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Choquet Expected Utility
A Choquet integral is a subadditive or superadditive integral created by the French mathematician Gustave Choquet in 1953. It was initially used in statistical mechanics and potential theory, but found its way into decision theory in the 1980s, where it is used as a way of measuring the expected utility of an uncertain event. It is applied specifically to membership functions and capacities. In imprecise probability theory, the Choquet integral is also used to calculate the lower expectation induced by a 2-monotone lower probability, or the upper expectation induced by a 2-alternating upper probability. Using the Choquet integral to denote the expected utility of belief functions measured with capacities is a way to reconcile the Ellsberg paradox and the Allais paradox. Definition The following notation is used: * S – a set. * \mathcal – a collection of subsets of S. * f : S\to \mathbb – a function. * \nu : \mathcal\to \mathbb^+ – a monotone set function. Assume t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subadditive
In mathematics, subadditivity is a property of a function that states, roughly, that evaluating the function for the sum of two elements of the domain always returns something less than or equal to the sum of the function's values at each element. There are numerous examples of subadditive functions in various areas of mathematics, particularly norms and square roots. Additive maps are special cases of subadditive functions. Definitions A subadditive function is a function f \colon A \to B, having a domain ''A'' and an ordered codomain ''B'' that are both closed under addition, with the following property: \forall x, y \in A, f(x+y)\leq f(x)+f(y). An example is the square root function, having the non-negative real numbers as domain and codomain, since \forall x, y \geq 0 we have: \sqrt\leq \sqrt+\sqrt. A sequence \left \, n \geq 1, is called subadditive if it satisfies the inequality a_\leq a_n+a_m for all ''m'' and ''n''. This is a special case of subadditive function, if a s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Allais Paradox
The Allais paradox is a choice problem designed by to show an inconsistency of actual observed choices with the predictions of expected utility theory. Statement of the problem The Allais paradox arises when comparing participants' choices in two different experiments, each of which consists of a choice between two gambles, A and B. The payoffs for each gamble in each experiment are as follows: Several studies involving hypothetical and small monetary payoffs, and recently involving health outcomes, have supported the assertion that when presented with a choice between 1A and 1B, most people would choose 1A. Likewise, when presented with a choice between 2A and 2B, most people would choose 2B. Allais further asserted that it was reasonable to choose 1A alone or 2B alone. However, that the same person (who chose 1A alone or 2B alone) would choose both 1A and 2B together is inconsistent with expected utility theory. According to expected utility theory, the person should choose e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Expected Utility
The expected utility hypothesis is a popular concept in economics that serves as a reference guide for decisions when the payoff is uncertain. The theory recommends which option rational individuals should choose in a complex situation, based on their risk appetite and preferences. The expected utility hypothesis states an agent chooses between risky prospects by comparing expected utility values (i.e. the weighted sum of adding the respective utility values of payoffs multiplied by their probabilities). The summarised formula for expected utility is U(p)=\sum u(x_k)p_k where p_k is the probability that outcome indexed by k with payoff x_k is realized, and function ''u'' expresses the utility of each respective payoff. On a graph, the curvature of u will explain the agent's risk attitude. For example, if an agent derives 0 utils from 0 apples, 2 utils from one apple, and 3 utils from two apples, their expected utility for a 50–50 gamble between zero apples and two is 0.5''u''(0 ... [...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] |
Subadditivity
In mathematics, subadditivity is a property of a function that states, roughly, that evaluating the function for the sum of two elements of the domain always returns something less than or equal to the sum of the function's values at each element. There are numerous examples of subadditive functions in various areas of mathematics, particularly norms and square roots. Additive maps are special cases of subadditive functions. Definitions A subadditive function is a function f \colon A \to B, having a domain ''A'' and an ordered codomain ''B'' that are both closed under addition, with the following property: \forall x, y \in A, f(x+y)\leq f(x)+f(y). An example is the square root function, having the non-negative real numbers as domain and codomain, since \forall x, y \geq 0 we have: \sqrt\leq \sqrt+\sqrt. A sequence \left \, n \geq 1, is called subadditive if it satisfies the inequality a_\leq a_n+a_m for all ''m'' and ''n''. This is a special case of subadditive function, if a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Superadditivity
In mathematics, a function f is superadditive if f(x+y) \geq f(x) + f(y) for all x and y in the domain of f. Similarly, a sequence \left\, n \geq 1, is called superadditive if it satisfies the inequality a_ \geq a_n + a_m for all m and n. The term "superadditive" is also applied to functions from a boolean algebra to the real numbers where P(X \lor Y) \geq P(X) + P(Y), such as lower probabilities. Properties If f is a superadditive function, and if 0 is in its domain, then f(0) \leq 0. To see this, take the inequality at the top: f(x) \leq f(x+y) - f(y). Hence f(0) \leq f(0+y) - f(y) = 0. The negative of a superadditive function is subadditive. Fekete's lemma The major reason for the use of superadditive sequences is the following lemma due to Michael Fekete. :Lemma: (Fekete) For every superadditive sequence \left\, n \geq 1, the limit \lim a_n/n is equal to \sup a_n/n. (The limit may be positive infinity, for instance, for the sequence a_n = \log n!.) For example, f( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nonlinear Expectation
