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Monday, June 8, 2020

Politics

From Wikipedia, the free encyclopedia
 
Women voter outreach (1935)
Politics (from Greek: Πολιτικά, politiká, 'affairs of the cities') is the set of activities that are associated with making decisions in groups, or other forms of power relations between individuals, such as the distribution of resources. The academic study of politics is referred to as political science.
Politics is a multifaceted word. It may be used positively in the context of a "political solution" which is compromising and non-violent, or descriptively as "the art or science of government", but also often carries a negative connotation. For example, abolitionist Wendell Phillips declared that "we do not play politics; anti-slavery is no half-jest with us." The concept has been defined in various ways, and different approaches have fundamentally differing views on whether the it should be used extensively or limitedly, empirically or normatively, and on whether conflict or co-operation is more essential to it.

A variety of methods are deployed in politics, which include promoting one's own political views among people, negotiation with other political subjects, making laws, and exercising force, including warfare against adversaries. Politics is exercised on a wide range of social levels, from clans and tribes of traditional societies, through modern local governments, companies and institutions up to sovereign states, to the international level. In modern nation states, people often form political parties to represent their ideas. Members of a party often agree to take the same position on many issues and agree to support the same changes to law and the same leaders. An election is usually a competition between different parties. 

A political system is a framework which defines acceptable political methods within a society. The history of political thought can be traced back to early antiquity, with seminal works such as Plato's Republic, Aristotle's Politics, Chanakya's Arthashastra and Chanakya Niti (3rd Century BCE), as well as the works of Confucius.

Etymology

The English word "politics" derives from the Greek word politiká (Πολιτικά, 'affairs of the cities'), the name of Aristotle's classic work, Politiká. In the mid-15th century, Aristotle's composition would be rendered in Early Modern English as "Polettiques", which would become "Politics" in Modern English

The singular politic first attested in English in 1430, coming from Middle French politique—itself taking from politicus, a Latinization of the Greek πολιτικός (politikos) from πολίτης (polites, 'citizen') and πόλις (polis, 'city').

Definitions

According to Harold Lasswell, politics is "who gets what, when, how".

For David Easton, it is about "the authoritative allocation of values for a society".

To Vladimir Lenin, "politics is the most concentrated expression of economics".

Bernard Crick argued that "politics is a distinctive form of rule whereby people act together through institutionalized procedures to resolve differences, to conciliate diverse interests and values and to make public policies in the pursuit of common purposes".

Adrian Leftwich gives the definition that "Politics comprises all the activities of co-operation, negotiation and conflict within and between societies, whereby people go about organizing the use, production or distribution of human, natural and other resources in the course of the production and reproduction of their biological and social life".

Approaches to politics

There are several ways in which approaching politics has been conceptualized.

Extensive and limited approaches

Adrian Leftwich has differentiated views of politics based on how extensive or limited their perception of what accounts as 'political' is. The extensive view sees politics as present across the sphere of human social relations, while the limited view restricts it to certain contexts. For example, in a more restrictive way, politics may be viewed as primarily about governance, while a feminist perspective could argue that sites which have been viewed traditionally as non-political, should indeed be viewed as political as well. This latter position is encapsulated in the slogan the personal is political, which disputes the distinction between private and public issues. Instead, politics may be defined by the use of power, as has been argued by Robert A. Dahl.

Moralism and realism

Some perspectives on politics view it empirically as an exercise of power, while other see it as a social function with a normative basis. This distinction has been called the difference between political moralism and political realism. For moralists, politics is closely linked to ethics, and is at its extreme in utopian thinking. For example, according to Hannah Arendt, the view of Aristotle was that "to be political . . . meant that everything was decided through words and persuasion and not through violence", while according to Bernard Crick "Politics is the way in which free societies are governed. Politics is politics and other forms of rule are something else". In contrast, for realists, represented by those such as Niccolò Machiavelli, Thomas Hobbes, and Harold Lasswell, politics is based on the use of power, irrespective of the ends being pursued.

