The paradox of the public choice theory was first described by the Marquis Condorcet in 1785, which was successfully generalized in the 50s of the last century by the American economist K. Arrow. Arrow's theorem answers a very simple question in collective decision theory. Let's say there are multiple choices in politics, public projects, or income distribution, and there are people whose preferences determine those choices.
The question is what procedures are in place to qualitatively determine choice. And how to learn about preferences, about the collective or social ordering of alternatives, from best to worst. Arrow's answer to this question surprised many.
Arrow's theorem says that there are no such procedures at all - in any case, they do not correspond to certain and quite reasonable preferences of people. Arrow's technical framework, in which he gave clear meaning to the problem of social contracting, and his rigorous response are now widely used to study problems in social economics. The theorem itself formed the basis of modern public choice theory.
Public Choice Theory
Arrow's theorem shows that if voters have at least three alternatives, then there is no electoral system that could transform the choice of individuals into public opinion.
The shocking statement came from economist and Nobel laureate Kenneth Joseph Arrow, who demonstrated this paradox in his Ph. D. thesis and popularized it in his 1951 book Social Choice and Individual Values. The title of the original article is "Difficulties in the Social Security Concept".
Arrow's theorem states that it is impossible to design an electoral system with order that would always meet fair criteria:
- When a voter chooses the alternative X over Y, then the community of voters will prefer X over Y. If the choices of each of voters X and Y remain unchanged, then the choice of society X and Y will be the same even if voters choose other pairs of X and Z, Y and Z, or Z and W.
- There is no "dictator of choice" because one voter cannot influence the choice of a group.
- Existing electoral systems do not cover the required requirements as they provide more information than ordinal rank.
State social management systems
Although the American economist Kenneth Arrow received the Nobel Prize in Economics, the work was more useful for the development of the social sciences, since Arrow's "Impossibility Theorem" marked the beginning of a completely new direction in economics - social choice. This industry is trying to mathematically analyze the adoption of joint decisions, in particular in the field of public social management systems.
Choice is democracy in action. People go to the polls and express their preferences, and in the end, the preferences of many people must come together to make a joint decision. This is why the choice of voting method is very important. But is there really a perfect vote? According to the results of Arrow's theory, obtained in 1950, the answer is no. If "ideal" means a preferential voting method that meets the criteria defined by reasonable voting methods.
The preferred voting method is ranking, where voters rate all candidates according to their preferences, and based on these ratings, the result is: another list of all candidates to be submitted by the common will of the people. According to Arrow's Impossibility Theorem, a reasonable voting method can be specified:
- No dictators (ND) - the result does not always have to match the assessment of one particular person.
- Pareto Efficiency (PE) - if each voter prefers candidate A to candidate B, then the result should indicatecandidate A over candidate B.
- Independence of Incompatible Alternatives (IIA) is the relative score of candidates A, B and should not change if voters change the score of other candidates, but do not change their relative scores of A and B.
Under Arrow's theorem, it turns out that in the case of elections with three or more criteria, there are no social choice functions that would simultaneously fit for ND, PE and IIA.
Rational selection system
The need for preference aggregation manifests itself in many areas of human life:
- Welfare economics uses microeconomic methods to measure welfare at the aggregate economic level. A typical methodology starts by deriving or inferring a welfare function, which can then be used to rank economically sound allocations of resources in terms of welfare. In this case, states are trying to find an economically viable and sustainable outcome.
- In decision theory, when a person must make a rational choice based on several criteria.
- In electoral systems, which are mechanisms to find a single solution from the preferences of many voters.
Under the conditions of the Arrow theorem, the order of preferences for a given set of parameters (results) is distinguished. Each unit in society, or each decision criterion, assigns a certain order of preference with respect to a set of outcomes. Society is looking for a systemrating-based voting, called the welfare function.
This preference aggregation rule transforms a preference profile set into one global public order. Arrow's statement states that if a governing body has at least two voters and three selection criteria, it is impossible to create a welfare function that will satisfy all of these conditions at once.
For each set of individual voter preferences, the welfare function must perform a unique and comprehensive public selection rating:
- This should be done in such a way that the result is a complete assessment of the audience's preferences.
- Should deterministically give the same score when voters' preferences appear to be the same.
Independence from Irrelevant Alternatives (IIA)
The choice between X and Y is connected solely with the individual's preferences between X and Y - this is independence in pairs (pairwise independence), according to Arrow's "Impossibility of Democracy" theorem. At the same time, a change in a person's assessment of irrelevant alternatives located outside of such groups does not affect the social assessment of this subset. For example, submitting a third candidate in a two-candidate election has no effect on the outcome of the election unless the third candidate wins.
