Preference (economics)

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A simple example of a preference order over three goods, in which an orange is preferred to a banana, but an apple is preferred to an orange Preference example.jpg
A simple example of a preference order over three goods, in which an orange is preferred to a banana, but an apple is preferred to an orange

In economics and other social sciences, preference is the order that a person (an agent) gives to alternatives based on their relative utility, a process which results in an optimal "choice" (whether real or theoretical). Preferences are evaluations, they concern matters of value, typically in relation to practical reasoning. [1] Instead of the prices of goods, personal income, or availability of goods, the character of the preferences is determined purely by a person's tastes. However, persons are still expected to act in their best (that is, rational) interest. [2] Rationality, in this context, means that when individuals are faced with a choice, they would select the option that maximizes self interest. Further, in every set of alternatives, preferences arise. [3]


The belief of preference plays a key role in many disciplines, including moral philosophy and decision theory. The logical properties that preferences possess also have major effects on rational choice theory which has a carry over effect to all modern economic topics. [4]

Using the scientific method, social scientists try to model how people make practical decisions in order to test predictions about human behavior. Although economists are usually not interested in what causes a person to have certain preferences, they are interested in the theory of choice because it gives a background to empirical demand analysis. [5]

Stability of preference is a deep assumption of most economic models. Gary Becker drew attention to this with his remark that "[t]he combined assumptions of maximizing behavior, market equilibrium, and stable preferences, used relentlessly and unflinchingly, form the heart of the economic approach as I see it." [6] More complex conditions of adaptive preference were explored by Carl Christian von Weizsäcker in his paper "The Welfare Economics of Adaptive Preferences" (2005), while remarking that: [7]

Traditional neo-classical economics has worked with the assumption that preferences of agents in the economy are fixed. This assumption has always been disputed, and, indeed, in the social sciences outside of neoclassical economics the assumption has never been accepted by anyone.


In 1926 Ragnar Frisch developed for the first time a mathematical model of preferences in the context of economic demand and utility functions. [8] Up to then, economists had developed an elaborated theory of demand that omitted primitive characteristics of people. This omission ceased when, at the end of the 19th and the beginning of the 20th century, logical positivism predicated the need of theoretical concepts to be related with observables. [9] Whereas economists in the 18th and 19th centuries felt comfortable theorizing about utility, with the advent of logical positivism in the 20th century, they felt that it needed more of an empirical structure. Because binary choices are directly observable, it instantly appealed to economists. The search for observables in microeconomics is taken even further by revealed preference theory, which holds consumers' preferences can be revealed by what they purchase under different circumstances, particularly under different income and price circumstances. [10]

Despite utilitarianism and decision theory, many economists have differing definitions of what a rational agent is. In the 18th century, utilitarianism gave insight into the utility-maximizing versions of rationality, however, economists still have no single definition or understanding of what preferences and rational actors should be analyzed by. [11]

Since the pioneer efforts of Frisch in the 1920s, one of the major issues which has pervaded the theory of preferences is the representability of a preference structure with a real-valued function. This has been achieved by mapping it to the mathematical index called utility. Von Neumann and Morgenstern 1944 book "Games and Economic Behaviour" treated preferences as a formal relation whose properties can be stated axiomatically. These type of axiomatic handling of preferences soon began to influence other economists: Marschak adopted it by 1950, Houthakker employed it in a 1950 paper, and Kenneth Arrow perfected it in his 1951 book "Social Choice and Individual Values". [12]

Gérard Debreu, influenced by the ideas of the Bourbaki group, championed the axiomatization of consumer theory in the 1950s, and the tools he borrowed from the mathematical field of binary relations have become mainstream since then. Even though the economics of choice can be examined either at the level of utility functions or at the level of preferences, to move from one to the other can be useful. For example, shifting the conceptual basis from an abstract preference relation to an abstract utility scale results in a new mathematical framework, allowing new kinds of conditions on the structure of preference to be formulated and investigated.