Conflict and co-operation

Agonism argues that politics essentially comes down to conflict between conflicting interests. Political scientist Elmer Schattschneider argued that "at the root of all politics is the universal language of conflict", while for Carl Schmitt the essence of politics is the distinction of 'friend' from foe'. This is in direct contrast to the more co-operative views of politics by Aristotle and Crick. However, a more mixed view between these extremes is provided by the Irish author Michael Laver, who noted that "Politics is about the characteristic blend of conflict and co-operation that can be found so often in human interactions. Pure conflict is war. Pure co-operation is true love. Politics is a mixture of both."

History of politics

The history of politics spans human history and is not limited to modern institutions of government.

Prehistoric

Frans de Waal argued that already chimpanzees engage in politics through "social manipulation to secure and maintain influential positions". Early human forms of social organization—bands and tribes—lacked centralized political structures. These are called stateless societies.

State formation

There are a number of different theories and hypotheses regarding early state formation that seek generalizations to explain why the state developed in some places but not others. Other scholars believe that generalizations are unhelpful and that each case of early state formation should be treated on its own.

Voluntary theories contend that diverse groups of people came together to form states as a result of some shared rational interest. The theories largely focus on the development of agriculture, and the population and organizational pressure that followed and resulted in state formation. One of the most prominent theories of early and primary state formation is the hydraulic hypothesis, which contends that the state was a result of the need to build and maintain large-scale irrigation projects.

Conflict theories of state formation regard conflict and dominance of some population over another population as key to the formation of states. In contrast with voluntary theories, these arguments believe that people do not voluntarily agree to create a state to maximize benefits, but that states form due to some form of oppression by one group over others. 

Some theories in turn argue that warfare was critical for state formation.

Early states

In ancient history, civilizations did not have definite boundaries as states have today, and their borders could be more accurately described as frontiers. Early dynastic Sumer, and early dynastic Egypt were the first civilizations to define their borders. Moreover, up to the twentieth century, many people lived in non-state societies. These range from relatively egalitarian bands and tribes to complex and highly stratified chiefdoms.

The first states of sorts were those of early dynastic Sumer and early dynastic Egypt, which arose from the Uruk period and Predynastic Egypt respectively at approximately 3000BCE. Early dynastic Egypt was based around the Nile River in the north-east of Africa, the kingdom's boundaries being based around the Nile and stretching to areas where oases existed. Early dynastic Sumer was located in southern Mesopotamia with its borders extending from the Persian Gulf to parts of the Euphrates and Tigris rivers.

Although state-forms existed before the rise of the Ancient Greek empire, the Greeks were the first people known to have explicitly formulated a political philosophy of the state, and to have rationally analyzed political institutions. Prior to this, states were described and justified in terms of religious myths.

Several important political innovations of classical antiquity came from the Greek city-states and the Roman Republic. The Greek city-states before the 4th century granted citizenship rights to their free population, and in Athens these rights were combined with a directly democratic form of government that was to have a long afterlife in political thought and history.

Modern states

The Peace of Westphalia (1648) is considered by political scientists to be the beginning of the modern international system, in which external powers should avoid interfering in another country's domestic affairs. The principle of non-interference in other countries' domestic affairs was laid out in the mid-18th century by Swiss jurist Emer de Vattel. States became the primary institutional agents in an interstate system of relations. The Peace of Westphalia is said to have ended attempts to impose supranational authority on European states. The "Westphalian" doctrine of states as independent agents was bolstered by the rise in 19th century thought of nationalism, under which legitimate states were assumed to correspond to nations—groups of people united by language and culture.

In Europe, during the 18th century, the classic non-national states were the multinational empires, the Austrian Empire, Kingdom of France, Kingdom of Hungary, the Russian Empire, the Spanish Empire, the Ottoman Empire, the British Empire. Such empires also existed in Asia, Africa and the Americas. In the Muslim world, immediately after Muhammad's death in 632, Caliphates were established which developed into multi-ethnic trans-national empires. The multinational empire was an absolute monarchy ruled by a king, emperor or sultan. The population belonged to many ethnic groups, and they spoke many languages. The empire was dominated by one ethnic group, and their language was usually the language of public administration. The ruling dynasty was usually, but not always, from that group. Some of the smaller European states were not so ethnically diverse, but were also dynastic states, ruled by a royal house. A few of the smaller states survived, such as the independent principalities of Liechtenstein, Andorra, Monaco, and the republic of San Marino.