Society is characterized by monotony and a positive combination of social and individual values. If a person changes their order of preference by promoting a certain option, then the ordersociety's preferences should correspond to the same option without change. A person should not be able to hurt an option by pricing it higher.
In the impossibility theorem, efficiency and justice in society are ensured through the sovereignty of the citizen. Every possible social order of preference must be achievable with some set of individual preference orders. This means that the welfare function is surjective - it has an unlimited target space. A later (1963) version of Arrow's theorem replaced the monotonicity and non-overlapping criteria.
Pareto. Efficiency or unanimity?
If each person prefers a particular option to another, then the order of social preference should also do so. It is essential that the welfare function be minimally sensitive to the preference profile. This later version is more general and has somewhat weaker conditions. The axioms of uniformity, no overlap, together with IIA, denote Pareto efficiency. At the same time, it does not imply IIA overlap and does not imply monotonicity.
IIA has three purposes:
- Standard. Irrelevant alternatives should not matter.
- Practical. Use of minimal information.
- Strategic. Providing the right incentives to truly identify individual preferences. Although Strategic Objective is conceptually different from IIA, they are closely related.
Pareto efficiency, named after the Italian economist and political scientist Vilfredo Pareto (1848-1923), is used in neoclassical economics along with the theoretical concept of perfect competition as a benchmark for evaluating the efficiency of real markets. It should be noted that none of the results are achieved outside of economic theory. Hypothetically, if perfect competition existed and resources were used as efficiently as possible, then everyone would have the highest standard of living, or Pareto efficiency.
In practice, it is impossible to take any social action, such as a change in economic policy, without worsening the situation of at least one person, so the concept of Pareto improvement has found wider application in economics. A Pareto improvement occurs when a change in distribution harms no one and helps at least one person, given the initial distribution of goods to a group of people. The theory suggests that Pareto improvements will continue to add value to the economy until a Pareto equilibrium is reached, when no more improvements can be made.
Formal statement of the theorem
Let A be the result set, N the number of voters or decision criteria. Denote the set of all complete linear orderings from A to L (A). The strict social security function (preference aggregation rule) is a function that aggregates the preferences of voters in a one-time order of preference byA.
N - a tuple (R 1, …, R N) ∈ L (A) N of voters' preferences is called a preference profile. In its strongest and simplest form, Arrow's impossibility theorem states that whenever the set of possible alternatives A has more than 2 elements, the following three conditions become inconsistent:
- Unanimity, or weak Pareto efficiency. If alternative A ranks strictly above B for all orders R 1, …, R N, then A ranks strictly above B on F (R 1, R 2, …, R N). At the same time, unanimity implies the absence of imposition.
- Non-dictatorship. There is no individual "I" whose strict preferences always prevail. That is, there is no I ∈ {1, …, N }, which for all (R 1, …, R N) ∈ L (A) N, ranks strictly higher than B from R. "I" ranks strictly higher than B over F (R 1, R 2, …, R N), for all A and B.
- Independence from irrelevant alternatives. For two preference profiles (R 1, …, R N) and (S 1, …, S N) such that for all individuals I, alternatives A and B have the same order in R i as in S i, alternatives A and B have the same order in F (R 1, R 2, …, R N) as in F (S 1, S2, …, S N).
Interpretation of the theorem
Although the Impossibility Theorem is mathematically proven, it is often expressed in a non-mathematical way with the statement that no voting method is fair, every ranked voting method has flaws, or the only voting method that is not wrong is a dictatorship. These statements are a simplificationArrow's result, which is not always considered correct. Arrow's theorem states that a deterministic preferential voting mechanism, i.e. one in which the order of preference is the only information in voting, and any possible set of votes produces a unique result, cannot satisfy all the conditions above at the same time.
Various theorists have suggested relaxing the IIA criterion as a way out of the paradox. Proponents of rating methods argue that the IIA is an unnecessarily strong criterion that is violated in most useful electoral systems. Proponents of this position point out that failure to meet the standard IIA criterion is trivially implied by the possibility of cyclic preferences. If voters vote like this:
- 1 vote for A> B> C;
- 1 vote for B> C> A;
- 1 vote for C> A> B.
Then the majority doubles group preference is that A beats B, B beats C, and C beats A, and this results in a scissors-rock-scissors preference for any pair comparison.
