Summary
The Mega Millions lottery jackpot reached a
near-world-record $380 million this week. Should you have
bought a ticket? Would any size jackpot make a
ticket worthwhile to buy?
Mega Millions
is a popular multi-state lottery in the United States
that uses what might be called a "progressive jackpot."
If no one wins the jackpot in a lottery drawing,
it is carried
forward into the next drawing's prize pool, combined with new
money from the previous drawing's sales.
Most people know that lotteries are losing propositions
under normal conditions, but
progressive lottery jackpots can, in theory, grow without bound.
Since the probability of winning the jackpot remains fixed,
an intriguing possibility emerges:
might it actually become profitable, in
expectation,
to buy a lottery ticket?
Many people have already observed that
a lottery ticket purchased in isolation will
have a positive expected value
if the post-tax jackpot grows larger than the (inverse) odds of
winning.
However, larger jackpots generate higher ticket sales,
and in the case of a tie the jackpot is split among all winners.
Some
analyses
account for the effect of ties, but none that I've found
have observed that as the jackpot grows,
ticket sales appear to increase super-linearly.
This means there is a point of diminishing return
where the negative expectation due to ties
outweighs the positive expectation due to having a larger jackpot.
Taking this into account,
it is unlikely that buying a lottery ticket is
ever profitable in expectation,
no matter how big the jackpot gets.
Winning If You're the Only Player
Let's start with the simplest case.
Imagine that you are the only person buying a
Mega Millions ticket.
A ticket costs $1, and
the probability of winning the jackpot is
1 in 175,711,536.
Therefore, if the jackpot is expected to yield exactly
$175,711,536, then your $1 ticket has an expected value of
$1 -- it's an even-money proposition, albeit one with very
high variance.
To simplify the calculations, assume for now that the
jackpot is the only prize; we'll consider the relatively
small effect of the smaller prizes in a moment.
I'm also assuming there are no intangible benefits to playing, such
as the thrill you get from watching the draw.
It would seem at first glance that any time the jackpot
grows beyond $175.7 million, it's economically
rational to buy a ticket. Buying a ticket for this week's
$380 million drawing would have been a smart move, right?
Slow down, big dreamer.
Unfortunately, an advertised jackpot of $175.7 million
doesn't actually yield $175.7 million in prize money. There
are two important deductions to take into account:
- Net present value. The advertised jackpot is
what Mega Millions will pay you in 26 yearly installments.
However, the dollar you use to buy your ticket is an
up-front cost today. For
the comparison to be valid, we have to use the net present
value of the jackpot -- that is, the immediate one-time cash
payout option. (Equivalently, we could consider the cost
of the ticket to be $1 invested for 26 years.)
Mega Millions pays cash up-front at 63% of the
advertised jackpot value.
- Taxes. The top marginal federal tax
rate is 35%. Not all states have an
income tax -- my home state of Washington does not, for
example -- but
states that do have an average tax rate of about
6%. Between the two, you keep about 59% of your
winnings, depending on your state.
So this week's $380 million jackpot wasn't really worth
$380 million
after all.
Taking tax and present value into account, it was really
only worth $380 x 0.63 x 0.59 = $141.2 million, which is
less than the 175.7 million to one odds against
winning. Assuming the jackpot was the only prize,
and that you were the only person who bought a ticket,
that ticket had an expected value of only
80.4 cents.
Let's now consider the effect of
other, non-jackpot prizes. They are paid immediately
(without the 26-year annuity), so we only need to
discount them for taxes, not present value. The table
below shows the other cash prizes, and their
contribution to the expected value of a ticket:
| Prize |
Post-Tax Prize |
Odds of
Winning |
Post-Tax
Expected Value |
| $250,000 |
$147,500. |
1:3,904,701 |
$0.0378 |
| $10,000 |
$5,900. |
1:689,065 |
$0.0086 |
| $150 |
$88.5 |
1:15,313 |
$0.0058 |
| $150 |
$88.5 |
1:13,781 |
$0.0064 |
| $7 |
$4.13 |
1:306 |
$0.0135 |
| $10 |
$5.9 |
1:844 |
$0.007 |
| $3 |
$1.77 |
1:141 |
$0.0126 |
| $2 |
$1.18 |
1:75 |
$0.0157 |
| Total |
|
|
$0.107 |
The non-jackpot prizes add a total of 10.7 cents of
expected value to a ticket, independent of the jackpot
size.
