A Theory of Risk without Expected Utility

Transcription

1 A Theory of Risk without Expected Utility By Hak Choi * Abstract This paper challenges the use of expected value concepts - including expected return, expected utility, non-expected utility and weighted utility - in the study of risk. It proves that all such concepts must lead to the rejection of any gambling or insurance proposal. Consequently, theories applying them become futile. Instead, this paper seeks to explain the risky behaviors by the Central Limit Theorem along with the indifference curve. It also corrects many paradoxes and biases, and shows that all gambling and insurance behaviors are rational. Keywords: Risk, Inter-temporal Decisions, Insurance, Betting. (JEL: D81, D91, G22) * Correspondence: Dr. Hak Choi, Department of International Business, Chienkuo Technology University, No.1, Chiehsou N. Rd., Changhua City, (500) Taiwan. Tel.: , hakchoi@gmail.com 1

2 A Theory of Risk without Expected Utility The literature on risk has two main streams of development. The first one is the expected utility (EU) theory, as started by Daniel Bernoulli some 250 years ago (Arrow, 1963, p.959; Starmer, 2000, p.333), and elevated by von Neumann and Morgenstern (1944/1947) through their famous book: Theory of Games and Economic Behavior. Friedman and Savage (1948) and Markowitz (1952) then develop the S-shape utility curve, while Mossin (1968) applies it to insurance specifically. Another stream of development is the non-expected utility (NEU) theory, as initiated by Allais (1953). It grows out of the dissatisfaction with the simple EU mathematics, which see some violations to the utility axioms. Though various calibrations of the so-called weighted utility (WU) function explain different versions of the Allais paradox, this theory looks more restrictive than the EU one in that it needs many kinds of arbitrary weights, e.g. utility weights, probability weights, decision weights and rank-dependent weights (Schoemaker, 1982, p.538; Birnbaum, 1999; Starmer, 2000). Since expected utility and weighted utility are mathematically equivalent, these two streams of development are also equivalent. No other field of studies has so many unsolved paradoxes and biases as the study of risk, despite its massive number of publications. There are at least two reasons for this disorder. In the first place, the existing theories use one-dimensional method: utility is a function of wealth. While explaining some phenomena, they also create more anomalies. Secondly, their decision rule is a simple number, EU or WU. Games or policies of extremely different payoffs may have the same number. They mean the same thing to the theorists, but they mean a lot of difference to the people concerned. It will be shown that such simple number actually goes against any gambling or insurance proposal. This renders these theories completely futile. Fortunately, the experiments of the NEU problems suggest a new explaining option. These experiments reveal some sporadic behaviors with percentage result. It is when a customer bets once in a while, or has one or two objects for insurance, when expected value is not applicable, only then is risk analysis required. This paper combines the Central Limit Theorem with standard indifference curve to provide an alternative, and perhaps better, explanation to the sporadic behaviors. Since no expected utility is involved, this explanation will be called the theory of risk without expected utility, or NoEU (no expected utility) for short-hand. 1 1 Although some NEU models also provide similar percentage solution, they require further assumption of a non-economic, logistic function, in addition to the weighted utility function (Birnbaum, 2005, p.270). Some other studies (Neilson,1994; Miyamoto and Wakker, 1996; and Nielsen and Jaffray, 2006) also name 2

3 The following section reviews the conventional EU and NEU theories and re-addresses them in a two-dimensional framework. Section II proves that the application of the EU or WU measure must lead to the rejection of any gambling or insurance proposal. Section III develops the main model of the paper, while Section IV corrects some anomalies. Some concluding remarks will be affixed in Section V. I. The EU and NEU Theories in Two Dimensions According to the conventional EU theory, a person, say John, will bet $k for the eventual win of $(q+1)k, where q is the per dollar net payoff, if his expected utility with betting is higher than his utility without betting. This result requires that he has increasing marginal utility of wealth. Conversely, another person, say Mary, will buy an insurance policy with $k premium to cover the eventual loss of $(q+1)k, if her expected utility without insurance is lower than her utility with insurance. This can happen if she possesses diminishing marginal utility. The EU theory looks very restrictive, because it requires two different cardinal utility functions for the two problems. In addition, the EU theory considers problems that involve only one choice between two alternatives. In contrast, the NEU theory starts with two choices, and shows that the pair of decisions according to the EU theory does not conform to experiment results. Consider the following example from Kahneman and Tversky (1979, p.266): Problem 3: A: (4,000,.80) or B: (3,000). N=95 [20] [80]* Problem 4: C: (4,000,.20) or D: (3,000,.25). N=95 [65]* [35] In problem 3, people face the choice between A (having $4,000 at a probability of 0.80) and B (having $3,000 for sure). Among the 95 participants, 20% of them choose A while 80% of them choose B. The choice for the certainty by the majority can be explained by decreasing marginal utility. Problem 4 can be similarly described. The probabilities of Problem 4 are obtained by dividing those of Problem 3 by 4. According to the NEU theory, if a person prefers B, he/she must also prefer D, for U(B)>0.8U(A) implies 0.25U(B)>0.2U(A). As the latter is not observed in Problem 4, this is regarded as a violation themselves theory without expected utility, but they actually mean non-expected utility. 3

