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. The concept is named after Vilfredo Pareto (1848–1923), Italian 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 it has a Pareto improvement.
• A situation is called Pareto optimal or Pareto efficient if no change could lead to improved satisfaction for all parties.

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 efficiency" is considered as a minimal notion of efficiency that does not necessarily result in a socially desirable distribution of resources: it makes no statement about equality, or the overall well-being of a society.[2][3]^:46–49 It is a necessary, but not sufficient, condition of efficiency.

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.[4]^: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

"Pareto optimality" is a formally defined concept used to describe when an allocation is optimal. An allocation is not Pareto optimal if there is an 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 reallocation is called a "Pareto improvement". When no further Pareto improvements are possible, the allocation is a "Pareto optimum".

The formal presentation of the concept in an economy is as follows: 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 ${\displaystyle \{x_{1}',...,x_{n}'\}}$ such that, 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 ${\displaystyle i}$.[5] 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.

In principle, a change from a generally inefficient economic allocation to an efficient one is not necessarily considered to be a Pareto improvement. Even when there are overall gains in the economy, if a single agent is disadvantaged by the reallocation, the allocation is not Pareto optimal. For instance, if a change in economic policy eliminates a monopoly and that market subsequently becomes competitive, the gain to others may be large. However, since the monopolist is disadvantaged, this is not a Pareto improvement. In theory, if the gains to the economy are larger than the loss to the monopolist, the monopolist could be compensated for its loss while still leaving a net gain for others in the economy, allowing for a Pareto improvement. Thus, in practice, to ensure that nobody is disadvantaged by a change aimed at achieving Pareto efficiency, compensation of one or more parties may be required. It is acknowledged, in the real world, that such compensations may have unintended consequences leading to incentive distortions over time, as agents supposedly anticipate such compensations and change their actions accordingly.[6]

Under the idealized conditions of the first welfare theorem, a system of free markets, also called a "competitive equilibrium", leads to a Pareto-efficient outcome. It was first demonstrated mathematically by economists Kenneth Arrow and Gérard Debreu.

However, the result only holds under the restrictive assumptions necessary for the proof: markets exist for all possible goods, so there are no externalities; all markets are in full equilibrium; markets are perfectly competitive; transaction costs are negligible; 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.[7]

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.[5]

Weak Pareto efficiency

Weak Pareto optimality is a situation that cannot be strictly improved for every individual.[8]

Formally, we define a strong pareto improvement 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-optimal 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.[9]

Constrained Pareto efficiency

Constrained Pareto optimality 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.[10]^: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".

The concept of constrained Pareto optimality assumes benevolence on the part of the planner and hence is distinct from the concept of government failure, which occurs when the policy making politicians fail to achieve an optimal outcome simply because they are not necessarily acting in the public's best interest.

Fractional Pareto efficiency

Fractional Pareto optimality is a strengthening of Pareto-optimality in the context of fair item allocation. An allocation of indivisible items is fractionally Pareto-optimal (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-optimality, which only considers domination by feasible (discrete) allocations.[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-optimal, since any other discrete allocation (without splitting items) makes someone worse-off.
• However, it is not fractionally-Pareto-optimal, 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).

Pareto-efficiency and welfare-maximization

Suppose each agent i is assigned a positive weight a_i. 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 x_a 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 x_a is Pareto-efficient: since all weights are positive, any Pareto-improvement would increase the sum, contradicting the definition of x_a.

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 W_a. 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 ${\displaystyle f:\mathbb {R} ^{n}\rightarrow \mathbb {R} ^{m}}$, 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. 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.

