Optimal Lottery Games: Implications for Preferences Under Risk

Transcription

1 Optimal Lottery Games: Implications for Preferences Under Risk A. Direr y April, 2009 Abstract This article investigates which form preferences of rank dependent expected utility players have to take to be consistent with pro t-maximizing lottery games. I begin by characterizing lottery s optimality conditions, and then reverse the problem by asking which preference patterns are compatible with the existence of optimal games. Compelling restrictions on RDU preferences are found which greatly vary with the continuous or discrete nature of the lottery. In the case of a continuous prize distribution, the value function is globally concave over the domain of prizes and the weighting function is globally convex over the probability domain. In the case of a lottery endowed with a nite number of prizes, the properties of the two functions are almost entirely reversed. The value function is convex in average within and across prizes. Conversely, the weighting function is concave in average within and across cumulative probabilities. JEL codes : D81, D21 Keywords : Decision-Making under Risk, Lottery Games, Firm Behavior 1 Introduction Most people buy lottery tickets that return in expectation less than the ticket price in money. Several behavioral explanations have been proposed to account for this fact. The propensity to gamble at unfavorable odds has been interpreted Paris School of Economics, INRA and University of Grenoble 2. Address : 48 Bd Jourdan Paris FRANCE. direr@ens.fr. y A previous version has circulated under the title "What do Lottery Games Reveal about Preferences Under Risk?". Useful comments by Nathalie Etchart-Vincent are gratefully acknowledged. Any errors or omissions are my own. 1

2 in terms of attitude towards wealth by Friedman and Savage (1948). They argue that people are risk-averse when stakes are small or negative but are risk-seeking with respect to prizes that are large enough to elevate their social standing. This inclination is modeled by a reverse S-shaped utility function and amounts to assuming increasing marginal utility for a broad range of wealth. Another line of explanation emphasizes the role of misperception of probabilities. Small probabilities of winning large prizes are overestimated while medium probabilities of loss are more accurately assessed. The consequences of this bias can be analyzed by rank dependent expected utility models developed among others by Quiggin (1982) or by the original prospect theory (Kahneman and Tversky, 1979) and the cumulative prospect theory (Tversky and Kahneman, 1992). By focusing on players preferences however, the supply side of the market has been generally overlooked. The market of lottery games has nowadays reached an industrial scale after several decades of steady growth all over the world (Garrett, 2001). Games are designed by pro t-oriented companies which are keen on making lottery games more attractive for players. They also regularly experiment new forms of games. In France, for instance, new scratch games with various prize structures have been launched every year for more than a decade by the state-owned lottery company. Clotfelter and Cook (1989, 1993) and Walker and Young (2001) provide many illustrations of the way lottery companies shape the payo distribution in order to enhance their pro t. 1 1 Friedman and Savage (1948) make a similar point: "[Lotteries] have been conducted in many countries and for many centuries, so that a great deal of evidence is available about them; there has been extensive experimentation with the terms and conditions that would make them attractive, and much competition in conducting them, so that any regularities they may show would have to be interpreted as re ecting corresponding regularities in human 2

3 This article builds on those observations by investigating which types of preference patterns are compatible with the existence of lottery games shaped by a pro t-maximizing company. Players are characterized by rank dependent expected utility preferences, which are commonly considered as a natural extension of the expected utility model (Machina, 1994). Such preferences are exible enough to allow for the presence of the two abovementioned explanations of gambling : risk loving through the convexity of the utility function and overestimation of small probabilities through the concavity of the weighting function. I begin by assuming that a monopolistic rm optimally chooses the value of prizes and their probabilities. This task can be made either by assuming the company o ers a lottery game characterized by a continuous prize distribution function or, more realistically, by assuming that the lottery game is endowed with a nite number of prizes. In the latter case, the total number of prizes is also set optimally. Once lottery s optimality conditions are characterized, the problem is reversed by asking which preference patterns are compatible with the existence of optimal games. Compelling restrictions on rank dependent expected utility preferences are found which greatly vary with the continuous or discrete nature of the lottery game. In the case of a continuous distribution of prizes, the value function is globally concave over the domain of prizes and the weighting function is globally convex over the probability domain. A dissemination of prizes is possible despite decreasing marginal utility due to increasing optimism about the occurrence of greater prizes. In the case of a discrete distribution of prizes, behavior". 3

4 the properties of the two functions are almost entirely reversed. The value function is convex in average within and across prizes. Conversely, the weighting function is concave in average within and across cumulative probabilities. The assumption of pro t maximization by rms is ubiquitous in all elds of economics but, paradoxically, has not permeates the analysis of lottery products despite its realism. A few articles exist however, which make lottery games endogenous by assuming that rms maximize their pro t. A notable reference is the seminal article by Friedman and Savage (1948) who rst suggest that optimal lottery games may reveal interesting information about preferences under risk. In particular, a concave utility function for the richest part of the population is justi ed by explicitly appealing to the operation of an entrepreneur conducting a lottery and seeking to maximize its pro t. While fruitful, their approach su ers from the assumption that risk preferences are solely explained by attitude toward wealth. As a result, they account for very limited patterns of gambling. Their conclusions about preferences have also been criticized on several grounds (Markowitz, 1952, Yaari, 1965, or Bailey et al., 1980, Machina, 1982). The present paper essentially extends their approach to a rank dependent utility framework, which allows to account for a richer set of preferences under risk. The model is also close to the one of Quiggin (1991). He considers the optimal structure of a lottery when agents preferences are ordered according to a rank dependent utility criterion. Assuming a concave value function and a reverse S-shaped probability transformation function, he shows that the optimal design involves at least two distinct prizes. The class of lotteries examined is however restricted by the fact that the number of prizes is not a dimension 4

