246		DECISION UNDER RISK AND UNCERTAINTY

History and Notes

Frank H. Knight was among the first economists to recognize the central role of risk in economic decision. In the 1920s he hypothesized that entrepreneurship resulted from a substantial tolerance for risk of the marketplace and the ambiguities of marketing new products. These early works by Knight led the way for later developments in the economics of risk by John von Neumann and Oskar Morgenstern, who proposed expected utility theory, and later Kenneth Arrow, who helped develop the theory. The behavioral work on risk owes much to this foundation. The overwhelming majority of behavioral models of decision under risk or uncertainty build directly on the foundation of expected utility. Many attempt to imitate the intuitive axiomatic definition proposed by von Neumann and Morgenstern, and others seek simple indices of risk behavior that closely resemble those proposed by Arrow in the expected utility context. Moreover, theories of ambiguity and ambiguity aversion were created in direct response to Knight’s distinction between risk and uncertainty. Knight’s own writings about risk and uncertainty clearly foreshadows the importance of psychology in economics.

Biographical Note

GABRIEL DUVAL/AFP/Getty Images

Maurice Felix Charles Allais (1911–2010)

École Polytechnique 1933; École Nationale Supérieure des Mines 1936; Ph.D Université de Paris, 1949; held faculty positions at École Nationale Supérieure des Mines, Université de Paris, Graduate Institute of International Studies in Geneva; held several positions at several national research centers

Maurice Allais obtained his college training in engineering and mathematics in the 1930s and began work in engineering examining issues com-

mon in mining. It was while working as an administrator in the Bureau of Mines Documentation and Statistics beginning in 1941 that Allais published his first academic research, a series of pieces examining, among other things, the foundational work of welfare economics. Paul Samuelson wrote that had these works been written in English “a whole generation of economic theory would have taken a

Thought Questions		247

different course.” Allais’ student Gerard Débreu won the Nobel prize for work extending the results derived by Allais. Allais’ academic work in the field of economics ranged widely, creating the foundation for many of the most widely used macroeconomic models, and it also examined issues of energy use, mining, and other areas of applied economics. Some believe it is odd that he is best recognized for the Allais’ paradox, because it was somewhat outside of the main body of his research. Nonetheless, his publication of the Allais’ paradox in many ways marked the first true behavioral economics publication, inspiring later authors such as Daniel Kahneman and Amos Tversky. Though he would probably not consider himself a behavioral economist, the question he posed sparked widespread interest in alternative theories of decision under risk and eventually in alternative theories of decision making in all contexts. His contributions were not limited to economics. Allais also published papers in engineering, theoretical physics, and history. In 1988 he was awarded the Nobel Prize in economics for his contributions to the study of efficient markets.

T H O U G H T Q U E S T I O N S

1.Consider the utility function given by Ux = lnx and the set of gambles with the possible outcomes $10, $20

and $30. For each exercise, it may be useful to use a spreadsheet or other numerical tools.

(a)Graph the indifference curves implied by expected utility in the Marschak–Machina triangle. What is the slope of the indifference curves?

(b)Now suppose that the decision maker maximizes probability weighted utility, with the weights given by

	p0.7
π p =	i			.
		1
	p0i .7 + 1 − pi	0.7	0.7

Graph a few examples of indifference curves. What does the probability weighting do to the shape of these curves relative to those in part a?

(c)Repeat the exercise in b, now assuming the decision maker maximizes rank-dependent expected utility. Thus, now the probability weighting function is applied to cumulative probabilities.

(d)Finally, consider the regret utility function given by

U x, y =	x − y 2	if	x ≥ y
	− x − y 2	if	x < y

Plot the implied indifference curves supposing the alternative choice would yield $19 with certainty and compare the shape of these curves to those of the

other models considered. How do these curves change when the foregone gamble is altered?

2.Suppose that there were three possible states of nature as represented in the table below:

States of Nature	1	2	3
Probability	0.35	0.4	0.25
Gamble 1	$1,000	$2,000	$3,000
Gamble 2	$1,800	$1,800	$1,800
Gamble 3	$2,500	$1,500	$1,500

(a)What conditions would be required for the regret theory utility function to predict preference cycling when choosing between pairs of the three possible gambles? Graph an example of a function that would satisfy these conditions.

(b)Is it possible for preference cycling to occur given the same choices under expected utility maximization with probability weights? Show why or why not.

(c)Would it be possible for preference cycling to occur given the same choices under rank-dependent expec- ted-utility maximization? Show why or why not.

