Abstract
Conventional wisdom points out that Toto bettors are ruled more by greed and unrealistic expectation of returns than anything else. Their investment decisions rest heavily on the size of jackpots than expected monetary rewards. However, the author finds evidence (at least in Singapore Toto market) suggesting otherwise and bettors do take into consideration their expected monetary returns. Though these bets do not have positive net returns, thus weak-form efficiency exists, the author finds the bettors' decision to play generates a level of sales that conforms to their original forecasts of expected return.

Introduction
Toto is unlike Singapore Sweep or 4-D because the expected monetary return of a ticket depends on the behaviour of other players. Like other betting games, Toto players make their bets and buy a corresponding number of tickets. But unlike the other gambles, Toto agency will take out a portion from the revenue of ticket sales and pay out to winners in the same drawing.

In the case of Singapore Toto, 55 per cent of revenue from sales of tickets goes into prizes of the current game. If there are more than one winner to each prize, the prize money will be split among them. In cases when there are no winners claiming the prizes, the amounts are rolled over to the next game. The rollover will enlarge the jackpot, increasing the bettors' expected rate of return. But the larger jackpot will attract more bettors and larger bets which will then lower the expected return since the probability of more than one winner to the jackpot increases and the winners have to share the prizes. For example, a jackpot of $1 million will usually attract a million tickets and most often only one winner will emerge from the game and claim the jackpot. However, a jackpot of $3 million will often bring in 3 million tickets and usually four players will end up sharing the prize which then amounts to less than $1m each. Hence, each bettor's expected return depends on the behaviour of other bettors and each bettor must project expected value based on what they think other bettors will do.

Since sales of tickets and prize money are public information and readily available, and according to the theory of rational expectation that people will make full use of available information, bettors should generally learn from experience and reduce the making of systematic errors. In stock markets when the bull and bear periods were separated by a relatively long time span, usually years, investors did not have sufficient time to learn from their errors; therefore bull and bear sentiments persisted for a much longer period. But in Toto, every drawing is independent of each other and with two games in every week, bettors should be quickly made to realise that a bigger jackpot affects their expected returns in two-fold. While bigger prize money increases their expected rewards, it also attracts more bettors and larger bets; hence lowering expected returns when the probability of having more than one winner to the prize increases, resulting the jackpot money being diluted among the winners.

The author in this paper seeks to examine whether bettors' demands for Toto tickets depend not only on the size of jackpots but also their expected monetary returns.

Methodology
Expected Monetary Return
The expected monetary return of a bet is given by

Expected Monetary Value = (Probability of Winning)*(Prize Money) - Cost of ticket

Toto, like all lottery games, is an "unfair" game which means its expected monetary value is usually negative. This is, of course, an outright violation of the economic assumption that human beings are generally risk adverse and avoid gambles whose odds are not in their favour. If so, how can we explain the Toto phenomenon? This phenomenon has baffled economists for centuries. To date, there are no satisfactory theories to put this puzzle to rest. Rather than allowing this problem to hinder our analysis, the author is inclined to believe non-pecuniary rewards make up the shortfall in the expected monetary value. By doing so as well as easing the computation process, the author defines expected monetary return as

Expected Monetary Value = (Probability of Winning)*(Prize Money)

The author does not believe this specification will undermine the theoretical foundation of this paper since he seeks only to examine the relationship between changes in expected monetary value with those in ticket sold (demand). Moreover, the cost of ticket remained unchanged throughout the sample period. The exclusion of the cost of ticket will not make the analysis less sound than was previously.

Following Cook and Clotfelter (1993: 636), the expected value of the jackpot from betting one combination is

EV = (probability of win)*((jackpot)*(expected share of jackpot if win).

For a large number of tickets and a small expected number of winners, the expected value per ticket is approximated by

EV = [1/N][J][1-e-pN] (1)

Where N is total ticket sales this drawing, J is total jackpot which include rollover amount if any and the jackpot amount this drawing, and p = (45C6)-1= 1/(8,145,060).

Toto Demands (Ticket Sales)
The bettor's demands depend on the expected returns, which in turn depend on the actual sales for that draw, which are only made known after the draw. Bettors must project expected value based on their projected sales for the next drawing. Fortunately, the Toto agency does make forecasts of the size of jackpot in the next drawing based on historical trends. Unfortunately, these forecasts are not readily available unless one conscientiously jots them down for each and every drawing. Therefore, the author uses ex post sales data and by doing so assumes "rational expectations" which means bettors do not make systematic errors in forecasting the total sales for the draws.

The author does not make a distinction between an individual bettor buying more than one ticket versus new bettors entering the market. Instead, the number of tickets sold is considered to represent overall market demand for Toto.

Regression Specification
Aforementioned, the objective of this paper is to examine the relationship between the demands for Toto versus the expected value as well as jackpot amount. 6 equations are specified and they are:

Model 1 ticketsi = 0 + 1jackpoti (2)
Model 2 ticketsi = 0 + 1(ereturnsi) (3)
Model 3 ticketsi = 0 + 1(jackpoti) + 2(ereturnsi) (4)
Model 4 lticketsi = 0 + 1(ljackpoti) (5)
Model 5 lticketsi = 0 + 1(lreturnsi) (6)
Model 6 lticketsi = 0 + 1(ljackpoti) + 2(lreturnsi) (7)

where tickets = number of tickets sold
jackpot = total jackpot, including rolled-over amount plus jackpot from current drawing
ereturns = expected monetary returns given by equation (1)
ltickets = ln(tickets)
ljackpot = ln(jackpot)
lreturns = ln(ereturns)

Performing regression on these equations, the author expects that if the coefficient of the expected returns to be both positive and significant, it means the bettors' demand for Toto tickets depends on their expected return and not merely on the size of jackpots. Moreover, it means that the market is quite efficient and bettors do not make systematic errors overtime.

