Testing for efficiency in lotto markets.

Scott, Frank A., Jr. ; Gulley, O. David

I. INTRODUCTION

When some agents behave irrationally or when some markets operate inefficiently, opportunities exist for others to profit. The profit motive tends to eliminate these opportunities so that markets will tend to be efficient. Interest in the efficiency of markets has led to much empirical testing. An enormous amount of work has been done in the field of finance investigating the efficiency of various financial markets.

Another area where individual rationality and market efficiency are prominent is wagering markets. While the economic significance of financial markets dwarfs that of wagering markets, gambling events offer excellent natural experiments for examining the same sort of economic behavior exhibited in financial markets. One gambling instrument, state-sponsored lotto games, is particularly interesting because of the way the mathematical expected value of a bet is determined.

Lotto differs from most other lottery products because the expected monetary value of a ticket depends on the behavior of other bettors. The expected value also depends on the amount of money rolled over (if any) from the previous drawing's jackpot. Repeated drawings of a lotto game thus present consumers with a range of betting opportunities, some more favorable than others. In deciding whether to purchase tickets, bettors must evaluate the expected monetary return, which requires them to forecast sales. A rational expectations equilibrium in the lotto market will not exist unless bettors' individual forecasts lead to an overall level of sales and ex post expected value that conform to their original expectations.

There are several aspects of lotto that make its study worthwhile. As with financial markets, a significant portion of the population participates. Just like investors, lotto players must formulate expectations about the return on their investment. Expected monetary return depends on the behavior of other players. Unlike investing in the stock market, however, the outcome of the purchase of a lotto ticket is based on objective probabilities. As Thaler and Ziemba [1988, 162] point out, the conditions for learning are optimal in lotto because there is quick and repeated feedback. If an efficient market equilibrium does not exist in lotto, we should not be optimistic about finding one in more complicated financial markets.

The next section introduces the notions of efficiency developed by Fama [1970] and applies them to wagering markets. The third section follows with an explanation of the lotto game and how the expected monetary value of a bet is determined. The fourth section contains tests of the weak form of market efficiency using data from the Kentucky, Massachusetts, and Ohio lotto games. Finally, in the fifth section strong-form efficiency is evaluated using these same games and applying the concept of a rational expectations equilibrium.

Unsurprisingly, we find that very rarely do lotto games offer a positive net expected return, thus meeting the requirements of weak-form efficiency. More importantly, we find general support for the existence of a rational expectations equilibrium in lotto markets. In most cases individual bettors' decisions to play generate a level of sales that conform to their original expectations of the expected value of a lotto ticket.

II. EFFICIENCY IN WAGERING MAVKETS

A capital market is efficient if there are no investment strategies that will yield abnormally high returns. Fama [1970] defines efficiency to mean that security prices reflect all the information contained in a given information set. If the information set is comprised of only historical prices, the market is weak-form efficient. If the information set includes all publicly available information, the market is semi-strong-form efficient. The inclusion of insider information as well makes the market strong-form efficient.

Interest in efficiency has led to much empirical testing of the efficiency of financial markets. Most work finds that markets are by and large efficient, but there are exceptions. LeRoy [1989] and Fama [1991] provide recent surveys of the prodigious research on capital market efficiency. Richard Thaler writes a regular column on "Anomalies" in the Journal Economic Perspectives. Many examples for the column come from the field of finance.

The notion of efficiency can be applied to other areas. Wagering markets present an excellent opportunity to test for efficiency. In considering horseracing and lotteries, Thaler and Ziemba [1988] modify Fama's definitions to fit wagering markets. They define weak efficiency to exist if there is no betting opportunity available that will yield a positive net expected return, i.e., there are no ex ante profitable betting opportunities. Strong efficiency exists if all bets have expected values equal to (1-t) times the amount bet, where t is the takeout rate, i.e. the proportion of each bet retained by the betting agency.