3.Confirm that the common ratio effect as found in Example 9.4 could be explained either by probability weighting or by regret theory. To do this, find utility

	248		DECISION UNDER RISK AND UNCERTAINTY

and weighting functions that satisfy the models and that lead to the choices found in Example 9.4.

4.Consider two gambles, each with outcomes $10, $20, $30,

and $40. Gamble	1 has probabilities p10, p20, p30,
1 − p10 − p20 − p30	for these outcomes and gamble 2

has probabilities q10, q20, q30, and 1 − q10 − q20 − q30. Suppose that Gamble 2 stochastically dominates Gamble 1. Thus, p10 > q10, p10 + p20 > q10 + q20, and p10 + p20 + p30 > q10 + q20 + q30. Show that anyone who maximizes rank-dependent expected utility must prefer Gamble 2 to Gamble 1.

5.Suppose that you are a policymaker considering instituting a tax to combat climate change. The current climate research is conflicting as to the probability that carbon emissions will lead to catastrophic climate

change. Suppose that the probability of a catastrophic climate change is equal to 0.3 × cϕ, where c represents

carbon dioxide emissions and, depending on which scientists you listen to, ϕ can be as low as 0.1 or as high as 0.9. As a policymaker you wish to maximize expected social welfare. If a catastrophic climate

change occurs, social welfare will be equal to 0, no matter how much other production occurs. If a

R E F E R E N C E S

Allais, P.M. “Le comportement de l’homme rationnel devant le risque: critique des postulats et axiomes de l’école Américaine.”

Econometrica 21(1953): 503–546.

Birnbaum, M., and J. Navarrete. “Testing Descriptive Utility Theories: Violations of Stochastic Dominance and Cumulative Independence.” Journal of Risk and Uncertainty 17(1998): 49–78.

Camerer, C.F., and T.-H. Ho. “Violations of the Betweenness Axiom and Non-Linearity in Probability.” Journal of Risk and Uncertainty 8(1994): 167–196.

Ellsberg, D. “Risk, Ambiguity, and the Savage Axioms.” Quarterly Journal of Economics 75(1961): 643–669.

Ghirardato, P., F. Maccheroni, and M. Marinacci. “Differentiating Ambiguity and Ambiguity Attitude.” Journal of Economic Theory 118(2004): 133–173.

Gilboa, I., and D. Schmeidler. “Maxmin Expected Utility with NonUnique Prior.” Journal of Mathematical Economics 18(1989): 141–153.

Guthrie, C. “Better Settle than Sorry: The Regret Aversion Theory of Litigation Behavior.” University of Illinois Law Review 1999 (1999): 43–90.

Kahneman, D., and A. Tversky. “Prospect Theory: An Analysis of Decision under Risk.” Econometrica 47(1979): 263–292.

catastrophic climate does not occur, then social welfare

is given	by the	profit of the emitting industry,
π = p − t	y − k y	= 3 − t y − 0.15y2, where y is

output, p is the output price, k. is the cost of production, and t is the tax you impose. The firm chooses y to maximize profits, according to y = 3 − t0.3. Carbon dioxide emissions are given by c = y.

(a)Suppose you display α-maxmin expected-utility preferences, with α = 1 (fully ambiguity averse).

What tax will you choose? What level of social welfare will be realized if a catastrophic climate change is not realized? You may use a spreadsheet to determine the answer if needed.

(b)Suppose α = 0 (fully ambiguity loving). What tax will you choose? What level of social welfare will be realized if a catastrophic climate change is not realized?

(c)Suppose that a definitive study shows that ϕ = 0.2. What is the expected social welfare–maximizing tax? What is the resulting expected social welfare and the social welfare resulting if no catastrophic climate change is realized?

Kahneman, D., and A. Tversky. “Rational Choice and the Framing of Decisions.” Journal of Business 59(1986): S251–S278.

Leland, J. “Generalized Similarity Judgments: An Alternative Explanation for Choice Anomalies.” Journal of Risk and Uncertainty 9(1994): 151–172.

Leland, J. “Similarity Judgments in Choice Under Uncertainty: A Reinterpretation of the Predictions of Regret Theory.” Management Science 44(1998): 659–672.

Lichtenstein, S., and P. Slovic. “Reversals of Preference Between Bids and Choices in Gambling Decisions.” Journal of Experimental Psychology 89(1971): 46–55.

Loomes, G., C. Starmer, and R. Sugden. “Observing Violations of Transitivity by Experimental Methods.” Econometrica 59(1991): 425–439.

Machina, M.J. “‘Expected Utility’ Analysis Without the Independence Axiom.” Econometrica 50(1982): 277–323.

Marschak, J. “Rational Behavior, Uncertain Prospects, and Measurable Utility.” Econometrica 18(1950): 111–141.

