The theorem according to which, in an eco-comic model involving several people, majority voting does not always generate an equilibrium situation. Let three persons, 1, 2 and 3, sequentially rank three situations, A, B and C, in order of preference. If person 1 ranks situations in the order A, B, C, person 2 - B, C, A, and person 3 - C, A, B, then when a non-strategic decision is made by a majority vote, it turns out that situation A is preferable to situation B, B is preferable to C, and C is preferable to A. Note, however, that this theorem says nothing about the inevitability of such a paradoxical situation and even about its probability, but it is simply stated that it is possible in principle.
Arrow's theorem- the theorem about the impossibility of “collective choice”. Formulated by American economist Kenneth Arrow in 1951.
The meaning of this theorem is that within the framework of the ordinal approach there is no method of combining individual preferences for three or more alternatives that would satisfy some completely fair conditions and would always give a logically consistent result.
The ordinal approach is based on the fact that an individual’s preferences regarding the alternatives offered for choice cannot be measured quantitatively, but only qualitatively, that is, one alternative is worse or better than another.
Within the framework of the cardinalist approach, which assumes the quantitative measurability of preferences, Arrow's theorem does not work in the general case.
AXIOMS of the ordinal approach
1. Axiom of completeness (complete orderliness, comparability). We assume that the economic subject we are studying has a preference relation such that he can compare any two alternatives: " x, yÎХ: XÊ y or yÊ X . If both occur, then y~x. The axiom is quite obvious, saying only that an individual is able to compare any two sets from the existing set; violation of the axiom is possible only in cases where the ranking of alternatives is extremely problematic, and when asked to compare 2 alternatives, the individual answers “I don’t know.” The axiom of completeness may not be satisfied due to the lack of completeness of information from the individual making the decision.
2. Axiom of reflexivity. We can always say that any set from a given set is at least as good as itself: " xÎH: XÊ X. That is, any product set is comparable to itself, it is no worse than itself. What is meant here is the following: let all this be unfolded in time, and today the individual likes this set, therefore, if this axiom is satisfied, then tomorrow the individual will also like this set, i.e. it is impossible to change preferences, since we believe that the relationship has already been determined. Situation of violation of the axiom: the child cannot choose between two absolutely identical objects.
3. Axiom of transitivity. " x, y, zÎХ: XÊ y, yÊ zÞ XÊ z. If a consumer believes that set X is at least as good as set Y, and set Y is at least as good as set Z, then he believes that set X is at least as good as set Z. In practical situations, the property of transitivity turns out to be difficult feasible. In practice, the following plays an important role: for transitivity to hold in reality, it is necessary that the set X was as narrow as possible; the smaller the set, the easier it is for an individual to form a truly transitive preference relation.
4. Axiom of consumer independence. Consumer satisfaction depends only on the amount of goods he consumes and does not depend on the amount of goods consumed by others. The axiom means that the consumer is not familiar with feelings of envy and compassion. This axiom is practically not applicable when analyzing externalities.
Consumer preferences are rational if they have the following two properties: completeness and transitivity.
Question 62.
Let's assume that one way or another we managed to identify public preferences. Unless these preferences are strictly liberalist, maximizing social welfare will require some redistribution of wealth among members of society. Moreover, as has been shown, the maximum social welfare is always, for any social preferences, achieved under conditions of a Pareto-optimal state of the economy. The task of society is, therefore, to redistribute wealth appropriately and at the same time achieve Pareto efficiency.
From a theoretical point of view, this problem can be solved simply. According to the second theorem of welfare economics, it is enough to properly redistribute the initial stock, the wealth that people already have, and then the market will provide a Pareto-efficient state of the economy in conditions of a fair distribution of wealth from a social point of view. The problem is that this requires using a redistribution mechanism that does not reduce economic efficiency. Beginning inventory consists of resources that can be used for sale. And we are talking, of course, not about the redistribution of resources in kind, but about the redistribution of the value of the initial stock.
