Expected Value of 540MM MegaMillions

[ferdowsi]

If you calculate a EV of more than the price of the ticket, you're doing it wrong. The lottery is a money-making endeavor for the state.

 

[APalley]

If you calculate a EV of more than the price of the ticket, you're doing it wrong. The lottery is a money-making endeavor for the state.

The lottery being a money maker for the state has no bearing on whether the EV is more than the price of the ticket and vice versa. Happens to be true that it's negative, but for other reason.

 

[aaronhotchner]

^ Also, the expected value can absolutely be positive, because unclaimed winners' pools are rolled into the next drawing. The state is raking in their profit and pushing the rest to another game, so when that happens a few times it is technically possible to buy a ticket for every combination and come out ahead. Where it fails is when you start including other people buying tickets -- if someone else had a winning ticket, you just lost a lot of money by having to split the pot, so it's likely not worth that kind of gamble.

 

[ferdowsi]

I expected better from a forum for quants.

APalley, how would the state make money if they gave out more than they received?

aaronhotcher, is your argument that the state only makes money when people don't claim winning lottery tickets? I did due diligence for a PE firm on a company that provided lottery services, and I can tell you that's definitely not the case. If on the other hand you're saying that the state can lose money in a given week because they are giving out the money collected from previous ticket sales, then you're right, but you're ignoring the question. We're talking about EV, not actual outcome. The expected outcome is for the state to make money, and the pool only got this high because they had several weeks of making higher than expected profits. The expected outcome this week will be for the state to make money (the higher the pool, the more people will buy tickets = no arbitrage condition); that may not turn out to be the outcome, but in probability, the actual outcome is rarely the expected outcome.

 

[aaronhotchner]

Dude, the jackpot is net state profits. The dollar sales of tickets sold far exceeds the dollar winnings. They get money simply by running the lottery. That does not preclude the possibility of a positive expected value for a particular drawing.

My argument has nothing to do with a winner not claiming and everything to do with nobody claiming because nobody won. The money available in the jackpot goes straight to the winner. When nobody wins, the money goes to the next jackpot. The state already took their cut from the first jackpot, so they can afford to put literally every dollar into the next jackpot and still make exactly as much money as they otherwise would. This is a very basic concept. When you do it over and over, you are effectively adding a net benefit to the expected return on a lottery ticket because the odds of a single ticket winning don't change while the payoff of winning increases. Do it enough and the expected return goes positive.

But what do I know? I'm just a disappointment on this forum for quants.

I'll make my frustration a little more obvious and say that I would expect better thinking from a five year old.

 

[aaronhotchner]

By the way, there is no "no-arbitrage condition" built into lotteries.

 

[purplyboy]

The Gov't already made a lot of money on this lottery. They are only paying up 347 million and collecting a lot more in the revenue that came in. I assume part of the 1.5 billion went to store fees and others fees, but the gov't already got their stake.

Americans spent nearly $1.5 billion for a chance to hit the jackpot, which amounts to a $462 million lump sum and around $347 million after federal tax withholding. With the jackpot odds at 1 in 176 million, it would cost $176 million to buy up every combination. Under that scenario, the strategy would win $171 million less if your state also withholds taxes.​

http://www.cnbc.com/id/46909877

I expected better from a forum for quants.

APalley, how would the state make money if they gave out more than they received?

aaronhotcher, is your argument that the state only makes money when people don't claim winning lottery tickets? I did due diligence for a PE firm on a company that provided lottery services, and I can tell you that's definitely not the case. If on the other hand you're saying that the state can lose money in a given week because they are giving out the money collected from previous ticket sales, then you're right, but you're ignoring the question. We're talking about EV, not actual outcome. The expected outcome is for the state to make money, and the pool only got this high because they had several weeks of making higher than expected profits. The expected outcome this week will be for the state to make money (the higher the pool, the more people will buy tickets = no arbitrage condition); that may not turn out to be the outcome, but in probability, the actual outcome is rarely the expected outcome.

 

[Andy Nguyen]

we have 3 winners and many who got 5/6 numbers correct.

 

[ferdowsi]

The Gov't already made a lot of money on this lottery. They are only paying up 347 million and collecting a lot more in the revenue that came in. I assume part of the 1.5 billion went to store fees and others fees, but the gov't already got their stake.

Americans spent nearly $1.5 billion for a chance to hit the jackpot, which amounts to a $462 million lump sum and around $347 million after federal tax withholding. With the jackpot odds at 1 in 176 million, it would cost $176 million to buy up every combination. Under that scenario, the strategy would win $171 million less if your state also withholds taxes.​

http://www.cnbc.com/id/46909877

Let's think about this: People spent 1.5 billion for 1.5 billion tickets. The state is giving out something like 347 million (let's round up to 500 million to include all the non-grand prize winners):

(0.5-1.5)/1.5 = -0.67

That is the ceiling for the EV of a ticket (you actually have to take into account the fact that the state may not pay out at all, in which case the EV is even lower).

