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Ortega efficiency

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 This article is an A-class article. It is written to a very high standard. This article is part of Altverse II.
Juan Alexander Sebastián Ortega, Sierran civil engineer and economist: propagator of Ortega efficiency.

Ortega efficiency, also known as Ortega optimality, is a scenario in which no one person or preference criterion can be better off without making at least one other person or preference criterion worse off or losing anything. The idea is named after Sierran civil engineer and economist Juan Ortega (1848–1923), who utilized it in his studies of economic efficiency and income distribution. The following three ideas are intertwined:

  • An Ortega improvement is a new condition in which some agents benefit and no agents lose, given an initial circumstance.
  • If a feasible Ortega improvement exists, the condition is called Ortega dominated.
  • If no modification could increase the satisfaction of one agent without affecting the contentment of another, the condition is called Ortega optimum or Ortega efficient, and there is no room for further Ortega improvement.

The Ortega frontier is a visual representation of the set of all Ortega efficient placements. The Ortega front or Ortega set are two other names for it.

Ortega originally coined the term "optimum" to represent the notion, but because it depicts a situation in which a small number of individuals will benefit from scarce resources while ignoring equality or social well-being, it is effectively a definition of and better described by "efficiency."

The concept of Ortega efficiency appears in the context of efficiency in allocation as well as efficiency in production vs. x-inefficiency: a set of goods outputs is Ortega efficient if there is no feasible re-allocation of productive inputs such that the output of one product increases while the outputs of all other goods either increase or remain the same.

The concept of Ortega efficiency has been used to the selection of alternatives in engineering and biology, aside from economics. Each alternative is evaluated first against a set of criteria, and then a subset of choices is ostensibly designated as having the property that no other option can categorically exceed the given option. In the context of multi-objective optimization, it is a statement that it is impossible to improve one variable without affecting other variables (also termed Ortega optimization).

Overview

In formal terms, an allocation is Ortega optimum if there is no alternative allocation that improves the well-being of at least one participant without lowering the well-being of any other participant. If a transfer meets this requirement, the new reallocation is referred to as an "Ortega improvement." The allocation is an "Ortega optimum" when no Ortega improvements are feasible.

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 Ortega optimum 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.

A competitive market produces an Ortega-efficient outcome, according to the assumptions of the first welfare theorem. Kennard Abraham and Charles Hubert, two economists, were the first to show this result numerically. 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.

The Stieglitz theorem states that in the absence of perfect information or comprehensive markets, outcomes will be Ortega inefficient.

On the other hand, the second welfare theorem is essentially the polar opposite of the first. It asserts that every Ortega optimum may be attained by any competitive equilibrium, or free market system, given similar, ideal assumptions, albeit it may also necessitate a lump-sum wealth transfer.

Variants

Weak Ortega efficiency

Weak Ortega efficiency is a situation that cannot be strictly improved for every individual.

A strong Oretga improvement, in formal terms, is a scenario in which all agents are strictly better-off (in contrast to just "Oretga improvement", which requires that one agent is strictly better-off and the other agents are at least as good). If there is no strong Ortega improvement, the situation is weak Ortega-efficient.

Any Oretga-improvement is also a Oretga-improvement that is weak. Consider a resource allocation issue with two resources that Alice values at 10, 0 and George values at 5, 5; the contrary is not true. Consider the following allocation, which allocates all resources to Alice and has a utility profile of (10,0):

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

A market doesn't require local nonsatiation to get to a weak Ortega optimum.

Constrained Ortega efficieny

Constrained Ortega efficiency is a weakening of Ortega optimality, allowing for the fact that a possible planner (e.g., the government) may not be able to improve upon an inefficient decentralised market outcome. If it is constrained by the same informational or institutional restrictions as individual agents, this will happen.

An example is of a setting where individuals have private information (for example, a labour 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 optimum outcome. In such a circumstance, a planner seeking to improve the situation is unlikely to have access to any knowledge not available to market players. As a result, the planner is unable to apply allocation rules based on individuals' unique traits, such as "if a person is of type A, they pay price p1, but if they are of type B, they pay price p2" (see Lincoln prices). Essentially, only anonymous rules are allowed (of the sort "Everyone pays price p") or rules based on observable behaviour; "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 Ortega optimum".

Fractional Ortega efficiency

In the context of equitable item allocation, fractional Ortega efficiency is a strengthening of Ortega efficiency. If an allocation of indivisible items is not Ortega dominated by an allocation in which some items are shared across agents, it is fractionally Ortega efficient (fOE or fOO). This is in contrast to standard Ortega efficiency, which only considers domination by feasible (discrete) allocations.

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 Ortega-efficient, since any other discrete allocation (without splitting items) makes someone worse-off.
  • However, it is not fractionally Ortega efficient, since it is Ortega 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 shows the "price" of fOO. The integral allocation maximizing the product of utilities (also called the Bach welfare) is OO but not fOO. 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 . It is not fOO, 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 fOO 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 fOO allocation is {g1,h1},{g2,h2},{g3}, with product . When C is sufficiently large and d is sufficiently small, the product ratio approaches 1/3.