Considerable work has been done on the efficiency of the horserace betting market.(1) Snyder [1978] finds that while one can earn above-average returns by following certain betting strategies, positive returns are not to be expected due to the take-out rate collected by the tracks. Ali [1977] finds that bettors tend to overbet longshots and underbet favorites, and in a later paper [1979] finds that different types of bets, which should be identically priced according to the efficient markets model, are in fact so priced. Asch and Quandt [1987] take a similar approach, but find some inefficiencies. Asch, Malkiel, and Quandt [1984] find that net profits may be possible in place and show betting.

Betting on football (American and English) differs slightly from horseracing in that it is not pari-mutuel. Zuber, Gandar, and Bowers [1985] examine betting on National Football League (NFL) games and suggest a strategy that would have produced net positive returns during their sample period. Sauer, Brajer, Ferris, and Marr [1988] apply that strategy to a later period and demonstrate that the inefficiencies disappear. Golec and Tamarkin [1991] also study NFL betting and find that bets on underdogs or home teams are potentially profitable, if transactions costs are low enough. Pope and Peel [1989] investigate betting on English football, where prices are fixed by bookmakers and differ across firms. They find that this market meets the most important criterion for efficiency, namely, no trading rule generates abnormal profits.

Because most lottery products have a negative and unchanging expected monetary return, only limited research has been done on the efficiency of lottery markets. Chernoff [1981], Thaler and Ziemba [1988], and Clotfelter and Cook [1989] evaluate the possibility of favorable investment opportunities arising from popular and unpopular numbers in lotto and numbers games. Cook and Clotfelter [1990] derive the relationship between the expected value of a lotto bet and sales and rollover, but then investigate economies of scale rather than testing for the possibility of abnormal returns.

III. EXPECTED VALUE OF A LOTTO BET

The three most common lottery products are instant (or scratch-off) games, numbers games, and lotto. To win the grand prize in a typical lotto game, a player buys a one-dollar ticket and must correctly match six numbers drawn randomly without replacement from, say, forty-four numbers. This is called a 6/44 game. The probability of any ticket winning the jackpot in a 6/44 game is 1 out of 7,059,052.(2) Lesser prizes are often awarded for matching fewer than six of the numbers. Lottery agencies take out from 40 to 50 percent of each dollar bet, some of which covers operating costs and the rest of which is turned over to the state.

Lottos have several interesting features. If the jackpot is not won on a given draw, the jackpot (minus prize payments for any partially correct tickets) is rolled over into the jackpot for the next drawing. These rollovers can create jackpots in the tens of millions of dollars. In addition, lottos are pari-mutuel games, which means that there can be multiple winners. Winning ticket holders share equally the grand prize. Finally, the grand prize usually is paid out over a twenty-year period. The advertised jackpot is naturally the undiscounted sum of the twenty annual payments.

The expected monetary value of a $1 lotto ticket thus depends on several factors, namely, the structure of the game, the value of previous jackpots (if any) rolled over into the current jackpot, and the number of tickets bought in the current drawing. Formally, expected monetary value is

(1) EV = [probability] x [jackpot] x [share] + [expected value of smaller prizes],

where

probability = probability that any given ticket matches six numbers drawn randomly without replacement from forty-four possible numbers, i.e., probability that any ticket wins the grand prize;

jackpot = dollar amount rolled over from previous unawarded jackpots plus proportion of current sales not retained by the lottery agency;

share = expected share of the jackpot if a winning ticket is held; and

expected value of smaller prizes = expected value of any smaller prizes awarded to players who correctly match fewer than six of the winning numbers.(3)

If bettors choose their numbers randomly, the probability distribution of winning tickets follows the binomial distribution.(4) Thus in a 6/44 game with one million players each randomly selecting their integer combination, the probability that, for example, exactly two players pick the winning combination is n[C.sub.k][p.sup.k][(1-p).sup.n-k], where n = 1,000,000, k = 2, and p = 1/7,059,052.(5) Since the probability of any ticket matching the six correct numbers is very small, and the number of tickets purchased is typically very large, then the Poisson probability distribution serves as a good approximation to the binomial distribution. As is shown by Saunders and Moody [1987] and also by Cook and Clotfelter [1990], the expected monetary value of a one-dollar ticket thus becomes

(2) EV = [1/N][R + (1 - t)N][1 - [e.sup.-pN]],

where N is total ticket sales this drawing, R is rollover, t is the takeout rate, and p is the above-defined probability of holding a winning ticket.

