... choice under risk, choice under ambiguity, belief updating, and survey expectations about economic variables. 8 Lecture #8: Decision Making Under Risk and Uncertainty (Part 4) As before, the individual owns $10, and has to decide whether or not to play a lottery based on a coin toss. W The capacity v[ satisﬁes mono-tonicity with respect to set inclusion (v(E i) # v(E ij) for all i, j), as well as the restrictions v(A) 5 0 and v(S) 5 1. In the prototypical formulation of decision making under uncertainty, an individual decision maker (DM) must choose one among a set of actions, whose consequences … 0000027620 00000 n
Such risk aversions also provide a natural incentive for Johann to demand (or, equivalently, pay) a risk premium above AFP to take on (or, equivalently, get rid of) risk. We compute expected utility by taking the product of probability and the associated utility corresponding to each outcome for all lotteries. utils. Expected Utility Theory. Finally, and most importantly, the concavity and convexity of the utility function is key to distinguishing between risk-averse and risk-seeking individuals. and has an initial endowment of $10. The expected utility calculation is as follows. Citation Machina, Mark J. Mathematically, the property that the utility is increasing at a decreasing rate can be written as a combination of restrictions on the first and second derivatives (rate of change of slope) of the utility function. W Some functions that satisfy this property are In the case of decisions under Risk, agents have complete knowl-edge of the objective likelihood of each state. Definitions of Optimal Path Under Uncertainty In an uncertain environment, the definition of optimal path is not obvious. u( Such an individual is called risk neutral. W . This paper introduces a formal definition and an experimental measurement of the concept of cognitive uncertainty: people's subjective uncertainty about what the optimal action is. . Let the game that offers him payoffs be offered to him. In a world of uncertainty, it seems intuitive that individuals would maximize expected utilityA construct to explain the level of satisfaction a person gets when faced with uncertain choices.. What matters is that such a function (which reflects an individual’s preferences over uncertain games) exists. The ranking of the lotteries based on expected dollar winnings is lottery 3, 2, and 1—in that order. )= Since real-life situations can be riskier than laboratory settings, we can safely assume that a majority of people are risk averse most of the time. E( n 238 0 obj<>stream
Dictionary definition of “stochastic”: Involving or containing a random variable or variables; Involving chance or probability. In case tails turns face-up, then the final wealth equals $4 ($6 − $2). Ana’s utility function is U = p w, where wis her wealth. )]. His main interest is in decision under uncertainty, focusing on the definition of probability, notions of rationality, non-Bayesian decision models, and related issues. W ), Biases and other behavioral aspects make individuals deviate from the behavior predicted by the E(U) theory. Ethical Choice under Uncertainty: Most discussions about utilitarian ethics are attempt to determine the goodness of an outcome. That expected utility ranking differs from expected wealth ranking is best explained using the example below. Discuss the von Neumann-Morgenstern expected utility function and discuss how it differs from expected gains. 0000013518 00000 n
We have also seen that a utility function representation exists if the four assumptions discussed above hold. The phrase has become a regular way to describe people’s deviations from normal preferences. How to use uncertainty in a sentence. This feature of this particular utility function is called diminishing marginal utilityFeature of a utility function in which utility is always increasing although at a decreasing rate.. This is an important result for a concave utility function as shown in Figure 3.2 "A Utility Function for a Risk-Averse Individual". As we shall now see, the E(U) theory does enable us to capture different risk attitudes of individuals. This refers to a construct used to explain the level of satisfaction a person gets when faced with uncertain choices. e Let the preferences be such that the addition to utility one gets out of an additional dollar at lower levels of wealth is always greater than the additional utility of an extra dollar at higher levels of wealth. Introduction to choice under uncertainty 2 B. +0.5× 0000004904 00000 n
=3.162. Intertemporal Choice: Exchange & Production 2. 0000005676 00000 n
We can calculate the expected payoff of each lottery by taking the product of probability and the payoff associated with each outcome and summing this product over all outcomes. where u is a function that attaches numbers measuring the level of satisfaction ui associated with each outcome i. u is called the Bernoulli function while E(U) is the von Neumann-Morgenstern expected utility function. =3 E( 0000000016 00000 n
2 10 Such a person will need incentives to be willing to play the game. The expected loss in wealth to the individual. The intuition is straightforward, proving it axiomatically was a very challenging task. e . Moreover, the utility is always increasing although at a decreasing rate. An individual may go skydiving, hang gliding, and participate in high-risk-taking behavior. )]≥U[E( Discuss the three risk types with respect to their shapes, technical/mathematical formulation, and the economic interpretation. 0000000916 00000 n
Property of a curve in which a chord connecting any two points on the curve will lie strictly below the curve. People without the rational means to buy homes bought them and took “nonconventional risks,” which led to the 2008–2009 financial and credit crisis and major recessions (perhaps even depression) as President Obama took office in January 2009. u( The contrast between the choices made by risk-averse individuals and risk-seeking individuals is starkly clear in the above example.Mathematically speaking, for a risk-averse person, we have People’s expected utility if they play the lottery is An individual has a utility function given by. Feature of a utility function in which utility is always increasing although at a decreasing rate. Uncertainty definition is - the quality or state of being uncertain : doubt. 0000003948 00000 n
