11 Jan 2016

## Powerball and the Lucas Critique

I realize a few commentators gave a nod to this consideration, but I don’t think they realized just how critical it is. An excerpt:

To see the point, suppose the Powerball official jackpot somehow rose to \$1.3 trillion, with a lump-sum payout of \$806 billion. Running through the same calculations as above, we might get a ballpark gross expected value of one ticket equal to \$1,700. That is far higher than the ticket price of \$2, making it a no-brainer to play. In fact, in order to eliminate any risk, a hedge fund might devote \$585 million to buying every combination of Powerball numbers. It would appear that by spending \$585 million on tickets, the hedge fund could guarantee itself the \$806 billion lump-sum payout. Who wouldn’t put up \$585 million to win a guaranteed \$806 billion?

Yet hold on a second. If one particular hedge fund sees this opportunity, why wouldn’t dozens more seize it? Yet it obviously can’t be the case that dozens of hedge funds can all guarantee themselves \$806 billion from the same pot of money. In this contrived scenario, what would happen is that the dozens of hedge funds would all buy every combination of Powerball ticket, and so whatever the winning number happened to be, there would be dozens of winners splitting the pot. Realizing this, some of the hedge funds might buy multiple tickets for each possible number…until the point at which it no longer made sense to buy a ticket.

#### 12 Responses to “Powerball and the Lucas Critique”

1. Tel says:

It seems like you are saying, that no one can beat the market.

http://newsfeed.time.com/2012/08/07/how-mit-students-scammed-the-massachusetts-lottery-for-8-million/

By 2005, the group had earned almost \$8 million with its system, according to an investigation by the Boston Globe. By 2010, it had figured out how to win the entire jackpot in a single drawing.

A recent report by the state’s inspector general reveals more details about the scheme, including the fact that the Massachusetts Lottery knew of the students’ ploy and for years did nothing to stop it. The inspector general’s report claims that lottery officials actually bent rules to allow the group to buy hundreds of thousands of the \$2 tickets, because doing so increased revenues and made the lottery even more successful.

Except, that we regularly find cases where at least some people do beat the market.

• Gene Callahan says:

“It seems like you are saying, that no one can beat the market.”

Bob, being an excellent writer, probably would have written that if he had meant that. So it “seems like” you misread him.

• skylien says:

Tel,

I think Gene is right. What matters is, who knows what and who acts based on which information in which way. Obviously the MIT students knew something others didn’t respectively couldn’t replicate (= the MIT students had skills which are not easy to learn) hence they beat the market.

And in Powerball it is still possible to beat the market. If all hedge funds don’t act because they read Bob’s post about what might happen, then you could gamble and actually do it, hoping no other one does because they are afraid everyone else might as well… 😉

• Harold says:

From the Time article ” because of a quirk in the way a jackpot was broken down into smaller prizes if there was no big winner.”

I think my suspicion as to why the rollovers had to be re-distributed in the UK lottery when the pot reached £50 million may be correct.

The mechanism for purchasing tickets is important – the article says the rules were bent to allow bulk purchase.

The lottery was poorly designed, but why did hedge funds not buy in like the students did? The information was public. Are they too stupid to spot a winner? EMT would suggest that enough people would have bought tickets until the excess profit was removed. This again shows us that behavior is not always as such theroies predict.

2. Harold says:

It is interesting that thi shas come up this week. The UK recently changed the rules so the chances of winning became 1 in about 45 million, instead of 1 in 14 million. The result was the expected increase in rollovers, ending up last week with a pot of £66million. However, the rules say that when the amount reaches £50M, it must be distributed in the next draw. If there is no outright winner it is split between the “5 numbers plus bonus ball”, at odds of 7.5 million to one.

The jackpot was shared by 2 people, the “5 plus bonus ball” had 5 winners.
There was a large number of extra tickets 70 million sold in total against the usual 45 million; about double the normal maximum amount. Interestingly, we had two winners from 70 million with odds of 45 million to one – presumably about what you would expect. It would cost £90 million to buy every number at £2 a ticket.

The jackpot was much more than double, and teh extra tickets was only double, so if you do play the lottery for the jackpot, it would have been a much better bet than normal, even taking the extra sales into account.

