I Can you turn investment in a lottery from subjective to objective?

[Arqane]

TL;DR Summary

Though buying a lottery ticket will almost always be a loss on average, the possible percentage return dwarfs any other type of investment. So when is the risk really worth it?

So let's say we have a lottery out there which costs $2 to buy a ticket, and currently has a $1B prize. Let's say the average return is $1.80 on that $2 (90%). Objectively, for every dollar spent, investing long-term in the stock market is mathematically better. But that doesn't take into account the maximum possible returns over a certain amount of time. A big lottery win absolutely dwarfs the amount of the greatest stock market return, so it ends up being high risk, but extremely high reward. The risk someone takes with an investment usually depends on a few circumstances, and often includes subjectivity. If you had $10,000 to invest, put only $9,998 into the stock market and bought a lottery ticket with the extra $2, the math would say that you're likely to lose $0.40 from my lottery example compared to investing the entire $10,000. With the possibility of winning $1B, the 5,500% return for Bitcoin one year looks tiny. So I guess you could do a probability graph, but what variables would be the best to include? And although I think it would still stay subjective even with the probabilities, could you come up with a reasonable rule of thumb to tell people where the probabilities cross (like saying that you should buy one lottery ticket for about every $10,000 invested in the market)? I find it an interesting exercise between the usual objectivity of math and dealing with subjective feelings of risk.

 

[fresh_42]

It is never worth it if you are acting according to the expectation value only. However, people have additional qualities. We take risks if they are only small enough, and the gains are high enough. Every person has a different individual function of standard abbreviation (risk) to expectation value (gain). This function defines whether you buy it or not, even though you are probably not aware of your amount of risk aversion.

You can also consider the price of a lottery ticket as the cost of the excitement and entertainment you get.

Economists deal with such topics in decision and game theory.

 

[Office_Shredder]

Actually, the small chance of a huge payout is bad. The utility of a dollar goes down as you have more of them. From the perspective of a regular person, 100 million dollars and 1 billion dollars are basically equally life changing. So having 1/10th the odds of winning in order to increase the jackpot 10x should not make you more excited about playing.

 

[phinds]

Actually, the small chance of a huge payout is bad. The utility of a dollar goes down as you have more of them. From the perspective of a regular person, 100 million dollars and 1 billion dollars are basically equally life changing. So having 1/10th the odds of winning in order to increase the jackpot 10x should not make you more excited about playing.

Yeah, but that's logical. Betting on the lottery is emotional. At least I think it is. I've never done it and have no intention of ever doing it but based on what people who do bet on it tell me, I think it's really an emotional (non-rational) decision so odds just don't come into it. Potential reward is all they see.

 

[DaveC426913]

Actually, the small chance of a huge payout is bad. The utility of a dollar goes down as you have more of them. From the perspective of a regular person, 100 million dollars and 1 billion dollars are basically equally life changing. So having 1/10th the odds of winning in order to increase the jackpot 10x should not make you more excited about playing.

Yeah. This was always my take. Why do lottery sales go up when the jackpot grows? Do people really think "well I wouldn't get outta bed for a million, but 10 million's a different story!"

 

[Arqane]

Yeah. This was always my take. Why do lottery sales go up when the jackpot gets grows? Do people really think "well I wouldn't get outta bed for a million, but 10 million's a different story!"

In my case that would be a distinction that matters. I'm starting up a company that will take $400k minimum, but really needs about $4 million for the full initial cost. So it would make a substantial difference between those two, especially after the taxes.
But, the amount does matter to a point in the mathematics calculation. To be worth the added risk that you're taking on the numbers have to be high. Winning a $10k jackpot from $1 is substantial, but most people can accumulate $10k and they can do it in multiple different ways. The higher the numbers go, the less options you have to make that much. But in this case, as investments are most commonly done through the stock market, it makes for a somewhat simpler comparison. It is definitely useful for someone that had $10k to invest to turn that into $1M or more quickly (as low as the chances are). If they already had multiple millions, it's likely less useful. Some of that can be quantified and factored into risk. But that's the main point of this post. Can someone put together a formula that would somehow show when it's a "smart" idea to invest in a lottery ticket. I don't think the average return per dollar really shows that well.

 

[hutchphd]

I love this story. Smart guys win.

 

[Arqane]

I love this story. Smart guys win.