Most theories see the nation state as a 19th-century European phenomenon, facilitated by developments such as state-mandated education, mass literacy and mass media. However, historians also note the early emergence of a relatively unified state and identity in Portugal and the Dutch Republic. Scholars such as Steven Weber, David Woodward, Michel Foucault and Jeremy Black have advanced the hypothesis that the nation state did not arise out of political ingenuity or an unknown undetermined source, nor was it an accident of history or political invention; but is an inadvertent byproduct of 15th-century intellectual discoveries in political economy, capitalism, mercantilism, political geography, and geography combined together with cartography and advances in map-making technologies.

Some nation states, such as Germany and Italy, came into existence at least partly as a result of political campaigns by nationalists, during the 19th century. In both cases, the territory was previously divided among other states, some of them very small. Liberal ideas of free trade played a role in German unification, which was preceded by a customs union, the Zollverein. National self-determination was a key aspect of United States President Woodrow Wilson's Fourteen points, leading to the dissolution of the Austro-Hungarian Empire and the Ottoman Empire after the First World War, while the Russian Empire became the Soviet Union after the Russian Civil War. Decolonization lead to the creation of new nation states in place of multinational empires in the third world.

Globalization

Political globalization began in the 20th century through intergovernmental organizations and supranational unions. The League of Nations was founded after World War I, and after World War II it was replaced by the United Nations. Various international treaties have been signed through it. Regional integration has been pursued by the African Union, ASEAN, the European Union, and Mercosur. International political institutions on the international level include the International Criminal Court, the International Monetary Fund, and the World Trade Organization.

Study of politics

The study of politics is called political science, or politology. It comprises numerous subfields, including comparative politics, political economy, international relations, political philosophy, public administration, public policy, and political methodology. Furthermore, political science is related to, and draws upon, the fields of economics, law, sociology, history, philosophy, geography, psychology/psychiatry, anthropology and neurosciences.

Comparative politics is the science of comparison and teaching of different types of constitutions, political actors, legislature and associated fields, all of them from an intrastate perspective. International relations deals with the interaction between nation-states as well as intergovernmental and transnational organizations. Political philosophy is more concerned with contributions of various classical and contemporary thinkers and philosophers.




Political science is methodologically diverse and appropriates many methods originating in psychology, social research and cognitive neuroscience. Approaches include positivism, interpretivism, rational choice theory, behavioralism, structuralism, post-structuralism, realism, institutionalism, and pluralism. Political science, as one of the social sciences, uses methods and techniques that relate to the kinds of inquiries sought: primary sources such as historical documents and official records, secondary sources such as scholarly journal articles, survey research, statistical analysis, case studies, experimental research, and model building.

Aspects of politics

Political system

Systems view of politics.

The political system defines the process for making official government decisions. It is usually compared to the legal system, economic system, cultural system, and other social systems. According to David Easton, "A political system can be designated as the interactions through which values are authoritatively allocated for a society".  Each political system is embedded in a society with its own political culture, and they in turn shape their societies through public policy. The interactions between different political systems are the basis for global politics.

Forms of government

Forms of government can be classified by several ways. The source of power determines the difference between democracies, oligarchies, and autocracies. In terms of the structure of power, there are monarchies (including constitutional monarchies) and republics (usually presidential, semi-presidential, or parliamentary). In terms of level of vertical integration, they can be divided into (from least to most integrated) confederations, federations, and unitary States. The separation of powers describes the degree of horizontal integration between the legislature, the executive, the judiciary, and other independent institutions. 

In a democracy, political legitimacy is based on popular sovereignty. Forms of democracy include representative democracy, direct democracy, and demarchy. These are separated by the way decisions are made, whether by elected representatives, referenda, or by citizen juries. Democracies can be either republics or constitutional monarchies. 


