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Dutch Book Argument
In decision theory, economics, and probability theory, the Dutch book arguments are a set of results showing that agents must satisfy the axioms of rational choice to avoid a kind of self-contradiction called a Dutch book. A Dutch book, sometimes also called a money pump, is a set of bets that ensures a guaranteed loss, i.e. the gambler will lose money no matter what happens. A set of bets is called coherent if it cannot result in a Dutch book. The Dutch book arguments are used to explore degrees of certainty in beliefs, and demonstrate that rational bet-setters must be Bayesian; in other words, a rational bet-setter must assign event probabilities that behave according to the axioms of probability, and must have preferences that can be modeled using the von Neumann–Morgenstern axioms. In economics, they are used to model behavior by ruling out situations where agents "burn money" for no real reward. Models based on the assumption that actors are rational are called r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dutch Language
Dutch ( ) is a West Germanic languages, West Germanic language of the Indo-European language family, spoken by about 25 million people as a first language and 5 million as a second language and is the List of languages by total number of speakers, third most spoken Germanic language. In Europe, Dutch is the native language of most of the population of the Netherlands and Flanders (which includes 60% of the population of Belgium). "1% of the EU population claims to speak Dutch well enough in order to have a conversation." (page 153). Dutch was one of the official languages of South Africa until 1925, when it was replaced by Afrikaans, a separate but partially Mutual intelligibility, mutually intelligible daughter language of Dutch. Afrikaans, depending on the definition used, may be considered a sister language, spoken, to some degree, by at least 16 million people, mainly in South Africa and Namibia, and evolving from Cape Dutch dialects. In South America, Dutch is the native l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Leonard Jimmie Savage
Leonard Jimmie Savage (born Leonard Ogashevitz; 1917 – 1971) was an American mathematician and statistician. Economist Milton Friedman said Savage was "one of the few people I have met whom I would unhesitatingly call a genius." Education and career Savage was born and grew up in Detroit. He studied at Wayne State University in Detroit before transferring to University of Michigan, where he first majored in chemical engineering, then switched to mathematics, graduating in 1938 with a bachelor's degree. He continued at the University of Michigan with a PhD on differential geometry in 1941 under the supervision of Sumner Byron Myers. Savage subsequently worked at the Institute for Advanced Study in Princeton, New Jersey, the University of Chicago, the University of Michigan, Yale University, and the Statistical Research Group at Columbia University. Though his thesis advisor was Sumner Myers, he also credited Milton Friedman and W. Allen Wallis as statistical mentors. Durin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Axioms
The standard probability axioms are the foundations of probability theory introduced by Russian mathematician Andrey Kolmogorov in 1933. These axioms remain central and have direct contributions to mathematics, the physical sciences, and real-world probability cases. There are several other (equivalent) approaches to formalising probability. Bayesians will often motivate the Kolmogorov axioms by invoking Cox's theorem or the Dutch book arguments instead. Kolmogorov axioms The assumptions as to setting up the axioms can be summarised as follows: Let (\Omega, F, P) be a measure space such that P(E) is the probability of some event E, and P(\Omega) = 1. Then (\Omega, F, P) is a probability space, with sample space \Omega, event space F and probability measure P. First axiom The probability of an event is a non-negative real number: :P(E)\in\mathbb, P(E)\geq 0 \qquad \forall E \in F where F is the event space. It follows (when combined with the second axiom) that P(E) is alwa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mutually Exclusive Events
In logic and probability theory, two events (or propositions) are mutually exclusive or disjoint if they cannot both occur at the same time. A clear example is the set of outcomes of a single coin toss, which can result in either heads or tails, but not both. In the coin-tossing example, both outcomes are, in theory, collectively exhaustive, which means that at least one of the outcomes must happen, so these two possibilities together exhaust all the possibilities. However, not all mutually exclusive events are collectively exhaustive. For example, the outcomes 1 and 4 of a single roll of a six-sided die are mutually exclusive (both cannot happen at the same time) but not collectively exhaustive (there are other possible outcomes; 2,3,5,6). Logic In logic, two propositions \phi and \psi are mutually exclusive if it is not logically possible for them to be true at the same time; that is, \lnot (\phi \land \psi) is a tautology. To say that more than two propositions a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
No-win Situation
A no-win situation or lose–lose situation is an outcome of a negotiation, conflict or challenging circumstance in which all parties are worse off. It is an alternative to a win–win or outcome in which one party wins. Arbitration or mediation may be used to avoid no-win outcomes and find more satisfactory results. In game theory In game theory, a "no-win" situation is a circumstance in which no player benefits from any outcome, hence ultimately losing the match. This may be because of any or all of the following: * Unavoidable or unforeseeable circumstances causing the situation to change after decisions have been made. This is common in text adventures. * '' Zugzwang'', as in chess, when any move a player chooses makes them worse off than before such as losing a piece or being checkmated. * A situation in which the player has to accomplish two mutually dependent tasks each of which must be completed before the other or that are mutually exclusive (a Catch-22). * Ignora ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axioms Of Probability