Preston, M.G., and P. Baratta. “An Experimental Study of the Auc- tion-Value of an Uncertain Outcome.” American Journal of Psychology 61(1948): 183–193.

References		249

Quiggin, J. “A Theory of Anticipated Utility.” Journal of Economic Behavior and Organization 3(1982): 323–343.

Rubinstein, A. “Similarity in Decision-Making Under Risk (Is There a Utility Theory Resolution to the Allais Paradox?)” Journal of Economic Theory 66(1995): 198–223.

Starmer, C. “Testing New Theories of Choice under Uncertainty using the Common Consequence Effect.” Review of Economic Studies 59(1992): 813–830.

Von Neumann, J., and O. Morgenstern. Theory of Games and Economic Behavior. Princeton: Princeton University Press, 1944.

Zeelenberg, M., and Pieters, R. “Consequences of Regret Aversion in Real Life: The Case of the Dutch Postcode Lottery.” Organizational Behavior and Human Decision Process 93(2004): 155–168.

Advanced Concept

The Continuity Axiom

Decision makers who adhere to the independence, order, and continuity axioms behave as if they maximize the expected (or mean) utility of wealth resulting from their choices. The order and independence axioms are defined in the text.

Continuity Axiom

If A B C then there is exactly one value r such that neither B nor the compound gamble that yields the gamble A with probability r and the gamble C with probability 1 − r are preferred (we write this as rA + 1 − rC B). Further, for any p > r the compound gamble that yields the gamble A with probability p and the gamble C with probability 1 − p is preferred to B (we write this pA + 1 − pC B), and for any q < r, B is preferred to the compound gamble that yields the gamble A with probability q and the gamble C with probability 1 − q, or B qA + 1 − qC.

The continuity axiom imposes the notion that increasing the probability of a preferred gamble (and thus lowering the probability of inferior gambles) will increase one’s preference for the gamble. Lowering the probability of the preferred gamble has the opposite effect. Suppose that A, B, and C are degenerate (in other words, there is probability 1 of receiving a particular outcome). For example, suppose A yields $100 with certainty, B yields $50 with certainty, and C yields $0 with certainty. Then continuity implies that I can find only one probability r, maybe 0.6, such that I am indifferent between the gamble yielding 0.6 probability of $100 and a 0.4 chance of $0 and the gamble that yields $50 with certainty. Raising the probability of getting $100 and lowering the probability of getting $0 will lead the gambler to take the compound gamble; lowering the probability of getting $100 and raising the probability of getting $0 will lead the gambler to choose the $50 instead.

		Prospect Theory and Decision
	10
		under Risk or Uncertainty

On October 12, 2007, American investors celebrated a new high-water mark as the Dow Jones Industrial Average hit an all-time high of 14,093. Stocks were booming, and many people were buying. Home values had also increased substantially over the previous decade, producing substantial wealth for those who had invested in home ownership. Many had taken out large loans to buy up real estate as an investment, counting on continuing increases in home prices to produce value. With a large cohort of baby boomers preparing for retirement, many had built substantial wealth through stock and real estate investments and felt well prepared financially for a long and comfortable retirement. That’s when things started to go sour.

Housing prices had begun to decline slightly between August and October, but by November the decline in home values had become too much for investors to ignore. Stock prices started a slow decline also. Although most prognosticators clearly predicted that further losses in real estate and stocks were ahead, optimistic investors began to reason among themselves, “I can’t get out now, I will lose money.” Sentiments such as “I don’t want to sell at a loss. I will wait until the price goes back up to where I purchased, and then get out,” were common. In January 2008, real estate prices started a steep decline. Many homes were now valued at less than the outstanding debt on the mortgage. Those who had counted on the money they could make upon selling their property to cover the mortgage would now be forced to default. Banks that had made these loans were in trouble. After a short period of stability, the housing market went into a steep decline, losing almost 30 percent between August 2008 and February 2009. The stock market took notice. By October 2008, stock prices began to crash, losing as much as 18 percent in a single day. Investors were finally ready to sell even at a loss. By March 2009, the Dow Jones Industrial Average was only 6,627. Stocks had shed nearly 53 percent of their value. Massive amounts of wealth had been destroyed, and baby boomers were left to come to grips with the fact that they might have to put off retirement for several years and perhaps cut back on their planned expenses.

