# The value of that ticket

Discussion in 'Math' started by nsaspook, Oct 22, 2018.

Aug 27, 2009
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2. ### WBahn Moderator

Mar 31, 2012
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Talk about a bunch of balderdash. "Ignoring tax implications" and then declaring it to be a "very good" deal? That's like saying, ignoring the fact that you died a horrible death, that tickets on the Titanic were really a very good deal because you got to keep your stateroom forever.

Even more to the point, they are ignoring the fact that you are paying out \$2 today for a \$5.53 payout spread over 29 years, or \$0.19/year, which would be an annual return of 3.4%. While that certainly beats the return on your "investments" in Social Security that you are forced to make, it's a pretty lousy return for just about any other kind of long-term investment. And that's assuming that you are the sole winner, something that becomes increasingly unlikely as the jackpot grows.

A much more honest analysis would look at the how much money you had to make versus how much money you get to keep. If you are in the 25% tax bracket federally and the 5% bracket in your state, then taking just the 7.63% that is your portion of FICA into account, you need to earn \$3.21 to buy the ticket. Then, taking the cash option, if you are the sole winner your payout of that \$5.53 is just \$3.12. But you don't get to keep that since you have to pay income tax and you are now certainly in the highest tax bracket, which is 39.6% federal and can be over 10% for state, but I'll keep it at 5%. That brings the expected value of your \$3.21 of labor down to \$1.73, or barely half what you invested.

Some "very good deal" that is.

wayneh and nsaspook like this.

Apr 30, 2011
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Dream crusher! If I win, I'm going to give you \$500k just to troll your belief in math.

nsaspook likes this.
4. ### Raymond Genovese Active Member

Mar 5, 2016
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I am not pro-lottery and I do not play the lottery. I am not, necessarily, anti-lottery either. When I looked, lotteries return (in winnings) about 50% of what they take in, making them a distinctly *bad* bet. I know, however, that it is not that simple because people are also buying something else with their ticket – hope, belonging, n activity perceived as meaningful, conversation topics for socializing and more. It is also true that dependence can result, with a whole host of negative consequences.

The analysis in the article is interesting. The point to be made is that with “progressive” gambling pools, a portion of the prize pool is rolled over into the next prize pool and that can result in a payout that is, functionally, above the 50% return mentioned previously. It can be the case that there is a positive expected ROI. If you only participate when the odds become favorable, it can indeed make sense mathematically, but not realistically or from a desirability standpoint.

In my view, it makes no sense realistically because of the incredibly low probability of occurrence and because of the mistaken value of suddenly obtaining large amounts of money.

While mathematically the expected value may be positive, that assumes an unlimited number of possible plays. In reality, our life span enforces a prohibition on unlimited plays (it is a bit like the realistic limit encountered when employing a Martingale system, except you run out of life instead of a ceiling on bet amount). That is, you have a limitation on opportunities because you are not likely to live long enough to realize even a .50 probability of seeing an event with a 1:3.2 million likelihood of occurrence, even if you bought a single ticket in every opportunity to *play* when there are favorable odds. That is, 1.6 million plays. In a lifetime (let’s say 75 years – remember minors cannot legally play – right?), that is more than 58 plays per day for everyday of a 75 year period – just to have a probability of .50, of at least one occurrence of the winning outcome. Moreover, it is also possible (but less likely if it is a positive ROI) that you would have spent more than the jackpot if/when you hit.

Of course, you could buy 20-30 thousand tickets at each opportunity, but that would be well into the dependence aspect and likely be undesirable for other reasons and, of course, blowing up the ROI. When you buy two tickets, you double your likelihood of a hit, but you can kiss that favorable bet idea goodbye unless it is unlike any jackpot lottery that we have ever seen so far – and you are the sole winner.

The second reason, which is undesirability, is because of the negative life-changing impact of a win that size. Long term, the data indicate that the negative outcomes (when we are talking hundreds of millions, not hundreds of thousands) are substantial. Most of us are simply ill-prepared for a change like that, no matter what we might think.

