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The supermarket checkout line is not a typically a place of inspiration for Fool posts, but a particularly slow line and a glowing lottery sign did the trick. Here's my question for all of you

At what point do you gamble on a lottery?

Putting up with my simplified examples below, I think both the expected value of a ticket and the odds of winning make a difference. For the purposes of this post, I am ignoring taxes, discount rates, and other winning numbers (other than that, how was the play, Mrs. Lincoln) but there are real world examples of lotteries with positive expected values:

http://www.holycross.edu/departments/economics/vmatheso/research/lottery2.pdf

Back to the checkout line…

I would not take this bet:

One school of thought suggests that you should take your chances when the expected value of a $1 ticket rises above $1. For example, if a SuperDoubleWhammyPowerball prize is $115 million, and sells 100 million tickets (yes, you know this in advance), the expected value of a $1 lottery ticket is $115M/100M, or $1.15. On average each of the lottery tickets will return $1.15, or a 15% return. The problem is the distribution of payment is so skewed. One person wins the entire $115 million pot, while all the rest of the bettors lost their $1. I would not take this bet, as I would effectively lose every time I play.

I might take this bet:

Now what if the lottery above were played for a pot of $11.50 with only 10 tickets available for purchase. The odds are the same. The expected value from a $1 ticket is the same $1.15, but I would be more likely to play this game as I stand a more reasonable chance of winning.

I would take this bet:

If I could play this game above over and over again, I would certainly play. I would receive a “lumpy” 15% on my efforts, losing 9 of 10 times on average but winning 1 of 10 for a 15% return overall. The game now becomes like a casino where the house loses and the odds favor the gambler.

As you are increasingly excluded from a greater number of hands/rounds at what point would you not take the bet? The expected value of a ticket remains the same at $1.15. Each time playing the lottery has the same favorable odds. But as the value of the pot rises, and the number of players increases, you stand an increasing chance of losing even through the expected value remains the same.

When would you play?





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When would you play?

I'd rather invest in the stock market - much better odds, and if you really find something worth finding, you can bet a whole heck of a lot and get a whole heck of a lot return (no skewing of results; if you bet right and bet big, you win very big), and investment knowledge ought to be cumulative to some degree. Fwiw, it was mentioned before whether anybody believes in EMT. I do (at least the weak version of random walk as I remember it). I believe very strongly in the idea that if a company reports good news on a consistent basis, the market will recognize those efforts (I believe this because it is almost always so - at least in the last 11 years in my experience). I don't believe the market can predict the future but I do believe there are limited times I can with some clarity...
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I like free money. Aside from your 'MegaMillions' example, it seems pretty free EV for those wagers.

Your post reminded me of this scene for some reason:

Blake: "Go and do likewise, gents. The money's out there-- you pick it up, it's yours, you don't, I got no sympathy for you. You want to go out on those sits tonight and close-- close-- it's yours. Not, you're gonna be shining my shoes."

Man, what a great movie.

Naj
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But as the value of the pot rises, and the number of players increases, you stand an increasing chance of losing even through the expected value remains the same.

I don't think this is correct. Since the prize is split if multiple players win, wouldn't the odds of having multiple winners (which becomes more likely as the number of participants goes up) have to be factored into the calculation of expected value?

Anyway, I'm with IM - I think it's a much better use of my time to, for example, try to figure out whether the easy comparisons for TJX make it a good buy, or whether their recent inventory issues are indicative of more fundamental problems with their business.

FWIW, I could never understand the mentality where people don't bother playing the lottery when it's "only" a $10 million jackpot but stand in line to buy tickets when it's $100 million. In both cases it's a life changing event with extremely long odds - my opinion is you should either just plunk down the buck every week knowing you probably won't get it back, or ignore the whole thing.
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When would you play?

I have never been in a situation where it would make sense for me.

The utility to me of more than a few million dolars is just not that high. Taking that into account, it just doesn't make sense to play even if the EV of a single ticket (based on a linear utility curve) is somewhat greater than $1.

--B+C
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let's skip the calculation of expectation value in any particular game; it's finicky, depends totally on detailed rules, and not very interesting.

One school of thought suggests that you should take your chances when the expected value of a $1 ticket rises above $1.

indeed, that is the ONLY valid school of thought.