4 to the von Neumann-Morgenstern ratio independence or common ratio axiom. The NEU way out of this paradox is to employ a subjective probability function, e.g. γ cp S( p) = γ γ cp + (1 p). For illustration, Birnbaum (1999) sets c=γ=0.4, to obtain, S(p) A =1, S(p) B =0.41, S(p) C =0.19, and S(p) D =0.20. Assuming u(x)=x, the observed preference reversal is explained with S(p)u(x). However, the NEU theory requires many non-economic, weighted utility functions, each for a limited number of problems. It is therefore more restrictive than the EU theory. The conventional EU and NEU theories can be re-addressed in a more general two-dimensional framework (Yaari, 1965; Rosett, 1967; Ehrlich and Becker, 1972; Rothschild and Stiglitz, 1976). Figure 1 is the indifference map of John, where the x-axis measures his present wealth and the y-axis his future wealth. Point B(x 1, y 1 ) is his initial wealth position without betting, while C(x 2, y 2 ) the position when he wins a game with betting. Specifically, x 2 x 1 = k with k>0 being the betting money, while y 2 y1 = ( q + 1) k the payoff. Subsequently, Point A(x 2, y 1 ) is the wealth position when he bets and loses. No interest rate is required, as the rate of exchange is determined by the nature of the game. A similar graph of insurance is Figure 2. Point B is the initial wealth position of Mary with insurance. Without insurance her present wealth will increase due to the premium saved by x 2 x1 = k > 0, and her wealth position becomes Point A when no accident happens. But, she also faces the probability of a larger loss in future wealth by y 2 y1 = ( q + 1) k < 0, and her wealth position will drop to Point C when an accident happens. It is obvious that k<0 in this case. Again no interest rate is involved, as the rate of exchange is also determined by the terms of the policy. However, insurance seekers must understand that the future may be relatively long. As indicated in Figures 1 and 2 respectively, John will choose to bet if his expected utility with betting is higher than his utility without betting, i.e. EU > U (B), while Mary will choose to be insured if her expected utility without insurance is lower than her utility with insurance, i.e. EU < U (B). EU is defined as: EU d 1 = U ( A) + U ( C), (1) d + 1 d + 1 where d is the odds-number or any other appropriate weight. This two-dimensional model uses exactly the same rationale as the conventional EU one, but it does not require any cardinal assumption. It also shows that people with diminishing marginal utility, as implied by the indifference curve, can also be gamblers. The definition of risk-loving or risk averse therefore needs reconsideration. 4

5 <Insert Figure 1 Here> <Insert Figure 2 Here> This 2-D approach can also reproduce the traditional one-dimensional models. In Figure 3 the horizontal axis measures the total wealth of John, which is made up of x plus y, while the vertical axis measures his utility level drawn from the two-dimensional model. Point B is the initial position without betting. With betting the utility change from B to A is negative due to the loss of the betting money, -k, while the change from A to C is positive to reflect the wining of the payoff, (q+1)k. Both changes are subject to diminishing marginal utility, and the curves are concave from below, but the degree of concavity may be different. Expected utility lies somewhere along the straight-line joining A and C (not shown). In order to induce John to gamble, it must be higher than U(B). <Insert Figure 3 here> Figure 3 shows the situation of a decision to bet. If the three utility positions are joined consecutively (A-B-C) together by the dotted lines, it looks as if there is increasing marginal utility of wealth, which is obviously a false image. A similar graph for insurance is Figure 4. Point B is the initial position with insurance. Without insurance Mary s wealth will increase by the premium saved, -k, but she also faces the probability of a greater loss, (q+1)k (k is negative in this case). Hence, her utility first increases from B to A, then drops from A to C. It is less of a dispute that both utility changes are diminishing, but again the concavity of the curves may be different. The expected utility without insurance lies somewhere along the straight-line joining A and C (not shown). She will choose to be insured, if it is lower than U(B). The situation of such a decision is depicted in Figure 4. Though the consecutive joining of A-B-C also shows diminishing marginal utility, it is not the true picture either. <Insert Figure 4 here> The two-dimensional model can also explain the NEU experiments. For example, Problem 3 of Kahneman and Tversky (1979) can be illustrated with an indifference curve cutting through the horizontal line, which represents the certainty. Most people prefer the certainty, B, to any amount of expected payoff in future wealth because Point A lies underneath the indifference curve due to its convexity property, as revealed in Figure 5. Whereas Problem 4 means a choice between two expected values on the same vertical line: the choices between two prospects upon giving up a certain amount of present wealth. The 5

