The Economics and Math Of Lotto

(Excerpted from True Odds by James Walsh, Merritt Publishing, 1996, Santa Monica, CA. 1-800-638-7597.) Reprinted with permission of Merritt Publishing. All rights reserved.

Considering horse racing and lotteries, economist Richard Thaler defined weak efficiency as existing if there is no betting opportunity available that will yield a positive net expected return. In other words, there are no profitable betting opportunities before the outcome is determined.

Thaler defined strong efficiency as existing if all bets have expected values equal to (1-t) times the amount bet, where t is the house's edge.

Thaler pointed out that the conditions for learning are optimal in Lotto games because there is quick and repeated feedback. If an efficient market equilibrium does not exist in Lotto, you shouldn't be optimistic about finding one in more complicated financial markets.

Lotto games differ from other lottery products because the expected monetary value of a ticket depends on the behavior of other players. 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 players with a range of betting opportunities, some more favorable than others.

The concept of a rational expectations equilibrium is useful in this context. If expectations are not correct on average, then they won't be confirmed by the outcomes of the game. Rational players will be caught once or twice in the wrong position and adjust their expectations--and possibly stop gambling.

For example: When a Lotto jackpot is headed for $10 million, a player might find the expected value of $0.30 for a lottery ticket attractive. He or she will buy tickets. If the actual jackpot only reaches $5 million, the expected value of a Lotto ticket will be something less--$0.20 or $0.15.

If the player is able to process this information, he or she will buy fewer tickets in subsequent games.

A rational expectations equilibrium occurs when expectations generate an outcome that conforms to those expectations. In the context of lotteries, equilibrium means that players' decisions to play generate a level of sales that conforms to their original expectations of jackpot size and, therefore, expected value of the lottery tickets.

In the ideal context, players aren't concerned about the size of the jackpots--what they care about is the expected value of a bet. The problem with this ideal is that is requires players to make a sales projection for the jackpot and combine it with their understanding of probability to generate a forecast of expected value.

However, buying patterns in lotteries suggest that players don't care about expected value as much as they should. More precisely, they value the pleasure factor of buying lottery tickets in such an erratic way that the results are effectively random--that is, they offer no discernible conclusions.

The Math Of Lotto

If players knew before they bought tickets 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 know with certainty. It is not so simple, however, because the expected value of a ticket depends on the behavior of other players and is only known with certainty after the drawing. Players must project expected value based on what they think other bettors will do.

But this doesn't mean you can't calculate the odds of winning a lottery.

Winning the Lotto is an extremely low probability event. Even if a given draw is characterized by a positive net expected return, the dollar expenditure and transaction costs of covering even a small proportion of the possible combinations would be prohibitive for most players.

Lottos have several other 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 twenty annual payments.

The expected monetary value of a $1 Lotto ticket 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.

In deciding whether to purchase tickets, players must evaluate the expected monetary return, which requires them to forecast sales. Like investors, Lotto players must project an expected return on their investment. And, like financial investments, expected monetary return in Lotto games 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.

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