Another historical turnpoint can be traced back to 1895, when Georg Cantor proved in a theorem that if a binary relation is linearly ordered, then it is also isomorphically embeddable in the ordered real numbers. This notion would become very influential for the theory of preferences in economics: by the 1940s prominent authors such as Paul Samuelson, would theorize about people having weakly ordered preferences. [13]

Historically, preference in economics as a form of utility can be categorized as ordinal or cardinal data. Both introduced in the 20th century, cardinal and ordinal utility take opposing theory and mindset to the application and analysis of preference in utility. Vilfredo Pareto introduced the concept of ordinal utility, while Carl Menger led the idea of cardinal utility. Ordinal utility in summation is the direct following of preference, where an optimal choice is taken over a set of parameters. A person is expected to act in their best interests, and dedicate their preference to the outcome with the greatest utility. Ordinal utility assumes that an individual will not have the same utility from a preference as any other individual, because they likely will not experience the same parameters which cause them to decide a given outcome. Cardinal utility is a function of utility where a person makes a decision based on a preference, and the preference decision is weighted based on a quantitative value of utility. This unit of utility is assumed to be universally applicable, and constant across all individuals. Cardinal utility assumes consistency across individuals' decision-making processes also, assuming all individuals will have the same preference, with all variables held constant. Marshall found that “a good deal of the analysis of consumer behavior could be greatly simplified by assuming that the marginal utility of income is constant” (Robert H. Strotz.), however, this cannot be held true to the utility of resources and decision making applied to income. Ordinal and cardinal utility theory provide unique viewpoints on utility, and can be used differently to model decision-making preferences, utilization development, and can be used across many different applications for economic analysis.


There are two fundamental comparative value concepts, namely strict preference (better) and indifference (equal in value to). [14] These two concepts are expressed in terms of an agents best wishes, however, they also express objective or intersubjectively valid betterness that does not coincide with the pattern of wishes of any individual person.

Suppose the set of all states of the world is and an agent has a preference relation on . It is common to mark the weak preference relation by , so that means "the agent wants y at least as much as x" or "the agent weakly prefers y to x".

The symbol is used as a shorthand to denote an indifference relation: , which reads "the agent is indifferent between y and x" , meaning they receive the same level of benefit from each.

The symbol is used as a shorthand to the strong preference relation: to symbolize that some alternative is "at least as preferred as" another one, which is just a binary relation on the set of alternatives. Therefore:

The former qualitative relation can be preserved when mapped into a numerical structure, if we impose certain desirable properties over the binary relation: these are the axioms of preference order. For instance: Let us take the apple and assign it the arbitrary number 5. Then take the orange and let us assign it a value lower than 5, since the orange is less preferred than the apple. If this procedure is extended to the banana, one may prove by induction that if is defined on {apple, orange} and it represents a well-defined binary relation called "at least as preferred as" on this set, then it can be extended to a function defined on {apple, orange, banana} and it will represent "at least as preferred as" on this larger set.


5 > 3 > 2 = u(apple) > u(orange) > u(banana)

and this is consistent with Apple Orange, and with Orange Banana.

Axiom of order (completeness):

In terms of preference completeness simply means that when a consumer is making a choice between two different options, the consumer can rank them so either, A is preferred to B, B is preferred to A or they are indifferent between the two. [15]

For all and we have or or .

Without completeness of preferences, consumers would not be able to come to a decision when given multiple options, as they wouldn't be able to rank them. Making completeness a necessity for the decision model.

In order for preference theory to be useful mathematically, we need to assume the axiom of continuity. Continuity simply means that there are no ‘jumps’ in people's preferences. In mathematical terms, if we prefer point A along a preference curve to point B, points very close to A will also be preferred to B. This allows preference curves to be differentiated. The continuity assumption is "stronger than needed" in the sense that it indeed guarantees the existence of a continuous utility function representation. Continuity is, therefore, a sufficient condition, but not a necessary one, for a system of preferences. [16]

Although commodity bundles come in discrete packages, economists treat their units as a continuum, because very little is gained from recognizing their discrete nature, the two approaches are reconcilable by this rhetorical device: When a consumer makes repeated purchases of a product, the commodity spaces can get converted from the discrete items to the time rates of consumption. Instead of, say, noting that a consumer purchased one loaf of bread on Monday, another on Friday and another the following Tuesday, we can speak of an average rate of consumption of bread equal to 7/4 loaves per week. There is no reason why the average consumption per week cannot be any real number, thus allowing differentiability of the consumer's utility function. We can speak of continuous services of goods, even if the goods themselves are purchased in discrete units.