They left some parts out of that story. It sounds like they were playing odds that returned greater than 100%, so basically when they 'rolled down' the prizes they ended up making a poor game that paid out more than it made. They didn't mention taxes, which surprises me. Maybe at the lower winnings they didn't charge taxes, because the values he gave minus the nearly 50% in taxes that are often mentioned on the big prizes wouldn't end up as a profit. I'm also a bit confused how the state also made money because if the game is paying out over 100%, then normally they would lose some. But I'm guessing that there was some insurance deal that was paying out the actual winnings that had been set at a certain rate, and they didn't update that rate when the lottery messed up the ratios. In that case the winners would have gained, the state could have gained, and it's the insurance company that would have lost out (which ultimately usually ends up back with the state and taxpayers bailing them out, anyway).
But in my example I was really considering games where the lottery was not 'broken' by incorrectly setting their prizes. Even if a lottery paid out <1% on every dollar (say a lottery only with the grand prize matching 6 numbers and no other prizes), it could still potentially be worth the investment in certain cases. That still deals with maximum potential returns, which are a real difference between investments.

 

[jedishrfu]

Here's a more detailed account:

https://www.insider.com/how-jerry-marge-selbee-won-the-lottery-2019-1

 

[Arqane]

Here's a more detailed account:

https://www.insider.com/how-jerry-marge-selbee-won-the-lottery-2019-1

I didn't really see any detail that wasn't in the clip. There are other potential options, but with the amount of numbers they played, I'm not sure that they used that strategy. You have much more consistent returns by making sure you don't play the same number combinations twice, but unless the machines that print them have that as a function, chances are they were probably randomly chosen. But that is one case where they could win their money and the state would still benefit as the people choosing separate numbers lowered their chances for failure, and other people choosing random numbers are just as likely to copy their wrong numbers as the correct ones, and they should end up right at the normal random chance. But like I said, while that's much more believable when they were playing somewhat smaller numbers of tickets, $6M in tickets would take too long to have a person input so you're relying on whether the computer let's you choose non-random combinations in bulk or not.

 

[mcastillo356]

Once my grandfather told my father to write down as much lottery numbers as he could. He wrote some ones. Then he (grandfather) told his son to check it with the list of that day winners.
My father took it serious. He ended up with a conclusion: not to mess with lottery. It's a unfortunate game

 

[PeroK]

I've not studied this in detail, but when there is a head-to-head sports contest (any soccer, tennis or cricket match or boxing match etc.) there are mutiple online betting sites that gives the odds for both sides. No individual site will give odds that add up to more than 1, but what if two different sites favour the different participants? Say online betting site A gives odds of 5/4 on team X and betting site B gives odds of 5/4 on team Y. Then, you could bet the same amount on each team on the different sites and guarantee positive return.

I have occasionally checked this out and it does seem that the different sites align with each other in this respect. I.e. the best odds on team X across the Internet plus the best odds on team Y across the internet always adds up to less than 1.

 

[jack action]

Can someone put together a formula that would somehow show when it's a "smart" idea to invest in a lottery ticket. I don't think the average return per dollar really shows that well.

Yes, it does. You are just not accepting the results.

Take your example of a lottery where everybody puts in 2$, and everybody, on average, receives 1.80$, and one person must receive 1B$. For this to work, there must be at least 5 billion people participating in your lottery ( = 1B / 0.20 ). This means that if everybody on Earth played your lottery, only one person would say «Damn! The system works!» A system where everybody on Earth participates, and only one person profits from it, is not a system that works. If you think the contrary, then I invite you - and everybody else on this forum - to PM me such that I can tell you how you can send me 1$ just for the fun of it.

The risk someone takes with an investment usually depends on a few circumstances, and often includes subjectivity. If you had $10,000 to invest, put only $9,998 into the stock market and bought a lottery ticket with the extra $2, the math would say that you're likely to lose $0.40 from my lottery example compared to investing the entire $10,000.

This is the kind of thinking that leads to gambling addiction.

If you follow that logic, here's what going to happen (We're talking about a certainty of 99.999...%): You will lose the 2$ and end up with 9998$. If your goal is to make money, then you have to reinvest that amount to compensate for your loss. How will you do it? Well, you have that great system where you could invest 9996$ in the stock market and 2$ in the lottery because it is not that much different from 9998$ and 2$. And you will lose 2$ again. And this cycle will work like so until the 2$ becomes a bigger portion of the total amount to invest. At that point, say 100$ and 2$, your logic will change to: «Now I don't have a choice, I MUST win the lottery to regain my lost. I can't quit now.» This will last until you will lose everything because the odds of losing 5000 times in a row are extremely high.

Your only hope in such a system is that the 9998$ you invest somewhere else gives you a gain of at least 2$. If you constantly gain 2$ and continue with your investing strategy, you will end up with 10000$, no matter when you stop. If you gain more than 2$, you will make money, but you would have made more if it wasn't for the lottery tickets you bought.

These are your odds. They are only based on luck. And if you're lucky, it only means that millions of other people have lost. You cannot share your "system" with millions of people and expect that most of them will gain from it. It is not a system. It is not an investment strategy.