With regard to calculating the amount of earned money needed to buy the \$2 ticket…

While it is true that you do get something from the amount that you pay in Federal and State income tax, and that could also be estimated in some manner, you do not get 100 times as much of that “something” if you pay \$100 tax versus \$1. So, it is not unreasonable to include Federal and State taxes when calculating the price of the ticket.

The analysis that includes the FICA portion of the money earned to buy the ticket, however, is fundamentally flawed. FICA cannot reasonably be treated in the same manner as a Federal or State income tax. It is much closer to a deferred annuity. In fact, you (including relevant survivors) can get more out of SS than is put in https://www.politifact.com/truth-o-...re-and-social-security-what-you-paid-what-yo/. Thus, you cannot treat it as a simple tax, but would have to calculate the value of that portion – which could, in fact, lower the amount of earned money to purchase the ticket.

5. ### WBahn Moderator

Mar 31, 2012
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Ah, what a wonderful way to be trolled!

My wife's planning to buy ten tickets. She used to ask me to buy them for her (on the rare occasions that it got high enough to get her interest) but she quickly discovered that I just couldn't bring myself to do it. I even went in with the intention of buy the tickets (with money she gave me) and couldn't do it. Show now she doesn't ask me to do it for her -- nor does she ask if she can do it. She simply tells me she's going to do it, knowing that I'm going to make some comment along the lines that her chances of winning will be just marginally better than mine. But I also tell her that, if she does win, I will be the first person to not laugh at her.

Shifting gears, I seems like a better (from a societal standpoint) would be to cap the size of a single prize but add more prizes. Let's say that the "grand prize" were capped at a level such that the 30 year annuity worked out to be \$100,000 (so a jackpot of something around \$3 million). For the overwhelming majority of American households (about 75% of them), that would be a prize level at which they could immediately retire and have a significant increase in their household income guaranteed for the next 30 years. So instead of a single jackpot prize of \$1.6 billion, they could have 533 \$3 million jackpots. I know that I, for one, would be MUCH more likely to buy a ticket if the odds of winning a \$3 million jackpot was 1 in 568,000 instead of a 1 in 303 million chance of winning \$1.6 billion. Whether "much" would actually translate into purchasing tickets is an unknown.

So why don't the big lotto games do this? The obvious likely answer is that, even if expected return on "investment" were much higher with multiple smaller jackpots, the bulk of the lottery-playing public are attracted by huge prize amounts regardless of how unlikely their chances of actually getting in on it are. After all, the average lottery-playing person is anything but mathematically savvy, otherwise they wouldn't be an average lottery=playing person (they might be, at best, a casual/occasional player). But I can't help but wonder whether they also figure that an occasional huge winner has virtually no impact on society while having ten or twenty thousand people a year leaving the workforce each year due to lottery winnings might.

6. ### WBahn Moderator

Mar 31, 2012
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While I will readily stipulate that you are getting something back for your FICA taxes, I think the article paints a too-rosy picture. For instance, it assumes that were you to invest the Social Security payments elsewhere you would only beat inflation by two percentage points. Yet the long-term average of the S&P is more like 7%-pts over inflation. To further underscore this, does anyone really think that if they invested 12.4% of their gross pay into an S&P 500 index fund over their entire working life that they would only get something in the neighborhood of \$1500 to \$2500 a month when they retired?

Another thing I couldn't tell for sure was whether or not the article was including the employer's portion of the contribution. Most pro-SS analyses don't on the basis that the employee didn't pay that other half. But some of the discussion in that article seems like it almost must be taking the full FICA tax burden, employer and employee, into account. I just don't know.

Calculating the value of what you get back from your FICA taxes would certainly be desirable, but I don't know how feasible it would be to come up with any truly defensible "average" fraction. I guess we could just WAG it and call it 50%. For many folks the value gained is very firmly zero. Your average monthly salary used for computing your benefit amount is not based on your entire work history, so if you are in a position where, for whatever reason, you aren't earning enough in a given year so that it will be one of your thirty highest, then your benefits will not change whether you pay anything in that year or not, so then it all should be taken into account when determining the price of the ticket. Similarly, if you are going to have over thirty years of contributions then even if the current year is going to be one of them, the marginal increase in your benefits is due only to the amount by which your current SS income exceeds the (adjusted) income in whichever year drops off the top-thirty list. But you are paying the full OASDI tax on the full income in BOTH years, which significantly dilutes the value of the contributions.