5 of choice for the company and by the constraint that all prizes are equally probable. In contrast, the present paper makes no prior restrictions on the probability distribution of prizes and on their number. The presentation is organized as follows. Section 2 outlines the main caracteristics of the model by way of a simple numerical example. Section 3 presents the full problem in which a rm must design a lottery game optimally. Section 4 derives implications for preferences under risk from the existence of an optimal lottery game. The last section provides concluding remarks. 2 Outline of the lottery problem Risk preferences consistent with pro t-maximizing lottery games are analyzed in two steps. First, optimal lottery games given players preferences are characterized. Then preferences are investigated by assuming the existence of optimal games. This section focuses on the rst step, which is complicated by the fact that not only the prizes values and probabilities are endogenous but also their number. It is however possible to catch the intuition of how this problem is handled by means of a simple numerical example. Let us consider a company that designs a lottery game so as to maximize its pro t under the constraint that players purchase the game. Let us assume further that the lottery is endowed with one negative prize 2 (the ticket price) and two positive prizes 10 and 50 (or 8 and 48 if the ticket price is a sunk cost). Their probabilities are respectively 0:9, 0:09 and 0:01. It is more convenient to frame the lottery in terms of cumulative probabilities. Those probabilities are respectively 0:9, 0:99 and 1. A necessary condition for this lottery to be optimal 5

6 is that all prizes and cumulative probabilities are optimal. We must also check that no alternative constellations of prizes and probabilities exist that would bring about higher pro t. There are actually two methods that allow to rule out such a possibility. The rst method is to expand the number of probabilities and to treat them as exogenous. Let us assume that the company must choose an optimal level of prize for every cumulative probabilities i=n, i = 1; 2; :::; n. More speci cally, consider the case n = 100. If the three-prize lottery is to be optimal, we must check that the payo 2 remains optimal for probabilities 1=100, 2=100 up to 90=100, the prize 10 for probabilities 91=100; :::; 99=100, and the top prize 50 for probability 1. However, the company overlooks potentially more pro table prize distributions if they are associated with probabilities that are not multiples of 1=100. This problem can be solved by applying the same method with n converging to in nity. The second method relies on the expansion of the number of prizes which are now exogenous variables. The company has to choose optimal cumulative probabilities over the set of prizes f w + (i=m) ; i = 1; :::; mg in which w is a lower bound and the interval s size of prizes. Note that this set can encompass any values, providing w, m and are taken large enough. For concreteness, consider the case w = 100, = 300 and m = 300. The threeprize lottery is now optimal if the (cumulative) probability 0 is optimal for prizes ranging from 99, 98 up to 3, the probability 0:9 is optimal for prizes 2; 1; :::; 49, the probability 0:99 for prizes 50; 51; :::; 99 and the probability 1 for the remaining prizes 100; :::; 200. This method only covers optimal lotteries 6

7 endowed with prizes that are in the speci ed set. The number of prizes m must tend to in nity in order to check that the lottery is the best one against all others candidates. It is crucial to note that the lottery problem with endogenous payo s is not distinct from the one with endogenous probabilities. They correspond to two alternative framings of the same problem. A rm can always think about an optimal lottery either by choosing a prize distribution over a continuum of probabilities, or by selecting an optimal probability schedule over a continuum of prizes. Both approaches are intended to characterize the same class of optimal lotteries. As a result, both problems will reveal (possibly) di erent but complementary aspects of players preference pattern. The remaining of the paper carefully applies those methods by assuming that players are characterized by rank dependent expected utility preferences. 3 Optimal lottery games Let us consider the problem faced by a risk-neutral monopoly which sells a lottery ticket endowed with n payo s (or prizes) so as to maximize its expected pro t. A prize x i, i = 1; :::; n is positive if a gain for the purchaser and negative if a cost. Let i, i = 1; :::; n denote the associated cumulative distribution of prizes : Pr(x x i ) = i. Consumers are characterized by rank dependent expected utility (RDU) preferences 2. The utility function (or value function) u(:) and the probability weighting function g(:) are both strictly increasing and twice di erentiable over [ w; 1[ and [0; 1] respectively. The lower bound w 2 Diecidue and Wakker (2001) o ers an introductory exposition of the theory of rank dependent expected utility. 7

8 re ects the maximum loss that a player can undergo. It can be her lifetime wealth or a fraction of it, in case of liquidity constraints for example. It is not necessary to be speci c about whether the utility function is de ned over wealth or changes in wealth. The weighting function satis es g(0) = 0 and g(1) = 1. The value derived from purchasing a lottery is: nx [g( i ) g( i 1 )]u(x i ) i=1 The company has to decide the value of each prize o ered by the game, their probability and their number, given the player s preferences. As outlined by the previous section, there are two alternative ways to characterize an optimal lottery. The company may choose optimal payo s over an exogenous and large set of probabilities. Or it may select optimal probabilities over a ne grid of exogenous prizes. I begin by studying the rst approach. 3.1 Lotteries with optimal payo s The company chooses which payo s x i, i = 1; 2; :::; n, to a ect to a given set of cumulative probabilities f 1 ; :::; n g. Probabilities are exogenous and equal to: i = i=n, i = 1; 2; :::; n Hence, all prizes have the same probability of occurrence 1=n. The corresponding lottery is denoted fx i : i=n; i = 1; :::; ng and its value for a RDU player is: nx [g(i=n) g((i 1)=n)]u(x i ) (1) i=1 8