The standard probability axioms are the foundations of probability theory introduced by Russian mathematician Andrey Kolmogorov in 1933. These axioms remain central and have direct contributions to mathematics, the physical sciences, and real-world probability cases. There are several other (equivalent) approaches to formalising probability. Bayesians will often motivate the Kolmogorov axioms by invoking Cox's theorem or the Dutch book arguments instead. Kolmogorov axioms The assumptions as to setting up the axioms can be summarised as follows: Let (\Omega, F, P) be a measure space such that P(E) is the probability of some event E, and P(\Omega) = 1. Then (\Omega, F, P) is a probability space, with sample space \Omega, event space F and probability measure P. First axiom The probability of an event is a non-negative real number: :P(E)\in\mathbb, P(E)\geq 0 \qquad \forall E \in F where F is the event space. It follows (when combined with the second axiom) that P(E) is alw ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Necessary And Sufficient Conditions
In logic and mathematics, necessity and sufficiency are terms used to describe a conditional or implicational relationship between two statements. For example, in the conditional statement: "If then ", is necessary for , because the truth of is guaranteed by the truth of . (Equivalently, it is impossible to have without , or the falsity of ensures the falsity of .) Similarly, is sufficient for , because being true always implies that is true, but not being true does not always imply that is not true. In general, a necessary condition is one (possibly one of several conditions) that must be present in order for another condition to occur, while a sufficient condition is one that produces the said condition. The assertion that a statement is a "necessary ''and'' sufficient" condition of another means that the former statement is true if and only if the latter is true. That is, the two statements must be either simultaneously true, or simultaneously false. In ordinary ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coherence (philosophical Gambling Strategy)
In decision theory, economics, and probability theory, the Dutch book arguments are a set of results showing that agents must satisfy the axioms of rational choice to avoid a kind of self-contradiction called a Dutch book. A Dutch book, sometimes also called a money pump, is a set of bets that ensures a guaranteed loss, i.e. the gambler will lose money no matter what happens. A set of bets is called coherent if it cannot result in a Dutch book. The Dutch book arguments are used to explore degrees of certainty in beliefs, and demonstrate that rational bet-setters must be Bayesian; in other words, a rational bet-setter must assign event probabilities that behave according to the axioms of probability, and must have preferences that can be modeled using the von Neumann–Morgenstern axioms. In economics, they are used to model behavior by ruling out situations where agents "burn money" for no real reward. Models based on the assumption that actors are rational are called ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frank P
Frank, FRANK, or Franks may refer to: People * Frank (given name) * Frank (surname) * Franks (surname) * Franks, a Germanic people in late Roman times * Franks, a term in the Muslim world for Franks#Crusaders and other Western Europeans as "Franks", all western Europeans, particularly during the Crusades Currency * Liechtenstein franc or frank, the currency of Liechtenstein since 1920 * Swiss franc or frank, the currency of Switzerland since 1850 * Westphalian frank, currency of the Kingdom of Westphalia between 1808 and 1813 * The currencies of the German-speaking cantons of Switzerland (1803–1814): ** Appenzell frank ** Aargau frank ** Basel frank ** Berne frank ** Fribourg frank ** Glarus frank ** Graubünden frank ** Luzern frank ** Schaffhausen frank ** Schwyz frank ** Solothurn frank ** St. Gallen frank ** Thurgau frank ** Unterwalden frank ** Uri frank ** Zürich frank Places * Frank, Alberta, Canada, an urban community, formerly a village * Franks, Illinois, United Sta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trifecta
Trifecta A trifecta is a parimutuel bet placed on a horse race in which the bettor must predict which horses will finish first, second, and third, in the exact order. Known as a trifecta in the US and Australia, this is known as a tricast in the UK, a tierce in Hong Kong, a triactor in Canada and a tiercé in France. A trio, offered in Hong Kong and France, is a variation in which the order of the horses is not relevant. Variations Boxed A "boxed" trifecta is where three horses are selected, and the player wins if these three horses finish first in any order. Boxed bets are effectively equivalent to placing standard trifecta bets on all six possible outcomes of the selected horses. For example, a boxed trifecta of horses numbered 6, 7 and 9, wins if horses finish in any of these combinations of outcomes: * 6, 7, 9 * 6, 9, 7 * 7, 6, 9 * 7, 9, 6 * 9, 6, 7 * 9, 7, 6. Banker One horse (the "banker") is chosen to win the race, and two or more selections are boxed to come sec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axiom Of Unitarity
Unit measure is an axiom of probability theory that states that the probability Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an e ... of the entire sample space is equal to one ( unity); that is, ''P''(''S'')=1 where ''S'' is the sample space. Loosely speaking, it means that ''S'' must be chosen so that when the experiment is performed, ''something'' happens. The term ''measure'' here refers to the measure-theoretic approach to probability. Violations of unit measure have been reported in arguments about the outcomes of eventsT. Oldberg and R. Christensen "Erratic measure" NDE for the Energy Industry 1995, American Society of Mechanical Engineers, New York, NY. under which events acquire "probabilities" that are not the probabilities of probability theory. In situations such as these ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bookmaker
A bookmaker, bookie, or turf accountant is an organization or a person that accepts and pays out bets on sporting and other events at agreed-upon odds In probability theory, odds provide a measure of the probability of a particular outcome. Odds are commonly used in gambling and statistics. For example for an event that is 40% probable, one could say that the odds are or When gambling, o .... History The first bookmaker, Harry Ogden, stood at Newmarket in 1795, although similar activities had existed in other forms earlier in the eighteenth century. Following the Gaming Act 1845, the only gambling allowed in the United Kingdom was at race tracks. The introduction of special excursion trains meant that all classes of society could attend the new racecourses opening across the country. Runners working for bookmakers would collect bets in Clock bag, clock bags. Cash flowed to the bookmakers who employed bodyguards against protection gangs operating within the vast cro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