6 decision for the higher one, C, is straightforward, as also illustrated in Figure 5. As revealed, the same graph can handle two, or more choices. It also indicates that weighted utility is not necessary for the NEU problems; expected payoff together with the indifference curve is enough to explain the results. In addition, most other anomalies created by the one-dimensional models will also disappear when they are displayed in two dimensions. 2 <Insert Figure 5 Here> II. The Futility of the conventional EU and NEU Theories The two-dimensional framework has made the EU and NEU theories more practical by discarding their restrictive assumptions. However, there is still a serious pitfall with these theories, one-dimensional or two: they all melt down to a simple number, EU or WU, but a simple number hides a lot of things. Games or policies of extremely different payoffs can have the same number. They might mean the same thing to the theorists, but they mean a lot of difference to the real people. Hence, these theories have only limited use, if any. This section first explains the relationship between expected return and expected utility, next it shows what expected value really means according to the law of large numbers, and then it proves that the expected measures cannot explain any risky behavior at all. The Expected Return Expected return, ER, the nature of a game or policy, can be defined as: q + 1 ER = k + k, (2) d + 1 or equivalently, d 1 ] ER = ( k) + qk. (2 ) d + 1 d + 1 According to (2) it is the balance of giving up $k of betting money for obtaining (q+1)k/(d+1) of expected payoff, where 1/(d+1) is the probability of the win; alternatively 2 The conventional EU and NEU theories look neo-classical. Neo-classical theory is characterized by constant returns to scale and the one-dimensional explanation of economic activities. A typical example is Solow (1956). This view is shared by Schoemaker (1982, p.533). 6

7 defined in (2 ) it is the weighted average of loss and win. It is also the balance of saving the premium against the expected loss in the case of insurance. If q=d, ER=0 for a fair game or policy. Most games are unfair with ER<0, while most policies are unfair with ER>0. This ER can also be expressed in the two-dimensional framework. For example, John bets (x-x 1 ) of present wealth in the expectation for an increase in future wealth by q + 1 ( x d + 1 wealth, E(y): x 1 ). Adding the latter to the initial future wealth yields the expected future ( q + 1) E( y) = y1 ( x x1). (3) ( d + 1) Depicted in the two-dimensional graph of Figure 6, this equation is represented by the line of the expected return in the case of gambling. In the case of insurance it is represented by the line of expected loss, as depicted in Figure 7. In both cases, the slope of the line is y ( q + 1) negative, =, whose absolute value measures the fairness of the game or policy. x ( d + 1) Most games and policies are unfair with the absolute slope less than one, for q<d. <Insert Figure 6 Here> <Insert Figure 7 Here> Expected Utility versus Expected Return In this two-dimensional framework, expected return and expected utility can be made comparable. Associated with any point on the line of expected return or loss there is a utility level. In particular, when x 2 =x 1 -k, U(ER)=U[x 2, E(y)]. For the same k, the expected utility of (1) can be expressed as: EU d 1 = U ( x2, y1) + U ( x2, y2). d + 1 d + 1 Then, by separability (Blackorby, et al., 1978), 3 d d 1 1 EU = U ( x2 ) + U ( y1) + U ( x2 ) + U ( y d + 1 d + 1 d + 1 d ) 7