It is possible to achieve a Pareto-efficient state of the economy only with such a mechanism of income redistribution, when the size of withdrawals (taxes) and subsidies depend on the cost of the initial stock and do not depend on how the resources that make up the initial stock are used. This refers to lump sum taxes and subsidies. For example, this is a tax on land or other types of real estate, which are paid even if the resources are not used at all. Such taxes and subsidies do not affect the amount of income generated from the use of resources and therefore do not encourage less efficient use of those resources.
However, it is usually impossible to determine the cost of the initial inventory in practice. The fact is that for the vast majority of people, the main component of the initial supply is their ability to work, or labor potential. What is this potential, what is its cost, i.e. income that can be obtained by selling all potentially possible quantities of labor on the market? The workers themselves usually do not know this.
Therefore, in practice, redistribution is carried out mainly through taxes and subsidies, the amount of which depends on the size of individual income, i.e. depends on the cost of labor and material resources used (sold on the market). Such taxes and subsidies encourage less intensive use of resources, in particular, a decrease in labor activity. As a result, available, potentially available resources are underutilized, and this indicates Pareto inefficiency.
However, if we consider society in development, we will come to the conclusion that lump-sum taxes and subsidies, strictly tied only to the cost of the initial stock, also create a tendency towards inefficiency. The point is that the amount of resources that makes up the initial stock of individuals can change. People study, improve their skills, and try to increase the amount of material resources at their disposal. Therefore, taxation and subsidies depending on the value of the initial stock would discourage the growth of labor potential and the entire resource base of society in the long term.
Apparently, any system of income redistribution has a disincentive effect on economic entities. This means that the contradiction between efficiency and social justice is, in principle, irremovable. For the sake of a more equitable distribution, efficiency must inevitably be sacrificed. The question is the size of the victim. Preference should be given to methods of income redistribution that have less of a disincentive effect on economic entities and lead to less efficiency losses.
A distortionary tax is a tax, upon the introduction of which an economic entity makes a different decision on the allocation of resources than before the introduction of the tax. A non-distorting tax does not have such an impact on the economy.
Income classification
| Theoretical classification | Budget classification |
| Non-distorting income | Value added tax (domestic goods); Excise taxes (domestic goods); Other taxes on goods and services; Total income tax; Payments for the use of natural resources; Export duties; Income of budget funds; Income of extra-budgetary (non-social) character funds |
| Distorting Income | Personal income tax; Corporate income tax; Payroll taxes, Other income taxes; Property taxes; Income from off-budget social funds. |
| Other income | Import duties; Value added tax on imported goods; Excise taxes on imported goods; Other taxes; Non-tax revenues. |
Lump-sum tax is a tax or fee paid in a fixed amount regardless of the duration and intensity of economic activity and its economic results. It is essentially a regressive tax, since its share in the entrepreneur’s costs falls with increasing sales volume. In Russian legislation, the term “cord tax” is more often used. For example, this type of tax is used by a simplified taxation system for individual entrepreneurs based on a patent (Tax Code of the Russian Federation, Article 346.25.1).
“The essence of this theorem is that any collective choice that satisfies quite reasonable axioms can provide the best alternative only if it contains features of coercion, or dictatorship. Impossibility theorem Arrow very sharply raised the question of the nature of economic science, and with it economic ethics. It is restrictive, because it reveals the limits of the viability of the economy.”
Kanke V.A., Philosophy of science: a short encyclopedic dictionary, M., “Omega-L”, 2008, p. 309.
Kenneth Arrow of Stanford University posed the most general question: Is it possible to create a voting system that is both rational (without contradictions), democratic (one person - one vote) and decisive (allows for a choice)?
Instead of attempting to invent such a system, Arrow proposed a set of requirements, axioms, that the system must satisfy. These axioms were intuitive, acceptable from the point of view of common sense, and could be expressed mathematically in the form of certain conditions.
Based on these axioms, Arrow tried to prove in general terms the existence of a voting system that simultaneously satisfies the three principles listed above: rational, democratic and decisive.