The expected monetary value of a lotto bet thus follows well-defined laws of probability and depends on readily available information. Since different values of sales and rollover cause expected value to vary from drawing to drawing, the responses of bettors can be analyzed. In addition to the monetary return of the bet, there also exists a nonmonetary return (i.e. the value derived from watching the numbers being drawn on television, thinking of how any prize money would be spent, etc.) The expected total return thus can be written as

(3) E[TR] = EV + E[PL],

where EV is as defined in equation (2) and E[PL] is the anticipated pleasure or nonmonetary return from betting.

As do previous authors who analyze efficiency in other betting markets, however, we do not attempt to formally incorporate such nonmonetary returns in our empirical analysis. Snyder [1978] points out the difficulty in disentangling, at least statistically, the pecuniary and nonpecuniary aspects of a bet. He concludes that the presence of these (unmodeled) nonpecuniary aspects merely strengthens his finding of efficiency in the horserace betting market.

We argue that this reasoning can be applied to our analysis as well. We seek to determine if pervasive market forces actually work toward efficiency in lotto markets. Before concluding that lotto markets are inefficient, caution must be exercised, because such a finding may be due to misspecification. The existence of a pleasure component that is not explicitly incorporated may cause us to think that we have discovered an inefficient market when in fact all that we have found is that the pleasure component swamps the financial component.

If, however, we find that even in the presence of a pleasure component which we are not able to incorporate explicitly, lotto markets tend to be efficient, then either of two conclusions is possible. One is that financial forces do not work towards efficiency, and that whatever direction they do work, they are exactly offset by pleasure forces working in the opposite direction. Alternatively, it may be that financial forces work towards efficiency, and pleasure forces are relatively insignificant. While we are willing to admit that the first alternative is possible, we find the second explanation much more convincing.

We conclude that lotto offers an opportunity to test for market efficiency that is perhaps superior to horseracing or football because ex ante probabilities are objective and not subjective, and because lotteries do not involve all the nonmonetary consumption benefits of attending a track and watching the horses race or watching a football game.

Winning the lotto is, however, an extremely low probability event. Even if a given draw is characterized by a positive net expected return, the dollar expenditure and transactions costs of covering even a small proportion of the possible combinations would be prohibitive for most players. This aspect of lotteries makes market efficiency tests all the more interesting, because we are most likely to reject efficiency in situations like lotto where the economic incentive to optimize expected return is so small.

IV. DATA

Before proceeding further, a brief discussion of data is in order. We employ data from four lotto games in three states. The Kentucky Lotto is a 6/42 game which started out with weekly drawings but soon moved to twice-weekly drawings. Actual jackpots have been announced at the beginning of each drawing period, and so are only indirectly driven by sales. The sample period runs from November 1989 until January 1991. Massachusetts Megabucks is a 6/36 game with twice-weekly drawings. A predicted jackpot is announced at the beginning of each drawing period, but the actual jackpot depends on rollover and sales during the drawing period. The sample period runs from July 1984 until December 1990.

Massmillions is a 6/46 game with weekly drawings. As with the Megabucks game, its jackpot is determined by rollover from previous draws and actual sales during the period. The sample period is from May 1987 until December 1990. The Ohio Super Lotto is a 6/44 game with twice-weekly drawings. Jackpots are announced at the beginning of each drawing period. The sample period is from August 1989 until September 1990.(6)

V. BETTOR BEHAVIOR AND A TEST OF WEAK EFFICIENCY

The take-out rate in lottery games is very high compared to other forms of gambling; therefore it is natural to ask why anyone would play lotto. As mentioned above, lotto obviously provides a thrill. Clotfelter and Cook [1989] offer several explanations of bettor behavior and use responses from surveys of players to back up their hypotheses. Some bettors play for fun, while others play hoping for financial gain. Quiggin [1991] attempts to reconcile the risk-seeking behavior implicit in lottery play with the observation that individuals in general display risk aversion. He modifies the Friedman-Savage expected utility model and is able to rationalize the observed structure of prizes in lottery games.