9 Restricting to uniform probability distributions does not lack generality, since a prize can be o ered several times in the same lottery. For instance, a lottery endowed with q prizes y of probability 1=n is equivalent for a RDU player to a lottery that o ers a single prize y with probability q=n. To see this, suppose that the lottery fx i : i=n; i = 1; :::; ng includes a sequence of q samevalue prizes x l+1 = x l+2 = ::: = x l+q = y with 1 l < l + q 1 n. Its RDU value is: ::: + [g((l + 1)=n) g(l=n)] u(y) + ::: + [g((l + q)=n) g((l + q 1)=n)] u(y) + ::: After pooling all prizes equal to y and aggregating associated probabilities, it becomes: ::: + [g((l + q)=n) g(l=n)] u(y) + ::: which is also the RDU value of a lottery o ering y with probability q=n. Conversely, a lottery where the prize y appears with probability p can be replicated by a lottery with a uniform probability distribution where y appears np times, providing np is an integer. The equivalence holds exactly as long as all probabilities of winning a prize can be expressed as multiples of 1=n. This is a very weak requirement as n can be raised as high as desired without changing the nature of the prize distribution. For instance a lottery fx i : i=n; i = 1; :::; ng in which the prize y appears q times is equivalent to the lottery fx i : i=rn; i = 1; :::; rng, with r a strictly positive natural integer, in which y is now duplicated rq times. Let us denote v the reservation utility that the consumer secures by abstaining. The company chooses prizes which maximizes its pro t, while meeting the player s participation constraint and a set of ordering constraints: 9

10 8 >< >: max fx 1;:::;x ng x i i=1 (1=n) n P P s.t. n [g(i=n) g((i 1)=n)]u(x i ) = v i=1 x i 1 x i ; i = 1; :::; n x 0 = w A Lagrange function is formed by appending the objective function and the constraints. The multipliers for the participation constraint, and the ordering constraints x i x i 1 0, are respectively denoted and i, i = 1; :::; n. A payo x i is optimal if it satis es the rst and second order conditions (with n+1 = 0): g(i=n) g((i 1)=n) 1=n + i+1 i u 0 (x i ) = 1=, i = 1; :::; n, (2) u 00 (x i ) 0; i = 1; :::; n. (3) To understand Eq. (2), rst assume that the boundary constraints do not bind : i+1 = i = 0. In that case, the ratio [g(i=n) g((i 1)=n)]=(1=n) measures by how much the chance of getting x i is distorted by the player. This ratio is equal to one if the probabilities are not distorted (i.e. the function g(:) is the identity function). The company must therefore increase the value of a prize wherever marginal utility is "high" or its true probability of occurrence is overestimated by the player (i.e. [g(i=n) g((i 1)=n)]=(1=n) is "high"). This rule o ers the best way for the company to relax the player s participation constraint and allows it to globally reduce the average payout of the lottery. The multipliers i+1 and i re ect the loss of pro t associated with the upper boundary constraint x i x i+1 and the lower boundary constraint x i 1 x i 10

11 respectively. If the upper constraint binds at the optimum ( i+1 > 0 and i = 0), x i is lowered compared to the case i+1 = i = 0 in order to alleviate the constraint. Conversely, x i is greater if the lower boundary constraint binds ( i > 0 and i+1 = 0). If the two boundary constraints bind simultaneously ( i ; i+1 > 0), there are two cases. If i > i+1, alleviating the constraint x i 1 x i by raising x i is more pro table for the rm than reducing the cost of the constraint x i x i+1 by lowering x i. The converse is true if i < i+1. 3 This section has derived the properties of a lottery endowed with a nite number of optimal prizes over an exogenous set of probabilities. This problem is reversed in the next section by assuming that the company now chooses optimal probabilities over a xed set of payo s. 3.2 Lotteries with optimal probabilities This section investigates a similar problem than the previous one. The main di erence lies in how the lottery problem is framed. Instead of decomposing probabilities into small units and nding associated optimal payo s, the prize set comprises a large number of payo s and corresponding probabilities are optimally chosen. Speci cally, let us consider a set of prizes fx 0 ; :::; x m g which are exogenous, increasingly ranked and equally spaced: x i = w + (i=m), i = 1; :::; m where w and w + are the lower and upper bound for admissible prizes. w is still interpreted as the maximum loss that the player may bear. 3 Note also that if a higher i induces a greater x i, raising x i mitigates in turn the cost of the constraint x i 1 x i and therefore the value of i. This feedback loop guarantees that i remains below 1=n + i in Eq. (2). 11

12 This lottery structure is exible enough to encompass any types of lottery. First, the company may be willing to o er less prizes than m + 1. If x i is not o ered, its probability of occurrence is simply zero, that is Pr(x x i ) = i = i 1. Second, the maximum prize x m may be taken as high as necessary to top all potential optimal prizes. Third, the company may also be willing to o er prizes that are not included in the set fx 0 ; :::; x m g. It is however able to pick up prizes within the set of prizes that approximate optimal ones with any degree of accuracy as long as m is taken su ciently large. The expansion of the number of prizes m does not alter the nature of the lottery. A lottery fx i : i ; i = 0; 1; :::; mg in which the di erence between any two consecutive prizes is =m can always be duplicated without loss of generality by another lottery x 0 j : 0 j ; j = 0; 1; :::; rm, with r an arbitrarily large integer, characterized by x 0 0 = x 0 and a constant increment (=m)=r of money between two consecutive prizes (that is x 0 j = x 0 + j(=m)=r). In this new lottery all previous prizes are still o ered. Their index is now j = ri instead of i (x 0 j = x0 0 + (ri)(=m)=r = x 0 + i(=m)). The newly created prizes are merely given a probability of occurrence equal to zero. Raising m does not modify the value of the lottery for a RDU player. All new prizes inserted between existing ones occur with probability zero ( j j 1 = 0). Their decision weights are consequently zero as well (g( j ) g( j 1 ) = 0). As a result, the increment =m between two prizes can be taken arbitrarily small without a ecting the characteristics of the initial lottery. The company chooses a set of probabilities f 1 ; :::; m g which maximizes its expected pro t, while meeting the player s participation constraint and a set of 12

