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"The Breakers", a gilded-age Victorian mansion in Newport, Rhode Island.

In American history, the Gilded Age refers to substantial growth in population in the United States and extravagant displays of wealth and excess of America's upper-class during the post-Civil War and post-Reconstruction era, in the late 19th century (1865-1901). The wealth polarization derived primarily from industrial and population expansion. The businessmen of the Second Industrial Revolution created industrial towns and cities in the Northeast with new factories, and contributed to the creation of an ethnically diverse industrial working class which produced the wealth owned by rising super-rich industrialists and financiers such as Cornelius Vanderbilt, John D. Rockefeller, Andrew Carnegie, Henry Flagler, and J.P. Morgan. Their critics called them "robber barons", referring to their use of overpowering and sometimes unethical financial manipulations. There was a small, growing labor union movement, led in part by Samuel Gompers, who created the American Federation of Labor (AFL), founded in 1886. It featured very close contests between the Republicans and Democrats, with occasional third parties. Nearly all the eligible men were political partisans and voter turnout often exceeded 90% in some states.

This period also witnessed the creation of a modern industrial economy. A national transportation and communication network was created, the corporation became the dominant form of business organization, and a managerial revolution transformed business operations. By the beginning of the twentieth century, per capita income and industrial production in the United States exceeded that of any other country except Britain. Long hours and hazardous working conditions led many workers to attempt to form labor unions despite strong opposition from industrialists and the courts.

The wealth of the period is highlighted by the American upper class's opulent self-indulgence, but also the rise of the American philanthropy (Andrew Carnegie called it the "Gospel of Wealth") that endowed thousands of colleges, hospitals, museums, academies, schools, opera houses, public libraries, symphony orchestras, and charities. The Beaux-Arts architectural idiom of the era clothed public buildings in Neo-Renaissance architecture.

This period overlaps with the nadir of American race relations, during which African Americans lost many of the civil rights obtained during the Reconstruction period. Increased racist violence, as well as exile of African Americans from the Southern states to the Midwest, started as soon as 1879. The end of the Gilded Age coincided with the Panic of 1893, a deep depression. The depression lasted until 1897 and marked a major political realignment in the election of 1896. After that came the Progressive Era.

The term "Gilded Age" was coined by Mark Twain and Charles Dudley Warner in their 1873 book, The Gilded Age: A Tale of Today.

[edit] Industrial and technological advances

The Gilded Age was rooted in industrialization, especially heavy industry like factories, railroads and coal mining. During the Gilded Age, American manufacturing production surpassed the combined total of Great Britain, Germany, and France. Railroad mileage tripled between 1860 and 1880, and tripled again by 1920, opening new areas to commercial farming, creating a truly national marketplace and inspiring a boom in coal mining and steel production. The voracious appetite for capital of the great trunk railroads facilitated the consolidation of the nation's financial market in Wall Street. By 1900, the process of economic concentration had extended into most branches of industry—a few large corporations, called "trusts", dominated in steel, oil, sugar, meatpacking, and the manufacture of agriculture machinery. Other major components of this infrastructure were the new methods for fabricating steel: the Bessemer and the Siemens steel making processes. The first billion-dollar corporation was United States Steel, formed by financier J. P. Morgan in 1901, who purchased and consolidated steel firms built by Andrew Carnegie and others.

Increased mechanization of industry is a major mark of the Gilded Age's search for cheaper ways to create more product. Frederick Winslow Taylor observed that worker efficiency in steel could be improved through the use of machines to make fewer motions in less time. His redesign increased the speed of factory machines and the productivity of factories while undercutting the need for skilled labor. This mechanization made some factories an assemblage of unskilled laborers performing simple and repetitive tasks under the direction of skilled foremen and engineers. Machine shops grew rapidly, and they comprised highly skilled workers and engineers. Both the number of unskilled and skilled workers increased, as their wage rates grew. Engineering colleges were established to feed the enormous demand for expertise. Railroads invented complex bureaucratic systems, using middle managers, and set up explicit career tracks. They hired young men at age 18-21 and promoted them internally until a man reached the status of locomotive engineer, conductor or station agent at age 40 or so. Career tracks were invented for skilled blue collar jobs and for white collar managers, starting in railroads and expanding into finance, manufacturing and trade. Together with rapid growth of small business, a new middle class was rapidly growing, especially in northern cities.

The United States became a world leader in applied technology. From 1860 to 1890, 500,000 patents were issued for new inventions—over ten times the number issued in the previous seventy years. George Westinghouse invented air brakes for trains (making them both safer and faster). Theodore Vail established the American Telephone & Telegraph Company. Thomas A. Edison invented a remarkable number of electrical devices, as well as the integrated power plant capable of lighting multiple buildings simultaneously; he founded General Electric corporation. Oil became an important resource, beginning with the Pennsylvania oil fields. Kerosene replaced whale oil and candles for lighting. John D. Rockefeller created Standard Oil Company to consolidate the industry.

[edit] Influential figures

Gilded age philanthropy: a table d'hôte dinner menu from March 4, 1893 for a meal in honor of symphony conductor Walter Johannes Damrosch.

Andrew Carnegie, John D. Rockefeller, and "Commodore" Cornelius Vanderbilt were amongst the most influential industrialists during the Gilded Age. Carnegie was born into a poor Scottish family; at age 18 he became an assistant to railroad superintendent Thomas A. Scott in Pittsburgh. In 1870, Carnegie erected his first blast furnace. Both Carnegie and Rockefeller gave away most of their wealth in large scale philanthropy. Carnegie created the Carnegie Institute of Technology (now part of Carnegie Mellon University) to upgrade craftsmen into trained engineers and scientists. Carnegie built hundreds of public libraries and several major research centers and foundations. Rockefeller retired from the oil business in 1897 and devoted the next 40 years of his life to giving away most of his money using systematic philanthropy, especially in the areas of education, medicine and race relations. "Commodore" C. Vanderbilt started out as a poor Staten Island farmer boy, then quickly through his sharp wit and lethal business policies built an enormous fortune in steamships and railroading to become the wealthiest man in the world in his day. His descendants and heirs would become famous for their ability to both increase and spend their wealth, building gigantic and lavish mansions and dominating Gilded Age high society.

[edit] Labor unions

[edit] The rise of unions

As American men, women, and children toiled away in the factories of the 1880s, many of them dreamed of one day living the Horatio Alger myth of moving from "rags to riches." While some Americans enjoyed unparalleled success as compared to their counterparts, many Americans still struggled to make ends meet. Workers began to organize labor unions to take responsibility for their own improvement. Modern labor unions emerged during the Civil War era. One of the earlier attempts at a national union was the National Labor Union, formed in Baltimore in 1866. The Knights of Labor, founded in 1869, had success in the late 1880s but then collapsed. The American Federation of Labor (AFL), a coalition of trades unions, became dominant in the 1890s, under Samuel Gompers. and the Industrial Workers of the World (IWW) emerged, encouraging all workers to unite.

[edit] The great strikes

While labor unions became increasingly popular in the late 1800s, their successes were also tempered by some crushing defeats. One of the earliest strikes following the Civil War was the Great Railroad Strike of 1877. It lasted for 45 days; it was finally ended when President Rutherford B. Hayes sent in federal troops. The number of deaths and the amount of damage caused by the strike scared many Americans. In 1886, the Knights of Labor were hit by two devastating events that eventually brought about an end to their union. The Great Southwest Railroad Strike of 1886 and the Haymarket Square Riot ended both internal and external support for the Knights. The Homestead Strike led by workers at the Homestead Steel Plant owned by Andrew Carnegie and Henry Clay Frick was an organized attempt at trying to secure better wages and working conditions than was ended with an assassination attempt on Frick. The final major strike of the late 1800s was the Pullman Strike which began at the company town of Pullman, Illinois. Owned by George Pullman, the town was the home to all the workers at the Pullman Palace Car Company. When Pullman refused to meet with workers to discuss intolerable conditions brought on by the Panic of 1893, a national rail strike was orchestrated by the American Railway Union led by Eugene V. Debs. This time the strike was halted by a court injunction and Debs was arrested by federal troops sent in by President Grover Cleveland. The Supreme Court upheld government action with their decision in the 1895 case In re Debs. With the Court decision coupled with the use of the Sherman Antitrust Act, labor unions were unable to regain much power until the 1930s.

[edit] Politics

A Group of Vultures Waiting for the Storm to "Blow Over"--"Let Us Prey." Cartoon of New York's Boss Tweed and other Tammany Hall figures, drawn in 1871 by Thomas Nast and published in Harper's Weekly.

Americans' sense of civic virtue was shocked by the scandals associated with the Reconstruction era: corrupt state governments, massive fraud in cities controlled by political machines, political payoffs to secure government contracts (especially the Crédit Mobilier of America scandal regarding the financing of the transcontinental railroad), and widespread evidence of government corruption during the Ulysses S. Grant Administration. This corruption divided the Republican party into two different factions, The Stalwarts led by Roscoe Conkling and the Half-Breeds led by James G. Blaine. Accordingly there were widespread calls for reform, such as Civil Service Reform led by the Bourbon Democrats and followed by the Republican Mugwumps, especially Samuel J. Tilden and Grover Cleveland. There was a sense that government intervention in the economy inevitably led to favoritism, bribery, kickbacks, inefficiency, waste, and corruption. The Bourbon Democrats led the call for a free market, low tariffs, low taxes, less spending and, in general, a Laissez-Faire (hands-off) government. They specifically denounced imperialism and overseas expansion. Many business and professional people supported this approach, although—to encourage rapid growth of industry and protect America's high wages against the low wage system in Europe—most Republicans advocated a high protective tariff. Labor activists and agrarians expressed the same spirit but focused their attacks on monopolies and railroads as unfair to the little man. Many Republicans also complained that high tariffs, for instance on British steel, benefited industrialists like Carnegie more than his employees who even at the time were regarded by many as being pitifully exploited.

In politics, the two parties engaged in very elaborate get-out-the vote campaigns that succeeded in pushing turnout to 80%, 90%, and even higher. It was financed by the "spoils system" whereby the winning party distributed most local, state and national government jobs, and many government contracts, to its loyal supporters. Large cities were dominated by political machines, in which constituents supported a candidate in exchange for anticipated patronage—favors back from the government, once that candidate was elected—and candidates were selected based on their willingness to play along. The best known example of a political machine from this time period is Tammany Hall in New York City, led by Boss Tweed. Presidential elections between the two major parties (the Republicans and Democrats), were closely contested, and Congress was marked by political stalemate. Mudslinging became an increasingly popular way of gaining advantage at the polls, and Republicans employed an election tactic known as "waving the bloody shirt". Candidates, especially when combating corruption charges, would remind voters that the Republican Party had saved the nation in the Civil War. During the 1870s, voters were repeatedly reminded that the Democrats had been responsible for the bloody upheaval, an appeal that attracted many Union veterans to the Republican camp. The Republicans consistently carried the North in presidential elections. The South, on the other hand, became the Solid South, nearly always voting Democratic. The political humiliations of Reconstruction were still fresh in many minds. Conversely, the Democrats invoked images of the "lost cause" and the glorious "stars and bars" in much the same way Republicans "waved the bloody shirt." The corruption of the Republican organization led to the defection of a group of reformers called the Mugwumps that supported Democrat Grover Cleveland in 1884. This victory gave Democrats control of the presidency for the first time since the Civil War (not counting the ascension of Andrew Johnson who was technically elected as part of the Union Party).

Overall, Republican and Democratic political platforms remained remarkably constant during the years before 1900. The negativity and ambiguity of politics began a shift in the press to yellow journalism, in which sensationalism and sentimental stories took as prominent a role as factual news.

[edit] Immigration

Prior to the Gilded Age, the time commonly referred to as the old immigration saw the first real boom of new arrivals to the United States. During the Gilded Age, approximately 10 million immigrants came to the United States in what is known as the new immigration, many in search of religious freedom and greater prosperity. The population surge in major U.S. cities as a result of immigration gave cities an even stronger impact on government, attracting power-hungry politicians and entrepreneurs. Pressuring voters or falsifying ballots was commonplace for politicians, who often sought power only to exploit their constituents. To accommodate the influx of people into the U.S., the federal government built Ellis Island in 1892 near the Statue of Liberty. After 1892, a short physical examination was given; those with contagious diseases were not admitted. Few immigrants went to the poverty-stricken South.

[edit] Chinese immigrants

The construction of the Central Pacific railroad in California and Nevada was handled largely by Chinese laborers. In the 1870 census there were 58 Chinese men and 4 women in the entire country; these numbers grew to 100,000 men and 40,000 women in the 1880 census.^[1] Labor unions such as the American Federation of Labor strongly opposed the presence of Chinese labor, by reason of both economic competition and race. Immigrants from China were not allowed to become citizens until 1950; however, their children born in the U.S. were full citizens.

