# Pareto optimum

Pareto optima (red) of a two-dimensional set of values ​​(blue). Such a Pareto front does not have to be continuous - it can have interruptions.
Pareto-efficient bundles of goods are on the production possibilities curve . An additional unit cannot be produced from either of the two goods if the production of the other good is not to be restricted.

A Pareto optimum (also Pareto efficient state ) is a (best possible) state in which it is not possible to improve one (target) property without having to worsen another at the same time.

The Pareto optimum is named after the economist and sociologist Vilfredo Pareto (1848–1923).

The set of all Pareto optima is also called the Pareto set (also Pareto front ). The Pareto criterion is the assessment of whether a condition improves by changing one target value ( Pareto superiority ) without even having to worsen another target value. Vilfredo Pareto did not originally refer to target values ​​/ characteristics / criteria (sometimes also called “characteristics”), but to individuals . With respect to individuals, a Pareto-optimal (Pareto-efficient) state denotes a state in which there is no way of improving one individual without simultaneously making another worse off.

Expressed mathematically, the - tuple is a Pareto optimum (here: maximum) of a set of -Tuples, if there is no other -Tuple that is at least as good in all parameters and really better in one, i.e. . h if there is no other tuples in there, so for all true: and for at least one applies: . ${\ displaystyle n}$ ${\ displaystyle x = (x_ {1}, x_ {2}, \ dotsc, x_ {n})}$${\ displaystyle A}$${\ displaystyle n}$${\ displaystyle A}$${\ displaystyle n}$${\ displaystyle n}$${\ displaystyle y = (y_ {1}, y_ {2}, \ dotsc, y_ {n})}$${\ displaystyle A}$${\ displaystyle i = 1,2, \ dotsc, n}$${\ displaystyle y_ {i} \ geq x_ {i}}$${\ displaystyle i}$${\ displaystyle y_ {i}> x_ {i}}$

Solving the problem of finding Pareto optima is called Pareto optimization .

## Definition according to set theory

Arbitrary quantities and the associated index quantity are given , where applies. Furthermore, let us now be a set of -Tuples. For the individual elements of any two tuples , a total order is given by the relation . With the index and the respective -th tuple elements and formally means that this is a true statement . In addition, at least two such tuple elements exist in their entirety , so that they are subject to a strict total order through the relation . This means that they satisfy the above total order, but may not be the same. If all these requirements are met and all elements are selected as described above, the following definition can now be made using the first-level predicate logic : ${\ displaystyle n}$${\ displaystyle A_ {1}, A_ {2}, \ dots, A_ {n}}$${\ displaystyle I: = \ {1,2, \ dots, n \}}$${\ displaystyle n \ in \ mathbb {N}}$${\ displaystyle A \ subseteq A_ {1} \ times A_ {2} \ times \ dots \ times A_ {n}}$${\ displaystyle n}$${\ displaystyle n}$${\ displaystyle x, y \ in A}$${\ displaystyle \ geq}$${\ displaystyle i \ in I}$${\ displaystyle i}$${\ displaystyle x_ {i} \ in x}$${\ displaystyle y_ {i} \ in y}$${\ displaystyle \ forall i: ((x_ {i} \ geq y_ {i}) \ vee (y_ {i} \ geq x_ {i}))}$${\ displaystyle A}$${\ displaystyle>}$

${\ displaystyle x}$is Pareto optimum . ${\ displaystyle: \ Leftrightarrow \ neg (\ exists y: (\ forall i: y_ {i} \ geq x_ {i}) \ wedge (\ exists i: y_ {i}> x_ {i}))}$

It should be noted that in the above definition only the existence of natural numbers and the definition of equality were assumed to be given without further explanations. This formulation is only to be regarded as a supplement to the above description in the introduction, since it presupposes fewer assumptions than implicitly true. The emphasis on the arbitrariness of the quantities at the beginning also makes it clear that this concept does not necessarily have to be limited to the use of numbers. If, for example, the sensations of individual test persons are to be taken into account as factors in an experiment in the social sciences, but these cannot be quantified precisely, it can still happen that a Pareto optimum can be found. The only condition here would be that these sensations can be compared with one another. ${\ displaystyle \ mathbb {N}}$

## Examples

### example 1

y-axis: strength
x-axis: "lightness" (= reciprocal of mass)
A component should be both resilient and light. Let it be characterized by the two properties of strength and mass . The higher the strength and the lower the mass, the better the component. If you enter the pairs of values ​​for many different components in a diagram that compares strength and lightness (reciprocal value of mass), you get the amount marked in blue in the adjacent graphic.