The problem is the distribution of payment is so skewed. One person wins the entire $115 million pot, while all the rest of the bettors lost their $1. I would not take this bet, as I would effectively lose every time I play.

while you'd be accruing a profit every time you played, the problem is that you'd need to play many many many times in order to have a good chance of a realized win.

if you play many many many times by buying one ticket apiece in many many many lotteries, then because of the delay between lotteries, your CAGR will go down to the point where you'd be better off sticking the money in a savings account instead.

if you instead play many many many times by buying up as much as you can of the entire issuance of a particular EV positive lottery.... now you're talking.

however, you run into the problem of "gambler's ruin" and wager sizing. if you're not a sure thing to win, then you have to control the percentage of your bankroll that you devote to each bet. otherwise you're likely to go bankrupt betting on favorable odds.

j.l. kelly's seminal paper on horsetrack betting is the right place to start for this line of thought, if you need to actually get down to numeric quantities on sizing your bets.

bottom line: EV positive lotteries can make sense if you control a large amount of money, and you can buy lots of tickets in several different lotteries. there exist several professional lottery pools that do just that.

otherwise, they're a waste of time.

trp
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I've never bought a lottery ticket. My personal opinion is that the number of times that a ticket might have an EV of >1 doesn't justify my time in trying to figure it out.

YMMV.

Steve
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Although I occasionally cheat a bit off these rules:

A) the EV has to be >$1 per $1 played. More specifically, that should be the lump sum after-tax EV, accounting for the possibility of multiple winners. I use a straight binomial distribution to calculate that.

I'll note that if one does this, the jackpots have to get VERY high indeed to be worthwhile in most situations.

B) Like B+C, I have marginal utility issues. Without getting into too many details, an after-tax lump sum of $1million would allow us to pay off all outstanding debts including mortgage, max out current college savings funds, use funds to take care of some desired projects, item purchases that we have been deferring, and park about $375K in savings. Using the 4% rule of thumb from REHP, this means we'd have 15K after-tax cash per year we could live off of. So we couldn't retire and live at the level we would like, but we could get by if we had to.

A second million would increase the annual after-tax to 55K, at which point we could retire if we wanted to and live rather well, although we couldn't go crazy. A third million would make that 95K/yr (that's actual spending cash available), at which point you'd have to really convince me it's worth working. ;-) Above that...

Well, 5M is 175K/yr. 10M is 375K/yr. Quite frankly, 10M would be insane for us, and over 10M becomes silly (gee, can we struggle by on 500K spending cash per year with no debts?). Heck, I dunno how much extra utility we get going over 5M. Yes, there's some, but not a lot.

Here in Illinois there are three lotteries (in addition to the "Pick 3", "Pick 4", and various scratch off things). "Little Lotto" has odds of about 575K-1 against, and the jackpot is awarded as a lump sum. It occasionally gets up above 600K, and I usually play when it hits 750K or higher. Not quite an after tax $1, but close.

"Lotto" has odds of about 10M to 1 against. It has a listed "annuitized" jackpot, the lump sum option currently is about 60% of that total. This one I cheat and play when it hits about 18-20M, which is only about 7M after-tax lump sum. (It's pretty rare that it hits that amount).

I can't bring myself to play Mega Millions (the third lottery, a multi-state one like Powerball) anymore. The odds against are about 135M to 1. Not only is it very rare for the jackpot to get anywhere near high enough, but even when it does the odds are so extremely long (as if the 10M to one of lotto wasn't long enough), and even if I were to win...well, as mentioned above, 70 or 80 or 130M would not mean much more in terms of "lifestyle change) than 10 or 15M. What, I could by 7 5+ million dollar houses instead of 2 or 3 houses worth $1 to $2 million each? Heck, we're not planning to buy a home in California; most of the places where we'd want to own (New Zealand, Greece) are pretty darn cheap and a 500K house would be a mansion.

So, although those "big jackpots" bring in most people, they don't work for me because of the flattening of our utility curve.

-synchronicity
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So, although those "big jackpots" bring in most people, they don't work for me because of the flattening of our utility curve.

The solution to your implied non-existent problem is simple: exposure to more marketing!
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"...Greece) are pretty darn cheap and a 500K house would be a mansion."

1. we are talking about a different country
2. one of us is wrong about prices (t)here (500K buys you a pretty good house/apartment in most places but far from a mansion)
3. I do not know what a mansion is
4. you discovered a hidden gem, either location-wise or a method to use people who do not get paid for the construction (I am interested in case 4 to read more about it)

Make your choice but I am afraid, you should restrict your plans towards New Zealand only, synchronicity.