Congress banned further Chinese immigration through the Chinese Exclusion Act in 1882; the act prohibited Chinese laborers from entering the United States, but some students and businessmen were allowed in. Subsequent to the act, the Chinese population declined to only 37,000 in 1940. Although many returned to China (a greater proportion than most other immigrant groups), most of them stayed in the United States. Chinese people were unwelcome in many areas, so they resettled in the "Chinatown" districts of large cities.

[edit] Urban life

Society itself underwent significant changes in the period following the Civil War. One of the most significant changes came in the further urbanization of the northern cities. As a result of increasing demand for factory workers as well as mass immigration from Europe, the population of cities began to swell. Major American cities such as New York and Chicago even saw populations grow in excess of one million people. These rapid changes in cities brought about modern architectural and transportation features. Louis Sullivan became a noted architect using steel frames to construct skyscrapers for the first time while pioneering the idea of "form follows function". One of his earliest works was the Wainwright Building in St. Louis, Missouri. Elisha Otis’ introduction of safety measures on elevators also helped building reach newer heights. American cities also expanded underground. In order to reduce congestion on the streets, major cities such as New York began to construct underground railroads, the forerunners to modern subways. As immigration increased in cities, poverty rose as well. New immigrants were forced to live in the poorest urban areas including the Five Points and Hell’s Kitchen in Manhattan. These areas were quickly overridden with crime gangs such as the Five Points Gang and the Bowery Boys rose to prominence. Families were forced into miserable living conditions in the so-called “dumbbell tenements” where multiple families would share rooms designed for single occupants. It was not until Danish photographer Jacob Riis published his photographic essay How the Other Half Lives that real attention was called to the unsafe and unsanitary conditions of tenement living.

[edit] Social changes

During the Gilded Age, many new social movements took hold in the United States. Many women abolitionists who were disappointed that the Fifteenth Amendment did not extend voting rights to them remained active in politics, this time focusing on issues important to them. Reviving the temperance movement from the Second Great Awakening, many women joined the Women’s Christian Temperance Union (WCTU) in an attempt to bring morality back to America. Other women took up the issue of women’s suffrage which had laid dormant since the Seneca Falls Convention With leaders like Susan B. Anthony the National American Woman Suffrage Association (NAWSA) was formed in order to secure the right of women to vote. Science also played an important part in social though as the work of Charles Darwin became popular. Following in Darwin’ s idea of natural selection, English philosopher Herbert Spencer proposed the idea of social Darwinism. This new concept justified the stratification of the wealthy and poor and coined the term “survival of the fittest.” Joining Spencer was Yale University professor William Graham Sumner whose book What Social Classes Owe to Each Other argued that assistance to the poor actually weakens their ability to survive in society. Sumner argued for a laissez faire and free market economy. Not everyone agreed with the social Darwinists and soon a whole movement to help the poor arose. Henry George proposed a “single tax” in his book Progress and Poverty. The tax would be leveled on the rich and poor alike, with the excess money collected used to equalize wealth and level out society. cIn Chicago, noted attorney Clarence Darrow made vocal arguments that poverty and not biology created crime. Wisconsin-born author Thorstein Veblen argued in his book The Theory of the Leisure Class that the “conspicuous consumption and conspicuous leisure" of the wealth had become the basis of social status in America. In Looking Backward author Edward Bellamy writes of a future America set in the year 2000 in which a socialist paradise has been established. The works of authors such as George and Bellamy became popular and soon clubs were created across America to discuss their ideas although these organizations rarely made any real social change. The Third Great Awakening which began before the Civil War returned and made a significant change in religious attitudes toward social progress. Followers of the new Awakening promoted the idea of the Social Gospel which gave rise to organizations such as the YMCA; Salvation Army, and settlement houses such as Hull House founded by Jane Addams in Chicago.

In computational mathematics, an iterative method attempts to solve a problem (for example an equation or system of equations) by finding successive approximations to the solution starting from an initial guess. This approach is in contrast to direct methods, which attempt to solve the problem by a finite sequence of operations, and, in the absence of rounding errors, would deliver an exact solution (like solving a linear system of equations Ax = b by Gaussian elimination). Iterative methods are usually the only choice for nonlinear equations. However, iterative methods are often useful even for linear problems involving a large number of variables (sometimes of the order of millions), where direct methods would be prohibitively expensive (and in some cases impossible) even with the best available computing power.

[edit] Attractive fixed points

If an equation can be put into the form f(x) = x, and a solution x is an attractive fixed point of the function f, then one may begin with a point x₁ in the basin of attraction of x, and let x_n+1 = f(x_n) for n ≥ 1, and the sequence {x_n}_n ≥ 1 will converge to the solution x. If the function f is continuously differentiable, a sufficient condition for convergence is that the spectral radius of the derivative is strictly bounded by one in a neighborhood of the fixed point. If this condition holds at the fixed point, then a sufficiently small neighborhood (basin of attraction) must exist.

[edit] Linear systems

In the case of a system of linear equations, the two main classes of iterative methods are the stationary iterative methods, and the more general Krylov subspace methods.

[edit] Stationary iterative methods

Stationary iterative methods solve a linear system with an operator approximating the original one; and based on a measurement of the error (the residual), form a correction equation for which this process is repeated. While these methods are simple to derive, implement, and analyse, convergence is only guaranteed for a limited class of matrices. Examples of stationary iterative methods are the Jacobi method and the Gauss–Seidel method.

[edit] Krylov subspace methods

Krylov subspace methods form an orthogonal basis of the sequence of successive matrix powers times the initial residual (the Krylov sequence). The approximations to the solution are then formed by minimizing the residual over the subspace formed. The prototypical method in this class is the conjugate gradient method (CG). Other methods are the generalized minimal residual method (GMRES) and the biconjugate gradient method (BiCG).

[edit] Convergence

Since these methods form a basis, it is evident that the method converges in N iterations, where N is the system size. However, in the presence of rounding errors this statement does not hold; moreover, in practice N can be very large, and the iterative process reaches sufficient accuracy already far earlier. The analysis of these methods is hard, depending on a complicated function of the spectrum of the operator.

[edit] Preconditioners

The approximating operator that appears in stationary iterative methods can also be incorporated in Krylov subspace methods such as GMRES (alternatively, preconditioned Krylov methods can be considered as accelerations of stationary iterative methods), where they become transformations of the original operator to a presumably better conditioned one. The construction of preconditioners is a large research area.

[edit] History

Probably the first iterative method for solving a linear system appeared in a letter of Gauss to a student of his. He proposed solving a 4-by-4 system of equations by repeatedly solving the component in which the residual was the largest.

The theory of stationary iterative methods was solidly established with the work of D.M. Young starting in the 1950s. The Conjugate Gradient method was also invented in the 1950s, with independent developments by Cornelius Lanczos, Magnus Hestenes and Eduard Stiefel, but its nature and applicability were misunderstood at the time. Only in the 1970s was it realized that conjugacy based methods work very well for partial differential equations, especially the elliptic type.

Gradient descent is a first-order optimization algorithm. To find a local minimum of a function using gradient descent, one takes steps proportional to the negative of the gradient (or the approximate gradient) of the function at the current point. If instead one takes steps proportional to the gradient, one approaches a local maximum of that function; the procedure is then known as gradient ascent.

Gradient descent is also known as steepest descent, or the method of steepest descent. When known as the latter, gradient descent should not be confused with the method of steepest descent for approximating integrals.

[edit] Description

Gradient descent is based on the observation that if the real-valued function is defined and differentiable in a neighborhood of a point , then decreases fastest if one goes from in the direction of the negative gradient of F at , . It follows that, if

for γ > 0 a small enough number, then . With this observation in mind, one starts with a guess for a local minimum of F, and considers the sequence such that

We have

so hopefully the sequence converges to the desired local minimum. Note that the value of the step size γ is allowed to change at every iteration.

This process is illustrated in the picture to the right. Here F is assumed to be defined on the plane, and that its graph has a bowl shape. The blue curves are the contour lines, that is, the regions on which the value of F is constant. A red arrow originating at a point shows the direction of the negative gradient at that point. Note that the (negative) gradient at a point is orthogonal to the contour line going through that point. We see that gradient descent leads us to the bottom of the bowl, that is, to the point where the value of the function F is minimal.

[edit] Examples

Gradient descent has problems with pathological functions such as the Rosenbrock function shown here. The Rosenbrock function has a narrow curved valley which contains the minimum. The bottom of the valley is very flat. Because of the curved flat valley the optimization is zig-zagging slowly with small stepsizes towards the minimum.

The gradient ascent method applied to :

[edit] Comments

Gradient descent works in spaces of any number of dimensions, even in infinite-dimensional ones. In the latter case the search space is typically a function space, and one calculates the Gâteaux derivative of the functional to be minimized to determine the descent direction.

Two weaknesses of gradient descent are:

1. The algorithm can take many iterations to converge towards a local minimum, if the curvature in different directions is very different.
2. Finding the optimal γ per step can be time-consuming. Conversely, using a fixed γ can yield poor results. Methods based on Newton's method and inversion of the Hessian using conjugate gradient techniques are often a better alternative.

A more powerful algorithm is given by the BFGS method which consists in calculating on every step a matrix by which the gradient vector is multiplied to go into a "better" direction, combined with a more sophisticated line search algorithm, to find the "best" value of γ.

Gradient descent is in fact Euler's method for solving ordinary differential equations applied to a gradient flow. As the goal is to find the minimum, not the flow line, the error in finite methods is less significant.

[edit] A computational example

The gradient descent algorithm is applied to find a local minimum of the function f(x)=x⁴-3x³+2 , with derivative f'(x)=4x³-9x². Here is an implementation in the C programming language.

#include <stdio.h>
#include <stdlib.h>
#include <math.h>
 
int main ()
{
	// From calculation, we expect that the local minimum occurs at x=9/4
	// The algorithm starts at x=6
 
	double xOld = 0;
	double xNew = 6;
	double eps = 0.01; // step size
	double precision = 0.00001;
	while (fabs(xNew - xOld) > precision)
	{
             xOld = xNew;
             xNew = xNew - eps*(4*xNew*xNew*xNew-9*xNew*xNew);
	}
 
	printf ("Local minimum occurs at %lg\n", xNew);
 
}

With this precision, the algorithm converges to a local minimum of 2.24996 in 70 iterations.

A more robust implementation of the algorithm would also check whether the function value indeed decreases at every iteration and would make the step size smaller otherwise. One can also use an adaptive step size which may make the algorithm converge faster.

In mathematics, the steepest descent method or saddle-point approximation is a method used to approximate integrals of the form

where f(x) is some twice-differentiable function, M is a large number, and the integral endpoints a and b could possibly be infinite. The technique is also often referred to as Laplace's method, which in fact concerns the special case of real-valued functions f admitting a maximum at a real point.

The idea of Laplace's method

Assume that the function f(x) has a unique global maximum at x₀. Then, the value f(x₀) will be larger than other values f(x). If we multiply this function by a large number M, the gap between Mf(x₀) and Mf(x) will only increase, and then it will grow exponentially for the function

As such, significant contributions to the integral of this function will come only from points x in a neighborhood of x₀, which can then be estimated.

General theory of Laplace's method

To state and prove the method, we need several assumptions. We will assume that x₀ is not an endpoint of the interval of integration, that the values f(x) cannot be very close to f(x₀) unless x is close to x₀, and that f'''(x₀) < 0.

We can expand f(x) around x₀ by Taylor's theorem,

Since f has a global maximum at x₀, and since x₀ is not an endpoint, it is a stationary point, the derivative of f vanishes at x₀. Therefore, the function f(x) may be approximated to quadratic order

for x close to x₀ (recall that the second derivative is negative at the global maximum f(x₀)). The assumptions made ensure the accuracy of the approximation

where the integral is taken in a neighborhood of x₀. This latter integral is a Gaussian integral if the limits of integration go from −∞ to +∞ (which can be assumed so because the exponential decays very fast away from x₀), and thus it can be calculated. We find

A generalization of this method and extension to arbitrary precision is provided by Fog (2008).

Steepest descent

In extensions of Laplace's method, complex analysis, and in particular Cauchy's integral formula, is used to find a contour of steepest descent for an (asymptotically with large M) equivalent integral, expressed as a line integral. In particular, if no point x₀ where the derivative of f vanishes exists on the real line, it may be necessary to deform the integration contour to an optimal one, where the above analysis will be possible. Again the main idea is to reduce, at least asymptotically, the calculation of the given integral to that of a simpler integral that can be explicitly evaluated. See the book of Erdelyi (1956) for a simple discussion (where the method is termed steepest descents).

Further generalizations

An extension of the steepest descent method is the so-called nonlinear stationary phase/steepest descent method. Here, instead of integrals, one needs to evaluate asymptotically solutions of Riemann-Hilbert factorization problems.