In this case, any aggregation rule that satisfies the basic majority requirement that the candidate with the most votes must win the election will fail the IIA criterion if social preferences must be transitive or acyclic. To see this, it is assumed that such a rule satisfies the IIA. Since the preferences of the majorityare observed, society favors A - B (two votes for A> B and one for B> A), B - C and C - A. Thus, a cycle is created that contradicts the assumption that social preferences are transitive.
So, Arrow's theorem does indeed show that any electoral system with the most wins is a non-trivial game, and that game theory should be used to predict the outcome of most voting mechanisms. This can be seen as a discouraging result because the game should not have efficient equilibria, for example, voting might lead to an alternative that no one really wanted but everyone voted for.
Social choice instead of preference
Rational collective choice of voting mechanism according to Arrow's theorem is not the goal of social decision making. Often it is enough to find some alternative. The alternative choice-focused approach explores either social choice functions that map each preference profile, or social choice rules, functions that map each preference profile to a subset of alternatives.
As for social choice functions, the Gibbard-Satterthwaite theorem is well known, which states that if a social choice function whose range contains at least three alternatives is strategically stable, then it is dictatorial. Considering the rules of social choice, they believe that social preferences stand behind them.
That is, they consider the rule as a choicemaximum elements - the best alternatives to any social preference. The set of maximal social preference elements is called the core. The conditions for the existence of an alternative in the core were studied in two approaches. The first approach assumes that preferences are at least acyclic, which is necessary and sufficient for preferences to have a maximum element in any finite subset.
For this reason, it is closely related to relaxing transitivity. The second approach drops the assumption of acyclic preferences. Kumabe and Mihara adopted this approach. They made the more consistent assumption that individual preferences matter the most.
Relative risk aversion
There are several indicators of risk aversion expressed by the utility function in Arrow Pratt's theorem. Absolute risk aversion - the higher the curvature u(c), the higher the risk aversion. However, since the expected utility functions are not uniquely defined, the necessary measure remains constant with respect to these transformations. One such measure is the Arrow-Pratt measure of absolute risk aversion (ARA), after economists Kenneth Arrow and John W. Pratt defined the absolute risk aversion ratio as
A (c)=- {u '' (c)}/ {u '(c)}, where: u '(c) and u '' (c) denote the first and second derivatives with respect to "c" of "u (c)".
Experimental and empirical data are generally consistent with a reduction in absolute risk aversion. relative measureArrow Pratt Risk Aversion (ACR) or Relative Risk Aversion Ratio is defined by:
R (c)=cA (c)={-cu '' (c)} /{u '(c) R (c).
As with absolute risk aversion, the respective terms used are constant relative risk aversion (CRRA) and decreasing/increasing relative risk aversion (DRRA/IRRA). The advantage of this quantity is that it is still a valid measure of risk aversion even if the utility function changes from risk propensity, i.e. utility is not strictly convex/concave across all "c". A constant RRA implies a reduction in the ARA of Arrow Pratt's theory, but the reverse is not always true. As a specific example of constant relative risk aversion, the utility function: u(c)=log(c), implies RRA=1.
Left graph: the risk-avoiding utility function is concave from below, and the risk-averse utility function is convex. The middle graph - in the space of expected standard deviation values, the risk indifference curves slope upwards. Right plot - with fixed probabilities of the two alternative states 1 and 2, the risk-averse indifference curves over state-dependent outcome pairs are convex.
Nominal Electoral System
Initially, Arrow rejected cardinal utility as an important tool for expressing social welfare, so he concentrated his claims on ranking preferences, but laterconcluded that a cardinal rating system with three or four classes is probably the best. According to the impossibility theorem, public choice assumes that individual and social preferences are ordered, that is, satisfaction with completeness and transitivity in various alternatives. This means that if preferences are represented by a utility function, its value is useful in the sense that it makes sense, since a higher value means a better alternative.
Practical applications of the theorem is used to evaluate broad categories of voting systems. Arrow's main argument argues that order voting systems must always violate at least one of the fairness criteria he outlined. The practical implication of this is that voting systems that are not in order need to be studied. For example, ranking voting systems where voters give each candidate a score can meet all of Arrow's criteria.
In fact, the voting mechanism, Arrow's Theorem rational collective choice and subsequent dialogue, was incredibly misleading in the field of voting. It is often believed by students and non-specialists that no voting system can meet Arrow's fairness criteria when, in fact, rating systems can and do meet all of Arrow's criteria.
