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# Pareto efficiency

Pareto efficiency or Pareto optimality is a situation where no individual or preference criterion can be better off without making at least one individual or preference criterion worse off or without any loss thereof. The concept is named after Vilfredo Pareto (1848–1923), Italian civil engineer and economist, who used the concept in his studies of economic efficiency and income distribution. The following three concepts are closely related:

• Given an initial situation, a Pareto improvement is a new situation where some agents will gain, and no agents will lose.
• A situation is called Pareto dominated if there exists a possible Pareto improvement.
• A situation is called Pareto optimal or Pareto efficient if no change could lead to improved satisfaction for some agent without some other agent losing or if there is no scope for further Pareto improvement.

The Pareto frontier is the set of all Pareto efficient allocations, conventionally shown graphically. It also is variously known as the Pareto front or Pareto set.[1]

Pareto originally used the word "optimal" for the concept, but as it describes a situation where a limited number of people will be made better off under finite resources, and it does not take equality or social well-being into account, it is in effect a definition of and better captured by "efficiency".[2]

In addition to the context of efficiency in allocation, the concept of Pareto efficiency also arises in the context of efficiency in production vs. x-inefficiency: a set of outputs of goods is Pareto efficient if there is no feasible re-allocation of productive inputs such that output of one product increases while the outputs of all other goods either increase or remain the same.[3]:459

Besides economics, the notion of Pareto efficiency has been applied to the selection of alternatives in engineering and biology. Each option is first assessed, under multiple criteria, and then a subset of options is ostensibly identified with the property that no other option can categorically outperform the specified option. It is a statement of impossibility of improving one variable without harming other variables in the subject of multi-objective optimization (also termed Pareto optimization).

## Overview

Formally, an allocation is Pareto optimal if there is no alternative allocation where improvements can be made to at least one participant's well-being without reducing any other participant's well-being. If there is a transfer that satisfies this condition, the new reallocation is called a "Pareto improvement". When no Pareto improvements are possible, the allocation is a "Pareto optimum".

The formal presentation of the concept in an economy is the following: Consider an economy with ${\displaystyle n}$ agents and ${\displaystyle k}$ goods. Then an allocation ${\displaystyle \{x_{1},...,x_{n}\}}$, where ${\displaystyle x_{i}\in \mathbb {R} ^{k}}$ for all i, is Pareto optimal if there is no other feasible allocation ${\displaystyle \{x_{1}',...,x_{n}'\}}$ where, for utility function ${\displaystyle u_{i}}$ for each agent ${\displaystyle i}$, ${\displaystyle u_{i}(x_{i}')\geq u_{i}(x_{i})}$ for all ${\displaystyle i\in \{1,...,n\}}$ with ${\displaystyle u_{i}(x_{i}')>u_{i}(x_{i})}$ for some ${\displaystyle i}$.[4] Here, in this simple economy, "feasibility" refers to an allocation where the total amount of each good that is allocated sums to no more than the total amount of the good in the economy. In a more complex economy with production, an allocation would consist both of consumption vectors and production vectors, and feasibility would require that the total amount of each consumed good is no greater than the initial endowment plus the amount produced.

Under the assumptions of the first welfare theorem, a competitive market leads to a Pareto-efficient outcome. This result was first demonstrated mathematically by economists Kenneth Arrow and Gérard Debreu.[5][citation needed] However, the result only holds under the assumptions of the theorem: markets exist for all possible goods, there are no externalities; markets are perfectly competitive; and market participants have perfect information.

In the absence of perfect information or complete markets, outcomes will generally be Pareto inefficient, per the Greenwald-Stiglitz theorem.[6]

The second welfare theorem is essentially the reverse of the first welfare-theorem. It states that under similar, ideal assumptions, any Pareto optimum can be obtained by some competitive equilibrium, or free market system, although it may also require a lump-sum transfer of wealth.[4]

## Variants

### Weak Pareto efficiency

Weak Pareto efficiency is a situation that cannot be strictly improved for every individual.[7]

Formally, a strong Pareto improvement is defined as a situation in which all agents are strictly better-off (in contrast to just "Pareto improvement", which requires that one agent is strictly better-off and the other agents are at least as good). A situation is weak Pareto-efficient if it has no strong Pareto-improvements.

