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Chris DeMuth Jr.
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Chris DeMuth Jr. is the founder of Rangeley Capital LLC. Rangeley is an investment firm that focuses on event driven, value-oriented investment opportunities. Prior to founding Rangeley Capital, Mr. DeMuth spent his career as a securities analyst for several hedge funds and proprietary trading... More
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• ##### A +EV Lottery? 21 comments
Dec 17, 2013 10:36 PM

Lotteries are typically very stupid. That is not because winning is unlikely, but because they typically have drastically negative expected values. However, over the next few minutes there is an American lottery, Mega Millions, that some claim has an anomalously positive expected value. According to the game's rules, Mega Millions tickets cost \$1.00 per play. Players may pick six numbers from two separate pools of numbers - five different numbers from 1 to 75 and one number from 1 to 15 - or select "Easy Pick". You win the jackpot by matching all six winning numbers in a drawing. The jackpots start at \$15 million and grow by a minimum of \$5 million per draw each time the jackpot rolls.

The drawing is scheduled for this evening at 11 PM eastern standard time. The jackpot is estimated at \$636 million. While winning is extremely unlikely, it appears to drive a positive expected value for each \$1 of cost:

(click to enlarge)

Why is this opportunity available? Is the demand for tickets dynamic enough to account for changes in the jackpot? Are huge jackpots such as tonight's occasionally undersubscribed? If you happen to read this, play, and win, I would recommend taking the annuity instead of the upfront cash option, based upon the following chart; the horizontal axis shows the number tickets sold, in millions, and the vertical axis shows the expected value in dollars of a ticket, based on the likelihood of a split, and the size of a split jackpot.

(click to enlarge)

Forget about luck. Instead, I would prefer to wish you a good expectancy!

Is there a catch? Sadly, there are catches. The first is taxation. Federal (often combined with state) taxes reduce the jackpot value by over a third.

In addition, lottery sales data indicates that large jackpots, which drive heightened press interest, are fully subscribed.

(click to enlarge)

Based on the hysteria around massive jackpots, increased demand actually lowers EV/ticket:

(click to enlarge)

Finally, according to computer scientist and mathematician Bill Butler, there is a significant impact from split pots:

(click to enlarge)

Since tonight's jackpot will probably not go to a single winner and any eventual winners will have to pay a large tax bill, the EV is reduced from \$1.63 to about \$0.64 after taxes. Also, the payoff structure is hugely propped up by tiny changes of huge windfalls, so that it is extremely likely that you will not win. If you play, the EV indicates that you are playing a losing game and the payoff structure indicates that you will not be able to play for long since your bankroll will be quickly consumed.

What do you do if you are comped a ticket? I recently attended a wedding at Lake Tahoe and we stayed at a hotel that had a casino. Various introductory bets came with the room. It might be nice to know how to make a less bad bet, even when it is not +EV. The key is to bet underused and unlucky numbers so as to reduce the chance of splitting a pot. Overused numbers include 1, 7, 13, 23, 32, 42, and 48. Underused numbers include 26, 34, 44, 45, and 46. Avoid number combinations that can form dates (pick two digits above 12, then two above 31, then two outside of common year ranges of 1900-2013). Despite this number system showing some merit, it still appears to be -EV in the US.

I am interested in doing more work on Chinese lucky numbers. Culturally, many Chinese are heavy gamblers and are also fascinated by lucky numbers (0, 6, 8, and 9 appear to be overused, while 4, 5, and 6 appear to be underused). Other number combinations sound like insults or vulgarities that may also be underused. This may be profitably exploitable in Chinese games.

For further reading on lotteries and their expectancies, you might like Finding Good Bets in the Lottery, and Why You Shouldn t Take Them.

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• Very cool article, enjoyed the read!
17 Dec 2013, 10:41 PM Reply Like
• Chris DeMuth Jr.
, contributor

18 Dec 2013, 06:12 AM Reply Like
• sheldond
, contributor

Chris,

If you use the unlucky numbers, you will be cursed. I like the theory never considered it and I don't play lotto.

D
17 Dec 2013, 10:56 PM Reply Like
• I love this stuff. Never saw it figured out but always thought that they had to be a catch in the circumstances when the jackpot was high.