Given a contour C in the complex sphere, a function f defined on that contour and a special point, say infinity, one seeks a function M holomorphic away from the contour C, with prescribed jump across C, and with a given normalization at infinity. If f and hence M are matrices rather than scalars this is a problem that in general does not admit an explicit solution.

An asymptotic evaluation is then possible along the lines of the linear stationary phase/steepest descent method. The idea is to reduce asymptotically the solution of the given Riemann-Hilbert problem to that of a simpler, explicitly solvable, Riemann-Hilbert problem. Cauchy's theorem is used to justify deformations of the jump contour.

The nonlinear stationary phase was introduced by Deift and Zhou in 1993, based on earlier work of Its. A (properly speaking) nonlinear steepest descent method was introduced by Kamvissis, K. McLaughlin and P. Miller in 2003, based on previous work of Lax, Levermore, Deift, Venakides and Zhou.

The nonlinear stationary phase/steepest descent method has applications to the theory of soliton equations and integrable models, random matrices and combinatorics.

Complex integrals

For complex integrals in the form:

with t >> 1, we make the substitution t = iu and the change of variable s = c + ix to get the Laplace bilateral transform:

We then split g(c+ix) in its real and complex part, after which we recover u = t / i. This is useful for inverse Laplace transforms, the Perron formula and complex integration.

Example: Stirling's approximation

Laplace's method can be used to derive Stirling's approximation

for a large integer N.

From the definition of the Gamma function, we have

Now we change variables, letting

so that

Plug these values back in to obtain

This integral has the form necessary for Laplace's method with

which is twice-differentiable:

The maximum of f(z) lies at z₀=1, and the second derivative of f(z) has at this point the value -1. Therefore, we obtain

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行为金融理论概述

　　现代金融理论是建立在资本资产定价模型(CAPM)和有效市场假说(EMH)两大基石上的。这些经典理论承袭经济学的分析方法与技术，其模型与范式局限在“理性”的分析框架中，忽视了对投资者实际决策行为的分析。随着金融市场上各种异常现象的累积，模型和实际的背离使得现代金融理论的理性分析范式陷入了尴尬境地。在此基础上，20世纪80年代行为金融理论悄然兴起，并开始动摇了CAPM和 EMH的权威地位。行为金融理论在博弈论和实验经济学被主流经济学接纳之际，对人类个体和群体行为研究的日益重视，促成了传统的力学研究方式向以生命为中心的非线性复杂范式的转换，使得我们看到了金融理论与实际的沟壑有了弥合的可能。行为金融理论将人类心理与行为纳入金融的研究框架，但是由于涉及人类心理与行为研究的难度，加上行为金融刚刚起步，因而其本身也存在很多缺陷。

行为金融理论产生背景和发展历史

(一)产生背景

　　行为金融理论是在对现代金融理论(尤其是在对EMH和CAPM)的挑战和质疑的背景下形成的。

　　在EMH理论形成的过程中，奥斯本(Osberne)和法玛(Fama)的贡献最大。奥斯本提出了关于股票价格遵循随机游走的主张，认为投资者是根据他们的期望价值或收益率来估计股票的，而期望价值是可能的收益率的概率加权平均值，所以投资者在奥斯本定义上的理性是以无偏的方式设定其主观概率。在此基础上，法玛建构并形成了有效市场假说(EMH)，EMH是由三个不断弱化的假说组成的：当投资者是理性时，投资者可以理性评估资产价值，市场是有效的；即使有些投资者不是理性的，但由于交易随机产生，也不会形成系统的价格偏差；即使投资者的非理性行为不是随机的，他们也将遇到理性的套期保值者，从而保证资产价格回归基本价值；

　　最后，即使非理性交易者在非基本价值的价格交易时，他的财富也将逐渐减少，以致不能在市场上生存。

　　法玛(1970)还进一步细分了三种有效市场，从而说明价格反映所有的公开信息，基本分析者的共识形成公平价格。所以EMH已经隐含了已知信息不能用来在市场上获利的命题。我们也可以说，到了法玛那里，EMH依赖于理性投资者。在EMH 产生与发展的同时，马科维茨(Markowitz)结合奥斯本的期望收益率分布，以其方差为度量，用以度量资产组合，得出投资者选择有效边界的风险和标准差给定水平上期望收益率最高的资产组合这个合意的结论。所以投资者在马科维茨定义上的理性是指他们是风险回避型的；在此基础上，夏普(Sharpe， 1964)、林特纳(Lintner，1965)和莫辛(Mossin，1966)将EMH和马科维茨的资产组合结合起来，以资本资产模型命名，建立了一个以一般均衡框架中的理性预期为基础的投资者行为模型CAPM。CAPM中的投资者有着同质的收益率预期，以相同的方式解读信息。在此假定下，CAPM得出：高风险的资产应为高收益率的补偿，投资者的最优投资决策应沿资本市场线进行的结论。

　　如果说EMH回答了已知的信息对获利没有价值的结论，那么CAPM则说明市场上的超额回报率是由于承担更大的风险才形成的结论，因而在一定程度上CAPM补充了EMH的理论空白和可能的漏洞。

　　自CAPM诞生后，20世纪七八十年代的研究一般集中在应用该模型进行经验研究和求证EMH的有效性上。但是随着后来研究的深入，逐渐发现了现代金融理论模型与投资者在证券市场上的实际投资决策行为是不相符合的。主要表现如下：

　　第一、现代金融理论认为人们的决策是建立在理性预期、风险回避、效用函数最大化等假设上。

　　实际投资决策并非如此。特韦尔斯基(1990)针对投资者准确无偏的奥斯本定义上的理性，指出投资者具有倾向于过分自信的心理特征；针对投资者如果接受更大的风险，他们就必须得到更高的收益率的补偿，即对马科维茨的投资者是风险回避型的修正，特氏研究表明，当牵涉到亏损时，投资者会倾向于追求风险，尤其是在追求风险有可能把他们的亏损减少到最低限度的时候；针对法玛和夏普意义上的投资者理性，研究发现投资者在决策中的预测是非贝叶斯预测，而且投资者会有回避损失和心理会计的偏差，还有减少后悔、推卸责任的心理。尤其值得提出的是研究表明，这种对理性决策的偏差是系统性的，并不能因为统计平均而消除 (Kahneman and Riepe，1998)。

　　第二，现代金融理论和EMH是建立在有效市场竞争的基础上。能够在市场竞争中幸存下来的只有理性投资者。证券市场投资行为是由理性的投资者主宰的。而Delong、Shleifer、Summers和Maldmann(1990、 1991)的研究表明，某些情况下，非理性投资者实际上可以获得比理性交易者更高的收益，非理性投资者仍然可以影响资产价格。

(二)发展历史

　　行为金融理论的发展历史可以简单概括为以下几个阶段：

　　1.早期阶段。19世纪Gustave Lebon的“The Crowd”和Mackey的“Extraordinary Poplular Delusion and the Madness Of Crowds”是两本研究投资市场群体行为的经典之作；凯恩斯是最早强调心理预期在投资决策中作用的经济学家，他基于心理预期最早提出股市“选美竞赛”理论和基于投资者“动物精神”而产生的股市“乐车队效应”；Purrell是现代意义上金融理论的最早研究者，在其《以实验方法进行投资研究的可能性》 (1951)论文中，开拓了应用实验将投资模型与人的心理行为特征相结合的金融新领域。后来的Paul、Slovic等人继续进行了一些人类决策过程的心理学研究。

　　2.心理学行为金融阶段(从1960年至80年代中期)。这一阶段的行为金融研究以 Tversky和Kahglelman为代表。Tversky研究了人类行为与投资决策模型基本假设相冲突的三个方面：风险态度、心理会计和过度自信，并将观察到的现象称为“认知偏差”。Kahneman和Tverskv(1979)共同提出了“期望理论”，使之成为行为金融研究中的代表学说。但是当时的行为金融的研究还没有引起足够重视，一方面是因为此时EMH风行一时，另一方面是因为人们普遍认为研究人的心理、情绪对金融研究是不科学的。

　　3.金融学行为金融阶段(从20世纪80年代中期至今)。市场不断发现的异常现象引起金融学界的注意，大量的证据表明许多金融理论还不完善；再加上期望理论得到广泛认可和经验求证，所以这个时期的行为金融取得了突破性的进展。这个时期行为金融理论以芝加哥大学的Thaler和耶鲁大学的Shiller为代表。Thaler(1987，1999)研究了股票回报率的时间序列、投资者心理会计等问题。Shiller(1981，1990a，1990b)主要研究了股票价格的异常波动、股市中的“羊群效应”(Herd Behavior)、投机价格和流行心态的关系等。此外，Orden(1998)对于趋向性效应(disposition effect)的研究，Ritter(1999)对于IPO的异常现象的研究，Kahneman等(1998)对反应过度和反应不足切换机制的研究都受到了广泛的关注。与上个时期相比，这个时期的行为金融理论研究是从投资策略上加以完善，注重把心理学研究和投资决策结合起来。

行为金融理论的理论内容

(一)理论基础

　　1.期望理论。期望理论是行为金融学的重要理论基础。Kahneman和 Tversky(1979)通过实验对比发现，大多数投资者并非是标准金融投资者而是行为投资者，他们的行为不总是理性的，也并不总是风险回避的。期望理论认为投资者对收益的效用函数是凹函数，而对损失的效用函数是凸函数，表现为投资者在投资帐面值损失时更加厌恶风险，而在投资帐面值盈利时，随着收益的增加，其满足程度速度减缓。期望理论成为行为金融研究中的代表学说，利用期望理论解释了不少金融市场中的异常现象：如阿莱悖论、股价溢价之迷(equity premium puzzle)以及期权微笑(option smile)等，然而由于Kahneman和Tversky在期望理论中并没有给出如何确定价值函数的关键——参考点以及价值函数的具体形式，在理论上存在很大缺陷，从而极大阻碍了期望理论的进一步发展。

　　2.行为组合理论(Behavioral Portfolio Theory，BPT)和行为资产定价模型(Behavioral Asset Pricing Model，BAPM)。一些行为金融理论研究者认为将行为金融理论与现代金融理论完全对立起来并不恰当。将二者结合起来，对现代金融理论进行完善，正成为这些研究者的研究方向。在这方面，Statman和Shefrin提出的BPT和BAPM引起金融界的注意。BPT是在现代资产组合理论(MPT)的基础上发展起来的。MAPT认为投资者应该把注意力集中在整个组合，最优的组合配置处在均值方差有效前沿上。BPT认为现实中的投资者无法作到这一点，他们实际构建的资产组合是基于对不同资产的风险程度的认识以及投资目的所形成的一种金字塔式的行为资产组合，位于金字塔各层的资产都与特定的目标和风险态度相联系，而各层之间的相关性被忽略了。BAPM是对资本资产定价模型(CAPM)的扩展。与CAPM不同，BAPM中的投资者被分为两类：信息交易者和噪声交易者。信息交易者是严格按CAPM行事的理性交易者，不会出现系统偏差；噪声交易者则不按CAPM行事，会犯各种认知偏差错误。两类交易者互相影响共同决定资产价格。事实上，在BAPM中，资本市场组合的问题仍然存在，因为均值方差有效组合会随时间而改变。

(二)投资行为模型

　　1.BSV模型(Barberis，Shleffer，and Vishny，1998)。BSV模型认为，人们进行投资决策时存在两种错误范式：其一是选择性偏差(representative bias)，即投资者过分重视近期数据的变化模式，而对产生这些数据的总体特征重视不够，这种偏差导致股价对收益变化的反映不足(under- reaction)。另一种是保守性偏差(conservation)，投资者不能及时根据变化了的情况修正自己的预测模型，导致股价过度反应(over -reaction)。BSV模型是从这两种偏差出发，解释投资者决策模型如何导致证券的市场价格变化偏离效率市场假说的。

　　2.DHS模型(Daniel，Hirsheifer and Subramanyam，1998)。该模型将投资者分为有信息和无信息两类。无信息的投资者不存在判断偏差，有信息的投资者存在着过度自信和有偏的自我归因(serf-contribution)。过度自信导致投资者夸大自己对股票价值判断的准确性；有偏的自我归因则使他们低估关于股票价值的公开信号。随着公共信息最终战胜行为偏差，对个人信息的过度反应和对公共信息的反应不足，就会导致股票回报的短期连续性和长期反转。所以Fama(1998)认为 DHS模型和BSV模型虽然建立在不同的行为前提基础上，但二者的结论是相似的。