Arrow's first axiom requires that the voting system be general enough to accommodate all possible distributions of popular votes. Intuitively, this requirement is quite obvious. It is impossible to predict the distribution of votes in advance. It is absolutely essential that the system work for all voter preferences. This axiom is called the axiom of universality.
Even more obvious from the point of view of common sense is Arrow’s second axiom: the axiom of unanimity, in accordance with it, it is necessary that the collective choice exactly repeat the unanimous opinion of all voters. If, for example, each voter believes that candidate A is better than candidate B, then the voting system should lead to this result.
Arrow's third axiom is called independence from unrelated alternatives.. Let a voter believe that of a pair of candidates A and B, A is the best. This preference should not depend on the voter’s attitude towards other candidates. The third axiom is quite attractive, but not so obvious from the point of view of everyday human behavior. Thus, one of the works provides a convincing example of violation of this axiom. A restaurant visitor initially compares dish A and B and wants to order A, because preparing dish B requires a highly qualified chef, and, in his opinion, such a cook is unlikely to be available in this restaurant. Suddenly he notices dish C on the menu - very expensive and also requiring high art of preparation. Then he chooses dish B, believing that the cook knows how to cook well.
Arrow's third axiom is often violated by figure skating judges. When giving comparative assessments to two strong singles skaters, they try to take into account the possibility of a good performance by the third strong candidate, leaving him with a chance to become the winner. An excellent performance in free skating by Skater C, who previously had a not very high result in the compulsory program, can affect the scores of Skaters A and B. If A had an excellent result in the compulsory program, judges sometimes rank him lower than Skater B with approximately equal performance in order to improve skater S's chances
Nevertheless, the very possibility of presenting the requirement of independence to the voting system as mandatory is beyond doubt.
Arrow's fourth axiom is called the axiom of completeness: the voting system must compare any pair of candidates to determine which is better. In this case, it is possible to declare two candidates equally attractive. The completeness requirement does not seem too strict for a voting system.
Arrow's fifth axiom is an already familiar condition - transitivity: if, according to voters, candidate B is no better than candidate A (worse or equivalent), candidate C is no better than candidate B, then candidate C is no better than candidate A. A voting system that does not violate transitivity is said to behave in a rational manner.
Having defined five axioms - the desirable properties of a voting system, Arrow proved that systems that satisfy these axioms have a disadvantage that is unacceptable from the point of view of democratic freedoms: each of them is the rule of a dictator - a person who imposes his preferences on all other voters.
The results revealed by Arrow were widely known. They dashed the hopes of many economists, sociologists, and mathematicians to find a perfect voting system. The requirement to exclude the dictator makes it impossible to create a voting system that satisfies all Arrow's axioms.
Therefore, Arrow's result is called the “impossibility theorem.”
Arrow’s theorem (also known as Arrow’s paradox) is a theorem about the impossibility of “collective choice.” Formulated by American economist Kenneth Arrow in 1951.
The meaning of this theorem is that within the framework of the ordinal approach there is no method of combining individual preferences for three or more alternatives that would satisfy some completely fair conditions and would always give a logically consistent result.
The ordinal approach is based on the fact that an individual’s preferences regarding the alternatives offered for choice cannot be measured quantitatively, but only qualitatively, that is, one alternative is worse or better than another.
Within the framework of the cardinalist approach, which assumes the quantitative measurability of preferences, Arrow's theorem does not work in the general case.
ARROW IMPOSSIBILITY THEOREM
(Arrow's impossibility theorem) The theorem according to which, in an economic model involving several people, majority voting does not always generate an equilibrium situation. Let three persons, 1, 2 and 3, sequentially rank three situations, A, by degree of preference. B and C. If person 1 puts situations in the order A, B, C, person 2 - B, C, A, and person 3 - C, A, B, then when a non-strategic decision is made by a majority vote, it turns out that situation A is preferable to situation B, B is preferable to C, and C is preferable to A. Note, however, that this theorem does not say anything about the inevitability of such a paradoxical situation or even about its probability, but only states that it is possible in principle.