13 ordering constraints (with 0 = 0): 8 >< >: s.t. mp max ( i i 1 )x i i=1 mp [g( i ) g( i 1 )]u(x i ) = v f 1;:::; mg i=1 i i 1 0, i = 1:::; m i 0, 1 i 0, i = 1; :::; m The rm must choose an upward sloping schedule of probabilities i over [0; 1] that maximizes its pro t while appealing enough to the player. The steeper the probability schedule from a prize to the next one, the less attractive the lottery from the player s point of view (the lower the probability of winning bigger prizes), and the higher the rm s pro t. A Lagrange function is formed by appending the objective function and the constraints. The multipliers for the participation constraint, the ordering constraints i i 1 0, the positivity constraints i 0 and the upper boundary constraints 1 i 0 are respectively, i, i and i, i = 1; :::; m. Let us de ne x t and x h the minimum and maximum prizes that both occur with a strictly positive probability at the optimum: t 1 = 0 < t and h 1 < h = 1 (with h m). The rst and second order conditions for interior solutions i 2 ]0; 1[ apply with i = i = 0: u(x i+1 ) u(x i ) =m + i i+1 g 0 ( i ) = 1= i = t; :::; h 1 (4) g 00 ( i ) 0 i = t; :::; h 1 (5) with t = h = 0 by de nition of x t and x h. Eq. (5) states that the weighting function must be convex in the close neighborhood of i. To understand why, 13

14 let us assume the converse case. If the marginal weight g 0 ( i ) were decreasing, the player would understate the deteriorating e ect on the lottery s value of a marginal increase of i or conversely, would overstate the bene t of a marginal reduction of i. In both cases, the rm would exploit this bias by shifting i in either direction. To understand Eq. (4), assume rst that i is free to move upward or downward (i.e. i = i+1 = 0). In that case, the rm must increase i if the player understates the true occurrence of winning at least x i (that is g 0 ( i ) is low) or if the utility gain from raising the prize by one unit of money is low ((u(x i+1 ) u(x i ))=(=m) is small). In both cases, the rm increases its pro t while minimizing the loss of attractivity for players. At the optimum, each probability must equally contribute at the margin to the relaxation of the player s participation constraint. If only the lower constraint binds (0 < i 1 = i < i+1 i.e. i > 0 and i+1 = 0), Eq. (4) shows that i is greater compared to the case with all the multipliers equal to zero. Raising i allows to alleviate the constraint. The converse is true if the upper constraint binds ( i+1 > 0). If the two boundary constraints simultaneously bind at the optimum ( i ; i+1 > 0 and i = i = 0), two cases are possible. If i > i+1, relaxing the constraint i 1 = i by raising i is more pro table for the rm than relaxing the constraint i = i+1 by lowering i. This means a higher probability i at the optimum compared to the case i = i+1 = 0, providing the marginal weights g 0 ( i ) are increasing, which is true according to Eq. 5). The inverse reasoning holds if i < i+1. Similar optimality conditions apply for boundary values i = 0 ( i > 0) and 14

15 i = 1 ( i > 0). If the positivity constraint binds ( i > 0), the rst and second order conditions are: u(x i+1 ) u(x i ) =m + i g 0 (0) = 1= i = 1; :::; t 1 (6) g 00 (0) 0 Likewise, the optimality conditions for i = 1 are: u(x i+1 ) u(x i ) =m i g 0 (1) = 1= i = h; :::; m 1 (7) u(x i ) x i + i g 0 (1) = 1= i = m g 00 (1) 0 Having characterized the optimality conditions that a lottery must satisfy, I now turn to the study of preference patterns that are compatible with. 4 Implications for preferences under risk I begin by the analysis of preferences consistent with the existence of a continuous distribution of prizes, which is the simplest case. The case of a lottery endowed with a nite number of prizes is studied next. 4.1 Continuous distributions This section deals with the special case of an optimal and continuous distribution of prizes. Let denote the cumulative probability of prizes and y() the prize function. y() is the inverse of the cumulative distribution function. For any probability 0, x 0 = y( 0 ) is such that Pr(x x 0 ) = 0. The existence of an optimal continuous distribution of prizes is assumed thereafter: 15

16 Assumption H1. An optimal prize function y() : [0; 1]! R exists where is the cumulative distribution of prizes and y(0) = lim y() when! 0. The prize function satis es y(0) > w, y 0 () > 0 and y 0 () < 1 over ]0; 1]. The company s expected pro t is non-negative : R 1 0 y()d 0. Optimal prizes are de ned over an exogenous domain of probabilities, as in Section 3.1. An optimal prize function satisfying H1 can emerge by adapting the results of this section and by assuming further that every optimal prize x i, i = 1; :::; n, takes a distinct value over the set of cumulative probabilities i = i=n, and that n converges to in nity (all proofs in Appendix): ]0; 1]. Proposition 1. Under H1, u 00 (y) 0 over ]y(0); y(1)] and g 00 () 0 over The concavity of the utility function is required by the second order condition of the lottery problem. The convexity of the probability weighting function means that the player systematically depreciates the probabilities attached to unfavorable events and exaggerates the occurrence of favorable outcomes. This is a strong form of optimism. The two results are closely linked. The inclusion of increasing prizes in the lottery game is impeded by decreasing marginal utility. As a consequence, those prizes must be matched with an increasing degree of optimism (or greater probability weights g 0 ()) to be optimal. The problem could also have been framed in terms of optimal probabilities 16