　　3.HS模型(Hong and Stein，1999)，又称统一理论模型(unified theory model)。统一理论模型区别于BSV和DHS模型之处在于：它把研究重点放在不同作用者的作用机制上，而不是作用者的认知偏差方面。该模型把作用者分为“观察消息者”和“动量交易者”两类。观察消息者根据获得的关于未来价值的信息进行预测，其局限是完全不依赖于当前或过去的价格；“动量交易者”则完全依赖于过去的价格变化，其局限是他们的预测必须是过去价格历史的简单函数，在上述假设下，该模型将反应不足和过度反应统一归结为关于基本价值信息的逐渐扩散，而不包括其他的对投资者情感刺激和流动性交易的需要。模型认为最初由于“观察消息者”对私人信息反应不足的倾向，使得“动量交易者”力图通过套期策略来利用这一点，而这样做的结果恰好走向了另一个极端——过度反应。

　　4.羊群效应模型(herd behavioral model)。该模型认为投资者羊群行为是符合最大效用准则的，是“群体压力”等情绪下贯彻的非理性行为，有序列型和非序列型两种模型。序列型由 Banerjee(1992)提出，在该模型中，投资者通过典型的贝叶斯过程从市场噪声以及其它个体的决策中依次获取决策信息，这类决策的最大特征是其决策的序列性。但是现实中要区分投资者顺序是不现实的。因而这一假设在实际金融市场中缺乏支持。非序列型则论证无论仿效倾向强或弱，都不会得到现代金融理论中关于股票的零点对称、单一模态的厚尾特征。

三、实证检验

　　进入20世纪80年代以来，与现代金融理论相矛盾的实证研究不断涌现，主要体现在投资策略的改变上。下面介绍几种典型的行为金融策略：

　　1.小公司效应。小公司效应是指小盘股比大盘股的收益率高。Banz(1981)发现股票市值随着公司规模的增大而减少的趋势。同一年，Reimganum(1981)也发现了公司规模最小的普通股票的平均收益率要比根据CAPM模型预测的理论收益率高出18%。最近Siegl(1998)研究发现，平均而言小盘股比大盘股的年收益率高出4.7%，而且小公司效应大部分集中在1月份。由于公司的规模和1月份的到来都是市场已知信息，这一现象明显地违反了半强式有效市场假设。Lakonishok等(1994)的研究发现，高市净盈率的股票风险更大，在大盘下跌和经济衰退时，业绩特别差。市盈率与收益率的反向关系对EMH形成严峻的挑战，因为这时已知的信息对于收益率有明显的预测作用。

　　2.反向投资策略(contrary investment strategy)。就是买进过去表现差的股票而卖出，过去表现好的股票来进行套利的投资方法。一些研究显示，如选择低市盈率(PE)的股票；选择股票市值与帐面价值比值低、历史收益率低的股票，往往可以得到比预期收益率高很多的收益，而且这种收益是一种“长期异常收益”(1ong-term anomalies)。Desia、Jain(1997)，Ikenberry、Rankine Stice(1996)也发现公司股票分割前后都存在着正的长期异常收益。行为金融理论认为反向投资策略是对股市过度反应的一种纠正，是一种简单外推的方法。

　　3.动量交易策略(momentum trading strategy)。即首先对股票收益和交易量设定过滤准则，当股市收益和交易量满足过滤准则就买入或卖出股票的投资策略。行为金融定义的动量交易策略源于对股市中间收益延续性的研究。Jegadeeshkg与Titman(1993)在对资产股票组合的中间收益进行研究时发现，以3至12个月为间隔所构造的股票组合的中间收益呈连续性，即中间价格具有向某一方向连续的动量效应。事实上，美国价值线排名(value line rankings)就是动量交易策略利用的例证。动量交易策略的应用其实就是对EMH的再次否定。

　　4.成本平均策略和时间分散化策略。成本平均策略指投资者根据不同的价格分批购买股票，以防不测时摊低成本的策略，而时间分散化指根据股票的风险将随着投资期限的延长而降低的信念，随着投资者年龄的增长而将股票的比例逐步减少的策略。这两个策略被认为与现代金融理论的预期效用最大化则明显相悖。Statman(1995)，Fisher、Statman(1999)利用行为金融中的期望理论、认知错误倾向、厌恶悔恨等观点对两个策略进行了解释，指出了加强自我控制的改进建议。

　　行为金融理论已经开始成为金融研究中一个十分引人注目的领域，它对于原有理性框架中的现代金融理论进行了深刻的反思，从人的角度来解释市场行为，充分考虑市场参与者的心理因素的作用，为人们理解金融市场提供了一个新的视角。行为金融理论是第一个较为系统地对效率市场假说和现代金融理论提出挑战并能够有效地解释市场异常行为的理论。行为金融理论以心理学对人类的研究成果为依据，以人们的实际决策心理为出发点讨论投资者的投资决策对市场价格的影响。它注重投资者决策心理的多样性，突破了现代金融理论只注重最优决策模型，简单地认为理性投资决策模型就是决定证券市场价格变化的实际投资决策模型的假设，使人们对金融市场投资者行为的研究由“应该怎么做决策”转变到“实际是怎样做决策”，研究更接近实际。因而，尽管现代金融理论依然是对市场价格的最好描述，但行为金融的研究无疑是很有意义的。

过度自信理论(overconfidence theory)

金融学的四大研究成果,即过度自信理论(Overconfidence Theory)、视野理论(Prospect Theory)、后悔理论(Regret Theory)及过度反应理论(Overreaction Theory)。

什么是投资者过度自信理论

　　大量的认知心理学的文献认为，人是过度自信的，尤其对其自身知识的准确性过度自信。人们系统性地低估某类信息并高估其他信息。Gervaris、Heaton和Odean(2002)将过度自信定义为，认为自己知识的准确性比事实中的程度更高的一种信念，即对自己的信息赋予的权重大于事实上的权重。关于主观概率测度的研究也发现确实存在过度估计自身知识准确性的情况。

　　心理学家们的研究还发现一些职业领域往往与过度自信相联系，如外科医生和护士、心理学家、投资银行家、工程师、律师、投资者和经理在判断和决策中会存在过度自信特征。Grifin和Amos Tversky(1992)发现人们在回答中等到极度困难的问题时，倾向于过度自信。在回答容易问题时，倾向于不自信；当从事的是可预测性较强，有快速、清晰反馈的重复性的任务时，倾向于仔细推算。如专业桥牌运动员、赌马者和气象学者在决策时都倾向于仔细推算。

　　Frank(1935)发现人们过度估计了其完成任务的能力，并且这种过度估计随着个人在任务中的重要，性而增强，人们对未来事件有不切实际的乐观主义。Kunda(1987)发现人们期望好事情发生在自己身上的概率高于发生在别人身上的概率，甚至对于纯粹的随机事件有不切实际的的乐观义。人们会有不切实际的积极的自我评价，往往认为自己的能力、前途等会比其他人更好。过度自信的人往往有事后聪明的特点，夸大自己预测的准确性，尤其在他们期望一种结果，而这种结果确实发生时，往往会过度估计自己在产生这种合意结果中的作用。Daniel、Hirshleifer和Subrahmanyam(1998)认为成功者会将自己的成功归因于自己知识的准确性和个人能力，这种自我归因偏差会使成功者过度自信。

　　过度自信的人在做决策时，会过度估计突出而能引人注意的信息，尤其会过度估计与其已经存在的信念一致的信息，并倾向于搜集那些支持其信念的信息，而忽略那些不支持其信念的信息。当某些观点得到活灵活现的信息、重要的案例和明显的场景支持的时候，人们会更自信，并对这些信息反应过度。而当某些观点得到相关性强的、简洁的、统计性的和基本概率信息支持的时候，人们通常会低估这些信息，并对这些信息反应不足。

过度自信理论的主要内容

　　人类倾向于从无序中看出规律，尤其是从一大堆随机的经济数据中，推出所谓的规律。Amos Tversky提供了大量的统计数据，来说明许多事件的发生完全是由于运气和偶然因素的结果，而人类有一种表征直觉推理(representative heuristic)特点，即从一些数据的表面特征，直觉推断出其内在的规律性，从而产生认知和判断上的偏差(biases of cognition and judgment)。投资者的归因偏好也加重了这种认知偏差，即将偶然的成功归因于自己操作的技巧，将失败的投资操作归于外界无法控制因素，从而产生了所谓过度自信(overconfidence)的心理现象 。过度自信是指人们对自己的判断能力过于自信。投资者趋向于认为别人的投资决策都是非理性的，而自己的决定是理性的，是在根据优势的信息基础上进行操作的，但事实并非如此。Daniel Kadmeman认为：过度自信来源于投资者对概率事件的错误估计，人们对于小概率事件发生的可能性产生过高的估计，认为其总是可能发生的，这也是各种博彩行为的心理依据；而对于中等偏高程度的概率性事件，易产生过低的估计；但对于90％以上的概率性事件，则认为肯定会发生。这是过度自信产生的一个主要原因。此外，参加投资活动会让投资者产生一种控制错觉(illusion of contro1)，控制错觉也是产生过度自信的一个重要原因。

　　投资者和证券分析师们在他们有一定知识的领域中特别过于自信。然而，提高自信水平与成功投资并无相关。基金经理人、股评家以及投资者总认为自己有能力跑赢大盘，然而事实并非如此。Brad Barber和Terrance Odean在此领域做了大量研究。男性在许多领域(体育技能、领导能力、与别人相处能力)中总是过高估计自己。他们在1991年至1997年中，研究了38000名投资者的投资行为，将年交易量作为过度自信的指标，发现男性投资者的年交易量比女性投资者的年交易量总体高出20％以上，而投资收益却略低于女性投资者。数据显示：过度自信的投资者在市场中会频繁交易，总体表现为年交易量的放大。但由于过度自信而频繁地进行交易并不能让投资者获得更高的收益。在另一个研究中，他们取样1991年至1996年中的78 000名投资者，发现年交易量越高的投资者的实际投资收益越低。在一系列的研究中，他们还发现过度自信的投资者更喜欢冒风险，同时也容易忽略交易成本。这也是其投资收益低于正常水平的两大原因。

过度自信对金融市场的影响

　　在传统的金融理论中都假设行为人是风险规避的，但现实中人往往是风险中 性甚至是风险寻求的。Friedman和Savage(1948)就发现，尽管赢得彩票的几率只有数百万分之一，但还是有很多人去买彩票，这种购买彩票的行为就表现为风险寻求。风险寻求的原因很可能是过度自信。

　　投资者的过度自信对金融市场也会造成影响。Odean(1998b)将市场参与者分为价格接受者、内部人和做市商，分析过度自信对金融市场的影响。在我国，价格接受者相当于中小散户投资者；机构投资者由于其较强的获得信息的能力，相当于内部人；我国不存在做市商，但是庄家的某些行为类似于做市商，可以作为做市商来分析其对金融市场的影响。这三类投资者在获取信息和价格决定上具有不同的机制。

1．过度自信对交易量的影响

　　当投资者过度自信时，市场中的交易量会增大。在无噪音的完全理性预期的市场中，如果不考虑流动性需求，交易量应该是零。如果理性是共识，当一个投资者买进股票时，另外的投资者卖出股票，买进者会考虑是否存在卖出者知道而买进者不知道的信息，这时就不会有交易产生。而现实中金融市场的交易量是非常大的。Dow和Gorton(1997)发现，全球外汇的日交易量大约是年度世界贸易总额和投资流动总额之和的四分之一。1998年纽约证券交易所的周转率超过75％。我国的情况更是 晾人，1996年上海证券交易所的换手率是591％，深圳证券交易所的换手率是902％。由于没有模型来说明在理性市场中交易量应该是多少，所以很难证明什么样的交易量是过多的。Odean(1998a)分析了投资者的买卖行为，发现在考虑了流动性需求、风险管理和税收影响后，投资者买进的股票表现差于卖出的股票，这些投资者交易过多，由于交易成本的原因，过多的交易损害了其收益，其解释是投资者是过度自信的，过度评价了其私人信息的准确性并错误地解释了这些信号，才导致了差的决策。Odean观察了166个投资俱乐部6年的交易，发现平均每年的周转量是65％，年净收益是14.1％，而作为基准的标准普尔500指数收益是18％。

2．过度自信对市场效率的影响

　　在理性市场中，只有当新的信息出来时，价格才会有变动。但是当投资者过度自信时，会对市场波动性产生影响。过度自信度对市场效率的影响取决于信息在市场中是如何散布的。如果少量信息被大量投资者获得，或者公开披露的信息被许多投资者做了不同的解释，过度自信会使这些信息被过度估计，导致价格偏离资产真实价值，这时过度自信损害了市场效率。如果信息仅为内部人所拥有，过度自信的内部人会过度估计其获得的私人信号，通过其过多的交易显示其私人信息，那么做市商、其他的投资者会迅速使得资产价格向其真实价值靠拢。如果内部人的信息对时间敏感，在其交易后会迅速成为公共信息，那么这种效率收益是短暂的，这时过度自信提高了市场效率。