In probability theory, a nonlinear expectation is a nonlinear generalization of the expectation. Nonlinear expectations are useful in utility theory as they more closely match human behavior than traditional expectations. The common use of nonlinear expectations is in assessing risks under uncertainty. Generally, nonlinear expectations are categorized into sub-linear and super-linear expectations dependent on the additive properties of the given sets. Much of the study of nonlinear expectation is attributed to work of mathematicians within the past two decades. Definition A functional \mathbb: \mathcal \to \mathbb (where \mathcal is a vector lattice on a probability space) is a nonlinear expectation if it satisfies: # Monotonicity: if X,Y \in \mathcal such that X \geq Y then \mathbb \geq \mathbb /math> # Preserving of constants: if c \in \mathbb then \mathbb = c The complete consideration of the given set, the linear space for the functions given that set, and the nonlinear expec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Daniel Kahneman
Daniel Kahneman (; he, דניאל כהנמן; born March 5, 1934) is an Israeli-American psychologist and economist notable for his work on the psychology of judgment and decision-making, as well as behavioral economics, for which he was awarded the 2002 Nobel Memorial Prize in Economic Sciences (shared with Vernon L. Smith). His empirical findings challenge the assumption of human rationality prevailing in modern economic theory. With Amos Tversky and others, Kahneman established a cognitive basis for common human errors that arise from heuristics and biases, and developed prospect theory. In 2011 he was named by '' Foreign Policy'' magazine in its list of top global thinkers. In the same year his book ''Thinking, Fast and Slow'', which summarizes much of his research, was published and became a best seller. In 2015, ''The Economist'' listed him as the seventh most influential economist in the world. He is professor emeritus of psychology and public affairs at Princeton U ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Amos Tversky
Amos Nathan Tversky ( he, עמוס טברסקי; March 16, 1937 – June 2, 1996) was an Israeli cognitive and mathematical psychologist and a key figure in the discovery of systematic human cognitive bias and handling of risk. Much of his early work concerned the foundations of measurement. He was co-author of a three-volume treatise, ''Foundations of Measurement''. His early work with Daniel Kahneman focused on the psychology of prediction and probability judgment; later they worked together to develop prospect theory, which aims to explain irrational human economic choices and is considered one of the seminal works of behavioral economics. Six years after Tversky's death, Kahneman received the 2002 Nobel Memorial Prize in Economic Sciences for the work he did in collaboration with Amos Tversky. (The prize is not awarded posthumously.) Kahneman told ''The New York Times'' in an interview soon after receiving the honor: "I feel it is a joint prize. We were twinned for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cumulative Distribution Function
In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Every probability distribution supported on the real numbers, discrete or "mixed" as well as continuous, is uniquely identified by an ''upwards continuous'' ''monotonic increasing'' cumulative distribution function F : \mathbb R \rightarrow ,1/math> satisfying \lim_F(x)=0 and \lim_F(x)=1. In the case of a scalar continuous distribution, it gives the area under the probability density function from minus infinity to x. Cumulative distribution functions are also used to specify the distribution of multivariate random variables. Definition The cumulative distribution function of a real-valued random variable X is the function given by where the right-hand side represents the probability that the random variable X takes on a value less tha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemann Integral
In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. It was presented to the faculty at the University of Göttingen in 1854, but not published in a journal until 1868. For many functions and practical applications, the Riemann integral can be evaluated by the fundamental theorem of calculus or approximated by numerical integration. Overview Let be a non-negative real-valued function on the interval , and let be the region of the plane under the graph of the function and above the interval . See the figure on the top right. This region can be expressed in set-builder notation as S = \left \. We are interested in measuring the area of . Once we have measured it, we will denote the area in the usual way by \int_a^b f(x)\,dx. The basic idea of the Riemann integral is to use very simple approximations for the area of . By taking better and be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Set Function
In mathematics, especially measure theory, a set function is a function whose domain is a family of subsets of some given set and that (usually) takes its values in the extended real number line \R \cup \, which consists of the real numbers \R and \pm \infty. A set function generally aims to subsets in some way. Measures are typical examples of "measuring" set functions. Therefore, the term "set function" is often used for avoiding confusion between the mathematical meaning of "measure" and its common language meaning. Definitions If \mathcal is a family of sets over \Omega (meaning that \mathcal \subseteq \wp(\Omega) where \wp(\Omega) denotes the powerset) then a is a function \mu with domain \mathcal and codomain \infty, \infty/math> or, sometimes, the codomain is instead some vector space, as with vector measures, complex measures, and projection-valued measures. The domain is a set function may have any number properties; the commonly encountered properties and categor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