8 d 1 = U ( x2 ) + U ( y1) + U ( y2 ). d + 1 d + 1 If the utility of y is linear, the last two terms can be expressed as: d 1 d 1 E [ U ( y)] = U ( y1) + U ( y2) = U[ y1 + y2 ] = U[ E( y)]. d + 1 d + 1 d + 1 d + 1 Along with U(x 2 ) it means EU = U (ER). However, as the indifference curve is convex, the marginal utility must be diminishing with respect to y. Hence, the weighted average of two utilities must be lower than the utility of the weighted average of two wealth levels, i.e. d 1 d 1 E [ U ( y)] = U ( y1) + U ( y2) < U[ y1 + y2 ] = U[ E( y)], d + 1 d + 1 d + 1 d + 1 and EU < U (ER). 4 Connecting the expected utility to the initial wealth position is the line of expected utility. Accordingly, this line may have a flatter slope than the line of expected return in the case of gambling, as depicted in Figure 6; whereas in the case of insurance this line may have a steeper slope than the line of expected loss, as shown in Figure 7. The Law of Large Numbers Expected value, return or utility, can be easily calculated from any two, or more, numbers, but its real meaning requires further investigation. For example, if John bets once or twice, the result is not necessarily an expected value. It is only when he bets many numerous times then he will end up with the expected value. This is the law of large numbers. This law states that when the number, n, of trials in an identical game with the v if successful binomial outcomes z =, where v=(q+1)k, approaches infinity, the 0 otherwise probability that the absolute difference between the sample mean, z, n and the expected value, E(z)=(q+1)k/(d+1), being greater than or equal to a given constant, ε, approaches zero (Grinstead and Snell, 1997, pp ), i.e. 3 Most utility functions are separable in their augments, e.g. the Cobb-Douglas and the CES functions. 4 For example, given y 1 =100, y 2 =200, d=1, E(y)=150, when U(y)=y 0.5, U(E[y])=4.50 but E[U(y)]=0.5* * =4.44, i.e. E[U(y)]<U(E[y]). 8

9 { E( z) ε} 0 Pr z n n. (4) In the business involving risk, the constant is represented by the profit of the game or d q policy, i.e. ε = πe(z), where π = is the profit rate. Hence, the law becomes: d +1 Pr { E( z) E( z) } 0 z n π. (5) Given large number and as long as d>q, this law is effective, which means there is no chance that the sample mean can be greater than the expected value by the profit, or there is no chance a customer can out-play the hosts or underwriters. If John bets very often, he must end up closely with the expected value, according to this law. For example, if the payoff of flipping a fair coin is $4, the sample mean of flipping it a thousand times will be close to the expected payoff of $2, as verified by the simulation in Figure 8. But then, if a host asks for a betting money of $3 for each flip, no gambler should play this game that often. Likewise, the insurance policies in large number with an expected loss of $2 requiring a premium of $3 each should also be turned down. <Insert Figure 8 Here> Since positive profit means q<d, the line of expected return in Figure 6 must have an absolute slope less than 1. Corresponding to the example of flipping coin, the slope is -2/3; if John bets $3, he gets $2 back on average. However, there is always a risk-free investment with absolute slope at least equal to one. It means that the rate of return of the risk-free investment is higher than that of gambling. Hence, all customers must reject any betting invitation in favor of the risk-free investment. Similarly, as far as expected loss is concerned, all customers should choose not to be insured. As indicated in Figure 7, if Mary chooses not to be insured, she saves $3 of present wealth and has an expected loss of $2. If she deposits the $3 in a bank, she will have more than $3 future wealth back. She should choose the risk-free investment too. Expected Utility and the Law of Large Numbers Expected return cannot explain gambling or insurance, but neither can expected utility. Since the line of EU has a flatter slope than that of the expected return in the case of 9

10 gambling as proved above, the preference for the risk-free investment is more evident. Although in the case of insurance the line of EU may have a steeper slope than that of the expected loss, the conclusion remains the same. Suppose the line of EU has a slope steeper than that of a risk-free investment, but it is still an expected result. Then, there always exists some other investment with even better expected result and its line will have even steeper slope. Consequently, Mary will prefer the other investment. WU may have different value from EU, but it is still a simple number. Again, it will also have the same consequences. Hence, neither ER, EU nor WU can explain gambling or insurance. The EU and NEU theories seem capable of explaining why people bet or choose to be insured, only if there is no comparison. Once an alternative is present, they become futile. This world does have a staggering variety of investment opportunities. The study of risk needs a total revision in its logic. Though the NEU models start with the percentage problems, their decisions are made on the simple number of some weighted utility. Once the criterion is met, all people must choose the indicated prospect. Hence, the percentage problems are not really explained. They are correct in the problem setting, but not in the solution. The succeeding passage will continue with the correct setting to search for the correct solution. III. The Theory of Risk for Activities without Expected Utility According to the preceding reasoning, the study of risk should abandon the expected value concepts, either ER, EU or WU. 5 Expected value is the approaching result of numerous repetitive trials. However, the daily problems of gambling and insurance do not allow many repetitions; they are sporadic behaviors with uncertain results, as faced by the experiment participants in the NEU models. It is such sporadic behaviors, not their expected results, that need a theory for explanation. An attempt to offer such a theory is explored here. The Gain/Loss Potential from Taking Risk If John bets only once in a while, he faces a gain potential (not potential gain) of future wealth with range from the horizontal line of y 1 up to the line of maximum payoff, as indicated by the shaded area in Figure 9. Similarly, if Mary has only one or two objects not being insured, she is exposed to the loss potential (not potential loss) marked by the shaded area in Figure 11. The loss potential 5 The author is indebt to Duncan Luce for pointing out the futility of expected utility and for inspiring him to 10