3．过度自信对波动性的影响

　　过度自信的价格接受者会过度估计他们的个人信息，这会导致总的信号被过度估计，使得价格偏离其真实价格。由于过度自信使投资者扭曲了价格的影响，使市场波动增加。过度自信的做市商会促使内部人揭示更多的私人信息，从而将价格设定的更接近其真实价格，这时过度自信使市场波动增加，同时当做市商过度自信时，其风险规避程度会小于其不具有过度自信特征的程度，会认为持有存货的风险不大，这增加了其存货量，存货量的增加降低了市场波动。过度自信对价格的影响取决于不同特征交易者的数量、财富、风险承受能力和信息。如果市场中价格接受者和内部人的数量和财富都较大，而做市商的数量较少、力量较小，则价格的波动性会更大。

4．过度自信对投资者期望效用的影响

　　当投资者过度自信时，其资产组合并没有完全分散化，集中的资产组合会降低其期望效用。如果信息是有成本的，过度自信的投资者会花费更大的成本去成为知情者，同时进行更频繁的交易，由于交易费用的原因，过多的交易会降低其净收益(Odean，1998a)。Lakonishok等(1992)发现在1983-1989年间，积极的基金经理的业绩差于标准普尔500指数的表现，扣除管理费，积极的管理减少了基金价值。这可能是由于过度自信使得基金经理在获取信息上花费太大，或者是对其选股能力过度自信所导致。但是De long，Shleifer，Summers和Waldmann(1990)证明了过度自信的交易者能够在市场中存活下来。Wang(1997)用双寡头模型证明了过度自信的基金经理不仅能够获得比他的理性竞争对手更高的期望收益和效用，而且也比他理性时的收益和效用更高，所以过度自信严格占优于理性。

From Wikipedia, the free encyclopedia

Cramer's rule is a theorem in linear algebra, which gives the solution of a system of linear equations or corresponding square matrices in terms of determinants. It is named after Gabriel Cramer (1704 – 1752), who published the rule in his 1750 Introduction à l'analyse des lignes courbes algébriques, although Colin Maclaurin also published the method in his 1748 Treatise of Geometry (and probably knew of the method as early as 1729).^[1]

Computationally, it is inefficient for large matrices and thus not used in practical applications which may involve many equations. However, as no pivoting is needed, it is more efficient than Gaussian elimination for small matrices, particularly when SIMD operations are used.

Cramer's rule is of theoretical importance in that it gives an explicit expression for the solution of the system.

Wikibooks has a book on the topic of Linear Algebra/Cramer's Rule

[edit] Explicit formulas for small systems

Consider the linear system

which in matrix format is

Then, x and y can be found with Cramer's rule as:

and

The rules for 3×3 are similar. Given:

which in matrix format is

x, y and z can be found as follows:

[edit] Finding solution of a system of linear equations in the general case

Consider a system of linear equations represented in matrix multiplication form as

where the square matrix A is invertible and the vector is the column vector of the variables.

Then the theorem states that:

where A_i is the matrix formed by replacing the ith column of A by the column vector b.

This formula is, however, of limited practical value for larger matrices, as there are other more efficient ways of solving systems of linear equations, such as by Gauss elimination or, even better, LU decomposition.

[edit] Finding inverse matrix

Let A be an n×n matrix. Then

where Adj(A) denotes the adjugate of A, det(A) is the determinant, and I is the identity matrix. If det(A) is invertible in R, then the inverse matrix of A is

If R is a field (such as the field of real numbers), then this gives a formula for the inverse of A, provided det(A) ≠ 0. The formula is, however, of limited practical value for large matrices, as there are other more efficient ways of generating the inverse matrix, such as by Gauss–Jordan elimination.

[edit] Abstract formulation

The rule holds in a commutative ring, not just in reals.

[edit] Applications to differential geometry

Cramer's rule is also extremely useful for solving problems in differential geometry. Consider the two equations and . When u and v are independent variables, we can define and .

Finding an equation for is a trivial application of Cramer's rule.

First, calculate the first derivatives of F, G, x, and y:

Substituting dx, dy into dF and dG, we have:

Since u, v are both independent, the coefficients of du, dv must be zero. So we can write out equations for the coefficients:

Now, by Cramer's rule, we see that:

This is now a formula in terms of two Jacobians:

Similar formulae can be derived for , , .

[edit] Applications to algebra

Cramer's rule can be used to prove the Cayley–Hamilton theorem of linear algebra, as well as Nakayama's lemma, which is fundamental in commutative ring theory.

[edit] Relevance to eigenvalues

Cramer's rule can be used to formulate the characteristic equation in eigenvalue problems. An eigenvector x is often expressed as

This can be viewed as a linear system of equations in which the coefficient matrix is the expression in the parentheses, the matrix of the unknowns is x, and the right hand side matrix is zero. According to Cramer's rule, solutions to x are expressed as a quotient. The numerator will always be zero, since here c=0. Hence to find the non-trivial solutions (not all zeros), the denominator must also equal zero; that is, the determinant of the coefficient matrix must be zero. Then the solutions of the equation are given by:

which is the familiar characteristic equation.

[edit] Applications to integer programming

Cramer's rule can be used to prove that an integer programming problem whose constraint matrix is totally unimodular and whose right-hand side is all integer has integer basic solutions. This makes the integer program substantially easier to solve.

Pythagoras

Pythagoras Pre-Socratic philosophy
Bust of Pythagoras of Samos in the Capitoline Museums, Rome
Full name	Pythagoras (Πυθαγόρας)
School/tradition	Pythagoreanism
Main interests	Metaphysics, Music, Mathematics, Ethics, Politics
Notable ideas	Musica universalis, Golden ratio, Pythagorean tuning, Pythagorean theorem
Influenced by[show] Thales, Anaximander, Pherecydes
Influenced[show] Philolaus, Alcmaeon, Parmenides, Plato, Euclid, Empedocles, Hippasus, Kepler

Pythagoras of Samos (Greek: Ὁ Πυθαγόρας ὁ Σάμιος, O Pūthagoras o Samios, "Pythagoras the Samian", or simply Ὁ Πυθαγόρας; born between 580 and 572 BC, died between 500 and 490 BC) was an Ionian Greek mathematician and founder of the religious movement called Pythagoreanism. He is often revered as a great mathematician, mystic and scientist; however some have questioned the scope of his contributions to mathematics and natural philosophy. Herodotus referred to him as "the most able philosopher among the Greeks". His name led him to be associated with Pythian Apollo; Aristippus explained his name by saying, "He spoke (agor-) the truth no less than did the Pythian (Pyth-)," and Iamblichus tells the story that the Pythia prophesied that his pregnant mother would give birth to a man supremely beautiful, wise, and beneficial to humankind.^[1]

He is best known for the Pythagorean theorem, which bears his name. Known as "the father of numbers", Pythagoras made influential contributions to philosophy and religious teaching in the late 6th century BC. Because legend and obfuscation cloud his work even more than with the other pre-Socratics, one can say little with confidence about his life and teachings. We do know that Pythagoras and his students believed that everything was related to mathematics and that numbers were the ultimate reality and, through mathematics, everything could be predicted and measured in rhythmic patterns or cycles. According to Iamblichus of Chalcis, Pythagoras once said that "number is the ruler of forms and ideas and the cause of gods and daemons."

He was the first man to call himself a philosopher, or lover of wisdom,^[2] and Pythagorean ideas exercised a marked influence on Plato. Unfortunately, very little is known about Pythagoras because none of his writings have survived. Many of the accomplishments credited to Pythagoras may actually have been accomplishments of his colleagues and successors.

Life

Pythagoras was born on Samos, a Greek island in the eastern Aegean, off the coast of Asia Minor. He was born to Pythais (his mother, a native of Samos) and Mnesarchus (his father, a Phoenician merchant from Tyre). As a young man, he left his native city for Croton, Calabria, in Southern Italy, to escape the tyrannical government of Polycrates. According to Iamblichus, Thales, impressed with his abilities, advised Pythagoras to head to Memphis in Egypt and study with the priests there who were renowned for their wisdom. He was also discipled in the temples of Tyre and Byblos in Phoenicia. It may have been in Egypt where he learned some geometric principles which eventually inspired his formulation of the theorem that is now called by his name. This possible inspiration is presented as an extraordinaire problem in the Berlin Papyrus. Upon his migration from Samos to Croton, Calabria, Italy, Pythagoras established a secret religious society very similar to (and possibly influenced by) the earlier Orphic cult.

Bust of Pythagoras, Vatican

Pythagoras undertook a reform of the cultural life of Croton, urging the citizens to follow virtue and form an elite circle of followers around himself called Pythagoreans. Very strict rules of conduct governed this cultural center. He opened his school to both male and female students uniformly. Those who joined the inner circle of Pythagoras's society called themselves the Mathematikoi. They lived at the school, owned no personal possessions and were required to assume a mainly vegetarian diet (meat that could be sacrificed was allowed to be eaten). Other students who lived in neighboring areas were also permitted to attend Pythagoras's school. Known as Akousmatikoi, these students were permitted to eat meat and own personal belongings. Richard Blackmore, in his book The Lay Monastery (1714), saw in the religious observances of the Pythagoreans, "the first instance recorded in history of a monastic life."

According to Iamblichus, the Pythagoreans followed a structured life of religious teaching, common meals, exercise, reading and philosophical study. Music featured as an essential organizing factor of this life: the disciples would sing hymns to Apollo together regularly; they used the lyre to cure illness of the soul or body; poetry recitations occurred before and after sleep to aid the memory.

Flavius Josephus, in his polemical Against Apion, in defence of Judaism against Greek philosophy, mentions that according to Hermippus of Smyrna, Pythagoras was familiar with Jewish beliefs, incorporating some of them in his own philosophy.

Towards the end of his life he fled to Metapontum because of a plot against him and his followers by a noble of Croton named Cylon. He died in Metapontum around 90 years old from unknown causes.

Bertrand Russell, in A History of Western Philosophy, contended that the influence of Pythagoras on Plato and others was so great that he should be considered the most influential of all western philosophers.

Pythagoreans

Pythagoras, the man in the center with the book, teaching music, in The School of Athens by Raphael

The so-called Pythagoreans, who were the first to take up mathematics, not only advanced this subject, but saturated with it, they fancied that the principles of mathematics were the principles of all things.

—Aristotle, Metaphysics 1-5 , cc. 350 BC

The organization was in some ways a school, in some ways a brotherhood, and in some ways a monastery. It was based upon the religious teachings of Pythagoras and was very secretive. At first, the school was highly concerned with the morality of society. Members were required to live ethically, love one another, share political beliefs, practice pacifism, and devote themselves to the mathematics of nature.

Pythagoras's followers were commonly called "Pythagoreans". They are generally accepted as philosophical mathematicians who had an influence on the beginning of axiomatic geometry, which after two hundred years of development was written down by Euclid in The Elements.

The Pythagoreans observed a rule of silence called echemythia, the breaking of which was punishable by death. This was because the Pythagoreans believed that a man's words were usually careless and misrepresented him and that when someone was "in doubt as to what he should say, he should always remain silent". Another rule that they had was to help a man "in raising a burden, but do not assist him in laying it down, for it is a great sin to encourage indolence", and they said "departing from your house, turn not back, for the furies will be your attendants"; this axiom reminded them that it was better to learn none of the truth about mathematics, God, and the universe at all than to learn a little without learning all. (The Secret Teachings of All Ages by Manly P. Hall).

In his biography of Pythagoras (written seven centuries after Pythagoras's time), Porphyry stated that this silence was "of no ordinary kind." The Pythagoreans were divided into an inner circle called the mathematikoi ("mathematicians") and an outer circle called the akousmatikoi ("listeners"). Porphyry wrote "the mathematikoi learned the more detailed and exactly elaborated version of this knowledge, the akousmatikoi (were) those who had heard only the summary headings of his (Pythagoras's) writings, without the more exact exposition." According to Iamblichus, the akousmatikoi were the exoteric disciples who listened to lectures that Pythagoras gave out loud from behind a veil.

The akousmatikoi were not allowed to see Pythagoras and they were not taught the inner secrets of the cult. Instead they were taught laws of behavior and morality in the form of cryptic, brief sayings that had hidden meanings. The akousmatikoi recognized the mathematikoi as real Pythagoreans, but not vice versa. After the murder of a number of the mathematikoi by the cohorts of Cylon, a resentful disciple, the two groups split from each other entirely, with Pythagoras's wife Theano and their two daughters leading the mathematikoi.

Theano, daughter of the Orphic initiate Brontinus, was a mathematician in her own right. She is credited with having written treatises on mathematics, physics, medicine, and child psychology, although nothing of her writing survives. Her most important work is said to have been a treatise on the philosophical principle of the golden mean. In a time when women were usually considered property and relegated to the role of housekeeper or spouse, Pythagoras allowed women to function on equal terms in his society.^[3]

The Pythagorean society is associated with prohibitions such as not to step over a crossbar, and not to eat beans. These rules seem like primitive superstition, similar to "walking under a ladder brings bad luck". The abusive epithet mystikos logos ("mystical speech") was hurled at Pythagoras even in ancient times to discredit him. The prohibition on beans could be linked to favism, which is relatively widespread around the Mediterranean.