Any strong Pareto-improvement is also a weak Pareto-improvement. The opposite is not true; for example, consider a resource allocation problem with two resources, which Alice values at 10, 0 and George values at 5, 5. Consider the allocation giving all resources to Alice, where the utility profile is (10,0):

• It is a weak-PO, since no other allocation is strictly better to both agents (there are no strong Pareto improvements).
• But it is not a strong-PO, since the allocation in which George gets the second resource is strictly better for George and weakly better for Alice (it is a weak Pareto improvement) - its utility profile is (10,5).

A market doesn't require local nonsatiation to get to a weak Pareto-optimum.[8]

### Constrained Pareto efficiency

Constrained Pareto efficiency is a weakening of Pareto-optimality, accounting for the fact that a potential planner (e.g., the government) may not be able to improve upon a decentralized market outcome, even if that outcome is inefficient. This will occur if it is limited by the same informational or institutional constraints as are individual agents.[9]:104

An example is of a setting where individuals have private information (for example, a labor market where the worker's own productivity is known to the worker but not to a potential employer, or a used-car market where the quality of a car is known to the seller but not to the buyer) which results in moral hazard or an adverse selection and a sub-optimal outcome. In such a case, a planner who wishes to improve the situation is unlikely to have access to any information that the participants in the markets do not have. Hence, the planner cannot implement allocation rules which are based on the idiosyncratic characteristics of individuals; for example, "if a person is of type A, they pay price p1, but if of type B, they pay price p2" (see Lindahl prices). Essentially, only anonymous rules are allowed (of the sort "Everyone pays price p") or rules based on observable behavior; "if any person chooses x at price px, then they get a subsidy of ten dollars, and nothing otherwise". If there exists no allowed rule that can successfully improve upon the market outcome, then that outcome is said to be "constrained Pareto-optimal".

### Fractional Pareto efficiency

Fractional Pareto efficiency is a strengthening of Pareto-efficiency in the context of fair item allocation. An allocation of indivisible items is fractionally Pareto-efficient (fPE or fPO) if it is not Pareto-dominated even by an allocation in which some items are split between agents. This is in contrast to standard Pareto-efficiency, which only considers domination by feasible (discrete) allocations.[10][11]

As an example, consider an item allocation problem with two items, which Alice values at 3, 2 and George values at 4, 1. Consider the allocation giving the first item to Alice and the second to George, where the utility profile is (3,1):

• It is Pareto-efficient, since any other discrete allocation (without splitting items) makes someone worse-off.
• However, it is not fractionally-Pareto-efficient, since it is Pareto-dominated by the allocation giving to Alice 1/2 of the first item and the whole second item, and the other 1/2 of the first item to George - its utility profile is (3.5, 2).

The following example[10] shows the "price" of fPO. The integral allocation maximizing the product of utilities (also called the Nash welfare) is PO but not fPO. Moreover, the product of utilities in any fPO allocation is at most 1/3 of the maximum product. There are 5 goods {h1,h2,g1,g2,g3} and 3 agents with the following values (where C is a large constant and d is a small positive constant):

Agents ↓ Goods ⇒ h1 h2 g1 g2 g3
A1 C C 1 1-d 1-d
A2 C C 1-d 1 1-d
A3 C C 1-d 1-d 1

A max-product integral allocation is {h1},{h2},{g1,g2,g3}, with product ${\displaystyle C^{2}\cdot (3-2d)}$. It is not fPO, since it is dominated by a fractional allocation: agent 3 can give g1 to agent 1 (losing 1-d utility) in return to a fraction of h1 that both agents value at 1-d/2. This trade strictly improves the welfare of both agents. Moreover, in any integral fPO allocation, there exists an agent Ai who receives only (at most) the good gi - otherwise a similar trade can be done. Therefore, a max-product fPO allocation is {g1,h1},{g2,h2},{g3}, with product ${\displaystyle (C+1)^{2}}$. When C is sufficiently large and d is sufficiently small, the product ratio approaches 1/3.