As discussed above, we do not formally model the nonmonetary returns of a lotto bet. Our purpose is to test whether financial forces act to move the market toward efficiency. It is sufficient for our test that bettors have different reservation prices, these being a function of bettors' attitudes toward risk and the utility (if any) derived from betting. The different reservation prices imply a downward-sloping demand curve for betting. The price of a dollar bet on the lotto is $1-EV, i.e., the purchase price minus the expected value of a ticket. As the expected value of a lotto bet rises, the price of a bet falls. Lower prices induce more players to participate in the game, and existing players are likely to purchase additional tickets.(7)

Weak-form efficiency exists if there are no betting opportunities that have a positive net expected value (Thaler and Ziemba [1988]). For a large enough rollover and a small enough level of sales, the expected value of a $1 ticket (from equation (2)) can be more than $1. The potential for profit from such a situation would likely encourage additional betting, which would reduce the expected value to no more than $1. Using data from the lotto games described above, we can test for weak-form efficiency.

The expected values of a one-dollar lottery ticket over repeated draws for each of these games are plotted in Figures 1-4. For the Kentucky Lotto in Figure 1, expected value varies from $0.09 to $0.58.(8) These figures assume that bettors realize the announced jackpot is the undiscounted sum of twenty annual payments and are able to calculate present value.(9) If that is so then the Kentucky Lotto market is weakly efficient. If bettors use the announced (undiscounted) jackpot the expected value would vary from $.18 to $1.17.

Figure 2 contains expected values for the Massmillions game. Discounted present values of the jackpot are used, rather than undiscounted values. Expected values range from $.28 to $.95. Expected values never exceed the dollar price of a ticket, hence the Massmillions market is also weakly efficient. Figure 3 contains the Megabucks expected values, also using discounted present value of the jackpot. Only once in six years did the game yield a positive net expected return. Figure 4 contains the Ohio Super Lotto expected values, which range from $0.16 to $0.76. Again, weak efficiency is indicated. That expected returns are consistently negative suggests risk-seeking behavior or nonpecuniary returns or both on the part of bettors.

Weak-form efficiency requires relatively simple bettor behavior: they must be able to recognize the potential for abnormal profits. Their collective response then eliminates this potential. We now turn our attention to strong-form efficiency, which involves more sophisticated bettor behavior: the ability to forecast the expected value of a lotto ticket given the information available to them.

VI. STRONG EFFICIENCY AND A TEST OF A RATIONAL EXPECTATIONS EQUILIBRIUM

If bettors knew ex ante what the expected value of a ticket would be in each drawing, then an analysis of lotto demand would be like any other commodity whose price is known with certainty. It is not so simple, however, because the expected value of a ticket depends on the behavior of other bettors and is only known with certainty ex post. Bettors must project expected value based on what they think other bettors will do.

Strong-form efficiency exists if all bets have expected values equal to one minus the takeout rate (Thaler and Ziemba [1988]). Both Kentucky and Ohio claim ultimately to return 50 percent of each bet to players in the form of prize money, and Massachusetts pays out 60 percent. From Figures 1-4 it is clear that expected values vary significantly from drawing to drawing, both exceeding and falling short of one minus the takeout rate. Strong form efficiency would seem not to be supported by the data in Figures 1-4.

With lotto games, however, such a simple test is not sufficient. As equation (2) indicates, the expected value of a lotto ticket depends on the structure of the game (i.e. the probability), the dollar amount rolled over from previous drawings, and the number of tickets purchased by bettors. The odds structure of the game does not change from drawing to drawing; however, rollover and sales do. The combined effect of rollover and sales on expected value can be seen if we plot the relationship between expected value and sales for different values of rollover. A different convergence path (i.e. the relationship between expected value and sales) exists for each value of rollover. Figure 5 illustrates the convergence path in the Massmillions game for selected rollover amounts.