The key here is that akousmata means "rules", so that the superstitious taboos primarily applied to the akousmatikoi, and many of the rules were probably invented after Pythagoras's death and independent from the mathematikoi (arguably the real preservers of the Pythagorean tradition). The mathematikoi placed greater emphasis on inner understanding than did the akousmatikoi, even to the extent of dispensing with certain rules and ritual practices. For the mathematikoi, being a Pythagorean was a question of innate quality and inner understanding.

There was also another way of dealing with the akousmata — by allegorizing them. We have a few examples of this, one being Aristotle's explanations of them: "'step not over a balance', i.e. be not covetous; 'poke not the fire with a sword', i.e. do not vex with sharp words a man swollen with anger, 'eat not heart', i.e. do not vex yourself with grief," etc. We have evidence for Pythagoreans allegorizing in this way at least as far back as the early fifth century BC. This suggests that the strange sayings were riddles for the initiated.

The Pythagoreans are known for their theory of the transmigration of souls, and also for their theory that numbers constitute the true nature of things. They performed purification rites and followed and developed various rules of living which they believed would enable their soul to achieve a higher rank among the gods.

Much of their mysticism concerning the soul seem inseparable from the Orphic tradition. The Orphics advocated various purificatory rites and practices as well as incubatory rites of descent into the underworld. Pythagoras is also closely linked with Pherecydes of Syros, the man ancient commentators tend to credit as the first Greek to teach a transmigration of souls. Ancient commentators agree that Pherekydes was Pythagoras's most intimate teacher. Pherekydes expounded his teaching on the soul in terms of a pentemychos ("five-nooks", or "five hidden cavities") — the most likely origin of the Pythagorean use of the pentagram, used by them as a symbol of recognition among members and as a symbol of inner health (ugieia).

Musical theories and investigations

Pythagoras was very interested in music, and so were his followers. The Pythagoreans were musicians as well as mathematicians. Pythagoras wanted to improve the music of his day, which he believed was not harmonious enough and was too hectic.

According to legend, the way Pythagoras discovered that musical notes could be translated into mathematical equations was when one day he passed blacksmiths at work, and thought that the sounds emanating from their anvils being hit were beautiful and harmonious and decided that whatever scientific law caused this to happen must be mathematical and could be applied to music. He went to the blacksmiths to learn how this had happened by looking at their tools, he discovered that it was because the anvils were "simple ratios of each other, one was half the size of the first, another was 2/3 the size, and so on." (See Pythagorean tuning.)

The Pythagoreans elaborated on a theory of numbers, the exact meaning of which is still debated among scholars. Pythagoras believed in something called the "harmony of the spheres." He believed that the planets and stars moved according to mathematical equations, which corresponded to musical notes and thus produced a symphony.^[4]

Influence

The Pythagorean theorem: The sum of the areas of the two squares on the legs (a and b) equals the area of the square on the hypotenuse (c).

Since the fourth century AD, Pythagoras has commonly been given credit for discovering the Pythagorean theorem, a theorem in geometry that states that in a right-angled triangle the square of the hypotenuse (the side opposite the right angle), c, is equal to the sum of the squares of the other two sides, b and a—that is, a² + b² = c².

While the theorem that now bears his name was known and previously utilized by the Babylonians and Indians, he, or his students, are often said to have constructed the first proof. It must, however, be stressed that the way in which the Babylonians handled Pythagorean numbers, implies that they knew that the principle was generally applicable, and knew some kind of proof, which has not yet been found in the (still largely unpublished) cuneiform sources.^[5] Because of the secretive nature of his school and the custom of its students to attribute everything to their teacher, there is no evidence that Pythagoras himself worked on or proved this theorem. For that matter, there is no evidence that he worked on any mathematical or meta-mathematical problems. Some attribute it as a carefully constructed myth by followers of Plato over two centuries after the death of Pythagoras, mainly to bolster the case for Platonic meta-physics, which resonate well with the ideas they attributed to Pythagoras. This attribution has stuck, down the centuries up to modern times.^[6] The earliest known mention of Pythagoras's name in connection with the theorem occurred five centuries after his death, in the writings of Cicero and Plutarch.

Today, Pythagoras is revered as a prophet by the Ahl al-Tawhid or Druze faith along with his fellow Greek, Plato. But Pythagoras also had his critics, such as Heraclitus who said that "much learning does not teach wisdom; otherwise it would have taught Hesiod and Pythagoras, and again Xenophanes and Hecataeus".^[7]

Religion and science

Pythagoras’ religious and scientific views were, in his opinion, inseparably interconnected. Religiously, Pythagoras was a believer of metempsychosis. He believed in transmigration, or the reincarnation of the soul again and again into the bodies of humans, animals, or vegetables until it became moral. His ideas of reincarnation were influenced by ancient Greek religion. He was one of the first to propose that the thought processes and the soul were located in the brain and not the heart. He himself claimed to have lived four lives that he could remember in detail, and heard the cry of his dead friend in the bark of a dog.

One of Pythagoras' beliefs was that the essence of being is number. Thus, being relies on stability of all things that create the universe. Things like health relied on a stable proportion of elements; too much or too little of one thing causes an imbalance that makes a being unhealthy. Pythagoras viewed thinking as the calculating with the idea numbers. When combined with the Folk theories, the philosophy evolves into a belief that Knowledge of the essence of being can be found in the form of numbers. If this is taken a step further, one can say that because mathematics is an unseen essence, the essence of being is an unseen characteristic that can be encountered by the study of mathematics.

Literary works

No texts by Pythagoras survive, although forgeries under his name — a few of which remain extant — did circulate in antiquity. Critical ancient sources like Aristotle and Aristoxenus cast doubt on these writings. Ancient Pythagoreans usually quoted their master's doctrines with the phrase autos ephe ("he himself said") — emphasizing the essentially oral nature of his teaching. Pythagoras appears as a character in the last book of Ovid's Metamorphoses, where Ovid has him expound upon his philosophical viewpoints. Pythagoras has been quoted as saying, "No man is free who cannot command himself."

Lore

There is another side to Pythagoras, as he became the subject of elaborate legends surrounding his historic persona. Aristotle described Pythagoras as a wonder-worker and somewhat of a supernatural figure, attributing to him such aspects as a golden thigh, which was a sign of divinity. According to Aristotle and others' accounts, some ancients believed that he had the ability to travel through space and time, and to communicate with animals and plants.^[8] An extract from Brewer's Dictionary of Phrase and Fable's entry entitled "Golden Thigh":

Pythagoras is said to have had a golden thigh, which he showed to Abaris, the Hyperborean priest, and exhibited in the Olympic games.^[9]

Another legend, also taken from Brewer's Dictionary, describes his writing on the moon:

Pythagoras asserted he could write on the moon. His plan of operation was to write on a looking-glass in blood, and place it opposite the moon, when the inscription would appear photographed or reflected on the moon's disc.^[10]

Other accomplishments

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The number is irrational.

One of Pythagoras's major accomplishments was the discovery that music was based on proportional intervals of the numbers one through four. He believed that the number system, and therefore the universe system, was based on the sum of these numbers: ten. Pythagoreans swore by the Tetrachtys of the Decad, or ten, rather than by the gods. Odd numbers were masculine and even were feminine. He discovered the theory of mathematical proportions, constructed from three to five geometrical solids. One member of his order, Hippasos, also discovered irrational numbers, but the idea was unthinkable to Pythagoras, and according to legend, Hippasos was executed. Pythagoras (or the Pythagoreans) also discovered square numbers. They found that if one took, for example, four small stones and arranged them into a square, each side of the square was not only equivalent to the other, but that when the two sides were multiplied together, they equaled the sum total of stones in the square arrangement, hence the name "Square Root"^[11]. He was one of the first to think that the earth was round, that all planets have an axis, and that all the planets travel around one central point. He originally identified that point as Earth, but later renounced it for the idea that the planets revolve around a central “fire” that he never identified as the sun. He also believed that the moon was another planet that he called a “counter-Earth” – furthering his belief in the Limited-Unlimited.

Groups influenced by Pythagoras

Influence on Plato

Pythagoras or in a broader sense, the Pythagoreans, allegedly exercised an important influence on the work of Plato. According to R. M. Hare, his influence consists of three points: a) the platonic Republic might be related to the idea of "a tightly organized community of like-minded thinkers", like the one established by Pythagoras in Croton. b) there is evidence that Plato possibly took from Pythagoras the idea that mathematics and, generally speaking, abstract thinking is a secure basis for philosophical thinking as well as "for substantial theses in science and morals". c) Plato and Pythagoras shared a "mystical approach to the soul and its place in the material world". It is probable that both have been influenced by Orphism.^[12]

Plato's harmonics were clearly influenced by the work of Archytas, a genuine Pythagorean of the third generation, who made important contributions to geometry, reflected in Book VIII of Euclid's Elements.

Roman influence

In the legends of ancient Rome, Numa Pompilius, the second King of Rome, is said to have studied under Pythagoras. This is unlikely, since the commonly accepted dates for the two lives do not overlap.

Influence on esoteric groups

Pythagoras started a secret society called the Pythagorean brotherhood devoted to the study of mathematics. This had a great effect on future esoteric traditions, such as Rosicrucianism and Freemasonry, both of which were occult groups dedicated to the study of mathematics and both of which claimed to have evolved out of the Pythagorean brotherhood. The mystical and occult qualities of Pythagorean mathematics are discussed in a chapter of Manly P. Hall's The Secret Teachings of All Ages entitled "Pythagorean Mathematics".

Pythagorean theory was tremendously influential on later numerology, which was extremely popular throughout the Middle East in the ancient world. The 8th-century Muslim alchemist Jabir ibn Hayyan grounded his work in an elaborate numerology greatly influenced by Pythagorean theory.^{[citation needed]}

From Wikipedia, the free encyclopedia

Independence of irrelevant alternatives (IIA) is a term for an axiom of decision theory and various social sciences. Although exact formulations of IIA differ, intentions of the usages are similar in attempting to provide a rational account of individual behavior or aggregation of individual preferences.

IIA is also known as "Chernoff's condition" or Sen's property α (alpha)

Under one common formulation, the axiom states that:

If A is preferred to B out of the choice set {A,B}, then introducing a third alternative X, thus expanding the choice set to {A,B,X}, must not make B preferable to A.

In other words, whether A or B is better should not be changed by the availability of X, which is irrelevant to the choice between A and B. This formulation appears in bargaining theory, theories of individual choice, and voting theory. It is controversial for two reasons: first, some theorists find it too strict an axiom; second, experiments by Amos Tversky, Daniel Kahneman, and others have shown that human behaviour rarely adheres to this axiom.

A distinct formulation of IIA is that of social choice theory:

If A is selected over B out of the choice set {A,B} by a voting rule for given voter preferences of A, B, and an unavailable third alternative X, then B must not be selected over A by the voting rule if only preferences for X change.

In other words, whether A or B is selected should not be affected by a change in the vote for an unavailable X, which is thus irrelevant to the choice between A and B. Kenneth Arrow (1951) shows the impossibility of aggregating individual preferences ("votes") satisfying IIA and certain other apparently reasonable conditions.

[edit] Voting theory

In voting systems, independence of irrelevant alternatives is often interpreted as, if one candidate (X) wins the election, and a new alternative (Y) is added, only X or Y will win the election.

Approval voting and range voting satisfy the independence of irrelevant alternatives criterion, largely because they are cardinal rather than ordinal voting systems and votes for one candidate do not restrict votes for another. Another cardinal system, cumulative voting, does not satisfy the criterion.

An anecdote which illustrates a violation of this property has been attributed to Sidney Morgenbesser:

After finishing dinner, Sidney Morgenbesser decides to order dessert. The waitress tells him he has two choices: apple pie and blueberry pie. Sidney orders the apple pie. After a few minutes the waitress returns and says that they also have cherry pie at which point Morgenbesser says "In that case I'll have the blueberry pie."

All voting systems have some degree of inherent susceptibility to strategic nomination considerations. Some regard these considerations as less serious unless the voting system specifically fails the (easier to satisfy) independence of clones criterion.

[edit] Local independence

A related criterion proposed by H. P. Young and A. Levenglick is called local independence of irrelevant alternatives. It says that if one candidate (X) wins an election, and a new alternative (Y) is added, X will still win the election if Y is not in the Smith set. In other words, the outcome of the election is independent of alternatives which are not in the Smith set. Note that this neither implies nor is implied by the independence of irrelevant alternatives criterion; in fact, the two are mutually exclusive.