Formulations
1951 formulation
Let there be N≥2 voters voting for n≥3 candidates (in terms of decision theory, candidates are usually called alternatives). Each voter has an ordered list of alternatives. The election system is a function that turns a set of N such lists (voting profile) into a common ordered list.
An election system may have the following properties:
Versatility
Monotone
If in all N lists some alternative x remains in place or rises higher, and the order of the others does not change, in the general list x must remain in place or rise higher.
Absence of a dictator
There is no voter whose preference determines the outcome of the election independently of the preferences of other voters.
1963 formulation
In the 1963 formulation, Arrow's conditions are as follows.
Versatility
Absence of a dictator
Independence from outside alternatives
Pareto efficiency, or the principle of unanimity
if each voter has alternative x ranked higher than y in the list, the final result must also be so.
Proof of Arrow's theorem
Let us introduce the following notation:
≻i - preferences of the i-th agent; [≻"] - preference profile (tuple whose elements are the preferences of all agents);
W: Ln → L - social welfare function; ≻W - collective preferences.
Let us denote by O the set of outcomes that each agent ranks in accordance with his preferences.
Let us give formal definitions:
Pareto efficiency
W is Pareto efficient if for any outcomes o1, o2 ∈ O, ∀i (o1 ≻i o2) ⇒ (o1 ≻W o2)
Independence from outside alternatives
W is independent of extraneous alternatives if for any outcomes o1, o2 ∈ O and for any two preference profiles [≻"] and [≻"] ∈ Ln, ∀i (o1 ≻i" o2 ⇔ o1 ≻i" o2) ⇒ ( o1 ≻W([≻"]) o2 ⇔ o1 ≻W([≻"]) o2)
Absence of a dictator
We assume that there is no dictator for W if there is no i such that ∀ o1, o2 ∈ O (o1 ≻i o2 ⇒ o1 ≻W o2)
Arrow's theorem
If |O| ≥ 3, then any Pareto efficient social welfare function W, independent of extraneous alternatives, has a dictator.
We carry out the proof in 4 stages.
Stage 1. Approval
If each agent places outcome b at the very top or bottom of his list of preferences, then in ≻W outcome b will also be either at the top or at the bottom of the list.
Let us take an arbitrary profile [≻] such that for all agents i, outcome b is located either at the top or at the bottom of the preference list ≻i. Now let's assume that our statement is false, i.e. there exist a,c ∈ O such that a ≻W b and b ≻W c. Let us then change the profile [≻] so that c ≻i a is satisfied for all agents, without changing the ranking of the remaining outcomes. Let us denote the resulting profile [≻"]. Since after such a modification the outcome b for each agent will still remain either at the top or at the bottom position in the list of his preferences, then from the independence of W from extraneous alternatives we can conclude that in the new profile a ≻W b and b ≻W c. Therefore, due to the transitivity of ≻W we obtain a ≻W c. But we assumed that for all agents c ≻i a, then due to Pareto efficiency there must be c ≻W a. proves the statement.
Stage 2. Approval
There is an agent who is central in the sense that by changing his vote he can move outcome b from the lowest position in the list ≻W to the highest position in that list.
Consider any preference profile in which all agents have ranked outcome b at the very bottom of their list of preferences ≻i. It is clear that in ≻W outcome b is in the lowest position. Let all agents begin to take turns rearranging outcome b from the lowest to the highest position in their preference lists, without changing the ranking of the remaining outcomes. Let n* be the agent who, by rearranging b in this way, changed ≻W. Let's denote [≻1] the preference profile just before n* moved b, and [≻2] the preference profile just after n* moved b. Thus, in [≻2], outcome b has changed its position in ≻W, and for all agents b is either at the top or at the bottom position ≻i. Therefore, by virtue of the statement proved in Stage 1, in ≻W outcome b occupies the top position.
Stage 3. Approval
Let's choose from a pair
Stage 4. Approval
n* - dictator over all pairs
Let's consider some outcome with. Due to Stage 2, there is some central agent n** for this outcome, who is also the dictator for all pairs
Methodological comments
Topic 10. Identifying individual preferences and public choice
Module 3 PUBLIC CHOICE THEORY
Integrated purpose of the module
This module substantiates the key positions of identifying individual preferences in the system of public choice, reveals the features of the formation of public choice in conditions of representative and direct democracy, and identifies the elements of the economic theory of a democratic state.