11 may include points of bankruptcy with y<0. When John is about to go for a game, he will move from Point B in Figure 9 along the y 1 -line horizontally up to a point determined by k. Starting from this new position, a range of gain potential in the vertical direction is opened up to him. Although the binomial outcome is either 0 or v, there is a subjective sampling distribution attached to the range reflecting John s average win. This distribution varies with the sample size. When the latter is large enough, the distribution approaches normal according to the Central Limit Theorem. Given the sampling distribution, the chance that John will bet depends on how minimum he is satisfied with an average win. Such minimum desired win is revealed by his indifference curve associated with the initial wealth position. For example, when this indifference curve happens to cut the vertical line at the expected return, there is 50% chance that John will bet. Corresponding to Figure 10, if John is satisfied with an average win of $2, there will be 50% chance that he will bet in a fair game with a payoff of $4. Alternatively stated, if all potential customers have the same sampling distribution and indifference curve, there will be 50% of them who will bet. Since people have different sampling distributions and indifference curves, the actual result may not be exactly 50%, but it must still be a percentage. This NoEU model thus reduces the number of gamblers substantially from the conventional EU/NEU solution. It also reproduces the NEU percentage setting. Following the same reasoning, when the indifference curve cuts the vertical line below (or above) the expected return, the chance is more (or less) than 50%. In the extreme, when the indifference curve is a horizontal line, which means John dislikes present wealth, the chance will be 100%. Contrarily, when the slope of his indifference curve is higher than that of the maximum payoff line, which means he loves present wealth very much, the chance becomes 0%. Given an indifference curve and a distribution function, exact figure of the chance can be calculated. 6 <Insert Figure 9 here> <Insert Figure 10 here> <Insert Figure 11 here> pursue territory beyond expected utility. 6 Now, gamblers have transformed the objective probability distribution of the game to the subjective probability distribution using the Central Limit Theorem. To certain extent, this approach is also a stochastic one. The author is indebted to Mark Machina for pointing this out during the 2007 Bayesian Research Conference. 11

12 Proposition 1 The chance that a person will bet is more than, equal to or less than 50%, if he/she can be satisfied with a minimum average win smaller than, equal to or larger than the expected win respectively. The percentage can also mean the percentage of people, if all potential customers have the same sampling distribution and indifference curve. Similar chance that Mary will purchase an insurance policy can also be worked out. In this case, the chance is represented by the vertical segment from the indifference curve down to the maximum loss, as marked by the bold line in Figure 11. The indifference curve now represents her maximum tolerable loss in return for the premium saved. When the indifference curve cuts the vertical line above (or below) the expected return, which means she can tolerate a smaller (larger) maximum loss, the chance is more (or less) than 50%. In the extreme, when the indifference curve is a horizontal line, the chance will be 100%. Contrarily, when it is a vertical line, the chance becomes 0%. Proposition 2 The chance that a person will choose to be insured is more than, equal to or less than 50%, if his/her maximum tolerable loss is smaller than, equal to or larger than the expected loss respectively. The percentage can also mean the percentage of people, if all potential customers have the same sampling distribution and indifference curve. Risk Attitude and Risky Product The preceding analyses indicate that the gambling and insurance seekers are the same type of people. They have the same flatter slope of the indifference curve. For the example of a Cobb-Douglas utility function, U = α ln x + β ln y, this slope can be expressed as: y yα =. A flatter slope can therefore be resulted from a smaller future wealth, a larger x xβ present wealth, a smaller present wealth coefficient or a larger future wealth coefficient. It is not necessary to label these people as risk-loving or risk-averse, instead they should be labeled as relatively rich in present wealth and relatively future wealth loving. Hence, John is more likely to bet, if his indifference curve has a flatter slope, which means that he (F1) has a larger present wealth, (F2) has a smaller future wealth, or (F3) prefers future to present wealth. 12

