程阳：美国超级百万 Mega Millions

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On October 18, 2013, matrix changes went into effect in the Mega Millions game that raised the top prize odds to about 1 in 259 million. At the same time, the annuity top prize was restructured in a graduated form, allowing a given value of current cash prize dollars to fund a larger advertised jackpot. At the time of these changes, the jackpot had survived five draws.

 

超级百万：Start Date Pick 5 out of Pick 1 out of Jackpot odds
1996年9月6日	50	25	1:52,969,000
1999年1月13日	50	36	1:76,275,360
2002年5月15日	52	52	1:135,145,920
2005年6月22日	56	46	1:175,711,536
2013年10月19日	75	15	1:258,890,850

 

 

Seventeen draws later, the jackpot had grown to $636 million, only $4 million shy of the record value established in March 2012! The jackpot was won by two tickets in its 22nd draw. Is this the “new normal” for Mega Millions? How often should we expect to see jackpots like this, given the new matrix and annuity structure?

 

This sort of question is being asked around the executive suites of many US lotteries in the early winter of 2014. I don’t have a definitive answer, but I can share some results from math modeling of the game that might be of interest. The kind of math model that is useful in this case is commonly called a “thousand-year model.” A thousand-year model aims to reveal the expected long-term variability of outcomes from a lottery game. It has basically three parts:

 

1) an account of how sales typically increase from draw to draw in response to an increasing jackpot,

 

2) an account of how the likelihood of paying the top prize increases as sales in the draw increase, and

 

3) a process that simulates the random drawing and its outcome: paying the top prize, or not. The model steps through thousands of draws and records the history of each “run”: how many successive draws without awarding the top prize, what total value of sales, what total value of prizes paid, and how much profit among the lotteries.

 

After a model like this has simulated a thousand years of play, we can tabulate the results and get an idea of the likelihood that we will see particular outcomes over the next few years, while we might hope to be managing the game.

 

This kind of model is particularly useful for projecting effects of changes in the game: not only structural changes to the game itself, but changes in how players respond to jackpots. There is considerable art in developing a good math description of how spending increases from one drawing to the next. The work I describe here is based on math that does a pretty good job of projecting how Mega Millions sales grow in response to jackpots up to $300 million. As of January 2014, there have only been 12 Mega Millions draws with jackpots  larger than $300 million, in nine separate runs. Beyond $300 million, each real-world instance we have seen has been a little different, and my current math models are not as good as human intuition in that range. Accordingly, I focus here on how frequently we might expect Mega Millions jackpots to reach $300 million under the current rules.

 

Clearly, all this is driven by how players respond to developing jackpots.

 

In the March 2012, the jackpot reached $290 million in draw 17. In December 2013, the jackpot reached $297 million in draw 19, despite having the benefit of a more favorable annuity factor after the first five draws. Adverse weather may have been a factor. However, a math model that does a good job accounting for sales between February 2012 (when Powerball raised its price to $2) and April 2013 (when California joined Powerball) does predict sales consistently higher than we actually saw in the latter part of 2013. Since I am using this same model in the work I report here, I take the precaution of reducing its sales projections by 15%. This brings the projected sales into line with what we experienced in late 2013.

 

So, with these key assumptions:

 

• top prize odds 1 in 259 million,

 

• annuity factor 1.80, and

 

• sales-for-jackpot about 85% of what we saw in the base period (Feb 2012 to April 2013),

 

my thousand-year model projects that:

 

• half of jackpots will be won in draw 10 or later,

 

• an “average” year might have 10 jackpots, and

 

• an “average” year might have one or two jackpots over $300 million.

 

The concept of an “average” year deserves caution. A year is not a long time for a game like Mega Millions. If an “average” year has one or two jackpots over $300 million, this does not mean that a year with none, or with three, is terribly unlikely.

It is precisely this year-to-year variability that thousand-year models help us appreciate. The distribution of jackpots over $300 million across years suggests the following likelihoods: Another way of thinking about this is in terms of sales dollars. Using my (admittedly imprecise) estimates of sales that come in for all jackpots, including those that grow above $300 million, I estimate that average annual sales of Mega Millions (base game only) might be about $2.9 billion. Year-to-year variability is such that the total would be between $2.3 billion and $3.5 billion about two-thirds of the time.

 

That’s a pretty broad “normal range”- plus or minus 20%. Yet that is the sort of thing we must deal with in projecting annual performance of our lotteries. We are fortunate to have two big, volatile games rather than just one - the joint variability of two independent games is less than that of either alone.

 

 

October 2013 format change

 

The final 5/56 + 1/46 Mega Millions drawing was held on October 18, 2013; its jackpot of $37 million was not won. The first drawing under the current 5/75 + 1/15 format, which saw the jackpot estimate "leap" to $55 million due to the change in the annuity structure, occurred on October 22. The minimum jackpot is now $15 million with rollovers of at least $5 million. Second prize (5+0) is now $1 million cash. Players now choose 5 of 75 white ball numbers, and 1 "Gold Ball" number out of 15.

