#### Lottery Purchases Explained by Curved Utility Functions

Lottery is a popular way for states to raise money for a variety of public projects. Unlike traditional taxes, they are generally seen as painless and not particularly onerous. Lottery revenue has allowed governments to provide a wide range of services without raising taxes too much for working people. At the time of the Revolutionary War, Alexander Hamilton argued that “the great bulk of the population will be willing to hazard trifling sums for the opportunity of winning a considerable gain.”

But there is something deeply wrong with this picture. Lotteries entice people to spend \$50 or \$100 a week, even though the odds are stacked against them, by giving them a sliver of hope that they will win. This is not just a bad economic idea, it is morally wrong.

The chances of winning the lottery depend on how many tickets you buy. If you buy all the tickets in a single roll, your odds of winning are 1 in 3. But as soon as you discard that ticket and buy another, your odds go up to 1 in 4. And so on. This effect is called the law of large numbers, and it was discovered by Romanian mathematician Stefan Mandel in the mid-1800s.

The purchase of a lottery ticket cannot be explained by decision models based on expected value maximization because the tickets cost more than they pay out in prizes. However, more general models based on the curvature of utility functions can capture risk-seeking behavior and explain lottery purchases. Moreover, the entertainment and non-monetary benefits of playing may be sufficient to offset the disutility of monetary loss.

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