Computation

Algorithms for computing the Pareto frontier of a finite set of alternatives have been studied in computer science and power engineering.[17] They include:

• "The maximum vector problem" or the skyline query.[18][19][20]
• "The scalarization algorithm" or the method of weighted sums.[21][22]
• "The ${\displaystyle \epsilon }$-constraints method".[23][24]

Use in biology

Pareto optimisation has also been studied in biological processes.[25]^: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.[26]^:166–169 Genes near the Pareto frontier were also shown to evolve more slowly (indicating that they are providing a selective advantage).[27]

Criticism

It would be incorrect to treat Pareto efficiency as equivalent to societal optimization,[28]^: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.[29]^: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.[30]^:95–132

Pareto efficiency does not require a totally equitable distribution of wealth.[31]^:222 An economy in which a wealthy few hold the vast majority of resources can be Pareto efficient. This possibility is inherent in the definition of Pareto efficiency; often the status quo is Pareto efficient regardless of the degree to which wealth is equitably distributed. 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.[32]^:18 The origin (and utility value) of the pie is conceived as immaterial in these examples. In such cases, whereby a "windfall" is gained that none of the potential distributees actually produced (e.g., land, inherited wealth, a portion of the broadcast spectrum, or some other resource), the criterion of Pareto efficiency does not determine a unique optimal allocation. Wealth consolidation may exclude others from wealth accumulation because of bars to market entry, etc.

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.[33]^:92–94

See also

• Admissible decision rule, analog in decision theory
• Arrow's impossibility theorem
• Bayesian efficiency
• Fundamental theorems of welfare economics
• Deadweight loss
• Economic efficiency
• Highest and best use
• Kaldor–Hicks efficiency
• Market failure, when a market result is not Pareto optimal
• Maximal element, concept in order theory
• Maxima of a point set
• Multi-objective optimization
• Pareto-efficient envy-free division
• Social Choice and Individual Values for the '(weak) Pareto principle'
• TOTREP
• Welfare economics

References

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25. 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.
26. Eiben, A. E., & Smith, J. E., Introduction to Evolutionary Computing (Berlin/Heidelberg: Springer, 2003), pp. 166–169.
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28. Drèze, J., Essays on Economic Decisions Under Uncertainty (Cambridge: Cambridge University Press, 1987), pp. 358–364
29. Backhaus, J. G., The Elgar Companion to Law and Economics (Cheltenham, UK / Northampton, MA: Edward Elgar, 2005), pp. 10–15.
30. Paulsen, M. B., "The Economics of the Public Sector: The Nature and Role of Public Policy in the Finance of Higher Education", in M. B. Paulsen, J. C. Smart, eds. The Finance of Higher Education: Theory, Research, Policy, and Practice (New York: Agathon Press, 2001), pp. 95–132.
31. Bhushi, K., ed., Farm to Fingers: The Culture and Politics of Food in Contemporary India (Cambridge: Cambridge University Press, 2018), p. 222.
32. Wittman, D., Economic Foundations of Law and Organization (Cambridge: Cambridge University Press, 2006), p. 18.
33. Sen, A., Rationality and Freedom (Cambridge, MA / London: Belknep Press, 2004), pp. 92–94.

Further reading

• Fudenberg, Drew; Tirole, Jean (1991). Game theory. Cambridge, Massachusetts: MIT Press. pp. 18–23. ISBN 9780262061414. Book preview.
• Bendor, Jonathan; Mookherjee, Dilip (April 2008). "Communitarian versus Universalistic norms". Quarterly Journal of Political Science. 3 (1): 33–61. doi:10.1561/100.00007028.CS1 maint: ref=harv (link)
• Kanbur, Ravi (January–June 2005). "Pareto's revenge" (PDF). Journal of Social and Economic Development. 7 (1): 1–11.CS1 maint: ref=harv (link)
• Ng, Yew-Kwang (2004). Welfare economics towards a more complete analysis. Basingstoke, Hampshire New York: Palgrave Macmillan. ISBN 9780333971215.
• Rubinstein, Ariel; Osborne, Martin J. (1994), "Introduction", in Rubinstein, Ariel; Osborne, Martin J. (eds.), A course in game theory, Cambridge, Massachusetts: MIT Press, pp. 6–7, ISBN 9780262650403 Book preview.
• Mathur, Vijay K. (Spring 1991). "How well do we know Pareto optimality?". The Journal of Economic Education. 22 (2): 172–178. doi:10.2307/1182422. JSTOR 1182422.CS1 maint: ref=harv (link)
• Newbery, David M.G.; Stiglitz, Joseph E. (January 1984). "Pareto inferior trade". The Review of Economic Studies. 51 (1): 1–12. doi:10.2307/2297701. JSTOR 2297701.CS1 maint: ref=harv (link)