Autocracies are either dictatorships (including military dictatorships) or absolute monarchies

  Federal states
  No government
The pathway of regional integration or separation

A federation (also known as a federal state) is a political entity characterized by a union of partially self-governing provinces, states, or other regions under a central federal government (federalism). In a federation, the self-governing status of the component states, as well as the division of power between them and the central government, is typically constitutionally entrenched and may not be altered by a unilateral decision of either party, the states or the federal political body. Federations were formed first in Switzerland, then in the United States in 1776, in Canada in 1867 and in Germany in 1871 and in 1901, Australia. Compared to a federation, a confederation has less centralized power.

The state

All the above forms of government are variations of the same basic polity, the sovereign state. The state has been defined by Max Weber as a political entity that has monopoly on violence within its territory, while the Montevideo Convention holds that states need to have a defined territory; a permanent population; a government; and a capacity to enter into international relations. 

A stateless society is a society that is not governed by a state. In stateless societies, there is little concentration of authority; most positions of authority that do exist are very limited in power and are generally not permanently held positions; and social bodies that resolve disputes through predefined rules tend to be small. Stateless societies are highly variable in economic organization and cultural practices.

While stateless societies were the norm in human prehistory, few stateless societies exist today; almost the entire global population resides within the jurisdiction of a sovereign state. In some regions nominal state authorities may be very weak and wield little or no actual power. Over the course of history most stateless peoples have been integrated into the state-based societies around them.

Some political philosophies consider the state undesirable, and thus consider the formation of a stateless society a goal to be achieved. A central tenet of anarchism is the advocacy of society without states. The type of society sought for varies significantly between anarchist schools of thought, ranging from extreme individualism to complete collectivism. In Marxism, Marx's theory of the state considers that in a post-capitalist society the state, an undesirable institution, would be unnecessary and wither away. A related concept is that of stateless communism, a phrase sometimes used to describe Marx's anticipated post-capitalist society.

Constitutions

Legislatures are an important political institution. Pictured is the Parliament of Finland.

Constitutions are written documents that specify and limit the powers of the different branches of government. Although a constitution is a written document, there is also an unwritten constitution. The unwritten constitution is continually being written by the legislative and judiciary branch of government; this is just one of those cases in which the nature of the circumstances determines the form of government that is most appropriate. England did set the fashion of written constitutions during the Civil War but after the Restoration abandoned them to be taken up later by the American Colonies after their emancipation and then France after the Revolution and the rest of Europe including the European colonies.

Constitutions often set out separation of powers, dividing the government into the executive, the legislature, and the judiciary (together referred to as the trias politica), in order to achieve checks and balances within the state. Additional independent branches may also be created, including civil service commissions, election commissions, and supreme audit institutions.

Political culture

Inglehart-Weltzel cultural map of countries.
 
Political culture describes how culture impacts politics. Every political system is embedded in a particular political culture. Lucian Pye's definition is that "Political culture is the set of attitudes, beliefs, and sentiments, which give order and meaning to a political process and which provide the underlying assumptions and rules that govern behavior in the political system".

Trust is a major factor in political culture, as its level determines the capacity of the state to function. Postmaterialism is the degree to which a political culture is concerned with issues which are not of immediate physical or material concern, such as human rights and environmentalism. Religion has also an impact on political culture.

Levels of politics

Macropolitics

Macropolitics describes political issues which affect the entire political system (e.g. the nation-state) or which relate to interactions between political systems (e.g. international relations).

Global (or world) politics covers all aspects of politics which affect multiple political systems, in practice meaning any political phenomenon crossing national borders. This may include cities, nation-states, multinational corporations, non-governmental organizations, or international organizations. An important element is international relations. The relations between nation-states may be peaceful, when they are conducted through diplomacy, or violent, which is described as war. States which are able to exert strong international influence are referred to as superpowers, while less powerful ones may be called regional or middle powers. The international system of power is called the world order, and it is affected by the balance of power which affects the degree of polarity in the system. Emerging powers are potentially destabilizing to it. 

