Mathemematicians, get your lottery tickets here.

Factual Questions
1

In the All you mathematicians, get your butts in here! thread, SingleDad said:

Now, SingleDad could be talking about two classes of misconception. The first, really a mathematical triviality, is the simple neglect to consider the effect of taxes and the time-value of an annuity pay-out (i.e. “a lottery ticket has a positive expectation where the net present value of the after-tax payout is more than X where the odds are 1/X”). The second, which raises interesting mathematical/statistical issues is the possibility that the payout will be shared among several winners.

My question is, given rational assumptions about net after-tax cash-value payout, chance of winning, number of participants, likelyhood of multiple winners and all other relevant factors, does the purchase of a single ticket (or low number of tickets relative to the number of combinations) have a positive net value.

I know that arbatraging the lottery (buying all numbers) will be a losing strategy (particularly if there are multiple arbatageurs).

However, let’s say that the odds of the NY Lottery are one in 13 million, and that the cash value payout is about half of the advertised payout, and that you’ll pay taxes of about half of the cash payout, so that (assuming you will be the only winner) a payout of $50 million would have a “positive” value.

Now, even if there is a possibility that there could be multiple winners, it seem that there are never (or exceedingly rarely) more than two winning tickets on the big lotters, and usually there is only one, so under most circumstances there would be a “positive” value to buying one ticket in a $100 million lottery.

(For the actual rules of Lotto in New York, go to http://www.nylottery.org/ .)

How would we determine what the amount of lottery payout is big enough that it’s a good bet to buy a ticket?

2

If Bill Gates has determined that it would be profitable to buy every number, how can he stop Ted Turner (or some syndicate of millionaire mathematicians) from doing the same, thereby lowering the payoff. I’ve heard rumors of someone buying every number, but I’ve never heard any specifics or corroborating evidence.

When jackpots get high, the probability that there will be multiple winners is greatly increases.

3

Well, as a practical measure, I think buying every number would be very difficult. I recall an Australian professional lottery group who decided to go after one of the huge jackpots a few years back.

They spent all week buying up numbers (I don’t think they could just go into the local liquor store and say “I’ll take 7 million lottery tickets, please.”), and only ended up getting about half the required tickets.

Fortunately for them, they hit the jackpot. I don’t remember if they had to split it with any other group, and I don’t remember what the individual take was once they split it amongst themselves (it was a very large group, IIRC).

To top it off, I believe lottery corporations tightened up the purchasing regulations to avoid instances like this in the future. They figured it might scare people off if it appeared the lottery was rigged in favor of large conglomerates that could come in and scoop up all the number combinations.

Back to the OP, I would agree that one would have a positive expectation as long as the net payout was greater than the odds on hitting, and you didn’t have to split the jackpot (which, of course, you could not predict–that’s why they call it gambling).

Personally, I am very much against state lotteries, but that’s another thread.

4

It’s actually fairly common for a lottery to grow large enough for a ticket purchase to have a positive expectation. To calculate this, you have to know the size of the jackpot, and do a statistical analysis of the possibility of splitting the jackpot with someone else, based on historical information.

But then the spectre of utility theory raises its ugly head. Just because something has a positive expectation doesn’t mean it’s a good bet. If a dollar is worth more to me than a 1 in 10 million chance of winning 11 million dollars, then it’s a bad bet. And if the expectation of a $1.00 lottery ticket is $1.05, then if it takes me 20 minutes to drive somewhere and buy one I’m only getting paid 15 cents an hour for my effort.

5

Well, dhanson, I suppose it’s silly to talk about lotteries without talking about expectation theory. (After all, that’s why people spend $1 for a one in 13 million chance to win $3 million before taxes over the next 26 years.)

But, I suppose my scientific question is whether a particular lottery purchase ticket has a positive value, and we can then each add our personal expectation theory afterward.

For instance, I’ll often spend a buck on a ticket when the jackpot is $15-20 million, even though it’s a “loser” bet. The few minutes I have of contemplating what I’d do with $20 million (or even the after-tax cash value of $5 million or so) is worth what I’m losing in “value.”

6

And therefore, buying a ticket has positive utility for you, even though it has a negative expectation. It is therefore rational to buy a ticket.

Utility Theory is very important when discussing extremes between the cost of a gamble and the potential payout. Expectation alone is a linear function, but in real life things do not work that way.

Consider this: A man gives you one hundred million dollars. Now he offers to flip a coin with you - if you win, he’ll give you 300 million more. If you lose, you lose the 100 million. Should you take the bet? It has a high expected value, and based on expectation alone you’d be an idiot not to.

However, the utility of an added 300 million is not worth a 50% chance of losing 100 million for the average person, and you’d be completely insane to take the bet. Unless you are Bill Gates. His utility curve is completely different, and it would be insane for him to not take the bet.

Lotteries work in much the same way. A person’s utility function may make a lottery ticket purchase a ‘good bet’, even though it has negative expectation. This explains why poor people tend to buy more lottery tickets than wealthy people. Utility also explains why more rich people buy insurance, which is always a negative expectation bet. The loss in utility from insurance payments is small for the wealthy, whereas the loss in utility from a fire or flood is very high. For the poor, the insurance payments carry enough loss in utility to make the insurance a ‘bad bet’. The expectation may be the same for both, yet both can come to opposite choices and still be correct.

People are generally rational about their own choices. Lotteries are not a tax on the stupid - even if everyone understood the math behind the lottery perfectly, some people would still quite rationally choose to play.

7

An excellent point, dhanson. If the loss of a dollar is trivial, then the bet has high utility.

But how about very poor people who spend hundreds of dollars per month on the lottery? That’s a fair amount of loss in terms of money available to support one’s lifestyle. Would you argue that’s rational utilitarian behavior?

Catapultam habeo. Nisi pecuniam omnem mihi dabis, ad caput tuum saxum immane mittam.

8

SingleDad, how many possible tickets are there? In Calif there are more than other states as I think we use 52 numbers.

Even if you bought every possible ticket number there is, the prize is way below that cost.

9

As Divemaster mentioned above, there was a group that won the Australian Lottery three times by buying every ticket. I saw a television show about their successful attempt to win the Virginia lottery. (The Australian Lottery changed its rules to prevent this from happening again) This happened about ten years ago and I’m pretty sure that I’m not the only one who saw it. Part of the show focused on the fact that they had to wait for the jackpot to get big enough to make it worthwhile. Their ideal number was something like $27 million. A lower number wouldn’t be worth it, and a higher number significantly increased the odds of splitting the jackpot, because a lot more tickets get sold when the jackpot is high. The $27 million figure is from my memory and almost surely incorrect, but the resoning behind the number was presented by a spokesperson for the group. Also, the reason the group chose to win the Virginia lottery was because at that time it paid out a lump sum instead of paying out over a period of twenty years.

Divemaster also said that the group bought only half the tickets. IIRC, the group had some problems and ended up with only about 90% of the tickets. That made for some tension during the drawing, but they did in fact buy the winning ticket. They had a computer record of all the tickets that they had purchased so they knew that they had won fairly soon after the drawing. However, it took them something like two days to find it in the stacks and stacks of tickets.

This is in response to the part of the OP that says that, “arbatraging the lottery (buying all numbers) will be a losing strategy.”

Am I supposed to believe that all this rain was suspended in mid-air until moments ago?

10

As soon as I hit submit I remembered that the television show was probably, “How’d they do that?”

Am I supposed to believe that all this rain was suspended in mid-air until moments ago?