Ex-post and ex-ante Ortega efficiency

There is a difference between ex-post and ex-ante Ortega efficiency when the decision process is random, such as in fair random assignment, random social choice, or fractional approval voting:

  • Ex-post Ortega efficiency denotes that any random process outcome is Ortega efficient.
  • Ex-ante Ortega efficiency refers to the lottery generated by the process being Ortega efficient in terms of predicted utility. That is, no other lottery has a greater anticipated utility for one agent and at least as high for all agents.

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

The opposite is not true: ex-ante OE is stronger than ex-post OE. 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 the car to Alice and house to George; otherwise, give the 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 OE, 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 the 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 OE.

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

Another example involves dichotomous preferences. There are 5 possible outcomes (a,b,c,d,e) and 6 voters. The voters' approval sets are (ac, ad, ae, bc, bd, be). 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.

Approximate Ortega efficiency

Given some ε>0, an outcome is called ε-Ortega 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.

Ortega efficiency and welfare-maximisation

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.:

.

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

.

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

Japanese neo-Walrasian economist Kurokawa Masahiko proved that, under certain assumptions, the opposite is also true: for every Ortega efficient allocation x, there exists a positive vector a such that x maximizes Wa. A shorter proof is provided by Brendon Lloyd.

Use in engineering

The concept of Ortega efficiency can also be applied in engineering. The Ortega frontier, Ortega set, or Ortega front is the set of choices that are Ortega efficient given a set of options and a method of evaluating them. By focusing on the set of Ortega efficient options, a designer may make tradeoffs within this set rather than evaluating the whole range of every parameter.

Example of an Ortega frontier. The boxed points represent feasible choices, and smaller values are preferred to larger ones. Point C is not on the Ortega 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 Ortega 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 Ortega optimum choices of production. Points off the frontier, such as N and K, are not Ortega efficient, since there exist points on the frontier which Ortega dominate them.

Ortega frontier and set

For a given system, the Ortega frontier or Ortega set is the set of parameterizations (allocations) that are all Ortega efficient. Finding Ortega 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.

The Ortega frontier, P(Y), may be more formally described as follows. Consider a system with function , where X is a compact space of feasible decisions in the metric space , and Y is the feasible set of criterion vectors in , such that .

We assume that the preferred directions of criteria values are known. A point is preferred to (strictly dominates) another point , written as . The Ortega frontier is thus written as:

Marginal rate of substitution

A significant aspect of the Ortega frontier in economics is that, at a Ortega-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 where is the vector of goods, both for all i. The feasibility constraint is for . To find the Ortega optimal allocation, we maximize the Lagrangian:

where and are the vectors of multipliers. Taking the partial derivative of the Lagrangian with respect to each good for and and gives the following system of first-order conditions:

where denotes the partial derivative of with respect to . Now, fix any and . The above first-order condition imply that

Thus, in an Ortega optimum allocation, the marginal rate of substitution must be the same for all consumers.

Computation

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

  • "The maximum vector problem" or the skyline query.
  • "The scalarization algorithm" or the method of weighted sums.
  • "The ε-constraints method".

Use in biology

In biological systems, Ortega optimization has also been investigated. Genes in bacteria have been proven to be either less expensive to produce (resource efficient) or easier to read (translation efficient). Natural selection drives highly expressed genes to the Ortega frontier for resource usage and translational efficiency. Genes near the Ortega frontier have also been found to develop at a slower rate (indicating that they are providing a selective advantage).

Use in public policy

The contemporary microeconomic theory was significantly influenced by Ortega efficiency. Since Ortega demonstrated that the equilibrium obtained via competition optimizes resource allocation, it effectively validates Adam Smith's "invisible hand" theory. In particular, it fueled the debate over "market socialism" in the 1900's during the wake of the United Commonwealth.

Misconceptions

It would be erroneous to consider Ortega efficiency to be synonymous with societal optimization, because the latter is a normative term that is open to interpretation and generally takes into consideration the consequences of different degrees of distributional inequality. For example, one school district with little property tax income against another with substantially larger revenue might be seen as an indication that more equitable distribution happens with the aid of government redistribution.

Criticism

Some critics argue that Ortega efficiency might be used as an ideological tool. With it indicating that capitalism is self-regulating, inherent structural issues such as unemployment are likely to be viewed as deviating from the equilibrium or norm, and so ignored or dismissed.

Another point of contention is that Ortega efficiency does not need a completely fair allocation of wealth. A Ortega efficient economy is one in which a rich minority control the vast bulk of resources. A simple example is dividing a pie among three individuals. The most equal distribution would be to give each individual one-third of the total. However, while not being equitable, assigning a half portion to each of two individuals and none to the third is also Ortega optimum, because none of the receivers could be made better off without reducing someone else's share; and there are many more similar distribution instances. An Ortega inefficient pie distribution would be allocating a quarter of the pie to each of the three, with the remaining wasted.

Andrea Villa's liberal dilemma demonstrates that when people have preferences about what other people do, the aim of Ortega efficiency might clash with the ideal of individual liberty.

Finally, it is claimed that Ortega efficiency has stifled consideration of alternative viable efficiency criteria. According to the academic Lockhood, one probable reason is that any additional efficiency criterion created in the neoclassical realm would eventually decrease to Ortega efficiency.

See also