Politics inside the limits of political systems, which in contemporary context correspond to national borders, are referred to as domestic politics. This includes most forms of public policy, such as social policy, economic policy, or law enforcement, which are executed by the state bureaucracy.

Mesopolitics

Mesopolitics describes the politics of intermediary structures within the political system, such as national political parties or movements.

A political party is a political organization that typically seeks to attain and maintain political power within government, usually by participating in political campaigns, educational outreach or protest actions. Parties often espouse an expressed ideology or vision bolstered by a written platform with specific goals, forming a coalition among disparate interests.

Political parties within a particular political system together form the party system. This may be a multiparty system, a two-party system, a dominant-party system, or a one-party system, depending on the level of pluralism. This is affected by characteristics of the political system, including its electoral system. According to Duverger's law, first-past-the-post systems are likely to lead to two-party systems, while proportional representation systems are more likely to create a multiparty system.

Micropolitics

Micropolitics describes the actions of individual actors within the political system. This is often described as political participation.

Political participation may take many forms, including:

Political corruption

Political corruption is the use of powers by government officials or their network contacts for illegitimate private gain. Forms of political corruption include bribery, cronyism, nepotism, and political patronage. Forms of political patronage in turn includes clientelism, earmarking, political machines, pork barreling, slush funds, and spoils systems.

A political system which operates for corrupt ends may be called a political machine.

When corruption is embedded in political culture, this may be referred to as patrimonialism or neopatrimonialism.

A form of government which is built on corruption is called a kleptocracy (rule of thieves).

Political ideas

Two-axis political compass chart with a horizontal socio-economic axis and a vertical socio-cultural axis and ideologically representative political colours, an example for a frequently used model of the political spectrum
 
Three axis model of political ideologies with both moderate and radical versions and the goals of their policies

Equality

Equality is a state of affairs in which all people within a specific society or isolated group have the same status in possibly all respects, possibly including civil rights, freedom of speech, property rights and equal access to certain social goods and social services. However, it may also include health equality, economic equality and other social securities. Social equality requires the absence of legally enforced social class or caste boundaries and the absence of discrimination motivated by an inalienable part of a person's identity. For example, sex, gender, ethnicity, age, sexual orientation, origin, caste or class, income or property, language, religion, convictions, opinions, health or disability must absolutely not result in unequal treatment under the law and should not reduce opportunities unjustifiably.

Left–right spectrum

Political analysts and politicians divide politics into left wing and right wing politics, often also using the idea of center politics as a middle path of policy between the right and left. This classification is comparatively recent (it was not used by Aristotle or Hobbes, for instance), and dates from the French Revolution era, when those members of the National Assembly who supported the republic, the common people and a secular society sat on the left and supporters of the monarchy, aristocratic privilege and the Church sat on the right.

The meanings behind the labels have become more complicated over the years. A particularly influential event was the publication of the Communist Manifesto by Karl Marx and Friedrich Engels in 1848. The Manifesto suggested a course of action for a proletarian revolution to overthrow the bourgeois society and abolish private property, in the belief that this would lead to a classless and stateless society.

The meaning of left-wing and right-wing varies considerably between different countries and at different times, but generally speaking, it can be said that the right wing often values tradition and inequality while the left wing often values progress and egalitarianism, with the center seeking a balance between the two such as with social democracy, libertarianism or regulated capitalism.

According to Norberto Bobbio, one of the major exponents of this distinction, the Left believes in attempting to eradicate social inequality—believing it to be unethical or unnatural while the Right regards most social inequality as the result of ineradicable natural inequalities, and sees attempts to enforce social equality as utopian or authoritarian. Some ideologies, notably Christian Democracy, claim to combine left and right wing politics; according to Geoffrey K. Roberts and Patricia Hogwood, "In terms of ideology, Christian Democracy has incorporated many of the views held by liberals, conservatives and socialists within a wider framework of moral and Christian principles." Movements which claim or formerly claimed to be above the left-right divide include Fascist Terza Posizione economic politics in Italy and Peronism in Argentina.