Although some authors include reflexivity as one of the axioms required to obtain representability (this axiom states that ), it is redundant inasmuch as the completeness axiom implies it already. [17]

Non-satiation of Preferences

A simple example of non-satiated preference, in which a large amount of oranges are preferred to a single orange. Non-Satiated preferences with oranges...jpg
A simple example of non-satiated preference, in which a large amount of oranges are preferred to a single orange.

Non-satiation refers to the belief any commodity bundle with at least as much of one good and more of the other must provide a higher utility, showing that more is always better, always wanting more is known as non-satiation. This assumption is believed to hold as when consumers are able to discard excess goods at no cost, then consumers can be no worse off with extra goods. [18]


Option A

Option B

In this Situation, utility from Option B > A, as it contains more apples and oranges with bananas being constant.


Transitivity of preferences is a fundamental principle shared by most major contemporary rational, prescriptive, and descriptive models of decision making. [19] Arguably the most discussed logical property of preferences is the following:

A≽B ∧ B≽C → A≽C (transitivity of weak performance) A∼B ∧ B∼C → A∼C (transitivity of indifference) A≻B ∧ B≻C → A≻C (transitivity of strict preference)

In order to have transitivity preferences, a person, player, or agent that prefers choice option B to A and A to F must prefer B to F. Claims of violations of transitivity by any decision maker (in particular individuals) requires evidence beyond any reasonable doubt, with the onus placed on the individual. [20]

When transitivity does not hold, it results in an endless loop of indecision, as the agent will always have an outcome that is preferred no matter what choice they make. Which is why the assumption of transitivity preference is believed to hold in most situations. Transitivity is one of the prerequisite for a rational consumer in the market.

Most commonly used axioms

Normative interpretations of the axioms

Everyday experience suggests that people at least talk about their preferences as if they had personal "standards of judgment" capable of being applied to the particular domain of alternatives that present themselves from time to time. [21] Thus, the axioms are an attempt to model the decision maker's preferences, not over the actual choice, but over the type of desirable procedure (a procedure that any human being would like to follow). Behavioral economics investigates inconsistent behavior (i.e. behavior that violates the axioms) of people. Believing in axioms in a normative way does not imply that everyone is asserted to behave according to them. Instead, they are a basis for suggesting a mode of behavior, one that people would like to see themselves or others following. [9]

Consumers whose preference structures violate transitivity would get exposed to being exploited by some unscrupulous person. For instance, Maria prefers apples to oranges, oranges to bananas, and bananas to apples. Let her be endowed with an apple, which she can trade in a market. Because she prefers bananas to apples, she is willing to pay, say, one cent to trade her apple for a banana. Afterwards, Maria is willing to pay another cent to trade her banana for an orange, and again the orange for an apple, and so on. There are other examples of this kind of irrational behaviour.

Completeness implies that some choice will be made, an assertion that is more philosophically questionable. In most applications, the set of consumption alternatives is infinite and the consumer is not conscious of all preferences. For example, one does not have to choose over going on holiday by plane or by train: if one does not have enough money to go on holiday anyway then it is not necessary to attach a preference order to those alternatives (although it can be nice to dream about what one would do if one would win the lottery). However, preference can be interpreted as a hypothetical choice that could be made rather than a conscious state of mind. In this case, completeness amounts to an assumption that the consumers can always make up their mind whether they are indifferent or prefer one option when presented with any pair of options.