17 over a domain of exogenous prizes as in Section 3.2. Assuming a distinct probability for every prize o ered and making m tend to in nity would produce identical results. The convexity of the probability weighting function would be directly derived from the second order condition. This type of lottery is not very realistic, however. Commercial lotteries do not o er negative prizes, except the ticket price, whose probability of occurrence is generally greater than 50 percent. This contradicts the property of a smooth prize distribution. This hypothesis is therefore amended by assuming that the lottery game o ers a nite number of prizes. 4.2 Discrete distributions This section is devoted to the analysis of a lottery game endowed with k prizes: Assumption H2. An optimal lottery y j : ' j ; j = 1; :::; k exists, with ' j = Pr(y y j ), k > 1, w y 1 < y 2 < ::: < y k, and 0 < ' 1 < ' 2 < ::: < ' k = 1. The company s pro t is nonnegative : P k j=1 (' j ' j 1 )y j 0. As already discussed in the introductory example, there are two ways two check that a distribution is optimal. The rst one is to consider a large number of exogenous probabilities which increment is made arbitrarily small. The company must choose optimal prizes over such a set that accords with the distribution posed in H2. The second one is to start from a ne and exogenous grid of prizes and to nd probabilities consistent with the distribution in H2. As we will see, the two approachs will shed a di erent light on the type of risk preferences implicit in those problems. 17

18 I begin by studying the case with exogenous probabilities and optimal prizes. The equivalence result emphasized in Section 3.1 states that any lottery can be replaced by a lottery fx i : i=n; i = 1; :::; ng where n is made arbitrarily large by duplicating the number of same-value prizes without altering the prospects for a RDU player. This equivalence property makes the number of prizes a true dimension of choice for the company while keeping a tractable problem. It leads to the following proposition: Proposition 2. Under H2: (a) u 00 (y j ) 0, j = 1; :::; k. If the wealth constraint is not binding: y 1 > w, then (with ' 0 = 0): (b) g(' j ) g(' j 1 ) ' j ' j 1 u 0 (y j ) = g(' l ) g(' l 1 ) ' l ' l 1 u 0 (y l ), j and l = 1; :::; k; (c) g 0 (' j 1 ) g 0 (' j ), j = 1; :::; k; (d) R ' j ' j 1 g 00 ()d 0, j = 1; :::; k. If the wealth constraint is binding (y 1 = w), results (b) (c) and (d) are still valid for j = 2; :::; k. For j = 1: (b ) g(' 1 ) ' 1 u 0 (y 1 ) < g(' j ) g(' j 1 ) ' j ' j 1 u 0 (y j ), j = 2; :::; k; (c ) g 0 (0) Q g 0 (' 1 ), (d ) R ' 1 0 g 00 ()d Q 0. 18

19 Proposition 2 displays restrictions on preferences that are only partially consistent with the restrictions found in the continuous distribution problem of Section 4.1. Results (a) and (b) are similar in spirit to the second and rst order conditions (??) and (8) of the continuous problem. The concavity result for utility only applies locally however. It does not inform about the curvature of the utility function between prizes. Results (c) and (d) contrast sharply with the global convexity of the weighting function found in Proposition 1. Marginal weights g 0 () are increasing from any probability to the next one (result (c)) and are decreasing in average between two consecutive probabilities (result (d)), which re ects a weak form of pessimism. They are the main di erences between a continuous prize distribution and a discrete one. The average concavity of the weighting function provides the mechanism that leads the company to choose a at prize schedule over a sequence of cumulative probabilities. Were this condition violated, the rm would have the incentive to scatter payo s across probabilities, which would multiply the number of prizes. Results (b ) to (d ) deal with the binding of the wealth constraint. The main result (d ) shows that the result of average concavity of the weighting function between 0 and the probability to win the minimum prize may no longer be true. The proof in Appendix shows that this case happens if the wealth constraint is costly enough for the rm. I now turn to the alternative framework studied in Section 3.2 in which prizes are exogenous and probabilities are set optimally. This section has assumed that prizes must fall in the set fx i = w + (i=m), i = 1; :::; mg. As 19

20 already explained, this domain of prizes may accomodate any types of lottery as long as the interval between two prizes is made arbitrarily small by raising m and the interval [ w; w + ] spans a wide space of potential prizes. This equivalence property allows to make the number of prizes endogenous and leads to Proposition 3: Proposition 3. Under H2 (with ' 0 = 0): (a) g 00 (' j ) 0, j = 0; 1; :::; k, (b) u(yj+1) u(yj) y j+1 y j g 0 (' j ) = u(y l+1) u(y l ) y l+1 y l g 0 (' l ); j and l = 1; :::; k 1, (c) u 0 (y j ) u 0 (y j+1 ), j = 1; :::; k 1, (d) R y j+1 y j u 00 (x)dx 0, j = 1; :::; k 1, (e) u 0 (y) g0 (' 1 ) u(y 2) u(y 1) g 0 (0) y 2 y 1 8y 2] w; y 1 ], (f) u 0 (y) u(y k) y k 8y y k. Results (a) to (d) are the symmetric of (a) to (d) in Proposition 2 in which the utility and weighting functions are exchanged. Result (a) states that the weighting function must be locally convex in the close neighborhood of the cumulative probabilities of the lottery. Result (d) shows that the utility function is concave in average between two probabilities. This property only applies to the interval of money bounded by the minimum prize y 1 and the maximum prize y k. Results (e) and (f) put additional restrictions on utility outside this interval. They ensure that cumulative probabilities equal zero for every payo s smaller 20