Again, I have to emphasize that if you get a positive result, you should think through your logic since the state is making money off of it. The state doesn't pay out 347 MM for each person who gets a winning combination. It splits it among all winners.

aaronhotchner, all efficient markets have a no arbitrage condition. Lottery systems seem to violate this since you'd expect people to be buying far closer to 500 MM tickets than 1.5 B. In that case, they are buying too many, and the state is making too much, but I guess that can partly be explained by JPAlonso's sentiment that people are doing it for fun plus the state has a near monopoly on gambling.

 

[ferdowsi]

I want to add one more argument: If the state expects to lose money on weeks when the payout is high, why have high payouts? Why not set the rules with a cap on the payout or some other mechanism to prevent it from losing money? I suppose they could be doing it as an advertisement for the rest of the year, but given lottery buying patterns, this doesn't seem likely.

 

[stillrivertrading]

ferdowski,

Do you by any chance work at the high rollers office in the Tropicana in Atlantic City? They recently made the same mistake you're making in this thread.

The expection value of a single ticket for the current jackpot is positive once the jackpot exceeds 1/odds_of_winning/fraction_left_after_taxes_and_lawyers, _if_ we make the (unfortately unrealistic assumption) that there will be only one winner. The thing that screws people who pile in when the jackpot gets large is that the expected number of winners is greater than 1.

 

[purplyboy]

http://www.cnn.com/video/#/video/bestoftv/2012/03/31/nr-kaye-lottery-benefits.cnn
12% goes to overhead
38% to education

 

[aaronhotchner]

I want to add one more argument: If the state expects to lose money on weeks when the payout is high, why have high payouts? Why not set the rules with a cap on the payout or some other mechanism to prevent it from losing money? I suppose they could be doing it as an advertisement for the rest of the year, but given lottery buying patterns, this doesn't seem likely.

You still don't seem to grasp that the state can't lose. A ticket is sold. $x goes to the pool, $y goes to the state. The pool is the sum of all the $x from each ticket sold. Only that money goes to winners. It is IMPOSSIBLE for the state to lose money this way. The state is not in the business of gambling; it collects guaranteed money to fund schools and fund the lottery operation itself. The only "expectation" for the state to lose is if not enough tickets are sold to fund the costs of running the lottery, but when is that ever going to happen?

Let's think about this: People spent 1.5 billion for 1.5 billion tickets. The state is giving out something like 347 million (let's round up to 500 million to include all the non-grand prize winners):

(0.5-1.5)/1.5 = -0.67

Not even close. Expected value is the money you win less the money you pay times the probability of winning.
EV = (1 / 176 million) * (jackpocket - 1) + (175,999,999 / 176 million) - 1 = jackpot / 176 million - 1
Set EV = 0
jackpot / 176 million = 1
jackpot = 176 million
If jackpot > 176 million, EV is positive, and it's in your interest to buy every ticket. But we already knew this, since it's so obvious.
However, this does not include the possibility of other people winning, which is a much messier problem and deals with binomial distributions that I don't really want to have to go through simply to prove why you're entirely wrong.

 

[APalley]

I expected better from a forum for quants.

APalley, how would the state make money if they gave out more than they received?

ferdowsi, here is a simple thought experiment to assist you with this basic probabilistic concept:

Say it is Tuesday, and the jackpot is at 460M. Say no one wins. Now, they base the next jackpot on projected ticket sales, and they for some reason only project a SINGLE ticket sale for Friday night's drawing. The Friday night jackpot is now $460,000,000.50. Say you happen to have the only ticket, and after cash value/tax the jackpot is 250M. What is your EV? 250,000,000/175711536 = 1.42 > 1!! Did the state make a profit? Of course! From the previous weeks combined they made 460M, and from this week they made .50, plus applicable taxes. Now extend this....

The flaw in your thinking is that you are equating the fact that the state will never pay out more than they bring in, with the EV of a ticket, which are 2 totally different things.

Of course, the actual EV is still ALWAYS way under $1 per ticket. But this has NOTHING to do with whether or not the state makes money! It has everything to do with the jackpot differential between consecutive lotteries. A higher jackpot differential implies more tickets sold. The number of tickets sold tends to increase super-linearly. It is a simple binomial distribution: If the jackpot differential is 50M, then ~100M tickets were sold. The odds of winning the entire jackpot alone is 1: 310,429,942.4. The number of outstanding tickets affects the odds of being the sole winner of the jackpot exponentially (the first derivative of the binomial distribution) :

Now for any of these cases, if you take the EV of each possibility (0 winners, 1 winner (you), 2 winners (you plus another)...) and add them you will never get > 1.

The only way it can ever happen is if the (previous jackpot + number of new tix sold/2) is greater than 1/Binomial(new number of tickets sold), which will never happen in practice due to the superlinear nature of ticket sales.

It would be prudent to refrain from personal attacks in your posts.