Under some extreme circumstances, there is no "rational" choice available. For instance, if asked to choose which one of one's children will be killed, as in Sophie's Choice, there is no rational way out of it. In that case, preferences would be incomplete, since "not being able to choose" is not the same as "being indifferent".

The indifference relation ~ is an equivalence relation. Thus we have a quotient set S/~ of equivalence classes of S, which forms a partition of S. Each equivalence class is a set of packages that is equally preferred. If there are only two commodities, the equivalence classes can be graphically represented as indifference curves. Based on the preference relation on S we have a preference relation on S/~. As opposed to the former, the latter is antisymmetric and a total order.

Factors which affect Consumer Preferences

1. Indifference Curve

An indifference curve is so named because the consumer would be indifferent between choosing any commodity bundles. [22] Indifference Curves explain an agent behaviour in terms of their preferences for different combinations of two goods, for example X & Y. An indifference curve can be detected in a market when the economics of scope is not overly diverse, or the goods and services are a part of a perfect market. Any bundles on the same indifference curve have the same utility level. One example of this is deodorant. Deodorant is similarly priced throughout a number of different brands. Deodorant also does not have any major differences in use; therefore, consumers really have no preference in which they should use.

2. Monopolised Markets

A monopolised market almost always has a direct effect on consumer preferences. A monopolised market refers to when a company and its product offerings dominate a sector or industry. [23] A monopolised market in turn, means that the business has complete control of supply and demand of a good or service. Businesses which have a monopoly can also use a number of strategies which can be used to ensure they remain control of the industry by refusing entry to the market, these are, blockaded entry, accommodated entry and deterred entry. [24] When a business has a monopoly, the business has a massive advantage in having a large majority of consumer preferences.

3. Changes in new technology

New changes in technology is a big factor in changes of consumer preferences. When an industry has a new competitor in the market who has found ways to make the goods or services work more effectively, has the ability to completely change the market. Some examples of changes in technology is Android phones. Five years ago, android was struggling to compete with Apple for market share. With the advances in technology throughout the last five years they have now passed the stagnant apple brand. Changes in technology examples are, but not limited to, increased efficiency, longer lasting batteries and new easier interface for consumers.

Types of Preferences

A simple graph showing convex preferences, as the indifference line, curves in Simple-indifference-curves.svg
A simple graph showing convex preferences, as the indifference line, curves in

Convex Preferences:

Convex preferences relate to averages between two points on an indifference curve. It comes in two forms, weak and strong. In its weak form, convex preferences states that if . Then the average of A and B is at least as good as A. Where as in its strong form the average of A and B would be preferred. Which is why in its strong form, the indifference line curves in, meaning that the average of any two points would result in a point further away from the origin, thus giving a higher utility. [25] One way to check convexity is to connect two random points on the same indifference curve and draw a straight line through these two points, and then pick one point on the straight line between those two points. If the utility level of the picked point on the straight line is greater than that of those two points, this is a strictly convex preference. Convexity is one of the prerequisites for a rational consumer in the market when maximizing his utility level under the budget constraint.

Concave Preferences

Concave preferences is the opposite of convex, where when , then the average of A and B is worst than A. This because concave curves slope outwards, meaning an average between two points on the same indifference curve would result in a point that is closer to the origin thus giving a lower utility. [26] To determine whether the preference is concave or not, one way is to still connect two random points on the same difference curve and draw a straight line through these two points, and then pick one point on the straight line between those two points. If the utility level of the picked point on the straight line is lower than that of those two points, this is a strictly concave preference.

Straight Line indifference

Straight line indifferences, occur when there are perfect substitutes. Perfect substitutes are goods and/or services that can be used in the same way as the good or service it replaces. Meaning when , then the average of A and B will fall on the same indifference line, and will give the exact same utility. [27]

An example of straight line indifference curves, where Good X and Good Y are perfect substitutes. Indifference-curves-perfect-substitutes.svg
An example of straight line indifference curves, where Good X and Good Y are perfect substitutes.