It is possible to get the +EV in lotteries..Rarely but +never.

http://bit.ly/1dMj8qz
18 Dec 2013, 12:52 AM Reply Like
• I have always wondered about the comparative expected value of lottery tickets versus the lottery's predecessor - the numbers "racket.
Of course, if you won playing the numbers, you probably didn't have to pay taxes.
18 Dec 2013, 01:55 AM Reply Like
• good guy3
, contributor

Yep. Big pots = split pots
18 Dec 2013, 02:03 AM Reply Like
• Chris DeMuth Jr.
, contributor

Author’s reply » That appears to be the big problem.
18 Dec 2013, 06:11 AM Reply Like
• Chris DeMuth Jr.
, contributor

Author’s reply » Mega-jackpot split by two tickets: http://bit.ly/1bOOlpR
18 Dec 2013, 06:35 AM Reply Like
• toddro
, contributor

I have a family member who plays the lottery. She will spend \$300 on tickets and have one \$100 winner and claim to all who will listen that she won \$100 today playing the lottery. "No, you lost \$200'" I will say. "But I won \$100." she will reply. And so it goes...
18 Dec 2013, 08:11 AM Reply Like
• Hi Chris,

I don’t understand the following:

The key is to bet underused and unlucky numbers so as to reduce the chance of splitting a pot. Overused numbers include 1, 7, 13, 23, 32, 42, and 48. Underused numbers include 26, 34, 44, 45, and 46. Avoid number combinations that can form dates (pick two digits above 12, then two above 31, then two outside of common year ranges of 1900-2013). Despite this number system showing some merit, it still appears to be -EV in the US.

The outcome of the jackpot combination is independent from the numbers you choose. So choosing underused numbers does not impact the chance of a split pot.

Only if the outcome of the jackpot is a combination of underused numbers, there is a greater chance that there is no split pot.

Right?
18 Dec 2013, 09:03 AM Reply Like
• Chris DeMuth Jr.
, contributor

Author’s reply » You are correct that the outcome of the jackpot combination is independent from the numbers you choose and that only if the outcome of the jackpot is a combination of underused numbers, there is a greater chance that there is no split pot. This tactic does not appear to get me to a +EV, but was more of a hypothesis that I'm still in the process of working out. I think that it is far more promising in China and might be especially relevant in perimutuel bets.
18 Dec 2013, 09:06 AM Reply Like
• jaginger
, contributor

Maarten,
The logic behind the key Chris mentions is this:

When you are picking numbers, you have exactly the same chance to win A POT, no matter what numbers you pick. Pick a bunch of common numbers, and you have a small chance of winning the pot. Pick a bunch of rare numbers, and you have exactly the SAME small chance of winning the pot.

Only difference is, if you do win after pick the rare numbers, you'll expect to win a LARGER pot because there is less chance you'll have to split the pot with someone.

So, you have a higher expected value of winnings when you use the rare numbers.
18 Dec 2013, 09:10 AM Reply Like
• Chris DeMuth Jr.
, contributor

Author’s reply » Exactly; couldn't (and didn't) say it better myself.
18 Dec 2013, 09:13 AM Reply Like
• johncworth
, contributor

About 70% of lottery tickets are generated via "quick pick" vs bettor designated numbers. That diminishes the edge of hand picking unpopular numbers quite a lot. Nevertheless, any edge is better than no edge.

On a mildly related topic, I will give you tip about why I am known to be WILDLY LUCKY at any kind of ticket stub drawing type of event. I bend any tickets I drop in in both directions (latitudinally & longitudinally) so that they create space between the other tickets. This gives you some edge, as yet to be determined how much. I certainly win a disproportionate number of drawings, I calculate that it increases my chances by 500% or so, though that calculation is entirely BS anectdata. Be that as it may, I offer it to all of you as a way to grab an edge. At the very least you can at least THINK you have an edge, which in and of itself can be an edge in certain situations. (Liability in others)

Happy holidays to all
kind regards
jcw
18 Dec 2013, 09:50 AM Reply Like
• Chris DeMuth Jr.
, contributor

Author’s reply » I love that idea so much. Thank you. An early Christmas present!
18 Dec 2013, 09:53 AM Reply Like
• toddro
, contributor