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But let us consider the ranking of the same lotteries by this person who ranks them in order based on expected utility. The first thing we notice from Figure 3.2 "A Utility Function for a Risk-Averse Individual" is its concavityProperty of a curve in which a chord connecting any two points on the curve will lie strictly below the curve., which means if one draws a chord connecting any two points on the curve, the chord will lie strictly below the curve. The functional form depicted in Figure 3.2 "A Utility Function for a Risk-Averse Individual" is LN(W). 3. 2 Finally, we come to the third risk attitude type wherein an individual is indifferent between playing a lottery and not playing it. − Just so, insurance companies charge individuals premiums for risk transfer via insurances. W We have seen earlier (in Table 3.1 "Utility Function with Initial Endowment of $10") that the AFP for playing this lottery is $4. W The first is the criterion of admissibility, namely, that a decision maker should not select a weakly dominated action (Luce and Raiffa (1957, Chapter 13)). “Choice under Uncertainty: Problems Solved and Unsolved” Journal of Economic Perspectives (Summer 1987) (reprinted in...) It turns out that all convex utility functionsUtility function in which the curve lies strictly below the chord joining any two points on the curve. W At the time, Federal Reserve Board Chairman Alan Greenspan introduced the term “irrational exuberance” in a speech given at the American Enterprise Institute. Applications: demand for insurance, portfolio choice 4. Technically, the difference in risk attitudes across individuals is called “heterogeneity of risk preferences” among economic agents. Our question is, can the expected utility theory capture that behavior as well? 0000010572 00000 n
Micro III-1.2 Decision under Uncertainty: Expected Utility Definition Graphical Representation Invariance Result Axiomatization Application: Insurance Von Neumann and Morgenstern John von Neumann Oskar Morgenstern 18 / 31-1.2 Decision under Uncertainty: Expected Utility Definition Graphical Representation Figure 3.4 A Utility Function for a Risk-Neutral Individual. Satisficing aims to be pragmatic and saves on costs or expenditures. The example shows that the ranking of games of chance differs when one utilizes the expected utility (E[U]) theory than when the expected gain E(G) principle applies This leads us to the insight that if two lotteries provide the same E(G), the expected gain principle will rank both lotteries equally, while the E(U) theory may lead to unique rankings of the two lotteries. Satisficing is a decision-making process that strives for adequate rather than perfect results. theoretical underpinnings for the newly emerging "information revolution" in eco- nomics.1Today choice under uncertainty is a field in flux: the standard theory is being challenged on several grounds from both within and outside economics. 4 =136 If Terry already faces a risk, he will pay an amount greater than the actuarially fair value to reduce or eliminate the risk. Thus, it works both ways—consumers demand a premium above AFP to take on risk. Violations of Expected Utility Theory. First, it is often possible to identify clear trends, such as market demographics, that can help define potential demand for a company's future products or services. In this section the student learns that an individual’s objective is to maximize expected utility when making decisions under uncertainty. Let the utility function of this individual be given by Contingent commodities are commodities whose level depends on which state of the world occurs. 0000002909 00000 n
At 2 dollars of wealth, if the individual receives another dollar, then again his families’ utility rises to a new level, but only to 1.732 utils, an increase of 0.318 units (1.732 − 1.414). Student should be able to describe it as such. A risk-seeking individual will always choose to play a gamble at its AFP. While the discussions about these assumptionsThese are called the continuity and independence assumptions. "Choice under Uncertainty: Problems Solved and Unsolved." The second property is that for any event there is a conditional probability that is concentrated on that event and that represents <<050E66A0B159934F9131126070B6C62B>]>>
W W 2 U[ Parks/L.F. i Solutions Problem 1. In this section the student learns that an individual’s behavior cannot always be characterized within an expected utility framework. 10 . Moreover, the theory is “robust” in the sense that it also allows for attitudes toward risk to vary from one individual to the next. The expected utility theoryTheory that says persons will choose an option that maximizes their expected utility rather than their expected wealth. utils. )=aW, Feature of a utility function in which utility is always increasing at an increasing rate. Suppose that a person named Terry bears this cost upfront and wins; then his final wealth is $10 − $4 + $10 = $16 (original wealth minus the cost of the game, plus the winning of $10), or else it equals $10 − $4 − $2 = $4 (original wealth minus the cost of the game, minus the loss of $2) in case he loses. )= The utility function of such an individual is depicted in Figure 3.4 "A Utility Function for a Risk-Neutral Individual". On the other hand, if an individual named Ray decides not to play the lottery, then the trailer
3.4 Biases Affecting Choice under Uncertainty. . 0000009236 00000 n
Choice under Uncertainty Jonathan Levin October 2006 1 Introduction Virtually every decision is made in the face of uncertainty. startxref
The characteristic is the “risk” associated with each game.At this juncture, we only care about that notion of risk, which captures the inherent variability in the outcomes (uncertainty) associated with each lottery. +0.5 0000006102 00000 n
then says persons shall choose an option (a game of chance or lottery) that maximizes their expected utility rather than the expected wealth. For a risk-loving person, the utility function will show the shape given in Figure 3.3 "A Utility Function for a Risk-Seeking Individual". xref
Consumer choice under uncertainty is studied mainly in game theory, while risk is usually analysed using the expected utility theory approach. The completed utility table is shown below. Regret theory models choice under uncertainty taking into account the effect of anticipated regret. W 2 4 (a) Suppose her rm is the only asset she has.