Figure 5 indicates that in the Massmillions game, if there is no rollover and sales equal $4,400,000, a one-dollar ticket has an expected value of $.19. If sales equal $8,800,000 then expected value rises to $.31. If $4,000,000 has been rolled over from previous drawings and sales equal $8,800,000, a one-dollar ticket has an expected value of $.58. In the limit, as sales approach infinity, expected value converges to one minus the takeout rate.(10)

Now, in practice which is more important in determining expected value, sales or rollover? During the period between May 1, 1987 and December 7, 1990, sales typically ranged from between one and five million dollars, with an average per draw of $2,055,000. Rollovers ranged from zero to $22,554,000. Since expected value only approaches one minus the takeout rate in the limit, and since bettors under ordinary circumstances are only willing to buy a limited number of tickets, in any given draw of an actual lotto game expected value will depend largely on the size of the rollover. Over the feasible range of sales there will still likely be a significant difference between expected value and one minus the takeout rate, even if bettors behave in the manner that we have described. A simple comparison of expected value with one minus the takeout rate thus does not constitute a meaningful test of strong-form efficiency.

Since a direct test of strong-form efficiency is not possible, we propose the following test of market efficiency instead. At the beginning of each drawing period, information is available to bettors on the amount of rollover, whether the draw will be held on a weekday or weekend, and previous sales trends. Bettors will decide whether to play the lotto based on their reservation price and their assessment of the expected return. The expected return depends on other bettors' behavior; therefore each bettor must generate his or her own forecast of total sales. The relevant question becomes: do lotto players make systematic errors in their forecasts of sales and hence expected value?

The concept of a rational expectations equilibrium is useful here. If lotto players make systematic forecast errors then the lotto market is not in equilibrium. If expectations are not correct on average, then expectations will not be confirmed by the outcomes of the game and rational players will adjust their expectations. Let us use Figure 5 to illustrate. Suppose that when rollover is zero the typical bettor forecasts sales to be $8.8 million. If this bettor finds the projected expected value of $0.31 attractive, he or she will buy lotto tickets. If actual sales when rollover is zero are only $4.4 million instead of $8.8 million, the realized return on a lotto ticket will only be $0.19, which is less than expected. If bettors comprehend and are able to process that information, then they will buy fewer tickets in subsequent drawings with similar rollovers.

A rational expectations equilibrium occurs when expectations generate an outcome that conforms to those expectations. In the context of lotto this means that bettors forecast sales and expected value, and then decide whether to play based on that forecast. Equilibrium means that in aggregate, bettors' decisions to play generate a level of sales that conform to their original expectations of sales and hence, expected value.

Testing for a rational expectations equilibrium in this context involves more than determining whether bettors' forecasts of sales are orthogonal to the information set available at the beginning of each drawing period, which is the standard test of rational expectations. In our model bettors are not concerned about sales per se, instead they care about the expected value of a bet. That requires them to take their sales forecast and combine it with their understanding of probability to generate a forecast of expected value, because it is their projection of expected value that determines whether and to what extent they purchase lotto tickets.(11)

To perform the test we regress the outcome of the lotto drawing, ex post expected value (computed from equation (2)), on the information available to bettors at the beginning of each drawing period. This includes rollover, whether a Wednesday or Saturday draw, the official prediction of the jackpot, and prizes in competing games. If bettors' predictions of expected value are fully captured by these items, then the errors that they make in forecasting expected value will contain no extractable information, i.e. they will be random. The residuals of the regression equation would then be uncorrelated with actual ticket sales. If bettors systematically misforecast expected value, then the residuals will be correlated with actual sales. A player could improve his or her expected return by playing the lotto only when other bettors have underforecast expected value.(12)

Results for the Massmillions game are contained in Columns A and B of Table I. Massmillions drawings are held on Friday nights. Rollover is known at the beginning of each drawing period. The lottery agency announces a projected jackpot based on its own forecast of sales. In separate regressions investigating lotto demand Massmillions players displayed some sensitivity to the Saturday Megabucks lotto.