Haha! Ok, someone who thinks like me and is always looking for the edge. I volunteer at a fundraiser every year and work in the kitchen. As soon as I get there, I buy 5 tickets for their raffle (of some really great prizes) and I fold them as you describe - I also put them in the kitchen's sub-zero freezer and keep them there all night until they call "last call" and then I then I hit them in the blast freezer and run them out to the box. Every year I hear cries of "not again!" ;>)
18 Dec 2013, 10:13 AM Reply Like
, contributor

My Hong Kong-born wife plays the lottery (at my expense), so I am invested in this discussion. She does hold to certain Cantonese cultural superstitions, but she always buys just one lottery ticket, with randomly-selected numbers. She always buys when the jackpot is very high. She has never won. This is all I have to offer, except the gut feeling that any further time invested in discussing lottery strategies probably has a payoff with negative expected value.
22 Dec 2013, 02:16 PM Reply Like
• TimeOnTarget
, contributor

Texas has a scratch off game called "Weekly Grand" That one is twenty years old --now they have "Bonus Weekly Grand" which is \$2,500 a week for 20 years.

Anyway, shortly after Weekly Grand came out I started carrying a ticket in my wallet, unscratched (now I carry the Bonus Weekly Grand). When I get annoyed with something at work, I ceremoniously pull it out and announce that "if I hit, I'm walking out of this place" and then proceed to scratch the stuff off.

Haven't hit yet, but it makes me feel better and usually gets a laugh.
18 Dec 2013, 09:33 AM Reply Like
• Ruerd Heeg
, contributor

Thanks for the article. It was great to scan through the pdf mentioned at the end of the article as well.

As we all know the more riskier an asset the less of it we should hold. As the pdf elegantly explains lotteries are so risky that we should not take these risks at all even if the returns are substantial.

Even if you planned to buy as many as 145000 tickets you shouldn't. Even then the trade-off between expected returns and risk is very bad compared to risks we take in the stock market.

An example of learning something by looking at extremes.
18 Dec 2013, 04:33 PM Reply Like
, contributor

Warren Buffett once said the lottery is a tax on stupidity. Many of the gas stations within three miles of my house in Connecticut sell lottery tickets. Whenever I go to pay for the gas, I notice that people are buying lottery tickets. I once read that poor people spend more on lottery tickets than the upper 1%.

I never buy lottery tickets. Occasionally family members who don't know what to give me for Christmas or my birthday give me lottery tickets. If I ever won the jackpot in Mega Millions, I would take the lump sum because of my age. I am 58 and doubt that I will be around to collect until age 87.

What would I do with a sudden lump sum payment of 250 million after taxes? I would deposit the check into the bank and then I would use the money to invest in the Dividend Aristocrats during substantial stock market corrections. I would not make any major changes to my lifestyle. Material possessions never interested me so I would not go on a spending spree. I would leave 90% of the money to charity in my will. I don't believe in leaving any child more than five million as an inheritance.
23 Dec 2013, 03:05 PM Reply Like
• connellybarnes
, contributor

On the Bitcoin front there was an interesting decentralized betting market recently being developed called Augur:

http://bit.ly/1OSTS8u

As is usual with decentralized networks the underlying technology looks very complicated. But hopefully they will simplify the user interface to the point that ordinary mortals can use it.

I had an idea since some years ago that one could avoid taking a position in stock as government-backed ownership of a company's assets. Instead, one could replicate a synthetic position in decentralized betting markets. Essentially, all one needs are betting markets that trade options on the company's future quoted exchange stock price. Then one could enter a synthetic long (buy call, short put) in the options chain. Assuming implied volatilities match actual volatilities, this would have the same payoff as being long the government-backed stock. Of course if the bet were placed in a cryptocurrency such as Bitcoin then there would be that additional risk, and there would be a risk that the parties determining contract outcomes need to be honest (Augur calls these parties 'referees').

But I thought it was an interesting idea, since it might be a way to decouple ownership from nation states.

Why would anyone want to take all these risks just to do that? Well, look at the U.S. dollar, it is constantly losing value. And then compare with the rate at which the Argentine Peso loses value, which is quite a bit more rapid. Thus, distributed betting markets could function as alternative stores of value when confidence in one's country's leaders is low.
19 Nov, 12:45 PM Reply Like

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