The total prize payout is 45%. Since the numbers playing only doubled and the jackpot improved by about 5, I was thinking this should get you in the realms of positive expected return – due to the fact that there had to be a payout. Yet if you bought every number and thus won the jackpot you would only get 2/3 your money back at best. How can this be? On digging further, only 15% of sales goes to the main prize – the other 30% is returned in smaller prizes: £25 wins for matching 3 balls, and other prizes for 4, 5 and 5 plus bonus ball.

If you were the only player and you bought every number in last saturdays draw, you would spend £90 million. You would get back 30% in small prizes (£27 million), 15% of the money you spent in this weeks contribution to the main prize (£13.5M), and about £50 million from the rolled-over prize, making a total of £90.5 million, or just above break-even. I wonder if that is why they picked £50M as the amount that cannot be further rolled over?

If someone else buys an equal number of tickets then the prize pool for this week doubles and your cances are reduced by two. Your expected return is the same for everything except the rolled-over amount which reduces by 2 to 25M, giving a total of £65.5M, or 73%. Since there were approximately this number of tickets sold, this represents a massive improvement in expected return to about as good as a slot machine, but still well short of positive.

Clearly the number of tickets actually sold does not increase in proportion to the prize money available. It seems possible that there will come a point where a positive expected return arises if the roll-over is allowed to build indefinitely. It could never build to the excess you use in the hypothetical, because someone would have bought all the numbers by then – if this is actually possible to do with the infrastructure for purchasing tickets. This would occur when someone estimated that the number of extra tickets sold will be just small enough to tip the expected return into positive territory – an estimate that will vary between people (or hedge funds).

That is, whilst the change in expected return affects behavior, the change is not fully compensated for by changes in behavior. Govt actions may shift the Phillips curve, but not by a smuch as such a simple assumption would predict.

3. RonB says:

If memory serves, one of the earliest large jackpots, several hundred million dollars, was in Florida. I forget the odds for that jackpot but for the sake of argument lets say the odds of winning were 1 in 250,000. There was an investment group that offered to pay the Florida lottery \$250,000,000 covering all possible number combinations and thus claim the jackpot. The lottery commission denied them saying they had to actually buy a ticket or in this case 250 million tickets. It would have taken multiple lottery machines printing 24hrs a day to print out the tickets, not to mention the time and number of people necessary to fill out the ticket request for each possible combination. That is the most likely reason why no group has “purchased” a win.

4. Josiah says:

You’ve convinced me not to buy a ticket. If I don’t win, it’ll be your fault.

• skylien says:

Actually not buying a ticket makes you a winner already. You won more than most who buy one, at least you broke even…

5. Guest says:

People are buying hope. A jar of invisible hope that last until the drawing is complete. I cant hardly blame anybody.

• Guest says:

I would like to add a few questions.

Why do prizes only equal 50% of all ticket sales. Why not 90%?

Why are grand prizes only 30 year annuities?

Why have we allowed government to hijack the peoples game?

• Guest says:

1 more thing. If state did not confiscate half of tickets up front and half of that half on the tail end, the odds and prizes would be way way better.

Wall-Street has more favorable regulations and tax code than lottery players.

Just think what the average Joe could do by cooperating in game such as Powerball if State had not commandeered the process. We have not even got to the reinvesting powerball.

• Harold says:

The national lottery in the UK is relatively recent (1994) and there was quite a lot of opposition. Some of that opposition was overcome by using 28% of the money to fund “good causes”. Buying a lottery ticket is a bit like giving to charity. From wiki ” 40% is awarded to health, education, environment and charitable causes, 20% to Sports, 20% to Arts and 20% to Heritage” This allows people to feel good about the lottery when they see local kids sports clubs etc “funded by lottery”.

5% of sales goes to the shop selling the ticket, 12% to Govt, 5% to the lottery operator.

An interesting thing fropm wiki: “academics Rachel E S Ziemba and William T Ziemba say with regard to 6/49 lotteries, “Random numbers have an expected loss of about 55%. However, six-tuples of unpopular numbers have an edge with expected returns exceeding their cost by about 65%.”
I don’t see how they get that result – seems to me that you must always be on a loser. There must be something I don’t understand going on.