21 than the minimal prize and equal one for every payo s equal or greater than the top prize. Those conditions are ful lled if the slope of the utility function is greater than a positive threshold below the minimum prize (result (e)) and smaller than another one above the top prize (result (f)). Put simply, the utility function cannot be too concave below the smallest prize and too convex above the top prize, otherwise the company would nd optimal to add additional prizes outside the interval of prizes. 4 As argued in the introductory example, Propositions 2 and 3 put restrictions on risk preferences of the same player. Indeed, for a lottery y j : ' j ; j = 1; :::; k to be optimal, two types of conditions must be full lled. First, all prizes fy j ; j = 1; :::; kg and probabilities ' j ; j = 1; :::; k must be locally optimal, which lead to conditions (a) and (b) in Proposition 2 and 3. Second, the company should not nd optimal to o er additional prizes or probabilities not speci- ed in the lottery. This is guaranteed by restrictions (c) and (d) in Propositions 2 and (c) to (f) in Proposition 3. Taken together, the last two propositions lead to new results: Proposition 4. Under H2, with y 1 > w (and ' 0 = 0): (a) g(' j ) g(' j 1 ) ' j ' j 1 g(' j+1 ) g(' j ) ' j+1 ' j, j = 1; :::; k 1, (b) u(yj+1) u(yj) y j+1 y j u(yj+1) u(yj) y j+1 y j, j = 1; :::; k 1. 4 Result (f) shares a similar interpretation with the proposition by Friedman and Savage (1948) in an expected utility setup, according to which the right-hand segment of the utility funtion is concave. 21

22 Proposition 4 states that the average slope of the weighting function is decreasing from any cumulative probability to the next one and the average slope of the utility function is increasing from any prize to the next one. Considered together, results (c) and (d) in Proposition 2, and (a) in Proposition 4 are consistent with a concave weighting function over the whole interval of probabilities except in the local neighborhood of the lottery s probabilities, where the function must be locally convex. Likewise, results (b) to (e) in Proposition 3 and (b) in Proposition 4 are compatible with a utility function which is convex below the top prize, except in the close neighborhood of prizes. Lastly, it is worthwhile remarking that the results of this section rest on the existence of an optimal lottery, which is taken as granted. In particular, conditions have been investigated for probabilities and prizes to be locally optimal. It has not been proved that they correspond to global optima as well. If they failed to be global optima, the company would better drop the corresponding prizes in a way that cannot be studied under H2. 5 Conclusion This article investigates which forms of preferences under risk are consistent with pro t-maximizing commercial games. The implications on preferences greatly di er in the cases of a continuous prize distribution and a discrete prize distribution. In the rst case, the weighting function is convex and the utility function is concave. This means that players gamble despite decreasing marginal utility of wealth as they are optimistic about the occurrence of big prizes. This con guration is almost entirely reversed when the company sells lottery games endowed 22

23 with a nite number of prizes. The weighting function is now concave and the utility function is convex both in an average sense. Those properties ensure that the rm bunches probabilities and prizes instead of spreading them over their corresponding domain.this does not preclude limited portions of concave utility and convex weighting function, in particular in the close neighborhood of the lottery s prizes and probabilities. Under such a pattern, players are now lured by winning high prizes that elevates their wealth level in the spirit of Friedman and Savage (1948). The model s results essentially rely on the assumption of pro t maximization as the type of lotteries considered remains general. More information on commercial lotteries could be exploited. For instance, popular games such as lotto games or scratch cards o er top prizes of large value but also several intermediate prizes and even prizes of small values. They never o er negative prizes, except the ticket price, which is generally small. The probability of losing is important (generally more than 50 percent) and the probability of winning falls sharply with the size of the price. Putting more structure to the shape of lotteries would bring about additional constraints on preference patterns. This could provide a useful test of the approach pursued by the paper. Lastly, the model has assumed a static model and homogeneous preferences through the ction of a representative player. It would be interesting to study to what extent the ndings of the model are modi ed when repeated play is allowed or when some heterogeneity in preferences is taken into account by the game company. 23

24 References BAILEY M. J., OLSON M. & WONNACOT P. (1980) "The Marginal Utility of Income Does Not Increase: Borrowing, Lending, and Friedman-Savage Gambles" American Economic Review 70 (3) CLOTFELTER C. T. & COOK P. J. (1989) "The Demand for Lottery products" NBER working paper COOK P. J. CLOTFELTER C. T. (1993) "The Peculiar Scale Economies of Lotto" American Economic Review 83 (3) DIECIDUE E., WAKKER P. P. (2001) "On the Intuition of Rank-Dependent Utility" Journal of Risk and Uncertainty 23 (3) FRIEDMAN M. & SAVAGE L. J. (1948) "The Utility Analysis of Choices Involving Risk" Journal of Political Economy 56 (4) GARRETT T. A. (2001) "An International Comparison and Analysis of Lotteries and the Distribution of Lottery Expenditures" International Review of Applied Economics, 15 (2), KAHNEMAN D. & TVERSKY A. (1979) "Prospect Theory: An Analysis of Decision Under Risk" Econometrica 47 (2) MACHINA M. (1982) ""Expected Utility Analysis" without the Independence Axiom" Econometrica 50 (2), MACHINA M. (1994) "Review: Generalized Expected Utility Theory: The Rank-Dependent Model" Journal of Economic Literature 32 (3) MARKOWITZ H. (1952) "The Utility of Wealth" Journal of Political Economy 60 (2), QUIGGIN J. (1982) "A Theory of Anticipated Utility" Journal of Economic Behavior and Organization, QUIGGIN J. (1991) "On the Optimal Design of Lotteries" Economica, TVERSKY A., KAHNEMAN D. (1992) Advances in prospect theory: Cumulative representation of uncertainty Journal of Risk and Uncertainty

