popularized intransitive cycles in unit weapons: swords and magic beats axes and bows, axes and bows beat lances and knives, and lances and knives beat swords and magic See also * Stochastic transitivity Stochastic transitivity models are stochastic versions of the transiti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
NP-hardness
In computational complexity theory, a computational problem ''H'' is called NP-hard if, for every problem ''L'' which can be solved in non-deterministic polynomial-time, there is a polynomial-time reduction from ''L'' to ''H''. That is, assuming a solution for ''H'' takes 1 unit time, ''H''s solution can be used to solve ''L'' in polynomial time. As a consequence, finding a polynomial time algorithm to solve a single NP-hard problem would give polynomial time algorithms for all the problems in the complexity class NP. As it is suspected, but unproven, that P≠NP, it is unlikely that any polynomial-time algorithms for NP-hard problems exist. A simple example of an NP-hard problem is the subset sum problem. Informally, if ''H'' is NP-hard, then it is at least as difficult to solve as the problems in NP. However, the opposite direction is not true: some problems are undecidable, and therefore even more difficult to solve than all problems in NP, but they are probably not NP- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Invertible Function
In mathematics, the inverse function of a function (also called the inverse of ) is a function that undoes the operation of . The inverse of exists if and only if is bijective, and if it exists, is denoted by f^ . For a function f\colon X\to Y, its inverse f^\colon Y\to X admits an explicit description: it sends each element y\in Y to the unique element x\in X such that . As an example, consider the real-valued function of a real variable given by . One can think of as the function which multiplies its input by 5 then subtracts 7 from the result. To undo this, one adds 7 to the input, then divides the result by 5. Therefore, the inverse of is the function f^\colon \R\to\R defined by f^(y) = \frac . Definitions Let be a function whose domain is the set , and whose codomain is the set . Then is ''invertible'' if there exists a function from to such that g(f(x))=x for all x\in X and f(g(y))=y for all y\in Y. If is invertible, then there is exactly one function s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gérard Debreu
Gérard Debreu (; 4 July 1921 – 31 December 2004) was a French-born economist and mathematician. Best known as a professor of economics at the University of California, Berkeley, where he began work in 1962, he won the 1983 Nobel Memorial Prize in Economic Sciences. Biography His father was the business partner of his maternal grandfather in lace manufacturing, a traditional industry in Calais. Debreu was orphaned at an early age, as his father committed suicide and his mother died of natural causes. Prior to the start of World War II, he received his baccalauréat and went to Ambert to begin preparing for the entrance examination of a grande école. Later on, he moved from Ambert to Grenoble to complete his preparation, both places being in Vichy France during World War II. In 1941, he was admitted to the École Normale Supérieure in Paris, along with Marcel Boiteux. He was influenced by Henri Cartan and the Bourbaki writers. When he was about to take the final examinati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Learning To Rank
Learning to rank. Slides from Tie-Yan Liu's talk at World Wide Web Conference, WWW 2009 conference aravailable online or machine-learned ranking (MLR) is the application of machine learning, typically Supervised learning, supervised, Semi-supervised learning, semi-supervised or reinforcement learning, in the construction of ranking function, ranking models for information retrieval systems. Training data may, for example, consist 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 possi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Debreu Theorems
In economics, the Debreu's theorems are preference representation theorems—statements about the representation of a preference ordering by a real-valued utility function. The theorems were proved by Gerard Debreu during the 1950s. Background Suppose a person is asked questions of the form "Do you prefer A or B?" (when A and B can be options, actions to take, states of the world, consumption bundles, etc.). All the responses are recorded and form the person's ''preference relation.'' Instead of recording the person's preferences between every pair of options, it would be much more convenient to have a single ''utility function'' - a function that maps a real number to each option, such that the utility of option A is larger than that of option B if and only if the agent prefers A to B. Debreu's theorems address the following question: what conditions on the preference relation guarantee the existence of a representing utility function? Existence of ordinal utility function ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Von Neumann–Morgenstern Utility Theorem
In decision theory, the von Neumann–Morgenstern (VNM) utility theorem demonstrates that rational choice under uncertainty involves making decisions that take the form of maximizing the expected value of some cardinal utility function. The theorem forms the foundation of expected utility theory. In 1947, John von Neumann and Oskar Morgenstern proved that any individual whose preferences satisfied four axioms has a utility function, where such an individual's preferences can be represented on an interval scale and the individual will always prefer actions that maximize expected utility. Neumann, John von and Morgenstern, Oskar, '' Theory of Games and Economic Behavior''. Princeton, NJ. Princeton University Press, 1953. That is, they proved that an agent is (VNM-)rational ''if and only if'' there exists a real-valued function ''u'' defined by possible outcomes such that every preference of the agent is characterized by maximizing the expected value of ''u'', which can then be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Preference
In psychology, economics and philosophy, preference is a technical term usually used in relation to choosing between alternatives. For example, someone prefers A over B if they would rather choose A than B. Preferences are central to decision theory because of this relation to behavior. Some methods such as Ordinal Priority Approach use preference relation for decision-making. As connative states, they are closely related to desires. The difference between the two is that desires are directed at one object while preferences concern a comparison between two alternatives, of which one is preferred to the other. In insolvency, the term is used to determine which outstanding obligation the insolvent party has to settle first. Psychology In psychology, preferences refer to an individual's attitude towards a set of objects, typically reflected in an explicit decision-making process. The term is also used to mean evaluative judgment in the sense of liking or disliking an object, as in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Luce's Choice Axiom
In probability theory, Luce's choice axiom, formulated by R. Duncan Luce (1959), states that the relative odds of selecting one item over another from a pool of many items is not affected by the presence or absence of other items in the pool. Selection of this kind is said to have " independence from irrelevant alternatives" (IIA). Overview Consider a set X of possible outcomes, and consider a selection rule P, such that for any a\in A \subset X with A a finite set, the selector selects a from A with probability P(a \mid A). Luce proposed two choice axioms. The second one is usually meant by "Luce's choice axiom", as the first one is usually called " independence from irrelevant alternatives" (IIA). Luce's choice axiom 1 (IIA): if P(a\mid A) = 0, P(b\mid A) > 0, then for any a, b \in B \subset A, we still have P(a\mid B) = 0. Luce's choice axiom 2 ("path independence"): P(a \mid A) = P(a\mid B)\sum_P(b\mid A)for any a \in B \subset A. Luce's choice axiom 1 is implied by choice ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