 

The Megaplier option remains; it now includes a 5x multiplier. The Megaplier now applies to all prizes except the jackpot; a 5+0 play with the Megaplier wins $5 million cash.

 

Former (through October 18, 2013) and current prize tiers, based on a $1 play:

 

•Match 5+0: $250,000/$1 million

•Match 4+MB: $10,000/$5,000

•Match 4+0: $150/$500

•Match 3+MB: $150/$50

•Match 3+0: $7/$5

•Match 2+MB: $10/$5

•Match 1+MB: $3/$2

•Match 0+MB: $2/$1

Payouts in California remain pari-mutuel.

 

The odds of winning the jackpot are decreased to 1 in about 258.9 million. The odds of winning a prize has increased to 1 in 14.71, but this also includes the "push" scenario (where the money won in the prize matches the amount of the wager, as is the case when a player not activating the Megaplier matches only the "Mega Ball"; or with the Megaplier if it is 2x); thus, the odds of turning a profit on any given bet is less than advertised. Matches & Odds

 

•5 numbers and the Mega Ball: 1 in 258,890,850

•5 numbers but not the Mega Ball: 1 in 18,492,204

•4 numbers and the Mega Ball: 1 in 739,688

•4 numbers but not the Mega Ball: 1 in 52,835

•3 numbers and the Mega Ball: 1 in 10,720

•3 numbers but not the Mega Ball: 1 in 766

•2 numbers and the Mega Ball: 1 in 473

•1 number and the Mega Ball: 1 in 56

•Only the Mega Ball 1 in 21[11]

 

The annuity, which was 20 annual payments (no cash option was available) when The Big Game began, changed from 26 equal yearly installments to 30 graduated annual payments (increasing 5% per annum) with the October 19, 2013 format change.

 

Record jackpots

 

The largest Mega Millions jackpot, advertised as $640 million at the time of the drawing (annuitized) or $462 million (cash value), was drawn on March 30, 2012. The initial estimate for that Friday's drawing (following the March 27 drawing, which was $363 million annuity) was $476 million (later increased to $500 million and again to $540 million); brisk ticket sales pushed the jackpot values, both annuitized (to $656 million) and the cash option ($474 million) higher. The amount spent on Mega Millions for drawings following its previous jackpot win, on January 24, 2012, was at least $1.5 billion.[12] Three jackpot-winning tickets had been confirmed (one each in Illinois, Kansas, and Maryland).[13]

 

Mega Millions' second-largest jackpot, $648 million, was for the December 17, 2013 drawing. Two winning tickets, one each from California and Georgia, were sold. The holder of the one sold in California potentially will share the jackpot, if claimed. The holder of the Georgia ticket claimed the next morning, selecting the cash option, which amounts to $173,819,742.50 before taxes. (If claimed, the California ticket holder(s) would receive an equal share, but receive a larger after-withholding amount, as California does not tax lottery winnings.)"[2][3][4][14]

 

Mega Millions' third-largest jackpot, $400 million, was for the March 18, 2014 drawing. Two tickets, one each from Florida and Maryland, split the prize; the winners have not yet chosen their payout option.[15]

 

Mega Millions' fourth-largest jackpot, $390 million, was for the March 6, 2007 drawing. Two tickets, one each from Georgia and New Jersey, split the then-record prize; both sets of winners chose the cash option, splitting $233 million (as noted below, interest rates change, resulting in different ratios between the cash values and annuity values of jackpots).

 

Mega Millions' fifth-largest jackpot annuity value ($380 million), and second-largest cash jackpot ($240 million), was for the January 4, 2011 drawing; two tickets, one each from Post Falls, Idaho and Ephrata, Washington, matched all six winning numbers, winning $190 million (annuity) each. The holders of each ticket also chose the cash option.

 

How the drawings are held

 

Two drawing machines are used in Mega Millions. The model used for Mega Millions is the Criterion II, manufactured by Smartplay International of Edgewater Park, New Jersey. The balls are moved around by means of counter-rotating arms which randomly mix the balls. One by one, the five white ball numbers drop through a hole in the bottom of the mixing drum. As of October 19, 2013, there are respectively 75 white balls in the first machine and the 15 gold-colored Mega Balls in the second machine.

 

Versions of (The Big Game) Mega Millions

 

Versions of The Big Game and Mega Millions have used different matrices:

 

Start Date Pick 5 out of Pick 1 out of Jackpot odds

September 6, 1996 50 25 1:52,969,000

January 13, 1999 50 36 1:76,275,360

May 15, 2002 52 52 1:135,145,920

June 22, 2005 56 46 1:175,711,536

October 19, 2013 75 15 1:258,890,850

 

超级百万：Start Date Pick 5 out of Pick 1 out of Jackpot odds
1996年9月6日	50	25	1:52,969,000
1999年1月13日	50	36	1:76,275,360
2002年5月15日	52	52	1:135,145,920
2005年6月22日	56	46	1:175,711,536
2013年10月19日	75	15	1:258,890,850