To pretend that we remain on topic, I buy lottery tickets whenever I feel like it (rarely/small amounts) either for fun or because I like the seller.
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1. we are talking about a different country
2. one of us is wrong about prices (t)here (500K buys you a pretty good house/apartment in most places but far from a mansion)


I'm not talking Athens, I'm talking NW Greece, near Kastoria. How much would 500K US$ (so just over 400K Euro) buy up there?

-synchronicity
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Utilizing the prizes page at www.powerball.com , the expected value of a $1 bet that does not include hitting Powerball is calculated as follows:

3/70.39 + 4/123.88 + 7/696.85 + 7/260.61 + 100/10685 + 100/12248.66 + 5000/502194.88 + 100000/2939677.32 = .0426 + .0323 + .0100 + .0269 + .0094 + .0082 + .0010 + .0340 = .1644.

Subtracting .1644 from 1.000 gives .8356, which is the expected value of the "hitting powerball" component of the powerball bet which will produce an expected value of $1 for a $1 bet. To get the lump sum payout powerball figure required, we solve for x in the equation:

x/120526770 = .8356

Solving for x gives the lump sum payout required of $100,712,169.

http://www.powerball.com/powerball/pb_prizes.asp

jkm929
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Solving for x gives the lump sum payout required of $100,712,169.

Which you would then have to adjust for taxes, annuitization, the possibility of multiple winners (modeld through a Poisson approximation or somesuch) and non-flat utility...

--B+C
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Trepanne? Was that you? I figgered y'all made your fortune & headed for the hills to play, drink, & fornicate.

Shoulda known you'd show up in thread with a gambling theme.

Good to see you're still around. Sad to say, I'm still slumming on the LVLT board.

Dan
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the possibility of multiple winners (modeld through a Poisson approximation or somesuch

That's what I feared. My use of a stright binomial distribution is bad, right?

-synchronicity, not much of a math guru
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I'm not talking Athens, I'm talking NW Greece, near Kastoria. How much would 500K US$ (so just over 400K Euro) buy up there?

My guess, it would buy you a nice house, more than enough to cover the needs of 4-5 persons, without excessive luxuries, not a mansion. (fyi a friend bought at noisy, middle class area downtown Athens, 5th floor apartment built 45 years ago without parking lot at area with huge parking problem but with good public transport, approx. 1050 ft², with minor problem at the titles, priced last December around 160,000€, approx. $190,000 paid almost all in cash, meaning no delays for mortgages, the seller wouldn't accept even a two week delay to get the money)

I knew already you were not talking about Athens, I missed the orientation, having read messages of yours in the past, thought you were looking north-east (Kavala area, islands of Thasos or Samothraki). I can't give you prices for any area outside of Athens but I can make a few remarks:

1. as you very well know, real estate prices can be extremely local. Do not know if you mean the city of Kastoria or the "county" but do not be surprised if the city is "unreasonably" expensive compared to your expectations. Kastoria used to be about 20-25 years ago one of, if not *the*, highest-income per capita area of the whole country, due to the fur trade & the rather low population. Their trade have been hit severely since then (similarities with textile industry), so I can not tell you how this has influenced the market price of land but it wouldn't surprise me to find inflated prices at the city.

2. it is, generally speaking, correct to think that land outside of Athens would be cheaper -especially for someone earning the significantly higher salaries of USA- but one has to calculate that -generally speaking again- construction costs can be much higher (small, inefficient markets, often illiquid, transportation costs, different labour costs etc).
Similar can apply for many daily items (groceries except those that are locally produced, gasoline etc). Gasoline is lately over 1€ the liter (I think 1 liter equals 0.264 US gal, about 3.78€, about US$4.50 the galon on the way to $5.50) at areas outside of Athens.

3. it does not apply in your case, since you only need to compare two areas, the one you currently live in and your "target", however for generic comparisons between here and there, I would not dare to compare Kastoria with any high-profile US location, not even mid-profile. Say, I would compare it (correct me if I am very off with this) with Montana or the Dakotas. From what I read at various TMF boards, prices are day and night between such places and locations like coastal CA or around NYC.

4. you also have to account for differencies at quality of the materials used, as well as the need to build a robust, antiseismic construction (meaning I would be skeptical to use prices of old buildings if not built with the strictest guidelines)

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I knew already you were not talking about Athens, I missed the orientation, having read messages of yours in the past, thought you were looking north-east (Kavala area, islands of Thasos or Samothraki).