13 On the other hand, according to (3) a higher profit rate means a flatter line of the expected return or loss. Then, the above determined chance of betting or being insured will be smaller. A higher profit rate also represents a more expensive product. Hence, a more (or less) expensive product has less (or more) betting or insurance business. This implies a demand schedule for a risky product. Accordingly, John is also more likely to choose to bet, if the game (F4) has a lower odds-ratio, d, (F5) has a higher per dollar net payoff, q, or (F6) the bet, k, is smaller. (F1) to (F6) are the factors affecting the betting decision. The factors that cause Mary to choose to be insured are almost identical, except the last one, which should be modified to: (F6 ) the object has a larger value. This factor means equivalently that the premium is larger. The Sample Size and Standard Deviation The sample size of the sampling distribution looks arbitrary, but it has one important implication. It represents the experience of the participants. Professional gambling or insurance seekers must have accumulated a larger sample, and the corresponding distribution must have a smaller standard error. They will make more accurate decisions. On the contrary, inexperienced people will make relatively inaccurate decisions, but they can still increase their accuracy by studying and collecting information. Figure 12 shows a sampling distribution when n=30. However, the distribution will start to look normal from n=7. Probability of the game or policy will also affect the standard error of the distribution. In particular, the standard error is largest when p=0.5, ceteris paribus. IV. Correcting the Anomalies This NoEU model can also explain many anomalies in the literature on risk, including the St. Petersburg s Paradox, the favorite-longshot bias, and various versions of the allais paradox. The St. Petersburg Paradox Why does John not continue to double his bet when he loses the preceding game? This 13

14 is the betting process of the St. Petersburg paradox or the Martingale betting strategy. With reference to Figure 9, doubling the bet means John moves twice as far leftward along the horizontal line y 1. Because of the convexity of the indifference curve, the bold line representing the chance shrinks drastically by such move. The chance may even be 0%, if the betting money is so large as to move the vertical line out of touch by the indifference curve. This also means that a person can always goes with a small bet; but as the stake gets larger, he/she had better have a second thought. There is also an insurance version of the St. Petersburg paradox: Mary will be more likely to seek insurance coverage when the value of the object gets larger. In that case she moves farther rightward along the y 1 -line of Figure 11. The vertical distance between the indifference curve and the maximum loss increases exponentially. This is, again, due to the convexity of the indifference curve. Mary may go without insurance for her dilapidated second-hand car, but she is more likely to have insurance to cover her brand new car. In this case of insurance, size does matter. However, it should not be confused with the large number problems, which must end up with the expected value and lead to the not-being-insured decision, as already proved in Section II. The Favorite-Longshot Bias Why are favorites under-bet and Longshots over-bet (Ziemba and Thaler, 1988)? Lotteries or longshots have very high maximum payoff, and the corresponding line in Figure 9 will have very steep slope. Hence, the chance or percentage may look enormous, especially when the ticket price is low. Such games attract a lot of customers. A traveler also has very large maximum loss potential, thus the chance he/she will buy a travel insurance policy is also large. Contrarily, favorites have very low payoff, the chance or percentage is correspondingly small. Hence, longshots, including lotteries and travel insurance, are over-bet or overbought, while favorites under-bet. Some financial products, like futures and warrants, are highly risky, i.e. they have larger fluctuation range or gain potential. They look like longshots and are over-invested. Contrarily, blue chips have smaller gain potential. They look like favorites and are under-invested. Selling short of securities is like not-being-insured. People are more cautious in selling short for the same reason proved in the insurance version of the St. Petersburg paradox. The Allais Paradox (Allais, 1953; Kaheman and Tversky, 1979) Although the standard two-dimensional model can already explain the Allais paradox 14

15 as demonstrated above with Figure 5, the NoEU can deliver more information. Problem 3 is quite straightforward, its elaboration will not be repeated here. The explanation for Problem 4 will be slightly different. The original problem states that: Problem 4: C: (4000,.20) or D: (3000,.25) N=95 [65]* [35] As C has an expected payoff of 800 while D 750, the expected return line of C is steeper than D, as also indicated Figure 12. Given identical standard error and indifference curve, the chance of choosing C is larger than choosing D, because of the higher expected payoff and as indicated by the respective vertical segments between the indifference curve and the respective payoffs at C and D. Alternatively stated, more participants will choose C than D according to Proposition 1. The present model thus delivers the similar percentage solution as the experiment does. This theory can also explain the negative version of this problem (Kahenman and Tversky, 1979, p.268): Problem 4 : C: (-4000,.20) or D: (-3000,.25) N=95 [42] [58]* The expected loss of D is -750, while that of C Since the line of expected loss of D lies higher than that of C as indicated in Figure 13, the chance that a person will choose to be insured is larger in C than in D according to Proposition 2. Without insurance, the person will be more likely to choose D than C, or more people will do so. <Insert Figure 12 Here> <Insert Figure 13 Here> The following problems involve choices with identical expected values (Kahenman and Tversky, 1979, p.268): Problem 7: (3000,.90) > (6000,.45) N=66 [86]* [14] Problem 8: (3000,.002) < (6000,.001) N=66 [27] [73]* 15