Types of Goods effecting Preferences

When a consumer is faced with a choice between different goods, the type of goods they are choosing between will affect how they make their decision process. To begin with, when there are normal goods, these goods have a direct correlation with the income the consumer makes, meaning as they make more money they will choose to consume more of this good and as their income decreases they will consume less of the good. The opposite to this however is inferior goods, these have a negative correlation with income, so as a consumer makes less money, they'll decide to consume more inferior goods as they are seen as less desirable meaning they come with a reduced cost and as they make more money, they'll consume less inferior goods as they'll have the money available to buy more desirable goods. [28] An example of a normal good would be branded clothes, as they are more expensive compared to their inferior good counterparts which are non-branded clothes. Goods that are not affected by income as referred to as a necessity good, which are product(s) and services that consumers will buy regardless of the changes in their income levels, these usually include medical care, clothing and basic food. Finally, there are also luxury goods, which are the most expensive goods and deemed the most desirable. Just like normal goods, as income is increased, so is the demand for luxury goods, however in the case for luxury goods, the greater the increase in income, the greater the increase in demand for luxury goods. [29]

Applications to theories of utility

In economics, a utility function is often used to represent a preference structure such that if and only if . The idea is to associate each class of indifference with a real number such that, if one class is preferred to the other, then the number of the first one is greater than that of the second one. When a preference order is both transitive and complete, then it is standard practice to call it a rational preference relation, and the people who comply with it are rational agents . A transitive and complete relation is called a weak order (or total preorder). The literature on preferences is far from being standardized regarding terms such as complete, partial, strong, and weak. Together with the terms "total", "linear", "strong complete", "quasi-orders", "pre-orders" and "sub-orders", which also have a different meaning depending on the author's taste, there has been an abuse of semantics in the literature. [21]

According to Simon Board, a continuous utility function always exists if is a continuous rational preference relation on . [30] For any such preference relation, there are many continuous utility functions that represent it. Conversely, every utility function can be used to construct a unique preference relation.

All the above is independent of the prices of the goods and services and of the budget constraints faced by consumers. These determine the feasible bundles (which they can afford). According to the standard theory, consumers chooses a bundle within their budget such that no other feasible bundle is preferred over it; therefore their utility is maximized.

Primitive equivalents of some known properties of utility functions

Lexicographic preferences

Lexicographic preferences are a special case of preferences that assign an infinite value to a good, when compared with the other goods of a bundle. [31]

Georgescu-Roegen pointed out that the measurability of the utility theory is limited as it excludes lexicographic preferences. Causing an amplified level of awareness placed upon lexicographic preferences as a substitute hypothesis on consumer behaviour. [32]

Strict versus weak

The possibility of defining a strict preference relation as distinguished from the weaker one , and vice versa, suggests in principle an alternative approach of starting with the strict relation as the primitive concept and deriving the weaker one and the indifference relation. However, an indifference relation derived this way will generally not be transitive. [8] The conditions to avoid such inconsistencies were studied in detail by Andranik Tangian. [31] According to Kreps "beginning with strict preference makes it easier to discuss noncomparability possibilities". [33]

Elicitation of preferences

The mathematical foundations of most common types of preferences — that are representable by quadratic or additive utility functions — laid down by Gérard Debreu [34] [35] enabled Andranik Tangian to develop methods for their elicitation. In particular, additive and quadratic preference functions in variables can be constructed from interviews, where questions are aimed at tracing totally 2D-indifference curves in coordinate planes. [36] [37]


Some critics say that rational theories of choice and preference theories rely too heavily on the assumption of invariance, which states that the relation of preference should not depend on the description of the options or on the method of elicitation. But without this assumption, one's preferences cannot be represented as maximization of utility. [38]

Milton Friedman said that segregating taste factors from objective factors (i.e. prices, income, availability of goods) is conflicting because both are "inextricably interwoven".

