The Multi-State Lottery Association has recently changed the odds for PowerBall, so I thought I’d update my earlier post. Summary of earlier statements: even though PowerBall, like most lotteries, is rigged to return fifty cents on the dollar, there is some jackpot size at which point it is theoretically a better than 100% return on a bet. (Presuming the ability to play an infinite number of times; in all tests of this kind, a small number of trials will show anomalous results. In PowerBall, that basically means that everyone’s lifetime play will be statistically anomalous.)

The new rules have made it even harder to win the big jackpot, and likewise for the smaller wins. The sole concession to the player is that the five-ball hit without the PowerBall has been increased to $200,000, so now it’s theoretically possible to win a million dollars without hitting the PowerBall draw (five out of five with a 5x PowerPlay multiplier).

In any case, if you play PowerBall without making the PowerPlay bet ($2 for a ticket versus $1), you are getting incorrect odds and hence contributing to the state when the jackpot is below $232,292,812. (Assuming cash payout in one lump sum; variations on the value of this prize in relation to the advertised jackpot not accounted for. There is also a bonus pool on the 5 ball win with no PowerBall when the jackpot increases more than $25m in any one game; also not included.)

The PowerPlay still improves the overall bet—a $2 bet increases the return on any non-jackpot win at least 2x, up to 5x. There used to be 380% return on this bet for any win; this has been lowered to 350%. Regardless, the jackpot needs to be “only” $189,520,398 to be an even game with this bet.

What I find interesting about this is that it shows the utter irrationality of PowerBall players. That is, the rule of design of most gambling is to provide selective reinforcement; provide the player with small wins on the road to taking the house edge. This is difficult when the house edge is as huge as it is with state lotteries, but you can still work this into the design. With PowerBall, every revision of the game makes it harder to win, therefore creating much larger jackpots at the expense of the small wins that the players used to receive.

In other words, people play PowerBall when they can win a sizeable fraction of a billion dollars, and they stay away when the win is “only” say, $30 million. This is actually mathematically correct, but I doubt most players apply the mathematics to this. From the player perspective, is a win of $15 million any less life-changing than a win of $100 million? Yet, that’s what draws people in.

Likewise, they don’t seem to mind a string of losses like most gamblers. In the current game, if you play every game (104 times a year), if you made even a small win more than twice a year, you’d be lucky. Back in 1999 or so I won enough in four drawings straight to pick up another ticket and a pack of smokes. That should happen now once every 17,300 years or so.

So my guess is that the PowerBall attracts even more strongly a sort of non-gambler gambler, which is to say that the people buying tickets aren’t really buying tickets to win per se, but they’re buying a ticket to dream about what they’d do with the money for the days until the drawing. This has been measured before, and also indicates that twice a week is about as often as you want to run a game of this kind. It also suggests that in ten years we’ll be seeing lotteries with billion-to-one odds against and regular demiannual prizes of a few hundred million. Which should be enough for a few tanks of gas.