16 Although both prospects have the same expected value in both problems, the standard errors associated with (3000,.9) and (6000,.01) are much smaller than those of the alternatives, as indicated in Figure 14. Hence, their chance of being chosen will be larger. As stated before, standard error is largest when p=0.5; and the more p is away from 0.5, the smaller the standard error. A more exact definition of standard error is given below in (6). <Insert Figure 14 Here> This NoEU theory explains more variations of the Allais paradox without calling upon weighted utility and produces the NEU percentage solution. Violation of Stochastic Dominance A more complicated NEU problem is the so-called violation of stochastic dominance. According to Birnbaum (2005, pp ): when the probability of winning prize x or greater given gamble A is greater than or equal to the probability to win x or more in gamble B, for all x, and if this probability is strictly higher for at least one value of x, gamble A stochastically dominates gamble B. In short, a game or prospect is said to be stochastically dominant, if its expected value is higher. Accordingly, the following prospect E stochastically dominates F (Birnbaum, 1999): Choice 5: E:.5 probability to win $100 F:.99 probability to win $100.5 probability to win $ probability to win $200 This is so because prospect E has a higher expected value with $150, than that of Prospect F with $101. Then, there will be violation of stochastic dominance, when most participants choose F over E. Another example of the violation is offered in Birnbaum (2005, p.262): A: 90 red marbles to win $96 B: 85 green marbles to win $96 05 blue marbles to win $14 05 black marbles to win $90 05 white marbles to win $12 10 yellow marbles to win $12 Again the expected value of prospect A is higher (87.7 vs. 87.3), it is therefore stochastically dominant. However, 70% of the participants choose B. Birnbaum uses a weighted utility, called transfer of attention exchange, to explain why they absolutely prefer B to A, and then converts the result to percentage with a logistic function. Such method is 16

17 still cardinal and arbitrary, as it requires assumption on a lot of parameters. It does not provide any economic meaning either. The reason for the alleged violations lies with the standard error, which in this situation is defined as: s n i i= 1 = n 2 p ( x x) i= 1 i x i. (6) Accordingly, s A =0.204 and s B = As revealed in Figure 15, prospect A has a higher expected value than prospect B, but the standard error associated with B is smaller. Hence, the percentage of choosing B becomes larger than that of choosing A. Then, there is no violation at all. The rationale is similar to that in Problem 3 of Kahneman and Tversky (1979) explained above: people prefer certainty to uncertainty. Prospect B is relative certain, though the expected value is slightly lower. The standard errors of prospects E and F are and respectively in Choice 5. The preference of F to E is not a violation either. <Insert Figure 15 Here> As a further verification, consider another example from Birnbaum (2005, p.272): 15 a G+: 90 red marbles to win $96 G-: 55 red marbles to win $96 05 blue marbles to win $14 05 blue marbles to win $90 05 white marbles to win $12 40 white marbles to win $12 The expected values and standard errors are respectively: G+(87.7, 0.204) and G-(62.1, 0.207). The preference for G+ is confirmed by Birnbaum s 42% violation, or 58% non-violation. V. Conclusion Expected return or utility can be easily calculated from any two, or more, numbers. However, the statistical meaning of it is the mean result from a large number of trials. The EU and NEU theories seem to have misunderstood this by applying it to a single, sporadic trial. 17

18 Instead of pursuing expected value, the present NoEU model tracks the whole normal distribution. Instead of relying on the one-dimensional utility function, it counts on the whole indifference curve. The combination of these two explains a lot of things. In addition to the general gambling and insurance behaviors, it also explains many alleged anomalies. Anomalies turn out to be normal rational behaviors. The major difference between the present theory and the conventional ones lies with the solution. The conventional EU and NEU theories produce absolute solution, whereas the present theory produces percentage solution. The present NoEU theory has the same percentage problem setting as the NEU theory, but produces the consistent percentage solution, as summarized in the following table. Theory Problem Setting Solution EU Absolute Absolute NEU Percentage Absolute NoEU Percentage Percentage 18

