Back in 2011 I saw a really interesting post at Kid Dynamite’s blog about positive expected value lotteries. In that particular case, the Inspector General investigated and found no ill-dealings from any insiders. I suggest you read the Inspector General’s report, because it’s a fascinating look at exactly what happened in that case.
After reading that post and the Inspector General’s report, I got curious. Surely this sort of thing must happen elsewhere? Maybe it even happens close to home?
Sure enough, the game is called Bullseye and it’s run by the New Zealand Lotteries Commission, an independent but government owned Crown Entity.
In Bullseye, you choose a 6-digit number. If that number is drawn from the 999,999 possible numbers, you win first prize.
You can work out how many tickets are sold in an average draw by looking at the “Results” tab on the link above. That will tell you how much has been paid out in each draw.
With a bit of basic mathematics, we can work out how many tickets were bought, and therefore how much was wagered.
Given the setup, one in ten tickets is expected to win a prize of any sort. So you can then just take the number of winning tickets (provided under “Results”) and multiply it by ten. For example, on 29 September 2012 (just a random date), 22,950 tickets were sold, at a total cost of $45,900 ($2 per ticket). No one won first or second division, and total payouts were $14,760. So the lottery returned 32.1% of the money it took in, for a return to bettors of -68.9%.
So far, so boring. But Bullseye has a special feature. When no one wins first division, first prize jackpots. When it gets to $400,000 it goes “must-be-won”. If no one wins first division on a “must-be-won” draw, the prize is pushed down to second division, and so on, until someone (or some people) hold a winning ticket – the $400 000 plus the prize for the division they are in is shared amongst them and any other winners in that division.
That feature grabbed my interest. I just googled “bullseye must be won” and found press releases from all the dates in the past that it has jackpotted. I then looked at the results for all the “must-be-won” draws (on the website above) and found something intriguing:
On a “must-be-won” draw, instead of a -68.9% return, your expected value is positive. The other interesting thing to note is that it has been declining over time:
This suggests that others might have picked up on this anomaly.
The way to guarantee the highest possible minimum return is to evenly space tickets. The other aspect of this is that the more tickets you buy, the higher percentage of your wagered money you guarantee getting back. Here’s a graph of the % of capital you are guaranteed to get back based on the number of tickets you can ensure winning, and historical prizes based on prior draws:
Extending the x-axis gives you a similar pattern; buying more tickets guarantees the minimum amount won, up to a certain point.
The problem is that guaranteeing division 1 costs $2 million, and at that level the lottery will pay out less than is wagered (it is the $400 000 jackpot that “must be won”, not the $400 000 + any additional money wagered).
At the current level of risk, my view is that the return doesn’t justify risking any real amount of capital, due to the top heavy nature of the prizes. If the distribution of prizes was more even (less for 1st division, more for 3rd, 4th and 5th) then the risk of this would come down significantly. But on current setup, you really need to win 1st division or 2nd division to achieve your expected value over the long term.
If you’re someone who likes to gamble, doing it on this is certainly a good place to do it, but returns are extremely volatile, and even in the long-term, if you bet $1000 on every single must be won draw, you would be unlikely to make money. Note that is a different question from whether it is positive expected value. It is, but when making money over the long term relies on winning division 1 or 2 on one of the draws, it is hard to achieve that positive expected value.