Dunno how specific I've been in the past re: areas of Greece. Heck, after talking with syncspouse a few days ago, it's only now that I'm "really" focusing on areas.

Do not know if you mean the city of Kastoria or the "county" but do not be surprised if the city is "unreasonably" expensive compared to your expectations. Kastoria used to be about 20-25 years ago one of, if not *the*, highest-income per capita area of the whole country, due to the fur trade & the rather low population. Their trade have been hit severely since then (similarities with textile industry), so I can not tell you how this has influenced the market price of land but it wouldn't surprise me to find inflated prices at the city.

Yup, we know all about what's happened to the fur trade. As to area, at least some of syncspouse's cousins live in/near Argos Orestikon (sp?). I suspect most of her relatives do not live in Kastoria but rather in the surrounding areas. (Since her grandfather grew tobacco, I'm thinking that wasn't in the actual city). And yes, at least some of her cousins are in the fur business. One actually lives in the States now and apparently does very well.

And yeah, syncspouse is Greek, and you've undoubtedly figured out. Her mom was born near Kastoria and lived there until shortly before syncposue was born.

As to to the rest of it, for us a "nice house, more than enough to cover the needs of 4-5 persons without excessive luxuries" would be a big 'ol house for us. Heck, 3K square feet, to me, is an AWFUL lot of house.

Thanks for the other comments, too. I was generally aware of some of them, but not all. I suspect we'd have a little advantage over some random Americans moving to Greece, as syncspouse is fluent in the language and has lots of relatives who are "locals". Me, I'll just nod my head and tell my wife "it's all Greek to me" (sorry, I know that's really bad).

Roughly where are you at in Greece, if I may ask? And have you lived there all your life?

Thanks again for the advice/information. I wish it was happening sooner rather than later.

-synchronicity
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If I could play this game above over and over again, I would certainly play.

Since there seems to be an opening for the job of provinding complicated theoretical papers to provide a different point of view in abstract discussions I am volunteering...

In 1963, Paul Samuelson wrote what is know a somewhat famous article entitled Risk and Uncertainty: A Fallacy of Large Numbers, about this very topic. A reprint may be found free of charge at:

www.casact.org/pubs/forum/94sforum/94sf049.pdf

Basically the idea is that if a single play is not acceptable, then no sequence should be acceptable, when the goal is to maximize the expected utility. The theory is basically that people, when thinking about the law of large numbers, tend to forget that even if you play 50 million or more you still don't have full certainty that you will win, but at the same time your potential losses grow accordingly (you could sequentially play 100 million times and still not win, losing a lot of money if the idea was to be risk averse).

Mathematically the expected value of return is the same no matter how many times you would play, it's a simple multiplication; but psychological humans behave very different. I don't remember the source off the top of my head, but there have been some experiments asking what people were willing to pay for the probability of winning something (i.e. how much they would pay to increase a single perentage point in their chance of winning). The increase from 0% to 1% (lottery) or from 99% to 100% (insurance, the assurance of winning) will always command much higher prices than, for example, paying from 34% to 35%; even though the expected return is the same in all cases.

It's quite interesting...
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hey dan,

I figgered y'all made your fortune & headed for the hills to play, drink, & fornicate.

alas, i'm still stuck down in the concrete canyons of the city, grubbing for filthy lucre, and doing far too much proxy-fornicating on the internet with the rest of you sad, sad wankers.

cheers, and LVLT p.o.s. to zero

trp
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i'm . . . doing far too much proxy-fornicating on the internet with the rest of you sad, sad wankers.

Hey, I resemble that remark!!!

jaloti
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Trepanne? Was that you? I figgered y'all made your fortune & headed for the hills to play, drink, & fornicate.

Shoulda known you'd show up in thread with a gambling theme.

Good to see you're still around. Sad to say, I'm still slumming on the LVLT board.

Dan


Yeah, ditto what Dan said. Although the OSTK board is fun reading while waiting for a LVLT post.

JuntoWisdom
State Lottery p.o.s. to zero
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alas, i'm still stuck down in the concrete canyons of the city, grubbing for filthy lucre, and doing far too much proxy-fornicating on the internet with the rest of you sad, sad wankers.

Ah, so that's what you've been up to. Good to see even in a simple reply my learning of vocabulary can increase. The need to "alt click" a word for a definition usually only happens while scanning a well written Treppane post. Thanks for the English as well as the financial education. Any good Labor Day picnic recipes you care to share? Having a small group of friends over tomorrow.

JuntoWisdom
LVLT pos to zero
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