7. ### Raymond Genovese Active Member

Mar 5, 2016
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This is not a simple question. That is, the payoff structure that maximizes purchasing tickets. From a strictly behavioral sciences point of view, early Psychology and then Business Departments of Universities were the places that these subjects have been researched the most (at least openly). Whether it is to determine how rich/lean a reinforcement schedule should be to maintain/fail to maintain a rat's lever-pressing for a food pellet or some MBAs setting up some equities situation to determine when people will risk the most to buy "tulip bulbs". Behavioral economics was much the rage for a while.

Historically, parimutuel betting on the horses has been studied quite a bit with respect to how to maximize the takeout while still maintaining the attendance (i.e., betting handle). Breakpoints were found where attendance would drop if the takeouts were too large (at least that was the presumption).The explanations had to do with risk of ruin in a practical sense (with higher takeouts bettors would lose all that they had to lose quicker and stop going). But it is more complicated since attendees don't only spend money on betting. As it stands now, the takeout is different for different types of bets and as you may suspect, the higher the potential payoff, the higher the takeout. But the data are confounded because horse racing betting is national everywhere it is allowed in the US, even intenrational sometimes (you can bet a race in England from a racetrack in Ohio)...and of course, you can do it legally over the internet as well.

I suspect that, in the case of State lotteries, it is an evolving and dynamic system and may even be too new and too unstable to have enough data to make the determination currently. For example, these super jackpots are linked to multi-state participation, which is relatively new.

Additionally, we are not set up to learn about extremely small probabilities, beyond the intellectual. So, for example, rats and people, will quickly learn 5 places with 5 distinct probabilities (say 0-.9) of receiving a reward, and their behavior will reflect the probabilities in a reasonably systematic fashion when costs are equal. Add another location that varies from another location by .0001 (even .01) and they will not be able to discriminate. When you think about it, why would we be good at something like that as it serves no real value from a natural selection perspective.

On the other hand, when one lotto is at a billion and the other is at 10 billion, you know which is going to get more plays - maybe not rationally so much as being a function of constant publicity and exposure. Since powerball was recently hit and megamillions wasn't, I would guess that you will see an increase in ticket purchases for the latter versus the former. The only reason that I know one was hit and not the other is because I have heard/read it multiple times in the last few days - including pretty much every news source that I access.

Finally, the opportunities to gamble are still increasing and we don't know the saturation point yet. Fifty-sixty years ago, outside of Nevada, legal betting in the US was limited to horses and dogs and maybe the occasional church bingo. Now, slots and varieties of casinos are available in many states. As sports betting becomes as prevalent as State lottos (within the next 10 years or so, I predict), we may be inching closer to finding out just how much people are willing to gamble.

8. ### WBahn Moderator

Mar 31, 2012
23,093
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Definitely agree that it is not simple. For one thing, people think and operate on many different behavioral and trying to equate how they make decisions to how rats make decisions involving levers and food is much too simplistic (but you've got to start somewhere). Among many other things, we have to consider how much information a person has available to them, how comprehensible is that information, and how well prepared is that person to actually comprehend the information they have access to. Then there's all the emotional motivations at play and whether that person is more driven by analytical arguments or emotional ones.

I've had some very interesting conversations over the years with people that were heavily involved with dog track betting and then a very illuminating discussion with the manager of one of the mountain casinos about how they set the payouts on their various slot machines (not just the mechanics of how they set the odds on a machine, but also the considerations that go into deciding what to set the odds at).

It is all far, far away from a purely quantitative "science". Even the big Vegas casinos and oddmakers understand that the best they can hope for is to have a firm handle on the major variables and how they affect their tiny little portion of the gambling world and that what they know about that portion only translates to other types of gaming very imperfectly