Freedom

Political freedom (also known as political autonomy or political agency) is a central concept in political thought and one of the most important features of democratic societies. Political freedom was described as freedom from oppression or coercion, the absence of disabling conditions for an individual and the fulfillment of enabling conditions, or the absence of life conditions of compulsion, e.g. economic compulsion, in a society. Although political freedom is often interpreted negatively as the freedom from unreasonable external constraints on action, it can also refer to the positive exercise of rights, capacities and possibilities for action and the exercise of social or group rights.

Authoritarianism and libertarianism

Authoritarianism and libertarianism disagree the amount of individual freedom each person possesses in that society relative to the state. One author describes authoritarian political systems as those where "individual rights and goals are subjugated to group goals, expectations and conformities," while libertarians generally oppose the state and hold the individual as sovereign. In their purest form, libertarians are anarchists, who argue for the total abolition of the state, of political parties and of other political entities, while the purest authoritarians are, by definition, totalitarians who support state control over all aspects of society.

For instance, classical liberalism (also known as laissez-faire liberalism) is a doctrine stressing individual freedom and limited government. This includes the importance of human rationality, individual property rights, free markets, natural rights, the protection of civil liberties, constitutional limitation of government, and individual freedom from restraint as exemplified in the writings of John Locke, Adam Smith, David Hume, David Ricardo, Voltaire, Montesquieu and others. According to the libertarian Institute for Humane Studies, "the libertarian, or 'classical liberal,' perspective is that individual well-being, prosperity, and social harmony are fostered by 'as much liberty as possible' and 'as little government as necessary.'" For anarchist political philosopher L. Susan Brown (1993), "liberalism and anarchism are two political philosophies that are fundamentally concerned with individual freedom yet differ from one another in very distinct ways. Anarchism shares with liberalism a radical commitment to individual freedom while rejecting liberalism's competitive property relations."

Information-based complexity

From Wikipedia, the free encyclopedia
 
Information-based complexity (IBC) studies optimal algorithms and computational complexity for the continuous problems which arise in physical science, economics, engineering, and mathematical finance. IBC has studied such continuous problems as path integration, partial differential equations, systems of ordinary differential equations, nonlinear equations, integral equations, fixed points, and very-high-dimensional integration. All these problems involve functions (typically multivariate) of a real or complex variable. Since one can never obtain a closed-form solution of the problems of interest one has to settle for a numerical solution. Since a function of a real or complex variable cannot be entered into a digital computer, the solution of continuous problems involves partial information. To give a simple illustration, in the numerical approximation of an integral, only samples of the integrand at a finite number of points are available. In the numerical solution of partial differential equations the functions specifying the boundary conditions and the coefficients of the differential operator can only be sampled. Furthermore, this partial information can be expensive to obtain. Finally the information is often contaminated by noise.

The goal of information-based complexity is to create a theory of computational complexity and optimal algorithms for problems with partial, contaminated and priced information, and to apply the results to answering questions in various disciplines. Examples of such disciplines include physics, economics, mathematical finance, computer vision, control theory, geophysics, medical imaging, weather forecasting and climate prediction, and statistics. The theory is developed over abstract spaces, typically Hilbert or Banach spaces, while the applications are usually for multivariate problems.

Since the information is partial and contaminated, only approximate solutions can be obtained. IBC studies computational complexity and optimal algorithms for approximate solutions in various settings. Since the worst case setting often leads to negative results such as unsolvability and intractability, settings with weaker assurances such as average, probabilistic and randomized are also studied. A fairly new area of IBC research is continuous quantum computing.

Overview

We illustrate some important concepts with a very simple example, the computation of
For most integrands we can't use the fundamental theorem of calculus to compute the integral analytically; we have to approximate it numerically. We compute the values of at n points
The n numbers are the partial information about the true integrand We combine these n numbers by a combinatory algorithm to compute an approximation to the integral. See the monograph Complexity and Information for particulars.

