Lottery (probability)

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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. (Rain:.70, No Rain:.30). [1] Much of the theoretical analysis of choice under uncertainty involves characterizing the available choices in terms of lotteries.

In economics, individuals are assumed to rank lotteries according to a rational system of preferences, although it is now accepted that people make irrational choices systematically. Behavioral economics studies what happens in markets in which some of the agents display human complications and limitations. [2]

Choice under risk

According to expected utility theory, someone chooses among lotteries by multiplying his subjective estimate of the probabilities of the possible outcomes by a utility attached to each outcome by his personal utility function. Thus, each lottery has an expected utility, a linear combination of the utilities of the outcomes in which weights are the subjective probabilities. [3] It is also founded in the famous example, the St. Petersburg paradox: as Daniel Bernoulli mentioned, the utility function in the lottery could be dependent on the amount of money which he had before the lottery. [4]

For example, let there be three outcomes that might result from a sick person taking either novel drug A or B for his condition: "Cured", "Uncured", and "Dead". Each drug is a lottery. Suppose the probabilities for lottery A are (Cured: .90, Uncured: .00, Dead: .10), and for lottery B are (Cured: .50, Uncured: .50, Dead: .00).

If the person had to choose between lotteries A and B, how would they do it? A theory of choice under risk starts by letting people have preferences on the set of lotteries over the three states of nature—not just A and B, but all other possible lotteries. If preferences over lotteries are complete and transitive, they are called rational. If people follow the axioms of expected utility theory, their preferences over lotteries will follow each lottery's ranking in terms of expected utility. Let the utility values for the sick person be:

In this case, the expected utility of Lottery A is 14.4 (= .90(16) + .10(12)) and the expected utility of Lottery B is 14 (= .50(16) + .50(12)), so the person would prefer Lottery A. Expected utility theory implies that the same utilities could be used to predict the person's behavior in all possible lotteries. If, for example, he had a choice between lottery A and a new lottery C consisting of (Cured: .80, Uncured: .15 Dead: .05), expected utility theory says he would choose C, because its expected utility is 14.6 (= .80(16) + .15(12) + .05(0)).

The paradox argued by Maurice Allais complicates expected utility in the lottery. [5] In contrast to the former example, let there be outcomes consisting of only losing money. In situation 1, option 1a has a certain loss of $500 and option 1b has equal probabilities of losing $1000 or $0. In situation 2, option 2a has a 10% chance of losing $500 and a 90% chance of losing $0, and option 2b has a 5% chance of losing $1000 and a 95% chance of losing $0. This circumstance can be described with the expected utility equations below:

Many people tend to make different decisions between situations. [5] People prefer option 1a to 1b in situation 1, and 2b to 2a in situation 2. However two situations have the same structure, which causes a paradox:

The possible explanation for the above is that it has a ‘certainty effect’, that the outcomes without probabilities (determined in advance) will make a larger effect on the utility functions and final decisions. [5] In many cases, this focusing on the certainty may cause inconsistent decisions and preferences. Plus, people tend to find some clues from the format or context of the lotteries. [6]

It was additionally argued that how much people got trained about statistics could impact the decision making in the lottery. [7] Throughout a series of experiments, he concluded that a person statistically trained will be more likely to have consistent and confident outcomes which could be a generalized form.

The assumption about combining linearly the individual utilities and making the resulting number be the criterion to be maximized can be justified of the grounds of the independence axiom. Therefore, the validity of expected utility theory depends on the validity of the independence axiom. The preference relation satisfies independence if for any three simple lotteries , , , and any number it holds that

if and only if

Indifference maps can be represented in the simplex.

Related Research Articles

Pareto efficiency or Pareto optimality is a situation where no action or allocation is available that makes one individual better off without making another worse off. 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:

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 consumer's ordinal preferences over a choice set, but is not necessarily comparable across consumers or possessing a cardinal interpretation. This concept of utility is personal and based on choice rather than on pleasure received, and so requires fewer behavioral assumptions than the original concept.

<span class="mw-page-title-main">Risk aversion</span> Economics theory

In economics and finance, risk aversion is the tendency of people to prefer outcomes with low uncertainty to those outcomes with high uncertainty, even if the average outcome of the latter is equal to or higher in monetary value than the more certain outcome. Risk aversion explains the inclination to agree to a situation with a more predictable, but possibly lower payoff, rather than another situation with a highly unpredictable, but possibly higher payoff. For example, a risk-averse investor might choose to put their money into a bank account with a low but guaranteed interest rate, rather than into a stock that may have high expected returns, but also involves a chance of losing value.

<span class="mw-page-title-main">Prospect theory</span> Theory of behavioral economics and behavioral finance

Prospect theory is a theory of behavioral economics, judgment and decision making that was developed by Daniel Kahneman and Amos Tversky in 1979. The theory was cited in the decision to award Kahneman the 2002 Nobel Memorial Prize in Economics.

The expected utility hypothesis is a foundational assumption in mathematical economics concerning human preference when decision making under uncertainty. It postulates that a rational agent maximizes utility, as formulated in the mathematics of game theory, based on their risk aversion. Rational choice theory, a cornerstone of microeconomics, builds upon the expected utility of individuals to model aggregate social behaviour.