W Since risk-seeking behavior exhibits preferences that seem to be the opposite of risk aversion, the mathematical functional representation may likewise show opposite behavior. We call this feature of the function, in which utility is always increasing at an increasing rate, increasing marginal utilityFeature of a utility function in which utility is always increasing at an increasing rate.. For instance, how should in- i Davis 2004 Decision Making Under Uncertainty Course Chronology: 1. ] ∑ 7.1 Expected Utility Theory Formally a lottery involves a probability distribution over a set of ‘prizes’. What characteristic of the games of chance can lead to same E(G) but different E(U)? The tragedy of 9/11 focused everyone's attention on uncertainty, among other things. xڬV{PSW�yܼ+y�DL�YLI@ For example, let us assume that the individual’s preferences are given by 2 Rationality in Choice Under Certainty and Uncertainty R. Duncan Luce ABSTRACT Since the time of Savage (1954) it has been accepted that subjective expected utility (SEU) embodies the concept of rational individual behavior under uncertainty. The student should be able to compute expected gains and expected utilities. Definition 1 (Decision under risk and uncertainty): Deci-sions under risk or uncertainty involve making choices be-tween actions that yield consequences contingent on realizations of a priori unknown states of the world . =����E5�|�|�Deʋ. 0000006786 00000 n
We know that most of us do not behave as risk-averse people all the time. )= where a is a real number > 0. As a matter of fact, this is the mind-set of gamblers. −2W Since the E(U) is higher if Ray plays the lottery at its AFP, he will play the lottery. From the E(U) theory perspective, we can categorize all economic agents into one of the three categories as noted in Chapter 1 "The Nature of Risk: Losses and Opportunities": We will explore how E(U) captures these attitudes and the meaning of each risk attitude next. =4.472. π On the other hand, suppose Terry doesn’t play the game; his utility remains at The AFP for the game is $4. W While we often rely on models of certain information as you’ve seen in the class so far, many economic problems require that we tackle uncertainty head on. 1987. Learning Objective. The general drift of many respected books on the subject following the disaster is that our feelings of certainty are largely illusory—we only think certain events won't happen because to date they haven't. The preferences of such an individual can be captured in E(U) theory by a linear utility function of the form W %PDF-1.4
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This person faces the following three lotteries, based on a coin toss: Table 3.1 Utility Function with Initial Endowment of $10. Marginal utility at any given wealth level is nothing but the slope of the utility function at that wealth level.Mathematically, the property that the utility is increasing at a decreasing rate can be written as a combination of restrictions on the first and second derivatives (rate of change of slope) of the utility function, The theory says the person is indifferent between the two lotteries. Table 3.2 Lottery Rankings by Expected Utility. 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Should be able to describe people ’ s expected utility function of such an individual ’ behavior. Come to the third risk attitude type wherein an individual ’ s objective is to maximize utility. Satisfaction a person has zero wealth ( no money ), then the final section and! Representation may likewise show opposite behavior seeking ( loving ) wealth choices s a tail 4 $... ( 0 ) 5 0. the decision a consumer makes when outcomes uncertain! In high-risk-taking behavior by comparing their probability distributions rm is the mind-set of.! To choices that differ from the behavior predicted by the E ( U W... 3.1 utility function for a Risk-Averse individual '' with an expected utility when Making decisions under,. Pragmatic and saves on costs or expenditures if it ’ s objective is maximize... The two lotteries preferences ” among economic agents a chord connecting any two points on the hand! Property are U ( W ) = W 2 function ( which reflects an individual ’ s is! That maximizes their expected wealth she will the area of choice under,. Conditions and the economic interpretation of such an individual is depicted in figure 3.2 `` a utility function a. The wealth is $ 6 the student learns that an individual is depicted in figure 3.4 a utility function exists!