### Ex-ante Pareto efficiency

When the decision process is random, such as in fair random assignment, there is a difference between ex-post and ex-ante Pareto-efficiency:

• Ex-post Pareto-efficiency means that any outcome of the random process is Pareto efficient.
• Ex-ante Pareto-efficiency means that the lottery determined by the process is Pareto-efficient with respect to the expected utilities. That is: no other lottery gives a higher expected utility to one agent and at least as high expected utility to all agents.

If some lottery L is ex-ante PE, then it is also ex-post PE. Proof: suppose that one of the ex-post outcomes x of L is Pareto-dominated by some other outcome y. Then, by moving some probability mass from x to y, one attains another lottery L' which ex-ante Pareto-dominates L.

The opposite is not true: ex-ante PE is stronger that ex-post PE. For example, suppose there are two objects - a car and a house. Alice values the car at 2 and the house at 3; George values the car at 2 and the house at 9. Consider the following two lotteries:

1. With probability 1/2, give car to Alice and house to George; otherwise, give car to George and house to Alice. The expected utility is (2/2+3/2)=2.5 for Alice and (2/2+9/2)=5.5 for George. Both allocations are ex-post PE, since the one who got the car cannot be made better-off without harming the one who got the house.
2. With probability 1, give car to Alice. Then, with probability 1/3 give the house to Alice, otherwise give it to George. The expected utility is (2+3/3)=3 for Alice and (9*2/3)=6 for George. Again, both allocations are ex-post PE.

While both lotteries are ex-post PE, the lottery 1 is not ex-ante PE, since it is Pareto-dominated by lottery 2.

### Approximate Pareto-efficiency

Given some ε>0, an outcome is called ε-Pareto-efficient if no other outcome gives all agents at least the same utility, and one agent a utility at least (1+ε) higher. This captures the notion that improvements smaller than (1+ε) are neglibigle and should not be considered a breach of efficiency.

## Pareto-efficiency and welfare-maximization

Suppose each agent i is assigned a positive weight ai. For every allocation x, define the welfare of x as the weighted sum of utilities of all agents in x, i.e.:

${\displaystyle W_{a}(x):=\sum _{i=1}^{n}a_{i}u_{i}(x)}$.

Let xa be an allocation that maximizes the welfare over all allocations, i.e.:

${\displaystyle x_{a}\in \arg \max _{x}W_{a}(x)}$.

It is easy to show that the allocation xa is Pareto-efficient: since all weights are positive, any Pareto-improvement would increase the sum, contradicting the definition of xa.

Japanese neo-Walrasian economist Takashi Negishi proved[12] that, under certain assumptions, the opposite is also true: for every Pareto-efficient allocation x, there exists a positive vector a such that x maximizes Wa. A shorter proof is provided by Hal Varian.[13]

## Use in engineering

The notion of Pareto efficiency has been used in engineering.[14]:111–148 Given a set of choices and a way of valuing them, the Pareto frontier or Pareto set or Pareto front is the set of choices that are Pareto efficient. By restricting attention to the set of choices that are Pareto-efficient, a designer can make tradeoffs within this set, rather than considering the full range of every parameter.[15]:63–65

Example of a Pareto frontier. The boxed points represent feasible choices, and smaller values are preferred to larger ones. Point C is not on the Pareto frontier because it is dominated by both point A and point B. Points A and B are not strictly dominated by any other, and hence lie on the frontier.
A production-possibility frontier. The red line is an example of a Pareto-efficient frontier, where the frontier and the area left and below it are a continuous set of choices. The red points on the frontier are examples of Pareto-optimal choices of production. Points off the frontier, such as N and K, are not Pareto-efficient, since there exist points on the frontier which Pareto-dominate them.

