By: Zack Halberstam
Government-run lotteries are pervasive in the United States, with almost every state administering lotteries. In 2019, U.S. lottery revenues exceeded 80 billion dollars. In Massachusetts alone, the lottery generated 1.1 billion dollars for local expenditures. Many Americans participate in lotteries, especially less well-off Americans, who spend perhaps 13 percent of their income on lottery tickets. Because lotteries are such popular games, it is worth analyzing the lottery from a game-theoretic perspective. Is playing the lottery worth it?
We begin the analysis by discussing the expected value (average return) of a typical lottery ticket. Lotteries are fundraising measures for the states administering them, so they must give out much less in prizes than their revenue. On average, only 60 percent of state lottery revenue is distributed in prizes. This means that, in most cases, the expected return of a ticket must be significantly less than its price, since states run the lottery with a sizable profit margin. To be more specific, we consider one of the most played lotteries in the United States: Powerball.
Powerball and Expected Value
To play Powerball, players pay 2 dollars and select five numbers from 1 to 69 and the titular Powerball, which varies from 1 to 26. Players win prizes based on the numbers they successfully predict. Predicting all 6 numbers results in a jackpot, shared if multiple players predict them correctly.
For the July 15th Powerball drawing, with a roughly typical jackpot of 137 million dollars, the expected value of a 2 dollar ticket is 79.8 cents, assuming the jackpot would not be shared and ignoring taxes. Jackpot sharing and taxation render the expected value even less, making typical Powerball drawings quite unfair, in terms of expected value.
However, for unusually large jackpots, the expected values of Powerball tickets may exceed their cost. For the drawing with the largest jackpot ever, in any U.S. lottery, 1.586 billion dollars, the expected value of a 2 dollar ticket would be a quite favorable $5.75, if the jackpot were not shared. While the increased likelihood of jackpot sharing with large jackpots and tax considerations generally keep the expected value of a ticket under 2 dollars, it is not inconceivable that Powerball would provide favorable expected values. The expected value of a Powerball ticket was occasionally more than its price before a series of rule changes beginning in 2009. So, would it be a good idea to play Powerball, if the expected value were favorable? The answer is still a resounding no.
Utility to income
To understand why even lotteries with high expected values of tickets are usually bad deals for players, we take a more nuanced view of the value of large prizes. The happiness of an individual with 10 million dollars is not 1 million times the happiness of one with 10 dollars, and thus it is crucial to understand how satisfaction changes with income to determine whether playing the lottery is a good idea.
Economists use the term “utility to income” to describe the satisfaction people derive when earning various amounts of income. The rate that utility to income changes as income increases is called “marginal utility to income,” and it is usually understood to decrease as income level increases. Consider an intuitive example: if you lost 1,000 dollars, you would be less upset if you were earning 1 million dollars per year than if you were making 10,000 dollars. The amount you value 1,000 dollars is less when you have more income, as is the amount you value each additional dollar.
This phenomenon, “diminishing marginal utility to income,” is extraordinarily relevant to lotteries, where players may win enormous sums of money. The difference in utility between winning 50 million dollars and 51 million dollars is negligible, though the difference between winning nothing and winning 1 million dollars is massive. Because expected value does not take this into account, explicitly considering utility to income is crucial in understanding the lottery. To quantify how much diminishing marginal utility to income affects the lottery, we use a model of utility to income proposed by Layard, Mayraz, and Nickell.
Layard, Mayraz, and Nickell suggest a model where utility to income is measured by a function of the form
Using results from surveys that show the variation in reported happiness over different income levels, they estimate the parameter .Taking their best estimate, =1.26, we can calculate the expected utility that individuals with various income levels have when playing Powerball (ignoring taxation and assuming no jackpot sharing, for simplicity) and the utility they have when not playing Powerball. Unsurprisingly, the expected utility for individuals at every income level is less when they play Powerball than when they do not.
To quantify the extent to which playing Powerball is a bad deal, we determine what a “fair ticket price” for the game would be. That is, we determine the ticket price at which the expected utility of playing Powerball is the same as the utility of not playing. We provide fair ticket prices for various income levels, at three different jackpot levels: the typical July 15, 2021 jackpot, the largest jackpot of all time, and an impossibly large jackpot of 1 quadrillion dollars (to demonstrate Powerball’s unfairness is not rectifiable by increasing the jackpot).
Interestingly, jackpot size barely affects fair ticket price. This is because, in the model, utility to income does not vary over many orders of magnitude, and thus the tiny chance of a jackpot barely impacts expected utility. The fair ticket prices are all between 19 and 27 cents, miniscule compared to the price of a ticket, as well as the expected value of the tickets (79 cents, $5.75, and over 3 million dollars, for the three considered jackpot sizes).