Pareto efficiency


In welfare economics, a Pareto improvement formalizes the idea of an outcome being "better in every possible way." A change is called a Pareto improvement if it leaves at least one person in society better off without leaving anyone else worse off than they were before. A situation is called Pareto efficient or Pareto optimal if all possible Pareto improvements have already been made; in other words, there are no longer any ways left to make one person better off without making some other person worse-off.
In social choice theory, the same concept is sometimes called the unanimity principle, which says that if everyone in a society prefers A to B, society as a whole also non-strictly prefers A to B. The Pareto front consists of all Pareto-efficient situations.
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.
Besides economics, the notion of Pareto efficiency has also been applied to selecting alternatives in engineering and biology. Each option is first assessed, under multiple criteria, and then a subset of options is 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.

History

The concept is named after Vilfredo Pareto, an Italian civil engineer and economist, who used the concept in his studies of economic efficiency and income distribution.
Pareto originally used the word "optimal" for the concept, but this is somewhat of a misnomer: Pareto's concept more closely aligns with an idea of "efficiency", because it does not identify a single "best" outcome. Instead, it only identifies a set of outcomes that might be considered optimal, by at least one person.

Overview

Formally, a state is Pareto-optimal if there is no alternative state where at least one participant's well-being is higher, and nobody else's well-being is lower. If there is a state change that satisfies this condition, the new state is called a "Pareto improvement". When no Pareto improvements are possible, the state is a "Pareto optimum".
In other words, Pareto efficiency is when it is impossible to make one party better off without making another party worse off. This state indicates that resources can no longer be allocated in a way that makes one party better off without harming other parties. In a state of Pareto Efficiency, resources are allocated in the most efficient way possible.
Pareto efficiency is mathematically represented when there is no other strategy profile s' such that ui ≥ ui for every player i and uj > uj for some player j. In this equation s represents the strategy profile, u represents the utility or benefit, and j represents the player.
Efficiency is an important criterion for judging behavior in a game. In zero-sum games, every outcome is Pareto-efficient.
A special case of a state is an allocation of resources. The formal presentation of the concept in an economy is the following: Consider an economy with agents and goods. Then an allocation, where for all i, is Pareto-optimal if there is no other feasible allocation where, for utility function for each agent, for all with for some. 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. 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.
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.

Pareto order

If multiple sub-goals exist, combined into a vector-valued objective function, generally, finding a unique optimum becomes challenging. This is due to the absence of a total order relation for which would not always prioritize one target over another target. In the multi-objective optimization setting, various solutions can be "incomparable" as there is no total order relation to facilitate the comparison. Only the Pareto order is applicable:
Consider a vector-valued minimization problem: Pareto dominates if and only if: : and We then write, where is the Pareto order. This means that is not worse than in any goal but is better in at least one goal. The Pareto order is a strict partial order, though it is not a product order.
If, then this defines a preorder in the search space and we say Pareto dominates the alternative and we write.
Image:Pareto order dominated.png|thumb| dominates in the Pareto order.
Image:Pareto order not dominated.png|thumb| does not dominate in the Pareto order and does not dominate in the Pareto order.

Variants

Weak Pareto efficiency

Weak Pareto efficiency is a situation that cannot be strictly improved for every individual.
Formally, a strong Pareto improvement is defined as a situation in which all agents are strictly better-off. 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, and George values at. Consider the allocation giving all resources to Alice, where the utility profile is :
  • It is a weak PO, since no other allocation is strictly better to both agents.
  • 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 its utility profile is.
A market does not require local nonsatiation to get to a weak Pareto optimum.

Constrained Pareto efficiency

Constrained Pareto efficiency is a weakening of Pareto optimality, accounting for the fact that a potential planner 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.
An example is of a setting where individuals have private information 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". Essentially, only anonymous rules are allowed 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 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 allocations.
As an example, consider an item allocation problem with two items, which Alice values at and George values at. Consider the allocation giving the first item to Alice and the second to George, where the utility profile is :
  • It is Pareto-efficient, since any other discrete allocation 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.

    Ex-ante Pareto efficiency

When the decision process is random, such as in fair random assignment or random social choice or fractional approval voting, 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 that 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 for Alice and 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 for Alice and 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.
Another example involves dichotomous preferences. There are 5 possible outcomes and 6 voters. The voters' approval sets are. All five outcomes are PE, so every lottery is ex-post PE. But the lottery selecting c, d, e with probability 1/3 each is not ex-ante PE, since it gives an expected utility of 1/3 to each voter, while the lottery selecting a, b with probability 1/2 each gives an expected utility of 1/2 to each voter.