25 323. WALKER I. and J. YOUNG (2001) "An Economist s Guide to Lottery Design" The Economic Journal 111, YAARI M. E. (1965) "Convexity in the Theory of Choice Under Risk" Quarterly Journal of Economics, 79 (2) Appendix Proof of Proposition 1. Let us de ne a discrete set of prizes fx i = y( i ); i = 1; :::; ng associated with a set of cumulative probabilities f i = i=n; i = 1; :::; ng where y() satis es H1. The property y 0 () > 0 means that x i+1 > x i 8i = 1; :::; n 1, and n arbitrarily large. Hence, the rst order conditions hold with i = 0, i = 1; :::; n: or, with n! 1: g(i=n) g(i=n 1=n) u 0 (x i ) = 1=, i = 1; :::; n, 1=n g 0 ()u 0 (y()) = 1=, 2]0; 1]. (8) The second order condition (3) is u 00 (y( i )) 0, i = 1; :::; n, or u 00 (y()) 0, 2]0; 1], once n! 1. y() is continuous over [0; 1] as y 0 () < 1 over the same interval. Hence, u 00 (y) 0 over ]y(0); y(1)]. Lastly, let us take the derivative of Eq. (8) with respect to and rearrange terms: g 00 () = u00 (y) u 0 (y) g0 ()y 0 (). Since u(y) and g() and y() are all strictly increasing functions over their respective intervals [y(0); y(1)], [0; 1] and [0; 1], the previous result u 00 (y) 0 over ]y(0); y(1)] implies g 00 () 0 over ]0; 1]. Proof of Proposition 2. 25

26 (a) As already proved, any lottery y j : ' j ; j = 1; :::; k can be reframed as a uniform probability distribution fx i : i=n; i = 1; :::; ng with n large enough, in which a prize of value y j is replaced by n(' j ' j 1 ) prizes x i of value y j, each occurring with probability 1=n (with ' 0 = 0). All x i must consequently meet the second order condition (3). Since prizes take only k distinct values, only k di erent second order conditions hold at the optimum, as in (a). (b) All prizes satisfying x i = y j are indexed by i 2 I j with: I j = n' j 1 + 1; n' j 1 + 2; :::; n' j, j = 1; :::; k. The rst order condition for x i, i 2 I j, j = 1; :::; k, are given by the n ' j ' j 1 conditions (2) which can be rewritten as (with n = 0 and given j = 1; ::; k): i+1 i = u 0 (x i ) [g(i=n) g((i 1)=n)] 1=n, i 2 I j. Written out in full: n'j 1 +2 n'j 1 +1 = u 0 (y j ) g(' j 1 + 1=n) g(' j 1 ) 1=n n'j 1 +3 n'j 1 +2 = u 0 (y j ) g(' j 1 + 2=n) g(' j 1 + 1=n) 1=n ::: (9) n'j n'j 1 = u 0 (y j ) g(' j 1=n) g(' j 2=n) 1=n n'j +1 n'j = u 0 (y j ) g(' j ) g(' j 1=n) 1=n. For j = 2; :::; k the multiplier n'j 1 +1 associated with the constraint x n'j 1 x n'j 1 +1 is equal to 0 in the rst equality since y j 1 = x n'j 1 < x n'j 1 +1 = y j. Likewise, for j = 1; :::; k 1 the multiplier n'j +1 = 0 in the last equality as y j = x n'j < x n'j +1 = y j+1. There are now two cases. If the wealth constraint is not binding: y 1 > w or equivalently x 1 > x 0 meaning that 1 = 0. Summing all the equalities in (9) and rearranging 26

27 terms lead to result (b) (recalling n+1 = 0): g(' j ) g(' j 1 ) 1 = ' j ' j 1 u 0 ; j = 1; :::; k. (10) (y j ) (c) Next, let us combine the rst and the last equalities in (9) (with n'j 1 = n'j = 0): g(' j 1 + 1=n) g(' j 1 ) 1=n + n'j 1 +2 This equality can be rewritten as: = g(' j) g(' j 1=n) 1=n n'j. 1=n n'j 1=n + n'j 1 +2 = g(' j) g(' j 1=n) = g(' j 1 + 1=n) g(' j 1 ) 1 1=n 1=n which is smaller than one since n'j n'j Taking the limit n! 1: g 0 (' j ) g 0 (' j 1 ) 0. (11) This inequality holds for any j = 1; :::; k, as in (c). (d) Straightforward from (c). (b ) Now assume that the wealth constraint is binding: y 1 = w or equivalently x 1 = x 0 meaning that 1 > 0. Summing the n' 1 conditions in (9) for j = 1 and rearranging terms, Eq. (10) becomes: which is equivalent to (b ). g(' 1 ) ' 1 1 u 0 (y 1 ) = 1= (c ) Next, let us combine the rst and the last equalities in (9) (the rst order conditions that x 1 and x n'1 must meet): This equality can be rewritten as: g(1=n) 1=n = g(' 1) g(' 1 1=n) 1=n n'1. 1=n n'1 = g(' 1) g(' 1 1=n) = g(1=n) 1=n =n 1=n. 27