The concept of transitivity is a highly debated topic, with many examples being presented to suggest that it does not hold in general. One of the most well-known being the Sorites paradox, which shows that indifference between small changes in value can be incrementally extended to indifference between large changes in values. [39]

Another criticism comes from philosophy. Even though the concept of preference is not controversial for economists, many philosophers find this concept problematic. Philosophers cast doubt that when most consumers share the same preference in the same market, which may lead to the result that the shared preference has become somewhat objective, whether the judgements of preferences for each individual will still depend on subjectivity or not.[ clarification needed ]

See also

Related Research Articles

As a topic of economics, utility is used to model worth or value. Its usage has evolved significantly over time. The term was introduced initially as a measure of pleasure or happiness as part of the theory of utilitarianism by moral philosophers such as Jeremy Bentham and John Stuart Mill. The term has been adapted and reapplied within neoclassical economics, which dominates modern economic theory, as a utility function that represents a single consumer's preference ordering over a choice set but is not comparable across consumers. This concept of utility is personal and based on choice rather than on pleasure received, and so is specified more rigorously than the original concept but makes it less useful for ethical decisions.

Indifference curve Concept in economics

In economics, an indifference curve connects points on a graph representing different quantities of two goods, points between which a consumer is indifferent. That is, any combinations of two products indicated by the curve will provide the consumer with equal levels of utility, and the consumer has no preference for one combination or bundle of goods over a different combination on the same curve. One can also refer to each point on the indifference curve as rendering the same level of utility (satisfaction) for the consumer. In other words, an indifference curve is the locus of various points showing different combinations of two goods providing equal utility to the consumer. Utility is then a device to represent preferences rather than something from which preferences come. The main use of indifference curves is in the representation of potentially observable demand patterns for individual consumers over commodity bundles.

In social choice theory, Arrow's impossibility theorem, the general possibility theorem or Arrow's paradox is an impossibility theorem stating that when voters have three or more distinct alternatives (options), no ranked voting electoral system can convert the ranked preferences of individuals into a community-wide ranking while also meeting a specified set of criteria: unrestricted domain, non-dictatorship, Pareto efficiency, and independence of irrelevant alternatives. The theorem is often cited in discussions of voting theory as it is further interpreted by the Gibbard–Satterthwaite theorem. The theorem is named after economist and Nobel laureate Kenneth Arrow, who demonstrated the theorem in his doctoral thesis and popularized it in his 1951 book Social Choice and Individual Values. The original paper was titled "A Difficulty in the Concept of Social Welfare".

In welfare economics, a social welfare function is a function that ranks social states as less desirable, more desirable, or indifferent for every possible pair of social states. Inputs of the function include any variables considered to affect the economic welfare of a society. In using welfare measures of persons in the society as inputs, the social welfare function is individualistic in form. One use of a social welfare function is to represent prospective patterns of collective choice as to alternative social states. The social welfare function provides the government with a simple guideline for achieving the optimal distribution of income.

The theory of consumer choice is the branch of microeconomics that relates preferences to consumption expenditures and to consumer demand curves. It analyzes how consumers maximize the desirability of their consumption as measured by their preferences subject to limitations on their expenditures, by maximizing utility subject to a consumer budget constraint.

Welfare economics is a branch of economics that uses microeconomic techniques to evaluate well-being (welfare) at the aggregate (economy-wide) level.

The expected utility hypothesis is a popular concept in economics that serves as a reference guide for decisions when the payoff is uncertain. The theory recommends which option rational individuals should choose in a complex situation, based on their risk appetite and preferences.

Utility maximization was first developed by utilitarian philosophers Jeremy Bentham and John Stewart Mill. In microeconomics, the utility maximization problem is the problem consumers face: "How should I spend my money in order to maximize my utility?" It is a type of optimal decision problem. It consists of choosing how much of each available good or service to consume, taking into account a constraint on total spending (income), the prices of the goods and their preferences.

In economics, an ordinal utility function is a function representing the preferences of an agent on an ordinal scale. Ordinal utility theory claims that it is only meaningful to ask which option is better than the other, but it is meaningless to ask how much better it is or how good it is. All of the theory of consumer decision-making under conditions of certainty can be, and typically is, expressed in terms of ordinal utility.