In decision theory, subjective expected utility is the attractiveness of an economic opportunity as perceived by a decision-maker in the presence of risk. Characterizing the behavior of decision-makers as using subjective expected utility was promoted and axiomatized by L. J. Savage in 1954 following previous work by Ramsey and von Neumann. The theory of subjective expected utility combines two subjective concepts: first, a personal utility function, and second a personal probability distribution.

<span class="mw-page-title-main">Cardinal utility</span>

In economics, a cardinal utility function or scale is a utility index that preserves preference orderings uniquely up to positive affine transformations. Two utility indices are related by an affine transformation if for the value of one index u, occurring at any quantity of the goods bundle being evaluated, the corresponding value of the other index v satisfies a relationship of the form

In decision theory, the Ellsberg paradox is a paradox in which people's decisions are inconsistent with subjective expected utility theory. Daniel Ellsberg popularized the paradox in his 1961 paper, "Risk, Ambiguity, and the Savage Axioms". John Maynard Keynes published a version of the paradox in 1921. It is generally taken to be evidence of ambiguity aversion, in which a person tends to prefer choices with quantifiable risks over those with unknown, incalculable risks.

The Allais paradox is a choice problem designed by Maurice Allais (1953) to show an inconsistency of actual observed choices with the predictions of expected utility theory. Rather than adhering to rationality, the Allais paradox proves that individuals rarely make rational decisions consistently when required to do so immediately. The independence axiom of expected utility theory, which requires that the preferences of an individual should not change when altering two lotteries by equal proportions, was proven to be violated by the paradox.

In accounting, finance, and economics, a risk-seeker or risk-lover is a person who has a preference for risk.

In decision theory and economics, ambiguity aversion is a preference for known risks over unknown risks. An ambiguity-averse individual would rather choose an alternative where the probability distribution of the outcomes is known over one where the probabilities are unknown. This behavior was first introduced through the Ellsberg paradox.

Info-gap decision theory seeks to optimize robustness to failure under severe uncertainty, in particular applying sensitivity analysis of the stability radius type to perturbations in the value of a given estimate of the parameter of interest. It has some connections with Wald's maximin model; some authors distinguish them, others consider them instances of the same principle.

In the fields of actuarial science and financial economics there are a number of ways that risk can be defined; to clarify the concept theoreticians have described a number of properties that a risk measure might or might not have. A coherent risk measure is a function that satisfies properties of monotonicity, sub-additivity, homogeneity, and translational invariance.

The rank-dependent expected utility model is a generalized expected utility model of choice under uncertainty, designed to explain the behaviour observed in the Allais paradox, as well as for the observation that many people both purchase lottery tickets and insure against losses.

In economics, Epstein–Zin preferences refers to a specification of recursive utility.

In probability theory and statistics, a stochastic order quantifies the concept of one random variable being "bigger" than another. These are usually partial orders, so that one random variable may be neither stochastically greater than, less than nor equal to another random variable . Many different orders exist, which have different applications.

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 they are 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.

Risk aversion is a preference for a sure outcome over a gamble with higher or equal expected value. Conversely, the rejection of a sure thing in favor of a gamble of lower or equal expected value is known as risk-seeking behavior.

In decision theory, a multi-attribute utility function is used to represent the preferences of an agent over bundles of goods either under conditions of certainty about the results of any potential choice, or under conditions of uncertainty.

References

  1. ↑ Mas-Colell, Andreu, Michael Whinston and Jerry Green (1995). Microeconomic theory. Oxford: Oxford University Press. ISBN   0-19-507340-1
  2. ↑ Mullainathan, Sendhil & Richard Thaler (2000) 'Behavioral Economics'. NBER Working Paper No. 7948, p. 2.
  3. ↑ Archibald, G (1959). "Utility, risk, and linearity". Journal of Political Economy. 67 (5): 438. doi:10.1086/258216. S2CID   154853936.
  4. ↑ Schoemaker, Paul J. H. (1980). Experiments on Decisions under Risk: The Expected Utility Hypothesis. Martinus Nijhoff Publishing. p. 12. doi:10.1007/978-94-017-5040-0. ISBN   978-94-017-5042-4.
  5. 1 2 3 Schoemaker, Paul J. H. (1980). Experiments on Decisions under Risk: The Expected Utility Hypothesis. Dordrecht. pp. 18–19. ISBN   978-94-017-5040-0. OCLC   913628692.{{cite book}}: CS1 maint: location missing publisher (link)
  6. ↑ Schoemaker, Paul J. H. (1980). Experiments on Decisions under Risk: The Expected Utility Hypothesis. Dordrecht. p. 89. ISBN   978-94-017-5040-0. OCLC   913628692.{{cite book}}: CS1 maint: location missing publisher (link)
  7. ↑ Schoemaker, Paul J. H. (1980). Experiments on Decisions under Risk: The Expected Utility Hypothesis. Dordrecht. p. 108. ISBN   978-94-017-5040-0. OCLC   913628692.{{cite book}}: CS1 maint: location missing publisher (link)

2) http://www.stanford.edu/~jdlevin/Econ%20202/Uncertainty.pdf

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