Added to the 80.4 cents of expected value from the
jackpot,
a ticket for a $380 million drawing would be worth
an expected 91.1 cents -- 8.9 cents short of the dollar
you paid for it, so it's still not a winning investment.
And we're still assuming you're the only one playing.
For a ticket to be an even-money bet, the
jackpot's expected value would have to reach 89.3 cents.
Added to the
10.7 cents of expected value from the other prizes,
that would make an even dollar.
For a ticket purchased in isolation,
taking taxes and present value into account,
the advertised jackpot would have to be 0.893 / (.63 x .59) = 2.40
times the odds against winning. The jackpot odds are
175.7 million to one, so
the advertised jackpot would have to grow to
$422 million before a ticket
purchased in isolation would have a neutral expected
value. So all you have to do is wait for the jackpot to
get just a little bit bigger, right?
Wait a second. I keep saying the tickets were puchased
"in isolation". What if you're not the only person playing?
The problem of ties
The analysis so far assumes you're the only one buying
a ticket. The problem is that you aren't.
If more than one ticket hits the jackpot, the prize is split
equally among all the winners, reducing its value to any one
winner.
To make the problem concrete, let's return to the case
of this week's $380 million jackpot.
According to
LottoReport.com,
a site that has recorded the sales figures for every Mega Millions
drawing since 2007,
there were 229.4 million tickets sold for this week's big
drawing.
Let's call this n.
Each ticket is independent and had a 1 in 175.7 million
chance of winning. Let's call this p. The
Poisson
distribution with λ = np = 1.31 tells us
how many winners we could expect in that drawing;
it's shown in the plot on the right
(click to enlarge).
There was a 27.1% chance no one would win,
a 35.4% chance exactly one ticket would win,
a 23.1% chance there would be a two-way tie,
and so on. (The actual result: a
two-way tie.)
However, the numbers we want are a little bit different:
we're trying to compute the expected value of
a winning
ticket -- that is, assuming I've hit the jackpot,
what's the probability I tied with someone else?
Or, mathematically, what's the probability there are
two winners, assuming there is at least one winner (me)?
We can turn to
Bayes'
Theorem, which tells us
we can compute this probability by dividing
the probability there are two winners by the
probability there is at least one winner
[which is 1 - P(no winners)].
This comes out to
0.231 / (1 - 0.271) = 0.317.
That is, the probability of a winner having to share
the jackpot with exactly one other winning ticket is
31.7%. In this case, each winner only gets half
the $380 million jackpot, or $190 million -- times
0.59 x 0.63 to account for those pesky taxes and
net present value.
Now let's put this all together. We can repeat the
computation above for each realistic number of winners who
might tie. In each scenario, we can compute the
probability of that kind of tie occurring, compute the
value of the jackpot per winner in that case, and multiply
those two numbers together to get that scenario's
contribution to the expected jackpot. Add them all up,
and we have the expected value of the jackpot, taking ties
into account, as shown in the table below:
Jackpot
Winners |
Probability |
Jackpot Share
($millions) |
Jackpot Share
Post-Tax, NPV |
Contribution to
Expected Jackpot Value
($millions) |
| 1 |
0.48535 |
380. |
141.25 |
68.55 |
| 2 |
0.316854 |
190. |
70.62 |
22.38 |
| 3 |
0.137902 |
126.67 |
47.08 |
6.49 |
| 4 |
0.0450136 |
95. |
35.31 |
1.59 |
| 5 |
0.0117546 |
76. |
28.25 |
0.33 |
| 6 |
0.00255793 |
63.33 |
23.54 |
0.06 |
| 7 |
0.000477116 |
54.29 |
20.18 |
0.01 |
| Total Expected Jackpot |
99.42 |
With
229.4 million tickets in play,
a 1 in 175.7 million chance of each ticket winning,
a 63% net-present value,
a 41% tax burden,
and an equal split among multiple winners,
the expected value of a $380 million jackpot to a winner
is just $99.4 million, making
the jackpot's contribution to the
expected value of a ticket only 56.6 cents.
Adding back the 10.7 cents of expected value from the non-jackpot prizes,
the total expected value of a ticket in this week's $380 million
drawing
was
67.3 cents.