With the same mass, the component that is stronger is better. With the same strength, the component that is lighter is better. If the improvement in one value meets the deterioration in the other, the components are not Pareto-comparable.

In relation to the graphic, the values ​​further to the right and above are Pareto-superior compared to the values ​​to the left and below. All components on the red curve are "the best". You are Pareto optimal. An increase in one value is then only possible if the other decreases. (On the red line: “Further to the right” forces you to “further down”; conversely, “further up” forces you to also have to go “further to the left”.)

An additional condition or requirement can reduce the Pareto front to a single “(very) best” component (with regard to all three requirements). This can also be a standard that converts strength and mass into one size and thereby makes the points on the red line comparable, which leads to a clearly optimal solution (with regard to the standard). Depending on whether the sizes are comparable, sometimes no standard can be found.

### Example 2

Suppose there are three individuals A, B and C who live on a street. A well must be drilled to provide drinking water. Everyone has to pay for the pipe from the well to their house. That is why everyone wants to have the well as close to their house as possible.
In the following sketch, the location of the three houses on the street is marked as A, B and C. Also, the five possible locations for the well are labeled as b1, b2, b3, b4, and b5. It is assumed that the vertical / horizontal distances to the nearest well or neighbor are each 50 m.
        Skizze der möglichen Orte für den Brunnen:

(b1)

(b2)    (b3)

(b4)    (b5)

=====|A|=====|B|=====|C|========Straße =====



Set A = {b1, b2, b3, b4, b5}

The parameters are the 3 tuple elements " Distance to A ", " Distance to B " and " Distance to C ":

• b1 (158.1 m, 150.0 m, 158.1 m)
• b2 (111.8 m, 100.0 m, 111.8 m)
• b3 (141.4 m, 111.8 m, 100.0 m)
• b4 (70.7 m, 50.0 m, 70.7 m)
• b5 (111.8 m, 70.7 m, 50.0 m)

For the first tuple entry (= “distance to A”) b4 is optimal, for the second tuple element b4 is also optimal, for the third tuple element b5 is optimal.

The Pareto optimum is thus {b4, b5}.

• The place b1 is not Pareto-optimal, because the place b2 is Pareto-wise superior to the place b1 (English: Pareto-superior). Location b2 represents an improvement for everyone involved compared to b1.
• But b2 is not Pareto-optimal either, because b4 is superior to b2 in terms of Pareto. Location b4 represents an improvement over b2 for everyone involved.
• The locations b2 and b3 are not comparable according to the Pareto criterion, since relocating the well from b2 to b3 puts one of the participants better as well as puts another participant worse off. The same applies to relocating the well from b3 to b2. It is not possible to weigh up the advantages and disadvantages of different people using the Pareto criterion.
• The location b3 is also not a Pareto optimum, because b5 represents an improvement for everyone compared to b3.
• The place b4 is Pareto-optimal, because to b4 there is no Pareto-wise superior alternative that puts (at least) one of the participants better without at the same time making another worse
• The location b5 is, however, also Pareto-optimal, because any relocation of the well to one of the other locations would put individual C in a worse position.
• The locations b4 and b5 are not comparable according to the Pareto criterion, since a relocation of the well from b4 to b5 puts one of the participants better as well as puts another worse off. The same applies to a move from b5 to b4.

## The Pareto criterion compared to the total utility criterion

In economic theory, the criterion of Pareto optimality superseded the utilitarian criterion of the “sum of individual benefits ” that had prevailed until then .

Under the influence of the positivistic philosophy of science, the idea of ​​utility as a numerically (cardinally) measurable and for different persons (interpersonal) comparable quantity was not accepted.

