# Some Probability Paradoxes in Choice from among Random Alternatives

01 Jun 1972-Journal of the American Statistical Association (Taylor & Francis Group)-Vol. 67, Iss: 338, pp 366-373

TL;DR: The problem of choosing from among possible random losses or payoffs has been studied in this paper, as in choosing from possible statistical decision procedures or from possible wagers, and the difficulties of choosing among possible losses or wagers have been discussed.

Abstract: The probability P(X>Y) can be arbitrarily close to 1 even though the random variable X is stochastically smaller than Yi the probabilities P(X

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TL;DR: The choice-under-uncertainty theory is a field in flux as mentioned in this paper and it is being challenged on several grounds from both within and outside economics, and the nature of these challenges, and of our profession's responses to them, is the topic of this paper.

Abstract: Ffifteen years ago, the theory of choice under uncertainty could be considered one of the “success stories” of economic analysis: it rested on solid axiomatic foundations, it had seen important breakthroughs in the analytics of risk, risk aversion and their applications to economic issues, and it stood ready to provide the theoretical underpinnings for the newly emerging “information revolution” in economics.1 Today choice under uncertainty is a field in flux: the standard theory is being challenged on several grounds from both within and outside economics. The nature of these challenges, and of our profession’s responses to them, is the topic of this paper.

1,058 citations

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TL;DR: This work abstracts from the usual numeric representations by encoding only qualitative relationships, which are inequality constraints on the joint probability distribution over the variables, and defines two types of qualitative relationship: Qualitative influences and Qualitative synergies.

415 citations

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TL;DR: In this article, a general method for determining Pitman Nearness is given in the case of univariate estimators, which is then applied to some estimation problems and generalized to the multivariate case.

Abstract: A general method for determining Pitman Nearness is given In the case of univariate estimators. This method is then applied to some estimation problems. The concept of Pitman Nearness is also generalized to the multivariate case. The James-Stein estimators are used to illustrate the multivariate comparison.

101 citations

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01 Jan 1944

TL;DR: Theory of games and economic behavior as mentioned in this paper is the classic work upon which modern-day game theory is based, and it has been widely used to analyze a host of real-world phenomena from arms races to optimal policy choices of presidential candidates, from vaccination policy to major league baseball salary negotiations.

Abstract: This is the classic work upon which modern-day game theory is based. What began more than sixty years ago as a modest proposal that a mathematician and an economist write a short paper together blossomed, in 1944, when Princeton University Press published "Theory of Games and Economic Behavior." In it, John von Neumann and Oskar Morgenstern conceived a groundbreaking mathematical theory of economic and social organization, based on a theory of games of strategy. Not only would this revolutionize economics, but the entirely new field of scientific inquiry it yielded--game theory--has since been widely used to analyze a host of real-world phenomena from arms races to optimal policy choices of presidential candidates, from vaccination policy to major league baseball salary negotiations. And it is today established throughout both the social sciences and a wide range of other sciences.

19,337 citations

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01 Jan 1958

TL;DR: In this paper, Dodgson's Third Pamphlet 'A Method...' (1876) was used to discuss the Elasticity of Committee Decisions with an Altering Size of Majority.

Abstract: I The Theory of Committees and Elections.- I. A Committee and Motions.- II. Independent Valuation.- III. Can a Motion be Represented by the same Symbol on Different Schedules?.- IV. A Committee using a Simple Majority: Single-peaked Preference Curves.- V. A Committee using a Simple Majority: other Shapes of Preference Curves.- 1. Curves either single-peaked or single-peaked with a plateau on top.- 2. Other classes of curves.- VI. A Committee using a Simple Majority: any Shapes of Preference Curves, Number of Motions Finite.- VII. Cyclical Majorities.- VIII. When the Ordinary Committee Procedure is in use the Members' Scales of Valuation may be Incomplete.- IX. Which Candidate ought to be Elected?.- X. Examination of some Methods of Election in Single-member Constituencies.- XI. Proportional Representation.- XII. The Decisions of a Committee using a Special Majority.- 1. When the members' preference curves are single-peaked.- 2. When the members' preference curves are subject to no restriction.- XIII. The Elasticity of Committee Decisions with an Altering Size of Majority.- 1. When the members' preference curves are single-peaked.- 2. When the members' preference curves are subject to no restriction.- XIV. The Elasticity of Committee Decisions with Alterations in the Members' Preference Schedules.- 1. When the members' preference curves are single-peaked.- 2. When the members' preference curves are subject to no restriction.- XV. The Converse Problem: the Group of Schedules to Correspond to a Given Voting Matrix.- XVI. A Committee using a Simple Majority: Complementary Motions.- XVII. International Agreements, Sovereignty and the Cabinet.- II History of the Mathematical Theory of Committees and Elections (Excluding Proportional Representation).- XVIII. Borda, Condorcet and Laplace.- 1. Jean-Charles de Borda (1733-1799).- 2. Marie Jean Antoine Nicolas Caritat, Marquis de Condorcet (1743-1794).- 3. Pierre-Simon, Marquis de Laplace (1749-1827).- 4. Conclusions.- XIX. E. J. Nanson and Francis Galton.- XX. The Circumstances in which Rev. C. L. Dodgson (Lewis Carroll) wrote his Three Pamphlets.- Appendix. Text of Dodgson's Three Pamphlets and of 'The Cyclostyled Sheet'.- A Discussion of the Various Methods of Procedure in Conducting Elections (1873).- Suggestions as to the Best Method of Taking Votes, Where More than Two Issues are to be Voted on (1874).- A Method of Taking Votes on More than Two Issues (1876) 'The Cyclostyled Sheet' (7 Dec. 1877).- Notes on Dodgson's Third Pamphlet 'A Method...' (1876).

2,575 citations