This is far worse than the 91.1 cents we computed
when we ignored the possibility of a tie.
Ticket sales as a function of jackpot size
Unfortunately, we are still not done dumping cold water on
the dreams of lottery winners. (It's a cruel hobby, I
know.) The analysis above took ties into account, but
assumed a fixed number of tickets in play -- 229.4 million,
to be exact. However, as the media delights in
reporting,
large jackpots incite a
ticket
buying
frenzy, and that makes ties increasingly likely.
Might the increased possibility of a tie
actually make larger jackpots less valuable to the winners?
The first step in answering that question
is to model how the size of the jackpot
will end up affecting the number of tickets that are
bought to try to win it.
Let's return to the excellent data curated by
LottoReport.com.
The red points in the graph on the right
(click to enlarge) are ticket sales
for 523 MegaMillions drawings since 2007, plotted against
the drawing's advertised jackpot.
The two points on the top right are this week's $380
million jackpot, which sold 229.4 million tickets,
and the $370 million jackpot from March of 2007,
which sold 212.8 million tickets.
The blue line is the best-fit third degree polynomial.
As you can see, with jackpots over about $250 million,
ticket sales really start to take off.
However, it's also important to acknowledge that the data
for jackpots above $350 million is sparse, and
non-existent above $380 million.
The curve may not be predictive for unprecedented
jackpots.
If the advertised jackpot grows to $500 million,
the model predicts over
$400 million in ticket sales.
It predicts a $750 million jackpot would generate
$1.1 billion in ticket sales.
But there's no way to know with certainty what
would actually
happen if a MegaMillions
jackpot grew to a half-billion dollars.
Mass hysteria?
The projected sales curve above also paints a grim picture for anyone still
holding out hope that a lottery ticket can ever be an
economically rational investment.
As the jackpot grows in
value,
the number of people who try to win it grows
super-linearly.
This human behavior has a mathematical consequence:
even though the
jackpot itself can theoretically grow without bound, there is
a point at which the consequent ticket-buying
grows to such a fever pitch that the expected value of the jackpot
actually starts going down again.
The graph below shows the effect, plotting
advertised jackpot size against expected jackpot value
if you win. It takes everything into account we've
discussed so far: net present value (63% of jackpot), taxes
(41% off), and the increasingly
devastating effect of ties as the number
of tickets grows super-linearly with jackpot size.
The right-hand axis in the above plot shows the expected
value of a ticket at each advertised jackpot size; it is
the expected jackpot value times the 1/175.7 million
probability of winning the jackpot,
plus the 10.7 cents of expected
value
that comes from the non-jackpot prizes.
It peaks at
69.3 cents
when the advertised jackpot reaches $420 million.
At that point the expected jackpot value is $103 million.
The expected value of a ticket never exceeds the break-even point of $1.
Thus, Mega
Millions tickets are never a rational investment, no
matter how big the jackpot grows.
Exceptions
Okay, I have to admit that I wasn't quite telling you the
whole story: Mega Millions is never a rational investment
for most people.
A friend of mine who is a professional
poker player pointed out an interesting loophole in my
analysis: for him, the cost of a ticket is tax-deductible!
Buying a lottery ticket is considered gambling, so the cost
of a losing ticket can be counted against any
gambling winnings in the same tax year -- and most
of his income is gambling winnings. For him,
a losing ticket only costs 60 cents.
Lucky guy! According to the curve above,
a MegaMillions ticket is profitable in expectation
for him (i.e., EV above 60 cents)
when the jackpot is between $275 million and
$575 million -- as long as he moves out of California,
where the marginal state tax rate over $1m is 10%.
It also bears repeating that this analysis rests
on our ability to predict future Mega Millions
sales based on previous sales. But with scant
evidence of what happens at $500 or $750 million,
who knows? Maybe the market will saturate --
perhaps by a
$500 million jackpot, everyone who knows how to
buy lottery tickets has bought
all the tickets they can afford
and the curve gets flat.
For the last word on this topic, however, I
cede the floor to
Durango Bill,
who
aptly observes
that driving to the store to buy a Mega Millions ticket
is more likely to be fatal than it is to make you rich.
That's why I plan to walk.
Jeremy Elson is a computer
science researcher
at Microsoft
Research, who spends his time outside of work
riding bicycles, flying airplanes, building
electronics, and playing with Mathematica.
