PARADOXES AND VIOLATIONS OF two options (original amounts were in NORMATIVE DECISION THEORY francs): JAY SIMON Choice 1. Defense Resources Management Institute, Naval Postgraduate A: $1 million for sure School, Monterey, California B: 0.10 probability of $2 million YITONG WANG 0.89 probability of $1 million L. ROBIN KELLER 0.01 probability of $0 The Paul Merage School of Business, University of California Irvine, Irvine, In the second survey, respondents were California asked to choose between the following two options: Traditional economic decision theory pro- Choice 2. poses that people behave in certain ways when faced with a well-formulated set C: 0.11 probability of $1 million of alternatives and information. It is a 0.89 probability of $0 normative theory; it suggests that people should act according to certain decision D: 0.10 probability of $2 million rules, but not that they will necessarily do so 0.90 probability of $0 in reality. The standard assumption is that a person has a utility function over all possible The majority of respondents preferred A outcomes, which defines the desirability or to B, and preferred D to C. usefulness of any given scenario. Traditional Choosing A and D appears to be reason- utility theory asserts that in a decision able. However, this example has been called without uncertainty, a person should choose the Allais Paradox, because these two choices the alternative resulting in the highest contradict the idea that people are maximiz- level of utility, and that in a decision with ing expected utility. If we let U0 represent uncertainty, a person should choose the the person’s utility for $0, U1 represent the alternative with the highest expected utility. person’s utility for $1 million, and U2 repre- Detailed explanations are given by von sent the person’s utility for $2 million, then Neumann and Morgenstern and Savage preferring A to B would imply that [1,2]. These ideas are generally sound and can U1 > 0.10U2 + 0.89U1 + 0.01U0, be valuable when used as a guide for decision makers. However, it was initially thought or equivalently, that they were also valuable as a descrip- tive theory. It has since been determined 10 1 U > U + U . that people violate normative decision theory 1 11 2 11 0 in many clear and predictable ways. In this article, we introduce and discuss several of If we examine the second survey choice, these interesting behavioral phenomena. we find that preferring D to C implies that One of the earliest researchers to demon- strate a deviation from expected utility 0.11U1 + 0.89U0 < 0.10U2 + 0.90U0. theory was Allais [3]. He distributed two surveys. In the first survey, respondents This may not seem like a problem were asked to choose between the following at first glance. However, with simple Wiley Encyclopedia of Operations Research and Management Science, edited by James J. Cochran Copyright © 2010 John Wiley & Sons, Inc. 1 2 PARADOXES AND VIOLATIONS OF NORMATIVE DECISION THEORY algebraic manipulation, we see that this is that preferences over gambles are linear in equivalent to probabilities, and in Allais’ example, they clearly are not. 10 1 Another major finding in this area was U1 < U2 + U0. 11 11 discussed by Ellsberg [4]. He conducted an experiment in which subjects were presented This is the exact opposite result of what with a jar. The subjects were told that the we saw in the first survey! It means that jar contains 30 red balls, and 60 balls which regardless of the utilities of the three out- are some unknown mix of yellow and black. comes, choosing both A and D (or both B and They were asked to choose between two C) contradicts the theory of expected utility gambles: maximization. Table 1 reveals the structure of the trans- formation of the first choice between A and B into the second choice between C and D.The A: Receive $100 if the ball is red final column corresponds to the 0.89 chance B: Receive $100 if the ball is black of the $1 million outcome received in either A or B in the first choice. This $1 million was replaced with a $0 in both C and D in The majority of participants chose gamble A. the second choice. Changing the amount of They were then asked to choose between the this ‘‘sure thing’’ of getting the same outcome following two gambles: of $1 million in the 0.89 probability outcome in choice 1 to a different sure thing of $0 in choice 2 should not change the preference C: Receive $100 if the ball is red or yellow order between the options. Savage (2, Pos- D: Receive $100 if the ball is black or yellow tulate 2) called this the sure-thing principle, which