No deterministic ranked methods satisfy IIA, but local IIA is satisfied by some methods which always elect from the Smith set, such as ranked pairs and Schulze method.

[edit] Criticism

Many voting theorists believe that IIA is too strong a condition. The majority criterion is incompatible with IIA. Consider

7 votes for A > B > C

6 votes for B > C > A

5 votes for C > A > B

Without loss of generality, suppose a voting system satisfying the majority criterion chooses A. If B drops out of the race, the remaining votes will be:

7 votes for A > C

11 votes for C > A

and C will win under the majority criterion, even though the change (B dropping out) concerned an "irrelevant" alternative candidate who did not win in the original circumstance. This is an example of the spoiler effect.

Some text of this article is derived with permission from http://condorcet.org/emr/criteria.shtml

[edit] IIA in social choice

From Kenneth Arrow (1951, pp. 15, 23, 27), each "voter" i in the society has an ordering R_i that ranks the (conceivable) objects of social choice—X, Y, and Z in simplest case -- from lowest on up (say candidates for public office or social states from different economic policies). The voting rule in turn maps the set of voter orderings (R₁, ...,R_n) onto a social ordering R that determines the social ranking of X, Y, and Z. In the first edition of his book, Arrow made a mistake in interpreting the IIA axiom. This mistake was later corrected but it reappears from time to time in various publications. Let's take a look at Arrow's original mistake. Among the objects of choice, he distinguished those that by hypothesis are specified as feasible and infeasible. Consider two possible sets of voter orderings (R₁, ...,R_n ) and (R₁', ...,R_n') such that the ranking of X and Y for each voter i is the same for R_i and R_i'. The voting rule generates corresponding social orderings R and R'. Now suppose that X and Y are feasible but Z is infeasible (say, the candidate is not on the ballot or the social state is outside the production possibility curve). Arrow required that the voting rule that R and R' select the same (top-ranked) social choice from the feasible set (X, Y), and that this requirement holds no matter what the ranking is of infeasible Z relative to X and Y in the two sets of orderings. In fact, the IIA axiom does not allow to "remove" an alternative from the available set (or a candidate from the ballot). It says nothing about what would happen in such a case. All alternatives are assumed "feasible." IIA requires that whenever a pair of alternatives is ranked the same way in two preference profiles (that are over the same sets of alternatives), then the voting rule must order these two alternatives identically. Paramesh Ray (1973) showed that Arrow's IIA does not imply an IIA similar to that at the top of this article nor conversely.

[edit] IIA in econometrics

The independence of irrelevant alternatives (IIA) in econometrics is a property of the multinomial logit (also called conditional logit) models in econometrics; outcomes that could theoretically violate this IIA (such as the outcome of multicandidate elections or any choice made by humans) may make multinomial logit and conditional logit invalid estimators.

IIA implies that adding another alternative or changing the characteristics of a third alternative does not affect the relative odds between the two alternatives considered. This implication is not realistic for applications with similar alternatives. Many examples have been constructed to illustrate this problem, including Beethoven/Debussy (Debreu 1960; Tversky 1972), Bicycle/Pony (Luce and Suppes 1965), and Red Bus/Blue Bus (McFadden 1974).

Consider the Red Bus/Blue Bus example. Commuters initially face a decision between two modes of transportation: car and red bus. Suppose that a consumer chooses between these two options with equal probability, 0.5, so that the odds equal 1. Now suppose a third mode, blue bus, is added. Assuming bus commuters do not care about the color of the bus, consumers are expected to choose between bus and car still with equal probability. But IIA implies that this is not the case: the probability of commuters that take each of the three modes equals one third. (Based on Wooldridge 2002, pp. 501-2.)

Many modeling advances have been motivated by a desire to alleviate the concerns raised by IIA. Generalized extreme value (McFadden 1978), multinomial probit (also called conditional probit) and mixed logit are alternative models for nominal outcomes which relax IIA, but these models often have assumptions of their own that may be difficult to meet or are computationally infeasible. The multinomial probit model has as a disadvantage that it makes calculation of maximum likelihood infeasible for more than five alternatives as it involves multiple integrals. IIA can also be relaxed by specifying a hierarchical model, ranking the choice alternatives. The most popular of these is called the nested logit model (McFadden 1984).

Generalized extreme value and multinomial probit models possess another property, the Invariant Proportion of Substitution (Steenburgh 2008), which suggests similarly counterintuitive individual choice behavior.

[edit] Examples

[edit] Borda count

In a Borda count election, 5 voters rank 5 alternatives [A, B, C, D, E].

3 voters rank [A>B>C>D>E]. 1 voter ranks [C>D>E>B>A]. 1 voter ranks [E>C>D>B>A].

Borda count (a=0, b=1): C=13, A=12, B=11, D=8, E=6. C wins.

Now, the voter who ranks [C>D>E>B>A] instead ranks [C>B>E>D>A]; and the voter who ranks [E>C>D>B>A] instead ranks [E>C>B>D>A]. Note that they change their preferences only over the pairs [B, D] and [B, E].

The new Borda count: B=14, C=13, A=12, E=6, D=5. B wins.

Note that the social choice has changed the ranking of [B, A], [B, C] and [D, E]. The changes in the social choice ranking are dependent on irrelevant changes in the preference profile. In particular, B now wins instead of C, even though no voter changed their preference over [B, C].

[edit] Instant-runoff voting

In an instant-runoff election, 5 voters rank 3 alternatives [A, B, C].

2 voters rank [A>B>C]. 2 voters rank [C>B>A]. 1 voter ranks [B>A>C].

Round 1: A=2, B=1, C=2; B eliminated. Round 2: A=3, C=2; A wins.

Now, the two voters who rank [C>B>A] instead rank [B>C>A]. Note that they only change their preferences over B and C.

Round 1: A=2, B=3, C=0; C eliminated. Round 2: A=2, B=3; B wins.

Note that the social choice ranking of [A, B] is dependent on preferences over the irrelevant alternatives [B, C].

[edit] Kemeny-Young method

Please help improve this article or section by expanding it. Further information might be found on the talk page. (August 2007)

[edit] Minimax Condorcet

Please help improve this article or section by expanding it. Further information might be found on the talk page. (August 2007)

[edit] Plurality voting system

In a plurality voting system 7 voters rank 3 alternatives [A, B, C].

3 voters rank [A>B>C] 2 voters rank [B>A>C] 2 voters rank [C>B>A]

The plurality winner is A, with 3 votes out of 7.

Now, the 2 voters who rank [C>B>A] instead rank [B>C>A]; note that they only change their preferences over [B, C].

The new plurality winner is B, with 4 votes out of 7.

The social choice has swapped B ahead of A, even though the only change in the preference profile was between B and C, which are irrelevant alternatives.

[edit] Range voting

Please help improve this article or section by expanding it. Further information might be found on the talk page. (August 2007)

[edit] Ranked Pairs

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[edit] Two-round system

A probable example of the two-round system's failing this criterion was the 1991 Louisiana gubernatorial election. Polls leading up to the election suggested that, had the runoff been Edwin Edwards v Buddy Roemer, Roemer would have won. However, in the actual election, David Duke managed to finish second and make the runoff instead of Roemer, a runoff which Edwards then won by a large margin. Thus, the presence of Duke in the election changed which of the non-Duke candidates won.

[edit] Schulze method

Please help improve this article or section by expanding it. Further information might be found on the talk page. (August 2007)

See Schulze method

維基百科，自由的百科全書

Logit模型（Logit model，也譯作「評定模型」，「分類評定模型」，又作Logistic regression，「邏輯回歸」）是離散選擇法模型之一，屬於多重變數分析範疇，是社會學、生物統計學、臨床、數量心理學、市場營銷等統計實證分析的常用方法。

[編輯] 邏輯分佈（Logistic distribution）公式

邏輯分佈函數圖像

其中參數β常用極大似然估計。

[編輯] IIA假設

「Independent and irrelevant alternatives」假設，也稱作「IIA效應」，指Logit模型中的各個可選項是獨立的不相關的。

[編輯] IIA假設示例

市場上有A，B，C三個商品相互競爭，分別佔有市場佔有率：60%，30%和10%，三者比例為：6:3:1

一個新產品D引入市場，有能力佔有20%的市場——

如果滿足IIA假設，各個產品獨立作用，互不關聯：新產品D佔有20%的市場佔有率，剩下的80%在A、B、C之間按照6:3:1的比例瓜分，分別佔有48%，24%和8%。

如果不滿足IIA假設，比如新產品D跟產品B幾乎相同，則新產品D跟產品B嚴重相關：新產品D奪去產品B的部分市場，佔有總份額的20%，產品B佔有剩餘的10%，而產品A和C的市場佔有率保持60%和10%不變。

[編輯] 滿足IIA假設的優點

• 可以獲得每個人其個性化的選擇集合的一致的參數估計
• 各個類別的子集的一般化的估計
• 大大節省時間
• 可選項數目很多的時候尤其如此

[編輯] IIA假設的檢驗

[編輯] Hausman檢驗

Hausman和McFadden提出的。

[編輯] 一般化模型的檢驗

[編輯] IIA問題的解決方法

[編輯] 多項式Probit模型

[編輯] 一般化極值模型

可以將可選項間的相關性建模

[編輯] 分簇Logit模型

分簇（Nested）表示可選項被分作不同的組，組與組之間不相關，組內的可選項相關，相關程度用1-λ_g來表示（1-λ_g越大，相關程度越高）

[編輯] 對偶組合Logit模型

[編輯] 一般化分簇Logit模型

[編輯] 混合Logit模型

[編輯] 二類評定模型（Binary Logit Model）

• 僅有兩個可選項：V_1n，V_2n

From Wikipedia, the free encyclopedia

See Michaelis-Menten kinetics for details

In statistics, nonlinear regression is a form of regression analysis in which observational data are modeled by a function which is a nonlinear combination of the model parameters and depends on one or more independent variables. The data are fitted by a method of successive approximations.

[edit] General

The data consist of error-free independent variables (explanatory variable), x, and their associated observed dependent variables (response variable), y. The ys are modeled as a random variable with mean a nonlinear function f(x,β). Systematic error may be present but its treatment is outside the scope of regression analysis. If the independent variables are not error-free, this is an errors-in-variables model, also outside this scope.

For example, the Michaelis-Menten model for enzyme kinetics

can be written as

where β₁ is the parameter V_max, β₂ is the parameter K_m and [S] is the independent variable, x. This function is nonlinear because it cannot be expressed as a linear combination of the βs.

Other examples of nonlinear functions include exponential functions, logarithmic functions, trigonometric functions, power functions, Gaussian function, and Lorentzian curves. Some functions, such as the exponential or logarithmic functions, can be transformed so that they are linear. When so transformed, standard linear regression can be performed but must be applied with caution. See Linearization, below, for more details.

In general, there is no closed-form expression for the best-fitting parameters, as there is in linear regression. Usually numerical optimization algorithms are applied to determine the best-fitting parameters. Again in contrast to linear regression, there may be many local minima of the function to be optimized. In practice, estimated values of the parameters are used, in conjunction with the optimization algorithm, to attempt to find the global minimum of a sum of squares.

For details concerning nonlinear data modeling see least squares and non-linear least squares.

[edit] Regression statistics

The assumption underlying this procedure is that the model can be approximated by a linear function.

where . It follows from this that the least squares estimators are given by

The nonlinear regression statistics are computed and used as in linear regression statistics, but using J in place of X in the formulas. The linear approximation introduces bias into the statistics. Therefore more caution than usual is required in interpreting statistics derived from a nonlinear model.

[edit] Ordinary and weighted least squares

The best-fit curve is often assumed to be that which minimizes the sum of squared residuals. This is the (ordinary) least squares (OLS) approach. However, in cases where the dependent variable does not have constant variance a sum of weighted squared residuals may be minimized; see weighted least squares. Each weight should ideally be equal to the reciprocal of the variance of the observation, but weights may be recomputed on each iteration, in an iteratively weighted least squares algorithm.

[edit] Linearization

Some nonlinear regression problems can be moved to a linear domain by a suitable transformation of the model formulation.

For example, consider the nonlinear regression problem (ignoring the error):

If we take a logarithm of both sides, it becomes

suggesting estimation of the unknown parameters by a linear regression of ln(y) on x, a computation that does not require iterative optimization. However, use of a linear transformation requires caution. The influences of the data values will change, as will the error structure of the model and the interpretation of any inferential results. These may not be desired effects. On the other hand, depending on what the largest source of error is, a linear transformation may distribute your errors in a normal fashion, so the choice to perform a linear transformation must be informed by modeling considerations.

For Michaelis-Menten kinetics, the linear Lineweaver–Burk plot

of 1/v against 1/[S] has been much used. However, it is very sensitive to data error and it is strongly biased toward fitting the data in a particular range of the independent variable, [S], its use is strongly deprecated.