The information set available to bettors thus contains Massmillions rollover, Megabucks rollover, the estimated jackpots for both Massmillions and Megabucks, and a time trend. Bettors are assumed to forecast expected value using a quadratic functional form. Hence ex post expected value is regressed on the items in the information set plus squared terms for Massmillions rollover and Megabucks rollover. When the residuals from this regression are regressed on Massmillions sales, there is no significant relationship. This result is consistent with the existence of a rational expectations equilibrium in the Massmillions game.

Megabucks drawings occur on Wednesday and Saturday nights. Rollover is known at the beginning of each drawing period. The lottery agency announces a projected jackpot based on its own forecast of sales. The expected value of a Megabucks ticket is regressed on these variables plus a time trend and rollover squared, and the results are contained in Column C of Table I.(13) Next, the residuals from this regression are regressed on ticket sales. These results are in Column D. The significant negative relationship between sales and the residuals indicates that bettors systematically underpredict expected value when rollover is small, and vice versa.(14)

The Kentucky Lotto differs somewhat from the Massachusetts lottos in that the jackpot is announced at the beginning of each drawing period and does not depend on sales during the period. The jackpot leads rather than lags sales. To estimate expected value bettors need to forecast sales only because the number of other bettors affects the probability that they will have to share the prize. The expected value regression for Kentucky thus includes jackpot and its square, a time trend, a weekly drawing dummy variable for the first twenty-four weeks when drawings were only held on Saturday nights, and a Wednesday drawing dummy for midweek draws after a twice-weekly format was adopted.

These results are contained in Column A of Table II. The significantly negative coefficient for the Wednesday drawing reflects [TABULAR DATA FOR TABLE 1 OMITTED] the lower sales associated with middle-of-the-week drawings. The significantly positive coefficient for the time trend is opposite that of the Massachusetts lottos. Since Kentucky uses a pre-announced jackpot, if sales decline over time then the probability of sharing the prize falls and hence the expected value for a given jackpot rises. This result is therefore consistent with the negative time trends in the two Massachusetts lottos.

When the residuals from this regression are regressed on sales there is a significant negative correlation. An analysis of the residuals, however, reveals an interesting occurrence. Shortly after the introduction of the Kentucky Lotto, the jackpot rolled over nine times without a winner and grew to $5,000,000. The accompanying media blitz led to a more than doubling of ticket sales from the ninth draw to the tenth, when the jackpot was finally won.

[TABULAR DATA FOR TABLE 2 OMITTED]

The residual associated with this drawing is an extreme outlier.(15) When that particular observation is not included in the regressions, the correlation between sales and the residuals is not statistically significant.(16) One possible explanation is that, given the novelty of the game, Kentucky lotto players had no basis for forming expectations about other players' behavior. As a group they overreacted to the large jackpot, driving expected value below what they had anticipated. More recent jackpots have grown to as large as $10 million without eliciting such responses.

The Ohio Super Lotto uses pre-announced jackpots just like the Kentucky Lotto. It is a game that has existed for a number of years, with fairly minor changes in the determination of the jackpot from time to time. Thus there should be no novelty effects in the sample period chosen. The expected value regression includes the announced jackpot and its square, a time trend, and a dummy for Wednesday draws. Results are in Column C of Table II, and they contain no surprises. Regressing the residuals of this regression on sales yields no significant correlation, which is consistent with a rational expectations equilibrium.

VII. SUMMARY AND CONCLUSIONS

Market efficiency is a critical concept in economics and finance. Only infrequently do situations arise where relatively clean tests of rational agent behavior or market efficiency can be conducted. State-sponsored lotto games do seem to offer such an opportunity, because the mathematical expectation of a bet depends on, among other things, bettors' aggregate behavior. In deciding whether to play each bettor must predict how others will behave and then act on that prediction.

We find that only on very rare occasions do lotto games offer a positive net expected monetary return, so that lotto markets are generally weak-form efficient. The strong-form efficiency requirement that expected value equal one minus the take-out rate is not meaningful for a game such as lotto where rollover is one of the primary determinants of expected value. Hence our application of the concept of a rational expectations equilibrium to lotto.