28 Taking the limit n! 1: 1=n n'1 1 = g0 (' 1 ) 1 if n'1 1=n g 0. (0) > 1 if 1 > 2 + n'1 (d ) The inequality in (d ) becomes apparent after integrating g 00 (') over [0; ' 1 ]. Hence, the binding of the wealth constraint may reverse the properties of the weighting function between 0 and ' 1 if the wealth constraint is costly enough for the rm, that is 0 is relatively high compared to n' Proof of Proposition 3. (a) As already explained, any lottery y j : ' j ; j = 1; :::; k can be reframed as a lottery endowed with a large number of evenly-spaced payo s fx i : i ; i = 1; :::; mg, with x i = w + i(=m), w y 1, x m y k, and m taken as high as desired. This lottery satis es i = 0 8 x i < y 1 and i = ' j 8x i = y j. Hence, any probability i must satisfy the second order condition (5) g 00 ( i ) 0, i = 1; :::; m. Once all redundant conditions have been discarded, k + 1 conditions remain that apply to f0; ' 1 ; :::; ' k g, which prove (a). (b) Consider interior optimal probabilities i 2]0; 1[. The rst order conditions are given by Eq. (4): u(x i+1 ) u(x i ) 1 = =m + i i+1 g 0 ( i ) i = t; :::; h 1. (12) Now, consider the largest string of consecutive and identical prizes fx l ; :::; x l+q 1 g. They satisfy x l = y j and x l+q = y j+1, j = 1; :::; k 1. Those prizes satisfy i = ' j and therefore g 0 ( i ) = g 0 (' j ), i = l; :::; l + q 1, j = 1; :::; k 1. 28

29 The q rst order conditions in (12) can be written out in full: l+1 l = =m g 0 (' j ) [u(x l+1 ) u(x l )] (13) l+2 l+1 = =m g 0 (' j ) [u(x l+2 ) u(x l+1 )] ::: l+q 1 l+q 2 = =m g 0 (' j ) [u(x l+q 1 ) u(x l+q 2 )] l+q l+q 1 = =m g 0 (' j ) [u(x l+q ) u(x l+q 1 )]. The multiplier l = 0 in the rst equation since ' j 1 = l 1 < l = ' j. Similarly, l+q = 0 in the last equality as ' j = l+q 1 < l+q = ' j+1. Summing the equalities, and replacing x l and x l+q by y j and y j+1, lead to result (b): u(y j+1 ) u(y j ) y j+1 y j g 0 (' j ) = 1=; j = 1; :::; k 1. (14) (c) Next, let us combine the rst and the last equalities in (13) (with l = l+q = 0) : u(x l+1) u(x l ) =m l+1 = u(x l+q) u(x l+q 1 ) =m + l+q 1. This can be rewritten as: =m + l+q 1 = u(x l+q) u(x l+q 1 ) = u(x l+1) u(x l ) =m l+1 =m =m = u(y j+1) u(y j+1 =m) = u(y j + =m) u(y j ) 1 =m =m 1. which is greater than one as l+q 1 ; l+1 0 and consequently l+q 1 l+1 (recalling that =m l+1 > 0). Taking the limit m! 1: u 0 (y j+1 ) u 0 (y j ) 0, j = 1; :::; k 1, which proves (c). (d) Straightforward from (c). 29

30 (e) Let us de ne the prize x t = y 1. For x i < x t, the rst order conditions in (6) are: Since i 0, i = 1; :::; t 1: u(x i+1 ) u(x i ) =m + i = 1 g 0 (0) u(x i+1 ) u(x i ) =m 1 g 0 (0) i = 1; :::; t 1 i = 1; :::; t 1 Recalling that x i+1 = x i + =m and taking m! 1: u 0 (y) 1 g 0 (0) 8y 2] w; y 1 ] Eq. (14) for j = 1 is: u(y 2 ) u(y 1 ) y 2 y 1 g 0 (' 1 ) = 1=. Combining the last two equations leads to (e): u 0 (y) g0 (' 1 ) u(y 2 ) u(y 1 ) g 0 8y 2] w; y 1 ]. (0) y 2 y 1 (f) Let us de ne the prize x h = y k (with h m). For x i x h or i h, i = 1. The rst order conditions in (7) are: u(x i+1 ) u(x i ) =m i = Since i 0, i = h; :::; m: u(x i ) x i + i = 1 g 0 (1) 1 g 0 (1) i = h; :::; m 1, i = m. u(x i+1 ) u(x i ) =m u(x i+1) u(x i ) =m i = u(x m) x m + m u(x m) x m i = h; :::; m 1. Recalling that x i+1 = x i + =m and taking m! 1: Integrating over [x h ; x m ]: u(x m ) u(x h ) = u 0 (y) u(x m) x m y 2 [x h ; x m [. (15) Z xm x h 30 u 0 (y)dy Z xm u(x m ) x h x m dy

31 and rearranging the terms: Combined with inequality (15): u(x h ) x h u(x m) x m. u(x h ) x h u 0 (y) y 2 [x h ; x m [. Replacing x h by y k and noting that x m can be arbitrarily large through the manipulation of lead to result (f). Proof of Proposition 4. (a) results from the combination of (b) in Proposition 2 and (c) in Proposition 3. (b) results from (b) in Proposition 3 and (c) in Proposition 2. 31