### Pareto frontier

For a given system, the Pareto frontier or Pareto set is the set of parameterizations (allocations) that are all Pareto efficient. Finding Pareto frontiers is particularly useful in engineering. By yielding all of the potentially optimal solutions, a designer can make focused tradeoffs within this constrained set of parameters, rather than needing to consider the full ranges of parameters.[16]:399–412

The Pareto frontier, P(Y), may be more formally described as follows. Consider a system with function ${\displaystyle f:X\rightarrow \mathbb {R} ^{m}}$, where X is a compact set of feasible decisions in the metric space ${\displaystyle \mathbb {R} ^{n}}$, and Y is the feasible set of criterion vectors in ${\displaystyle \mathbb {R} ^{m}}$, such that ${\displaystyle Y=\{y\in \mathbb {R} ^{m}:\;y=f(x),x\in X\;\}}$.

We assume that the preferred directions of criteria values are known. A point ${\displaystyle y^{\prime \prime }\in \mathbb {R} ^{m}}$ is preferred to (strictly dominates) another point ${\displaystyle y^{\prime }\in \mathbb {R} ^{m}}$, written as ${\displaystyle y^{\prime \prime }\succ y^{\prime }}$. The Pareto frontier is thus written as:

${\displaystyle P(Y)=\{y^{\prime }\in Y:\;\{y^{\prime \prime }\in Y:\;y^{\prime \prime }\succ y^{\prime },y^{\prime }\neq y^{\prime \prime }\;\}=\emptyset \}.}$

### Marginal rate of substitution

A significant aspect of the Pareto frontier in economics is that, at a Pareto-efficient allocation, the marginal rate of substitution is the same for all consumers.[17] A formal statement can be derived by considering a system with m consumers and n goods, and a utility function of each consumer as ${\displaystyle z_{i}=f^{i}(x^{i})}$ where ${\displaystyle x^{i}=(x_{1}^{i},x_{2}^{i},\ldots ,x_{n}^{i})}$ is the vector of goods, both for all i. The feasibility constraint is ${\displaystyle \sum _{i=1}^{m}x_{j}^{i}=b_{j}}$ for ${\displaystyle j=1,\ldots ,n}$. To find the Pareto optimal allocation, we maximize the Lagrangian:

${\displaystyle L_{i}((x_{j}^{k})_{k,j},(\lambda _{k})_{k},(\mu _{j})_{j})=f^{i}(x^{i})+\sum _{k=2}^{m}\lambda _{k}(z_{k}-f^{k}(x^{k}))+\sum _{j=1}^{n}\mu _{j}\left(b_{j}-\sum _{k=1}^{m}x_{j}^{k}\right)}$

where ${\displaystyle (\lambda _{k})_{k}}$ and ${\displaystyle (\mu _{j})_{j}}$ are the vectors of multipliers. Taking the partial derivative of the Lagrangian with respect to each good ${\displaystyle x_{j}^{k}}$ for ${\displaystyle j=1,\ldots ,n}$ and ${\displaystyle k=1,\ldots ,m}$ and gives the following system of first-order conditions:

${\displaystyle {\frac {\partial L_{i}}{\partial x_{j}^{i}}}=f_{x_{j}^{i}}^{1}-\mu _{j}=0{\text{ for }}j=1,\ldots ,n,}$
${\displaystyle {\frac {\partial L_{i}}{\partial x_{j}^{k}}}=-\lambda _{k}f_{x_{j}^{k}}^{i}-\mu _{j}=0{\text{ for }}k=2,\ldots ,m{\text{ and }}j=1,\ldots ,n,}$

where ${\displaystyle f_{x_{j}^{i}}}$ denotes the partial derivative of ${\displaystyle f}$ with respect to ${\displaystyle x_{j}^{i}}$. Now, fix any ${\displaystyle k\neq i}$ and ${\displaystyle j,s\in \{1,\ldots ,n\}}$. The above first-order condition imply that

${\displaystyle {\frac {f_{x_{j}^{i}}^{i}}{f_{x_{s}^{i}}^{i}}}={\frac {\mu _{j}}{\mu _{s}}}={\frac {f_{x_{j}^{k}}^{k}}{f_{x_{s}^{k}}^{k}}}.}$

Thus, in a Pareto-optimal allocation, the marginal rate of substitution must be the same for all consumers.[citation needed]