Revealed preference theory, pioneered by economist Paul Anthony Samuelson in 1938, is a method of analyzing choices made by individuals, mostly used for comparing the influence of policies on consumer behavior. Revealed preference models assume that the preferences of consumers can be revealed by their purchasing habits.

Local nonsatiation

The property of local nonsatiation of consumer preferences states that for any bundle of goods there is always another bundle of goods arbitrarily close that is preferred to it.

Pairwise comparison generally is any process of comparing entities in pairs to judge which of each entity is preferred, or has a greater amount of some quantitative property, or whether or not the two entities are identical. The method of pairwise comparison is used in the scientific study of preferences, attitudes, voting systems, social choice, public choice, requirements engineering and multiagent AI systems. In psychology literature, it is often referred to as paired comparison.

In economics and consumer theory, quasilinear utility functions are linear in one argument, generally the numeraire. Quasilinear preferences can be represented by the utility function where is strictly concave. A useful property of the quasilinear utility function is that the Marshallian/Walrasian demand for does not depend on wealth and is thus not subject to a wealth effect; The absence of a wealth effect simplifies analysis and makes quasilinear utility functions a common choice for modelling. Furthermore, when utility is quasilinear, compensating variation (CV), equivalent variation (EV), and consumer surplus are algebraically equivalent. In mechanism design, quasilinear utility ensures that agents can compensate each other with side payments.

In decision theory, the von Neumann–Morgenstern (VNM) utility theorem shows that, under certain axioms of rational behavior, a decision-maker faced with risky (probabilistic) outcomes of different choices will behave as if he or she is maximizing the expected value of some function defined over the potential outcomes at some specified point in the future. This function is known as the von Neumann–Morgenstern utility function. The theorem is the basis for expected utility theory.


In order theory, a branch of mathematics, a semiorder is a type of ordering for items with numerical scores, where items with widely differing scores are compared by their scores and where scores within a given margin of error are deemed incomparable. Semiorders were introduced and applied in mathematical psychology by Duncan Luce (1956) as a model of human preference. They generalize strict weak orderings, in which items with equal scores may be tied but there is no margin of error. They are a special case of partial orders and of interval orders, and can be characterized among the partial orders by additional axioms, or by two forbidden four-item suborders.

In expected utility theory, a lottery is a discrete distribution of probability on a set of states of nature. The elements of a lottery correspond to the probabilities that each of the states of nature will occur, e.g.. Much of the theoretical analysis of choice under uncertainty involves characterizing the available choices in terms of lotteries.

In economics, the Debreu theorems are several statements about the representation of a preference ordering by a real-valued function. The theorems were proved by Gerard Debreu during the 1950s.

In utility theory, the responsive set (RS) extension is an extension of a preference-relation on individual items, to a partial preference-relation of item-bundles.

In theoretical economics, an abstract economy is a model that generalizes both the standard model of an exchange economy in microeconomics, and the standard model of a game in game theory. An equilibrium in an abstract economy generalizes both a Walrasian equilibrium in microeconomics, and a Nash equilibrium in game-theory.

Stochastic transitivity models are stochastic versions of the transitivity property of binary relations studied in mathematics. Several models of stochastic transitivity exist and have been used to describe the probabilities involved in experiments of paired comparisons, specifically in scenarios where transitivity is expected, however, empirical observations of the binary relation is probabilistic. For example, players' skills in a sport might be expected to be transitive, i.e. "if player A is better than B and B is better than C, then player A must be better than C"; however, in any given match, a weaker player might still end up winning with a positive probability. Tighly matched players might have a higher chance of observing this inversion while players with large differences in their skills might only see these inversions happen seldomly. Stochastic transitivity models formalize such relations between the probabilities and the underlying transitive relation.


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[1] [2] [3]

  1. Board, S. (2021). Competitive Strategy: Week 8 Entry. Retrieved 26 April 2021, from
  2. Kenton, W. (2021). Monopoly. Retrieved 26 April 2021, from
  3. University of Southern Indiana. (2021). Retrieved 26 April 2021, from