Because we have only partial information we can use an adversary argument to tell us how large n has to be to compute an -approximation. Because of these information-based arguments we can often obtain tight bounds on the complexity of continuous problems. For discrete problems such as integer factorization or the travelling salesman problem we have settle for conjectures about the complexity hierarchy. The reason is that the input is a number or a vector of numbers and can thus be entered into the computer. Thus there is typically no adversary argument at the information level and the complexity of a discrete problem is rarely known.

The univariate integration problem was for illustration only. Significant for many applications is multivariate integration. The number of variables is in the hundreds or thousands. The number of variables may even be infinite; we then speak of path integration. The reason that integrals are important in many disciplines is that they occur when we want to know the expected behavior of a continuous process. See for example, the application to mathematical finance below.

Assume we want to compute an integral in d dimensions (dimensions and variables are used interchangeably) and that we want to guarantee an error at most for any integrand in some class. The computational complexity of the problem is known to be of order (Here we are counting the number of function evaluations and the number of arithmetic operations so this is the time complexity.) This would take many years for even modest values of The exponential dependence on d is called the curse of dimensionality. We say the problem is intractable.

We stated the curse of dimensionality for integration. But exponential dependence on d occurs for almost every continuous problem that has been investigated. How can we try to vanquish the curse? There are two possibilities:
  • We can weaken the guarantee that the error must be less than (worst case setting) and settle for a stochastic assurance. For example, we might only require that the expected error be less than (average case setting.) Another possibility is the randomized setting. For some problems we can break the curse of dimensionality by weakening the assurance; for others, we cannot. There is a large IBC literature on results in various settings; see Where to Learn More below.
  • We can incorporate domain knowledge. See An Example: Mathematical Finance below.

An example: mathematical finance

Very high dimensional integrals are common in finance. For example, computing expected cash flows for a collateralized mortgage obligation (CMO) requires the calculation of a number of dimensional integrals, the being the number of months in years. Recall that if a worst case assurance is required the time is of order time units. Even if the error is not small, say this is time units. People in finance have long been using the Monte Carlo method (MC), an instance of a randomized algorithm. Then in 1994 a research group at Columbia University (Papageorgiou, Paskov, Traub, Woźniakowski) discovered that the quasi-Monte Carlo (QMC) method using low discrepancy sequences beat MC by one to three orders of magnitude. The results were reported to a number of Wall Street finance to considerable initial skepticism. The results were first published by Paskov and Traub, Faster Valuation of Financial Derivatives, Journal of Portfolio Management 22, 113-120. Today QMC is widely used in the financial sector to value financial derivatives. 

These results are empirical; where does computational complexity come in? QMC is not a panacea for all high dimensional integrals. What is special about financial derivatives? Here's a possible explanation. The dimensions in the CMO represent monthly future times. Due to the discounted value of money variables representing times for in the future are less important than the variables representing nearby times. Thus the integrals are non-isotropic. Sloan and Woźniakowski introduced the very powerful idea of weighted spaces which is a formalization of the above observation. They were able to show that with this additional domain knowledge high dimensional integrals satisfying certain conditions were tractable even in the worst case! In contrast the Monte Carlo method gives only a stochastic assurance. See Sloan and Woźniakowski When are Quasi-Monte Carlo Algorithms Efficient for High Dimensional Integration? J. Complexity 14, 1-33, 1998. For which classes of integrals is QMC superior to MC? This continues to be a major research problem.

Brief history

Precursors to IBC may be found in the 1950s by Kiefer, Sard, and Nikolskij. In 1959 Traub had the key insight that the optimal algorithm and the computational complexity of solving a continuous problem depended on the available information. He applied this insight to the solution of nonlinear equations which started the area of optimal iteration theory. This research was published in the 1964 monograph Iterative Methods for the Solution of Equations.
 
The general setting for information-based complexity was formulated by Traub and Woźniakowski in 1980 in A General Theory of Optimal Algorithms

Introduction to entropy

From Wikipedia, the free encyclopedia https://en.wikipedia.org/wiki/Introduct...