Totomania or Expected Returns? an Econometric Investigation on Singapore Toto Market

Totomania or expected returns?

An econometric investigation on Singapore Toto Market

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

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