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How To Design A National Lottery To Maximise Income To The Public Purse

Governments should make the best use of national or state lotteries to support the public finances. And according to research by Akira Maeda, there is an optimal combination of prizes and winning probabilities that will maximise a government's revenue regardless of the way the lottery is organised.

The study, published in the Economic Journal, draws on a survey by the Japan Lottery Association in 2002, which shows that for many ticket buyers, playing the national lottery is neither a charitable donation nor a realistic investment. For lottery designers, this suggests that the more willing customers are to buy lottery tickets for ''fun'', the lower the value of the prizes needs to be.

A lottery originator pays out prizes to winners from total sales, and uses the remainder of the proceeds for the benefit of the public or the non-profit enterprise sponsoring the lottery. Lotteries effectively work as a substitute for taxation and donations.

Lottery games have a history as public fundraising instruments nearly as long as that of charitable donations. But most economists in academia have shied away from the subject. While lotteries have long played a role in public financing practices, the reasons that they actually work, and their raison d'être remain unclear.

There are two rational explanations for why lottery tickets are purchased:

  • First, it is possible that people voluntarily buying lottery tickets are doing so because they wish to contribute to the public finances.
  • Second, there may be a desire to make money. Ticket buyers supposedly think that the ticket is worth the investment.

Observing lottery games in the real world, however, leaves the author unconvinced that all people buying lottery tickets are motivated by either of these two reasons.

The reasons offered by respondents from the Japan Lottery Association survey include: ''expectation of winning the prize'' (60%), ''having a big dream'' (53%), ''just for fun'' (42%) and ''to have fun regardless of winning or losing'' (40%).

These results indicate that for most ticket buyers, playing the national lottery is neither a charitable donation nor a realistic investment. Moreover, from the viewpoint of standard investment theory, the behaviour of lottery ticket buyers makes no sense.

Since a lottery originator pays out prizes to lottery winners from the total sales, it should be clear to lottery ticket buyers that their likely return will be less than their investment.

This study represents a new approach to lotteries, offering practical guidance on optimal lottery design that will maximise total revenue. Two important findings are that:

  • First, ''lottery games should be designed such that the ratio of the winning prize amount in each prize class to total lottery sales be equalised to the elasticity of demand for lottery ticket purchases, with respect to the winning prize amount in each prize class''. In other words, the more willing customers are to buy lottery tickets the lower the value of the prize is required to be.
  • Second, ''given that lotteries are optimally designed, the fundraising potential of a lottery is independent of its type (specifically, of whether it is an ordinary type with fixed prizes, or a pari-mutuel or lotto type with fixed probabilities of winning)''.

''Optimal Lottery Design for Public Financing'' by Akira Maeda is published in the October 2008 issue of the Economic Journal.

Akira Maeda

associate professor at Graduate School of Energy Science, Kyoto University,