Purpose of the module consists in identifying the features of a non-market type of decision-making (based on the action of political institutions), in understanding public choice as a collective way of making decisions regarding the production of public goods and redistribution of income, as well as in substantiating the features of the formation of public choice in conditions of representative and direct democracy and determining essential elements of the economic theory of a democratic state.
Collective choice. Pareto optimum and unanimous decisions. Optimal majority. Simple majority rule. May's theorem. The paradox of voting. Median voter theorem. The nature of decisions and selection procedures. Alternative rules for making collective decisions.
K. Arrow believed that collective decision-making is based on the preferences of the individuals who make up society. It is necessary to find a mechanism according to which the ordinal preferences of individuals will be aggregated into the ordinal preferences of society. It is also important that public preferences reflect any change in individual preferences, i.e. the aggregation mechanism must be perfect. Arrow first raised the problem of the dependence of social welfare on the mechanism for identifying and aggregating individual preferences.
If we assume that society can act as one aggregate individual and make decisions that satisfy each member of society, then there must be criteria for ordering social alternatives. Such logical and ethical criteria that a system of reducing individual preferences into one comprehensive indicator of social welfare must meet were proposed by Arrow in his work “Public Choice and Individual Values.”
Arrow believed that a democratic "social welfare function" linking individual preferences and social choice must meet four requirements:
Transitivity (if social choice A is preferable to choice B, and choice B is preferable to choice C, then choice A is preferable to choice B),
Pareto efficiency (an alternative solution cannot be chosen if there is another feasible alternative that improves the lives of some members of society and does not worsen anyone,
Absence of dictatorship (the choice of individuals between alternatives is independent and does not depend on the preferences of others),
Independence of extraneous alternatives (if an individual chooses between alternatives A and B, then his choice between A and B does not depend on how he evaluates C).
Arrow proved that the four conditions are in conflict, so no social welfare scheme can meet all the requirements simultaneously. The simplest example of Arrow's "impossibility theorem" is known as Condorcet's paradox, named after the famous French mathematician who lived in the 18th century.
a) main:
1. Akhinov A.G., Zhiltsov E.N. Economics of the public sector: Textbook. – M.: INFRA-M, 2008.- 345 p.
2. Economics of the public sector: a textbook for university students studying in the direction of "Economics" and economic specialties / [V. D. Andrianov and others]; edited by Doctor of Economics Sciences P.V. Savchenko, Doctor of Economics. Sciences I. A. Pogosova, Doctor of Economics. Sciences E. N. Zhiltsova; Moscow State University named after M. V. Lomonosova, Institute of Economics RAS. - Moscow: Infra-M, 2010. - 763 p.
3. Economics, organization and management of the public sector: a textbook for university students studying in the direction of “Economics” and economic specialties / N.A. Voskolovich, E.N. Zhiltsov, S.D. Enikeeva; under. ed. N.A. Voskolovich. – M..: UNITY-DANA, 2008. – 367 p.
4. Electronic textbook “Gallery of Economists” Theory of Public Choice http://gallery.economicus.ru/cgi-bin/g_framen.pl?type=school&search=pubchoice -
b) additional:
1. Blumer H. Collective Behavior. Chapt. XIX -XXII / New Outline of the Principles of Sociology. N. Y., 1951. P. 167-221
2. Atkinson E.B.. Stiglitz J.E. Lectures on the economic theory of the public sector: Textbook / Trans. Ed. L.L. Lyubimova. M., 1995. L.10.
3. Ahinov G.A. Fundamentals of public sector economics: Course of lectures. M.: Faculty of Economics of Moscow State University, TEIS, 2003. pp. 122-128.