Often the plan to buy low and sell high doesn’t quite work out that way. Why are investors so willing to hold on to a losing investment when it becomes clear it will not perform? Although few would consider making new investments in what appears to be a losing investment, many are loath to shed losing investments if it means realizing a loss. Loss aversion has profound effects on the way decision makers deal with risky decisions. In many ways, the introduction of loss aversion to the economics of risky choice has been the

250

Prospect Theory and Decision under Risk or Uncertainty		251

cornerstone of behavioral economics. In 1979, Daniel Kahneman and Amos Tversky introduced a model of risky choice based on loss aversion, which they called prospect theory. For many economists, prospect theory was their first exposure to behavioral models. Prospect theory makes powerful predictions regarding behavior under risk, providing some of the most compelling evidence of behavioral biases in economic choice.

EXAMPLE 10.1 The Reflection Effect

Suppose you were given $1,000 in addition to whatever you own. Then choose between the following two gambles:

Gamble A:	Gamble B:
$1,000 with probability 0.5	$500 with certainty
$0 with probability 0.5

Daniel Kahneman and Amos Tversky asked 70 participants to decide between these two options, and 84 percent chose option B. Choosing option B is not terribly surprising. The expected value of Gamble A is 0.5 × 1,000 + 0.5 × 0 = 500, the value of Gamble B. Thus, anyone who is risk averse should choose Gamble B over Gamble A. A different set of 68 participants were asked the following question:

In addition to whatever you own, you have been given $2,000. You are now asked to choose between

Gamble C:	Gamble D:
− $1,000 with probability 0.5	− $500 with certainty
$0 with probability 0.5

In every respect, this is the same choice as that between Gamble A and Gamble B.

Applying expected utility theory, the expected utility of Gamble A is given by

0.5 × U w + 1000 + 1000	+ 0.5 × U w + 1000 , where w represents the amount of wealth
you currently possess.	Alternatively, Gamble C is represented by 0.5 × U w +

2000 − 1000 + 0.5 × Uw + 2000. These are clearly identical. Moreover, the expected utility of Gamble B is Uw + 1000 + 500, which is identical to the expected utility of Gamble D, Uw + 2000 − 500. Thus, anyone who behaves according to expected utility theory and chooses Gamble B must also choose Gamble D. Yet, 69 percent of participants chose Gamble C when given the choice. On average, participants were risk loving when choosing between C and D but risk averse when choosing between A and B. This is a special case of the reflection effect, the observation that risk preferences over gains tend to be exactly opposite of those over identical-magnitude losses.

Kahneman and Tversky discovered this effect by asking participants to choose between a series of similar gambles with the money outcomes reflected around the origin (i.e., positive values made negative), though without the wealth adjustment

	252		PROSPECT THEORY AND DECISION UNDER RISK OR UNCERTAINTY

presented in the previous problem. In each case, the majority of participants were risk averse when choosing between gains but risk loving when choosing between losses. The results of these choices are presented in Table 10.1. Without the wealth adjustment, these gambles do not truly demonstrate a violation of expected utility. For example, in the choice between the first two positive gambles in Table 10.1, choosing B implies that 0.8 × Uw + 4,000 + 0.2 × Uw < Uw + 3,000. Choosing A in the corresponding negative gamble implies 0.8 × Uw − 4,000 + 0.2 × Uw > Uw − 3,000. Because the utility function is evaluated at different points in the negative and positive gambles, no contradiction is implied. However, if this holds at every wealth level (as the results suggest), a contradiction is implied. Utility of wealth functions cannot be both concave and convex over the same range of outcomes (the violation noted in the first example in this section, which makes the wealth adjustment). Nonetheless, this is what the reflection effect implies.

Table 10.1 Reflected Gambles*

Gamble A			Gamble B			Percent Choosing A
Probability	Outcome		Probability	Outcome		Positive Gamble	Negative Gamble
0.800	±$4,000		1.000	±$3,000		20	92
0.200	±$4,000		0.250	±$3,000		65	42
0.900	±$3,000		0.450	±$6,000		86	8
0.002	±$3,000		0.001	±$6,000		27	70

*See Example 10.1. For each gamble, the remaining probability is assigned to the outcome of $0. Gambles were originally denominated in shekels.

EXAMPLE 10.2 The Isolation Effect

Kahneman and Tversky also discovered another curious anomaly illustrated in the following choice problem.

Consider the following two-stage game. In the first stage, there is a probability of 0.75 to end the game without winning anything and a probability of 0.25 to move into the second stage. If you reach the second stage you have a choice between

Gamble E:	Gamble F:
$4,000 with probability 0.8	$3,000 with certainty
$0 with probability 0.2

Your choice must be made before the game starts, that is, before the outcome of the first stage is known.

Of the 141 participants who faced this problem, 78 percent chose Gamble F. If we choose Gamble E, we have a 0.25 probability of moving to the second stage and then an