Monday, January 05, 2004
Is it rational to play PowerBall?
On the assumption that the only relevant values are monetary, then it will be rational to play PowerBall whenever the expected utility of the ticket is higher than the price. It turns out that it becomes rational to play PowerBall whenever the jackpot is over $100 million.
Like all lotteries, PowerBall pays out for a number of prizes other than the jackpot. For example, PowerBall pays $3 if you match just the powerball, and the odds of doing so are 1 in 70.39. That chance of winning adds to the value of the ticket. In this case, the added value is about four cents. [ (1/70.39)*3 ]
The total value of the chance to win all prizes short of the jackpot is a little over 17 cents (to be exact, it's $0.17331097110494888).
Once we know this, it's easy to determine how large the jackpot must be to make buying a ticket rational. The odds of winning the jackpot are 1 in 120,526,770. When we multiply the jackpot by these odds, we get the value of the chance of winning the jackpot. Since we've already got over 17 cents in our pocket from the subsidiary payouts, it will be rational to buy a lottery ticket if this number is above 83 cents. When we crunch the numbers, we find that this occurs whenever the jackpot rises to at least $99,638,158.45.
Last week, the PowerBall jackpot was $210,000,000.00. This meant that a ticket was worth nearly $1.92. Since tickets sold for only a dollar, a lottery ticket was a good value. Investors who took advantage of this opportunity would, briefly, have shown a very healthy return.
A caveat: The preceding analysis assumes that the value of the jackpot can be known. In fact, this assumption is false. There is a non-negligible chance that more than one winning ticket will be sold. In such cases the jackpot is split equally among all winners. Thus, the expected utility of a ticket is significantly lower than one would expect given the published value of the jackpot.