4. Buchanan JM. The Constitution of Economic Policy. Calculation of consent. Boundaries of freedom. boundaries of politics. economicsTEIS, 1999. Ch. what a theory. t lectures. historical experience, lessons for Russia // Economic Issues. ie Essays. Series: “Nobel Laureates in Economics” T. 1. M.: “Taurus Alpha”, 1997. Ch. 3.4.
5. Milchakova N. Game by the rules: “Public choice” by James Buchanan // Questions of Economics. - 1994.- No. 6.
6. Nureyev R.M. Public choice theory. Course of lectures. M.: ed. House State University Higher School of Economics, 2005. pp. 144-217.
7. Olson M. Logic of collective actions. M., 1995. Ch. 6.
8. Stiglitz J. Yu. Economics of the public sector. M.: MSU, 1997. Ch. 6.
9. Electronic textbook. Henry Bloomer. Collective behavior// http://www.fidel-kastro.ru/sociologia/blumer.htm#16
10. Yakobson L.I. Public sector of the economy: economic theory and policy. M.: State University Higher School of Economics, 2000. Ch. 4.
Arrow's axioms In 1951, Kenneth Arrow from Stanford University wondered about the possibility of creating a voting system that would simultaneously satisfy three principles: rationality (no contradictions, no intransitivity), democracy (one person - one vote) and decidability (allowing for choice) . He did not propose such a system, but Arrow developed a set of requirements, axioms, that it must satisfy. Based on the above axioms, Arrow tried to prove in general terms the existence of a voting system that simultaneously satisfies the three principles listed above. Let's consider these axioms. Axiom 1 - the axiom of universality - requires that the voting system be effective for any possible distribution of votes, for any preferences of voters. Axiom 2 is the axiom of unanimity, according to which the unanimous opinion of all those voting for the choice of a particular candidate should lead to the collective choice of the same candidate. Axiom 3 - the axiom of independence from unrelated alternatives - says that in group ordering the order of certain candidates should not change when voters' attitudes towards other candidates change. Axiom 4 is an axiom of completeness, according to which the voting system must compare any pair of candidates. Axiom 5 - the condition of transitivity suggests that the voting system should not violate the transitivity of relations between voters, there should be no contradictions in it. Having defined the five axioms of the desired voting system, Arrow at the same time showed that systems that satisfy these axioms have an unacceptable drawback from the point of view of democratic freedoms: in order to fulfill the axiomatic requirements, they require the participation of a person (dictator) who imposes his preferences on all other voters. The requirement to exclude the dictator leads to the impossibility of creating a voting system that satisfies all Arrow’s axioms. Therefore, Arrow's result is called the "impossibility theorem."
Under conditions of certainty, the decision maker knows everything about the possible states of the essence of the phenomena influencing the decision, and knows what decision will be made. The decision maker simply selects the strategy, course of action, or project that will produce the greatest return.
In general, the development of decisions under conditions of certainty is aimed at finding the maximum return, either in the form of maximizing benefits (income, profit or utility) or minimizing costs. This search is called optimization analysis. Three optimization methods are used by the decision maker: marginal analysis, linear programming and incremental profit analysis.
Certainty is understood as a state of knowledge when the decision maker knows in advance the specific outcome for each alternative. In other words, the decision maker has comprehensive knowledge of the state of the environment and the results of each possible decision.
Certainty occurs in most arithmetic and algebraic problems, as well as in many linear and nonlinear programming models. Such models are used to find an option for allocating resources that gives the greatest return on a certain indicator (such as profit or cost), or the smallest value of some other criterion (such as costs) under given constraints.
In reality, only a little can remain certain over a sufficiently long time interval. Therefore, strategic decisions are made in conditions that are very far from complete knowledge. Accordingly, they are accepted under conditions of either risk or uncertainty.
Certainty
The decision is made under conditions of certainty, when the manager can accurately determine the result of each alternative decision possible in a given situation. Relatively few organizational or personal decisions are made under conditions of certainty. However, they still occur. In addition, the elements of complex large decisions can be viewed as certain. The level of certainty in decision making depends on the external environment. It increases when there is a strong legal framework that limits the number of alternatives and reduces the level of risk.