### Computation

Algorithms for computing the Pareto frontier of a finite set of alternatives have been studied in computer science and power engineering.[18] They include:

• "The maximum vector problem" or the skyline query.[19][20][21]
• "The scalarization algorithm" or the method of weighted sums.[22][23]
• "The ${\displaystyle \epsilon }$-constraints method".[24][25]

## Use in public policy

The modern microeconomic theory drew inspirations heavily from Pareto efficiency. Since Pareto showed that the equilibrium achieved through competition would optimize resource allocation, it is effectively corroborating Adam Smith's "invisible hand" notion. More specifically, it motivated the debate over "market socialism" in the 1930s.[2]

## Use in biology

Pareto optimisation has also been studied in biological processes.[26]:87–102 In bacteria, genes were shown to be either inexpensive to make (resource efficient) or easier to read (translation efficient). Natural selection acts to push highly expressed genes towards the Pareto frontier for resource use and translational efficiency.[27]:166–169 Genes near the Pareto frontier were also shown to evolve more slowly (indicating that they are providing a selective advantage).[28]

## Common misconceptions

It would be incorrect to treat Pareto efficiency as equivalent to societal optimization,[29]:358–364 as the latter is a normative concept that is a matter of interpretation that typically would account for the consequence of degrees of inequality of distribution.[30]:10–15 An example would be the interpretation of one school district with low property tax revenue versus another with much higher revenue as a sign that more equal distribution occurs with the help of government redistribution.[31]:95–132

## Criticism

This section will introduce criticisms from the most radical to more moderate ones.

Some commentators contest that Pareto efficiency could potentially serve as an ideological tool. With it implying that capitalism is self-regulated thereof, it is likely that the embedded structural problems such as unemployment would be treated as deviating from the equilibrium or norm, and thus neglected or discounted.[2]

Pareto efficiency does not require a totally equitable distribution of wealth, which is another aspect that draws in criticism.[32]:222 An economy in which a wealthy few hold the vast majority of resources can be Pareto efficient. A simple example is the distribution of a pie among three people. The most equitable distribution would assign one third to each person. However the assignment of, say, a half section to each of two individuals and none to the third is also Pareto optimal despite not being equitable, because none of the recipients could be made better off without decreasing someone else's share; and there are many other such distribution examples. An example of a Pareto inefficient distribution of the pie would be allocation of a quarter of the pie to each of the three, with the remainder discarded.[33]:18

The liberal paradox elaborated by Amartya Sen shows that when people have preferences about what other people do, the goal of Pareto efficiency can come into conflict with the goal of individual liberty.[34]:92–94

Lastly, it is proposed that Pareto efficiency to some extent inhibited discussion of other possible criteria of efficiency. As the scholar Lockhood argues, one possible reason is that any other efficiency criteria established in the neoclassical domain will reduce to Pareto efficiency at the end.[2]

## References

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24. ^ "On a Bicriterion Formulation of the Problems of Integrated System Identification and System Optimization". IEEE Transactions on Systems, Man, and Cybernetics. SMC-1 (3): 296–297. 1971. doi:10.1109/TSMC.1971.4308298. ISSN 0018-9472.
25. ^ Mavrotas, George (2009). "Effective implementation of the ε-constraint method in Multi-Objective Mathematical Programming problems". Applied Mathematics and Computation. 213 (2): 455–465. doi:10.1016/j.amc.2009.03.037. ISSN 0096-3003.
26. ^ Moore, J. H., Hill, D. P., Sulovari, A., & Kidd, L. C., "Genetic Analysis of Prostate Cancer Using Computational Evolution, Pareto-Optimization and Post-processing", in R. Riolo, E. Vladislavleva, M. D. Ritchie, & J. H. Moore, eds., Genetic Programming Theory and Practice X (Berlin/Heidelberg: Springer, 2013), pp. 87–102.
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32. ^ Bhushi, K., ed., Farm to Fingers: The Culture and Politics of Food in Contemporary India (Cambridge: Cambridge University Press, 2018), p. 222.
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