If the coefficient of the expected returns is insignificant, it means the bettors' demand for Toto tickets is independent on their expected return and they based their decision merely on the size of jackpots.

If the coefficient of the expected returns is both negative and significant, it means the bettors' demand for Toto depends on the expected value but the wrong way. This means the market is relatively inefficient and one can secure a consistent higher expected return by betting on games with a smaller jackpot amount.

The Game and the Data
Our sample ranges from April 2000 to February 2004. Data on each drawing consist of the six combination of winning numbers plus one additional winning number, the prize money for the first, second, third, four, fifth and sixth category, and the number of winners for each category. All these information are readily obtained from the Singapore Pools website at "http://www.singaporepools.com.sg".

To play Toto, bettor must choose at least six numbers from 1 to 45 and match them against the winning numbers drawn. Toto draws take place on Mondays and Thursdays. Six numbers plus one additional number will be drawn. Bettor wins a prize if his Toto bet matches at least three numbers and the additional number. One Ordinary Entry for one draw costs $0.50. Since the minimum bet for the Toto Game is $1.00, bettors have to buy two Ordinary Entries for one draw or one Ordinary Entry for two draws. In Singapore Toto, 54 per cent of Toto sales goes into the prize pool, distributed among the six prize categories. The prizes include the jackpot or the first prize (6 out of 6 numbers), the second prize (5 out of 6 plus the additional number), the third prize (5 out of 6 numbers), the fourth prize (4 out of 6 plus the additional number), the fifth prize (4 out of 6 numbers), and the sixth prize (3 out of 6 plus the addition number). Their corresponding ratios are 33 per cent, 13 per cent, 13 per cent, 13 per cent, $30 per winner, and $20 per winner of pool prize. The last two categories account for the remaining 28 per cent of the pool prize. Since the fifth and sixth prizes are guaranteed at $30 and $20 per winner, depending on the number of winners, the ratios do fluctuate a little.

From the above information, the author is able to work out the unclaimed jackpot amount, the rolled-over amount, and total tickets sold.

The total number of drawings during this period were 402. Because data from one drawing were unavailable and data from two drawings were lost due to computation needs, we are left with 399 observations.

Results and Conclusion
From Table 1, the six models and their coefficients are all significant at 1 per cent significant. From the Dubin-Watson test and VIP test, there is little evidence of autocorrelation with the exception of model 3 and 6.

For model 2 and 4, when the expected value is the sole independent variable, its coefficient is significant and positive. This implies that the bettors' demand for Toto tickets depends on their expected return and not merely on the size of jackpot. Moreover, it means that the market is quite efficiency and bettors do not make systematic errors overtime.

However, in model 3 and 6 when the expected return is being regressed together with independent variable jackpot. The coefficient of the expected return is significant but negative.The bettors' demand for Toto depends on the expected value but the wrong way. This also implies the market is relatively inefficient and one can secure a consistent higher expected return by betting on games with a smaller jackpot amount. But the author suspects the results to be unreliable because the VIF tests indicate the existence of multicollinearity in the models. Multicollinearity causes estimator standard errors to be large; thus leading to reliable t-test for the estimates.

Limitations of the Study
The constraint of time and the author's incompetence in both the subject matter and econometrics knowledge have made this report far from complete and rigorous. The results obtained may be objective, but the interpretation of the findings and inference is rather subjective in nature. In addition the way the study is conducted and the way the data are obtained can also contribute to the invalidity of the results as explained below:

First, aforementioned, the author uses the ex post sales data and by doing so assumes "rational expectations" and therefore bettors do not make systematic errors in forecasting the total sales for the draw. This assumption is far from reasonable and realistic. A better specification is:

Sales Forecastt = Fn(Actual Salest-1 , Sales Forecastt-1) and Fn(Agency's Forecastt) where the sales forecast for period t is expressed and estimated using the function of actual sales and sales forecast for period t-1 as well as the agency's forecast for period t.

Second, the different weights of the total jackpot size and the probability of winning to the expected value given by equation (1) may affect the regression results greatly. The sign of the expected value is negative when regressed with independent variable jackpot but positive when it is the sole regressor. A better specification is to isolate the probability of winning from the expected value.

Third, as mentioned earlier, the ratios are of the six prize categories are fixed at 33 per cent, 13 per cent, 13 per cent, 13 per cent, $30 per winner, and $20 per winner of pool prize. The last two categories account for the remaining 28 per cent of the pool prize. Since the fifth and sixth prizes are guaranteed at $30 and $20 per winner, depending on the number of winners, the ratios do fluctuate. However, the author for ease of computation assumes the ratios are fixed at the official stated ratios. Moreover, the author could not work backward to obtain the exact ratios due to the information is not publicly available.

Fourth, the White's Test rejects the null hypothesis of no heteroscedasticity. This implies that the standard errors of the parameter estimates are incorrect and, thus, any inferences derived from them may be misleading.

Leo Kee Chye

Saturday, April 24, 2004