Our results offer some support for the existence of a rational expectations equilibrium. Systematic forecast errors by bettors are not evident in the Massmillions game and the Ohio Super Lotto. Individual bettors' decisions to play generate a level of sales that conform to their original expectations of expected monetary value. That conclusion is not supported by the Massachusetts Megabucks lotto and the earliest days of the Kentucky Lotto. It is interesting to note that, unlike the other three games studied, the structure of Megabucks lends itself to relatively few rollovers. In Massmillions, Kentucky Lotto, and Ohio Super Lotto where rollovers occur more frequently and bettors have more opportunities to learn, bettors' expectations appear on average to be correct.

1. For a thorough review, see Thaler and Ziemba [1988].

2. The number of possible combinations is given by 44!/6!38!

3. In Massachusetts, for example, these smaller prizes are not Pari-mutuel and do not vary from drawing to drawing.

4. If players do not choose their numbers randomly, then the binomial distribution does not describe the probability distribution of bets. Cook and Clotfelter [1990], however, show that under plausible assumptions, expected value is asymptotic to the relative popularity of the particular numerical combination times the proportion of each bet allocated to the jackpot. They also note that the correlation between actual "coverage" of combinations bet and the coverage if all numbers were chosen randomly is close to one.

5. The notation n[C.sub.k] denotes n!/k!(n-k)!

6. Sample periods were chosen such that the structure of each lotto game remained the same. Each of the games have recently changed formats, adopting longer odds that lead to more rollovers. The marketing philosophy seems to be that more rollovers create bigger jackpots and disproportionate increases in subsequent sales.

7. See Gulley and Scott [1993] for a discussion and estimation of lotto demand curves.

8. The expected values of a Kentucky Lotto ticket may seem somewhat low, however, Kentucky offered volume discounts on lotto tickets that were not available in the other two states. In addition to single tickets for a dollar, three tickets could be purchased for $2.00 and eight tickets for $5.00. The average price of a lotto chance in Kentucky was therefore less than a dollar, and in fact, weighted by the proportion of each option selected by players, was around $0.66.

9. The present values of the jackpots are computed using the yield-to-maturity on the twenty-year Treasury bond on the day of the drawings.

10. The behavior of expected value in the presence of large rollovers can be understood by inspecting equation (2). Increases in rollover influence the middle term of the equation, causing expected value to increase. However, the first and third terms force expected value to decrease and converge to (1-t).

11. This testing procedure is consistent with the more general model of betting behavior described earlier in equation (3) that incorporates both monetary and nonmonetary returns from betting if the two sources of returns are uncorrelated with each other. We are grateful to the editors for pointing this out to us.

12. Exploiting such a profit opportunity is difficult in practice because of the logistical problems encountered in buying a large number of lotto tickets. Such a strategy is not impossible, however, as was demonstrated recently when an Australian syndicate attempted to cover all the numbers in the Virginia state lotto.

13. The time trend variable is included to pick up any factors that systematically affect sales and expected value over time.

14. Note that as in all tests of rational expectations, ours is a joint test of rational expectations and the adequacy of our specific model. It is possible that our model does not fully capture the behavior of bettors.

15. In the sixteenth week of the newly introduced Kentucky Lotto the jackpot grew after several rollovers to $3.5 million dollars, before it was finally won. On the thirty-fourth draw the jackpot grew to $5.0 million dollars. Both were record highs to those respective points in time, and generated sales frenzies. The residuals associated with these two observations in the expected value regression are -.017 and -.031, respectively, by far the largest residuals in the sample.

16. The t-statistic falls from 2.22 to 1.57, which is not significant at the 10-percent level.

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FRANK A. SCOTT, JR. and O. DAVID GULLEY, Associate Professor, Economics, University of Kentucky and Assistant Professor, Economics, Bentley College. The authors would like to thank Steve Holland, Mark Toma, members of the Applied Microeconomics Workshop at the University of Kentucky, and two anonymous referees for helpful comments. Gail Antonellis, Jennifer Bishop, Thomas O'Heir, Susie Chin, Peter Ramsey, and Dominic Cypriano provided assistance with collecting the data. Don Brown provided assistance with the figures. Gulley's research was supported by a Bentley College Summer Research Grant.