19 References Allais, M. (1953) Le Comportement de l Homme Rationnel devant le Risque: Critique des Postulats et Axiomes de l Ecole Americaine. Econometrica, 21, Arrow, K.J. (1963) Uncertainty and the Welfare Economics of Medical Care. American Economic Review, 53(5), Birnbaum, Michael H. (1999) The Paradoxes of Allais, Stochastic Dominance, and Decision Weights. in J. C. Shanteau, B. A. Mellers, & D. Schu, Decision Science and Technology: Reflections on the contribution of Ward Edwards, MA: Kluwer Academic Press, Birnbaum, Michael H. (2005) A Comparison of Five Models that Predict Violations of First-Order Stochastic Dominance in Risky Decision Making. Journal of Risk and Uncertainty, 31(3), Blackorby, Charles, Primont, D., and Russel, R.R. (1978) Duality, Separability and Functional Structure, NY: Elsevier. Ehrlich, I., and Becker, G. (1972) Market Insurance, Self-Insurance and Self-Protection. Journal of Political Economy, 80(4), Friedman, Milton, and Savage, L.J. (1948) The Utility Analysis of Choices Involving Risks, Journal of Political Economy, 56(4), Grinstead, Charles M., and Snell, J. Laurie. (1997) Introduction to Probability, 2 nd Edition, American Mathematical Society. Kahneman, Daniel, and Amos Tversky. (1979) Prospect Theory: An Analysis of Decision under Risk, Econometrica, 47 (2), Machina, Mark J. (1982) Expected Utility Analysis without the Independence Axiom. Econometrica, 50(2), Markowitz, H. (1952) The Utility of. Journal of Political Economy, 60 (2), Miyamoto, John M., and Wakker, Peter. (1996) Multiattribute Utility Theory without Expected Utility Foundation. Operation Research, 44(2), Mossin, Jan. (1968) Aspects of Rational Insurance Purchasing. Journal of Political Economy, 76, Neilson, William S. (1994) Second Price Auctions without Expected Utility. Journal of Economic Theory, 62(1), Nielsen, Thomas D., and Jaffray, Jean-Yves. (2006) Dynamic Decision Making without Expected Utility. European Journal of Operational Research, 169(1), Rosett, R.N. (1967) The Friedman-Savage Hypothesis and Convex Acceptance Sets: A Reconciliation. Quarterly Journal of Economics, 81(3), Rothschild, Michael, and Stiglitz, Joseph. (1976) Equilibrium in Competitive Insurance Markets. Quarterly Journal of Economics, 90(4),

20 Schoemaker, Paul. (1982) The Expected Utility Model: Its Vairants, Purposes, Evidence and Limitations."Journal of Economic Literature, 20(2), Solow, R.M. (1956) A Contribution to the Theory of Economic Growth. Quarterly Journal of Economics, 70, pp Starmer, Chris. (2000) Developments in non-expected utility theory: The Hunt for a Descriptive Theory of Choice under Risk Journal of Economic Literature, 38(2), von Neumann, J., and Morgenstern, O. (1947), Theory of Games and Economic Behavior, 2nd Ed., Princeton: Princeton University Press. Yaari, Menahem E. (1965) Convexity in the Theory of Choice under Risk. Quarterly Journal of Economics, 79(2), Ziemba William T., and Thaler R.H. (1988) Anomalies: Parimutuel Betting Markets, Racetracks and Lotteries. Journal of Economic Perspectives, 2,

21 Graphs Future y 2 C EU y 1 A B U(B) x 2 x 1 Present Figure 1. The 2-D Expected Utility Model for Betting 21

22 y y 1 B A U(B) EU y 2 C x 1 x 2 x Figure 2. The 2-D Expected Utility Model for Insurance 22

23 Figure 3. The Reconstruction of the Friedman-Savage Hypothesis for Betting Utility C B A (x + y) The lines joining AB and BC look as if there is increasing marginal utility of wealth. 23

24 Figure 4. The Reconstruction of the Friedman-Savage Hypothesis for Insurance Utility B A C (x + y) Though the joining of A-B-C shows diminishing marginal utility of wealth, it is not the true picture. 24

25 y A 3200 C D B 3000 x Figure 5. A NEU Experiment with Two Choices (Not to Scale) 25

26 Future The Risk-Free Investment ER EU B y 1 x 1 Present Figure 6. The Expected Return of Risk and Risk-Free Product 26

27 Future y 1 B EL The Risk-Free Investment EU x 1 Present Figure 7. The Expected Loss of Risk and Risk-Free Product 27

28 Figure 8. The Law of Large Numbers with the Constant of ε/v=0.25. Note: If v=4, the expected value will be 2, and the dotted line1 or 3. 28

29 Future B y 1 U -k x 1 Present Figure 9. The Chance that a Person will Bet 29

30 y 1 E(y) y 1 +(q+1)k Figure 10. The Binomial Distribution with d=1, q=1, k=2, and n=30 30

31 Future y 1 B k x 1 Present Figure 11. The Chance that a Person Will Choose to Be Insured 31

32 Future D C D C B y 1 U -k x 1 Present Figure 12. Explaining the Allais Paradox 32

33 Future y 1 B D C D x 1 C Present Figure 13. Explaining the Negative Version of the Allais Paradox 33

34 Future B y 1 U -k x 1 Present Figure 14. The Allais Paradox and Standard Error 34

35 Future B y 1 U -k x 1 Present Figure 15. Explaining Violation of Stochastic Dominance 35