From Wikipedia, the free encyclopedia

In statistics and econometrics, the term panel data refers to two-dimensional data. In marketing, panel data refers to data collected at the point-of-sale (also called scanner data).

Data is broadly classified according to the number of dimensions. A data set containing observations on a single phenomenon observed over multiple time periods is called time series. In time series data, both the values and the ordering of the data points have meaning. A data set containing observations on multiple phenomena observed at a single point in time is called cross-sectional. In cross-sectional data sets, the values of the data points have meaning, but the ordering of the data points does not. A data set containing observations on multiple phenomena observed over multiple time periods is called panel data. Alternatively, the second dimension of data may be some entity other than time. For example, when there is a sample of groups, such as siblings or families, and several observations from every group, the data is panel data. Whereas time series and cross-sectional data are both one-dimensional, panel data sets are two-dimensional.

Data sets with more than two dimensions are typically called multi-dimensional panel data.

[edit] Example

balanced panel:		unbalanced panel:

In the example above, two data sets with a two-dimensional panel structure are shown. Individual characteristics (income, age, sex) are collected for different persons and different years. In the left data set two persons (1, 2) are observed over three years (2003, 2004, 2005). Because each person is observed every year, the left-hand data set is called a balanced panel, whereas the data set on the right hand is called an unbalanced panel, since Person 1 is not observed in year 2005 and person 3 only in 2004.

[edit] Analysis of panel data

A panel has the form

where i is the individual dimension and t is the time dimension. A general panel data regression model is written as y_it = α + β'X_it + u_it. Different assumptions can be made on the precise structure of this general model. Two important models are the fixed effects model and the random effects model. The fixed effects model is denoted as

y_it = α + β'X_it + u_it,

u_it = μ_i + ν_it.

μ_i are individual-specific, time-invariant effects (for example in a panel of countries this could include geography, climate etc.) and because we assume they are fixed over time, this is called the fixed-effects model. The random effects model assumes in addition that

and

that is, the two error components are independent from each other.

前景理论(Prospect Theory)

　　金融学的四大研究成果,即前景理论(或视野理论)(Prospect Theory)、后悔理论(Regret Theory)、过度反应理论(Overreaction Theory)及过度自信理论(Over confidence Theory)。

前景理论概述

　　很多学者研究风险以及不确定性条件下的决策，提出的模型非常多，其中最常用的被接受的理性选择模型是Von Neumann和Morgenstern(1953)发展的财富预期效用理论。该理论提供了数学化的公理，是一个标准化的模型(解决了当人们面对风险选择时他们应该怎样行动的问题)，应用起来比较方便。但是在最近的几十年，该理论遇到了很多问题，它不能解释众多的异象，它的几个基础性的公理被实验数据所违背，这些问题也刺激了其它的一些试图解释风险或者不确定性条件下个人行为的理论的发展。前景理论(Prospect Theory)就是其中比较优秀的一个。

　　前景理论认为人们通常不是从财富的角度考虑问题，而是从输赢的角度考虑，关心收益和损失的多少。

前景理论的产生

　　前景理论（PT）首先由国外学者Kahneman和阿莫斯·特沃斯基（Amos Tversky）(1979)(用KT代表两个作者)明确的提出，他们认为个人在风险情形下的选择所展示出的特性和VonNeumann—Morgenstem的效用理论的基本原理是不相符的。

　　一是他们发现和确定性的结果相比个人会低估概率性结果，他们称之为确定性效应(Certainty Effect)。KT还指出确定性效应导致了当选择中包含确定性收益时的风险厌恶以及当选择中包含确定性损失的风险寻求。

　　二是他们还发现了孤立效应(Isolation Effect)，即当个人面对在不同前景的选项中进行选择的问题时，他们会忽视所有前景所共有的部分。孤立效应会导致当一个前景的描述方法会改变个人决策者决策的变化。

　　三是KT发现了反射效应(Reflection Effect)，当正负前景的前景绝对值相等时，在负前景之间的选择和在正前景之间的选择呈现镜像关系。为了补偿这些VMUT所不能解释的关于个人行为的特征，KT提出了新模型PT。

前景理论二定律

　　面对风险决策，人们会选择躲避还是勇往直前？这当然不能简单绝对的回答，因为还要考虑到决策者所处的环境，企业状况等情况，我们先抛开这些条件来研究在只考虑风险本身的时候，人们的心理对决策的影响。这时候我们会得出很有意思的结论。

卡尼曼的前景理论有两大定律：

　　（1）人们在面临获得时，往往小心翼翼，不愿冒风险；而在面对损失时，人人都变成了冒险家。

　　（2）人们对损失和获得的敏感程度是不同的，损失的痛苦要远远大于获得的快乐。

　　来看两个好玩的试验：一是有两个选择，A是肯定赢1000元，B是50%可能性赢得2000元，50%可能性什么也得不到。你会选择哪一个呢？大部分人会选A，这说明人是风险规避的。二是有这样两种选择，A是你肯定损失1000元，B是50%可能性损失2000元，50%可能性什么也不损失。结果大部分人会选B，这说明他们是风险偏好的。

可是我们仔细分析一下，这两个实验其实是一样的，只是玩了个文字游戏而已。第一个实验中假设你刚刚赢了2000元（以此为参照），那么如果你选了A就相当于肯定损失了1000元；选B50%可能性赢2000元就相当于50%可能性不损失钱，50%可能性什么也得不到就相当于50%可能性损失2000元。

这个似乎玩文字游戏的实验结论对管理决策是很有意义的。举个例子，一家公司面临两个决策，投资方案A肯定盈利200万元，投资方案B有50%的可能盈利300万元，50%的可能盈利100万元。这时候如果公司定的盈利目标比较低，不如说100万元，那么A方案看起来好像多赚了100万元，而B则要么刚好达到目标，要么多赚200万元。A和B都是获得，那么大多数人会选A方案。但要是公司定的目标比较高，比如是300万元，那么大多数人会选B方案，员工会抱着说不定会达到目标的心理，去拼一下。这说明老板（决策者）完全可以通过改变盈利目标来改变员工对待风险的态度。

前景理论还可以用来解释一些决策现象。比如群体在奖惩决策时，奖励往往是就低不就高，惩罚往往是就宽不就严。为什么会有这种向中间回归的趋向呢?可以这么解释：获奖励者并非大多数人，多少有奖金就可以了，重要的是名誉，多几块钱少几块钱人们不会太在意；受到惩罚者也非大多数人，名誉上已经惩罚了，物质上象征性地惩罚一下, 给人改过的机会。这就像中国古代的中庸之道，得饶人处且饶人。

前景理论对营销的启示

　　一、对营销措施的启示

　　把前景理论引入现实生活，我们可以得到非常合拍的验证，可以举出生动的例子。把这个理论引入营销学，应该注意各项营销措施，例如：

　　1、在推出营销措施时，我们的措施要明确而实在，因为人们在面临获得时会注意规避风险，恐怕上当受骗，以为会中圈套。有时营销徒有虚名，有时好事不被看好。所以我们的营销方案要实在，我们对方案的宣传要明明白白。

　　2、不要轻易取消以前的营销措施，因为人们在面临损失时是风险偏爱的，他们会感到吃了亏，来的不是时候，因此而对酒店的一切感到不信任，使酒店蒙受生意和信誉的双重损失。如果确实要有新的变动，可以采取改变营销办法的变通措施。

　　3、在决定一项营销措施时，要考虑是否能够持续下去，因为人们对损失比对获得更敏感，如果取消以往的营销好处，人们会非常不满。把曾经有过的优惠去掉，还不如当初就根本没有。

　　二、对营销宣传的启示

　　营销宣传也是大有学问的。根据前景理论，分别经历两次获得所带来的高兴程度之和要大于把两个获得加起来一次所经历的高兴程度；而两个损失结合起来所带来的痛苦要小于分别经历这两次损失所带来的痛苦程度之和。因此，在营销宣传中要注意如下操作。

　　1、如果有几项营销措施出台和宣传，应该把它们分别推出和宣传。

　　2、如果有几个价格上调的收费项目，应该把它们放在一起推出和宣布。

　　3、如果有一个大的好消息和一个小的坏消息，就应该一起推出，使其带来的快乐超过带来的痛苦。

　　4、如果有一个大的坏消息和一个小的好消息，就应该分别推出，使好消息带来的快乐不致以被坏消息的痛苦所淹没。

前景理论的缺陷

　　一是作为一个描述性的模型，前景理论具有描述性模型共有的缺点，和规范性模型(具有严格数学推导的模型)相比，它缺乏严格的理论和数学推导，只能对人们的行为进行描述，因此前景理论的研究也只能使其描述性越来越好，换句话说它只是说明了人们会怎样做，而没有告诉人们应该怎样做。

　　二是前景理论的应用研究，尤其在我国的应用研究还不足，前景理论作为风险下决策的描述性模型，其应用价值非常大，应用范围也非常广，而目前的应用研究主要集中在金融市场上，因此应用范围方面还有待拓展。

前景理论的实例

　　卡尼曼在做诺贝尔演讲时，特地谈到了一位华人学者的研究成果，他就是芝加哥大学商学院终身教授、中欧国际工商学院行为科学中心主任奚恺元教授。奚教授用心理学来研究经济学、市场学、决策学等学科的问题，是这个领域的主要学者之一。

　　看上去很美---在差的物品上花更多的钱

　　如果有两个匹萨，他们的配料和口味等其它方面完全相同，只不过一个比另外一个更大一点，你是不是愿意为大的匹萨支付更多的钱？

　　答案似乎毫无疑问是肯定的。人应该都是理性的，对于好的东西和坏的东西，人们总是愿意为好的东西支付更多的钱。可是，在现实生活中，人的决策却并不总是如此英明。

　　来看一个奚教授于1998 年发表的冰淇淋实验。现在有两杯哈根达斯冰淇淋，一杯冰淇淋A有7盎司，装在5盎司的杯子里面，看上去快要溢出来了；另一杯冰淇淋B是8盎司，但是装在了10盎司的杯子里，所以看上去还没装满。你愿意为哪一份冰淇淋付更多的钱呢？

　　如果人们喜欢冰淇淋，那么8盎司的冰淇淋比7盎司多，如果人们喜欢杯子，那么10盎司的杯子也要比5盎司的大。可是实验结果表明，在分别判断的情况下（评点：也就是不能把这两杯冰淇淋放在一起比较，人们日常生活中的种种决策所依据的参考信息往往是不充分的），人们反而愿意为分量少的冰淇淋付更多的钱。实验表明：平均来讲，人们愿意花2.26 美元买7盎司的冰淇淋，却只愿意用1.66 美元买8盎司的冰淇淋。

　　这契合了卡尼曼等心理学家所描述的：人的理性是有限的。人们在做决策时，并不是去计算一个物品的真正价值，而是用某种比较容易评价的线索来判断。比如在冰淇淋实验中，人们其实是根据冰淇淋到底满不满来决定给不同的冰淇淋支付多少钱的。

　　再来看一个奚教授做的餐具的实验。比方说现在有一家家具店正在清仓大甩卖，你看到一套餐具，有8个菜碟、8个汤碗和8个点心碟，共24件，每件都是完好无损的，那么你愿意支付多少钱买这套餐具呢？如果你看到另外一套餐具有40件，其中24件和刚刚提到的完全相同，而且完好无损，另外这套餐具中还有8个杯子和8个茶托，其中2个杯子和7个茶托都已经破损了。你又愿意为这套餐具付多少钱呢？结果表明，在只知道其中一套餐具的情况下，人们愿意为第一套餐具支付33美元，却只愿意为第二套餐具支付24美元。

　　虽然第二套餐具比第一套多出了6个好的杯子和1个好的茶托，人们愿意支付的钱反而少了。因为到底24件和31件算是多，还是少，如果不互相比较是很难引起注意的，但是整套餐具到底完好无缺还是已经破损，却是很容易判断的。

　　瞧，人们还是依据比较容易判断的线索做出判断的，尽管这并不划算。其实，大到联合国的公共决策都可能发生这种偏差。来看这样一个实验。比如说太平洋上有小岛遭受台风袭击，联合国决定到底给这个小岛支援多少钱。假设这个小岛上有1000 户居民，90％居民的房屋都被台风摧毁了。如果你是联合国的官员，你以为联合国应该支援多少钱呢？但假如这个岛上有18000 户居民，其中有10％居民的房子被摧毁了（你不知道前面一种情况），你又认为联合国应该支援多少钱呢？从客观的角度来讲，后面一种情况下的损失显然更大。可实验的结果显示，人们觉得在前面一种情况下，联合国需要支援1500 万美元，但在后面一种情况下，人们觉得联合国只需要支援1000 万美元。据说，这个实验从一般的市民到政府官员，屡试不爽。

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简介

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生平

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哲学思想

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主要著作

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后世影响

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客观评价

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名言语录