Ordinal evaluations in the form of preferences ( is better / equally good / worse than / not decidable) now take the place of additive, cardinal utility variables . As a rule, ranking orders (preference orders) can be formed from this (1st rank , 2nd rank , 3rd rank etc. or briefly ). There is no need for a measure of utility that can be applied to interpersonal relationships, as these are individual orders of preference. The weighting of the individuals with their interests is implicit in the Pareto criterion. The individuals with their interests are weighted equally insofar as it does not matter which of the individuals is better or worse off. ${\ displaystyle x}$${\ displaystyle y}$${\ displaystyle x}$${\ displaystyle y}$${\ displaystyle z}$${\ displaystyle x \ succ y \ succ z}$

## The Pareto criterion in connection with a status quo regulation

Taken in isolation, the Pareto criterion is a plausible and unproblematic criterion for social decisions. It advocates all changes that are beneficial to anyone and not harmful to anyone.

However, it becomes ethically problematic if the optimality or efficiency so defined remains the only point of view.

As has been shown, there are u. U. a large number of Pareto Optima, which are not comparable in terms of value. In economic reality, however, a selection takes place, because - as is usual with legal systems - the existing situation, the status quo, remains if there are no decisions. As a result, there is no change in the existing structure as long as it creates a disadvantage for any owner. By combining the criterion of Pareto optimality with a status quo clause, the Pareto criterion works in favor of the existing conditions.

## Use case economic theory

A social situation is then referred to as economically efficient or Pareto optimal when it is not possible welfare of an individual by an Re- allocation to increase the resources without simultaneously reducing the other individual. In other words: a condition in which there is no way to make one individual better off without making another worse off at the same time. Since a Pareto optimum represents a social optimum, such a state is always worth striving for. In contrast, a condition is said to be Pareto inefficient when there is another allocation that makes one individual better off without making another individual worse off.

### Conditions for efficiency (Pareto optimality)

Pareto optimality of an economy means that the production factors are put to optimal use. This is the case if the following conditions are met:

1. Exchange optimum
The marginal utility gains of all goods that an individual consumes are identical. It is said that the marginal rates of substitution are the same ( Gossen's second law ). In this case, the individual consumes the goods that maximize his utility .
2. Optimal use of factors
The marginal productivities of the factors used must be the same. This condition ensures that the greatest possible quantity of goods is produced.

In modern economies, deviations from several conditions of Pareto optimality occur regularly. Monopolies, externalities , information asymmetries and the availability of public goods can all impair the functioning of the market mechanism. In this case, according to the theory of the second best, it is unclear whether an isolated measure to create the conditions increases efficiency.

### Pareto improvement

A condition (e.g. an allocation) is a Pareto improvement compared to another condition when a consumer is better off without worsening another consumer.

## criticism

The Pareto criterion is controversial in economics, especially in the context of social choice theory .

In an article published in 1970, Amartya Sen claimed the " impossibility of a Pareto liberal ". Using assumptions that are similar to those made by Arrow for his famous impossibility theorem , but less strict, he demonstrated that there are situations in which a “liberal attitude” (formalized as a social preference, which in certain situations is strictly related to preferences of the individual concerned) conflict with the Pareto criterion. He illustrated this with an example in which a prudish person wished that his neighbor did not read Lawrence ' Lady Chatterley ' s Lover and would rather read the book himself, even if he hated it. The neighbor would like to read the book himself, but he would even prefer the prude to read it. Sen showed that it would be liberally optimal to opt for the latter when choosing between the prude or no one reading the book, and when choosing between the libertine and no one to choose the libertine while Pareto- it would be optimal if the prude reads it. From this he concluded that the Pareto criterion should be questioned.

In practice, there will rarely be an opportunity for government action or a change in law that actually does not put anyone in a worse position. Guido Calabresi even argued that the criterion of Pareto-optimality could not be a guideline for the state because rational individuals under the assumptions of the Coase theorem have always found Pareto-optimal solutions among themselves in private negotiations. State decisions would necessarily always have a distributive effect, and some of the citizens affected would always be worse off.