must be obeyed under expected utility maximization. Given this choice, the majority of partici- This was one of the first clear demon- pants selected gamble D. As in the Allais’ strations of a systematic violation of the example, choosing both A and D seems rea- principles of expected utility theory. That in sonable. However, this has been called the itself is a very influential finding. However, Ellsberg Paradox, because it is also a viola- we may want to ask why it happened. What tion of expected utility theory. It may not be element of people’s underlying preferences is immediately clear how expected utility is vio- not being captured by expected utility theory? lated, so let us write a participant’s estimated It turns out there are multiple explana- fraction of black balls as B, and observe the tions for Allais’ results, but the main reason following contradiction: they occurred is a phenomenon often referred to as the certainty effect. Reducing the prob- Choosing A implies that ability of a negative outcome from 0.01 to 0 1 2 is usually judged to be a greater improve- U($100) + U($0) > BU($100) ment than reducing the probability of that 3 3 same negative outcome from, say, 0.90 to +(1 − B) U($0). 0.89. Expected utility maximization requires Assuming that U($100) > U($0), this sim- plifies to B < 1/3. Table 1. Allais Paradox Choosing D implies that 0.10 0.01 0.89 A $1 million $1 million $1 million 2 (1 − B) U($100) + BU($0) < U($100) B $2 million $0 $1 million 3 C $1 million $1 million $0 1 D $2 million $0 $0 + U($0). 3 PARADOXES AND VIOLATIONS OF NORMATIVE DECISION THEORY 3 Assuming that U($100) > U($0), this sim- the second question, the expected value of plifies to B > 1/3. losing $4000 with probability 0.8 is −$3200. Thus, there is no consistent set of beliefs It has greater risk than the sure loss, and a (about the fraction of black balls and the frac- lower expected value. Preferring the gamble tion of yellow balls) which allows us, using would imply that the person is risk-seeking. traditional subjective expected utility theory, Utility curves formed over total wealth to prefer both gamble A and gamble D.That are almost always concave, implying risk is, no single probability of drawing a black aversion. ball would lead to this pair of responses. The answers to these two questions were However, there is a simple reason that most startling. People (in aggregate) overwhelm- people choose both gamble A and gamble D: ingly preferred receiving $3000 for sure in these are the choices for which we can eas- question 1, and overwhelmingly preferred ily compute the exact probability of receiving losing $4000 with probability 0.8 in question $100. Ellsberg showed that not only do peo- 2. That is, they were clearly risk-averse in ple display aversion to risk, they also display question 1, and clearly risk-seeking in ques- aversion to ‘‘ambiguity.’’ People tend to pre- tion 2. Kahneman and Tversky also asked fer gambles for which they are confident that similar types of questions in different con- they know the exact probabilities involved. texts, using different probabilities and out- Table 2 shows the structure of this para- comes. In all cases, they found that the major- dox. The choice between A and B is trans- ity’s preferred answer to the question dealing formed by changing the sure thing of $0 when with losses was the opposite of that for the a yellow ball is chosen to a sure thing of $100 question dealing with gains. They referred to when yellow is chosen. So, if A is preferred this phenomenon as the reflection effect. over B,thenC should be preferred over D. Since these people did not all have the Possibly the most groundbreaking work same level of wealth, it would be impossible to bring violations of normative utility the- to construct a utility function over wealth ory to a wide audience is from Kahneman that could incorporate the reflection effect and Tversky [5]. They highlighted two major and accurately represent these subjects’ behavioral violations of expected utility max- responses. Instead, Kahneman and Tversky imization, and demonstrated them using sim- proposed that people evaluate outcomes ple survey results. as gains and losses rather than using the Their first major violation resulted from a resulting total wealth levels. Based on their very simple pair of questions: survey data, they postulated that people tend to be risk-averse over gains and risk-seeking 1. Would you prefer to receive $3000 for over losses. Numerous subsequent studies sure, or $4000 with probability 0.8? have demonstrated this phenomenon, yet it 2.