Real Life Example

Not long ago, it just so happen that I studied such odds. A local lottery had a 70 million $ jackpot with 1 in 33 million odds for a 5 $ ticket. But I could join a group with 70 people in it. So now my odds are about 1: 500 000 for a 1 million $ jackpot. These odds are a lot better and 1 million $ is still pretty great.

But there is more. They are other smaller winnings also available with the same ticket. The smallest one - still making me a winner once split 70 ways - was 13 $ for a 1:531 odd. That is 8 $ in my pocket! And look how ridiculous those odds are compared to 33 million! I can't pass up this opportunity!

It sounded way too good, so I tried to visualize what 1:531 meant. So there is this guy in front of me who I give 5 $ and then asks me: «I'm thinking of a number between 1 and 531. If you guess it right, I'm giving you 13 $. You only have one chance.» So what? I say «287» or «53», the guy says «Nope, that wasn't it» and I just lost 5 $. I would feel so stupid playing this game. And playing that lottery would have been exactly that.

I did not buy any tickets.

 

[fresh_42]

Once my grandfather told my father to write down as much lottery numbers as he could. He wrote some ones. Then he (grandfather) told his son to check it with the list of that day winners.
My father took it serious. He ended up with a conclusion: not to mess with lottery. It's a unfortunate game

I like the following comparison. The odds at the roulette table are far better. You can always bet red or black and the zero (Europe) or zeros (USA) are only a little disturbance: $1/37$ or $1/19.$
The odds are therefore $0.\overline{486}$ (Europe) and $0,47368421\ldots$ (USA). Yet, casinos seem to be a profitable business.

 

[hutchphd]

Can someone put together a formula that would somehow show when it's a "smart" idea to invest in a lottery ticket. I don't think the average return per dollar really shows that well.

Its smart when your expectation value is positive and the risk of losing a substantial amount is small enough.
The Selbees (in the video) made more than $27 million over a period of nine years. They relied on the small denomination wins when nobody had won the jackpot (the game rewarded this circumstance) and their probability of a losing fluctuation got smaller the more tickets they could purchase. After they had won some money they could aford to buy large numbers of tickets. As I recall they spent ten-hour days manually buying the tickets when the circumstances aligned. There were several states with such lotteries and they would do road trips!
The other stuff you are talking about is all calculable, but I frankly do not understand your point.

/

 

[Arqane]

This is the kind of thinking that leads to gambling addiction.

If you follow that logic, here's what going to happen (We're talking about a certainty of 99.999...%): You will lose the 2$ and end up with 9998$.

While I'm aware of the slippery slope of gambling addiction, you completely skipped the point and simply ran the numbers that are absolutely going to lose. You're correct in that. I also kept this part of the quote because the certainty isn't anywhere near 99.999...% that you'll lose $2. Most lotteries seem to give about 94% returns, and this one I mentioned gives 90%, so you only lose $0.20 on average, and each single ticket depends on the different odds at different prize values.
It is already standard practice by investment firms to diversify, and in certain cases that diversity uses riskier assets. Dipping into certain assets that are only expected to return 90%, but have also showed high yield returns in the past is something that some people will do. Venture capitalists are another good example where they expect many of their investments to fail (some will recoup the initial investments, others will not), but through diversity and a few winners that return very large percentage gains they can thrive. That is what I'm focusing on in this scenario, and why there can be "smart" choices given certain risk criteria.

Well, you have that great system where you could invest 9996$ in the stock market and 2$ in the lottery because it is not that much different from 9998$ and 2$. And you will lose 2$ again. And this cycle will work like so until the 2$ becomes a bigger portion of the total amount to invest. At that point, say 100$ and 2$, your logic will change to: «Now I don't have a choice, I MUST win the lottery to regain my lost. I can't quit now.» This will last until you will lose everything because the odds of losing 5000 times in a row are extremely high.

No, that is not the logic that I'm using. Though it is a danger of people who don't run the math or manage their risks. Let's use the example I set of 90% returns from a lottery and let's say the stock market returns 110% per year (pretty close to the average). If you fully invested in the stock market, you would get 110% returns, fully investing in the lottery gives you 90% returns. The ratio changes depending on how much you spend on each. In this case if you spent 50% on each then you would expect right around 100% returns, neither gaining or losing (you know, not counting inflation always making you lose at 100% returns). While that is your average returns, your potential returns vary substantially. Let's use the Bitcoin example as the theoretical maximum return for stocks (I couldn't find a higher yearly return, but feel free to add one). Therefore the absolute maximum return you could get from 100% investment in the stock market is 5500% in one year. But if you put 99.9% in the stock market and 0.01% in the lottery, all of a sudden your maximum possible return spikes to 50,000,000,000% while the average return only drops by a miniscule amount (.004% in my $10,000 example). Any additional benefit sharply drops after that 1 ticket because of the low odds and the maximum potential return not really changing. But the question is, at what value is that 1 ticket a smart investment?

 

[Arqane]

The other stuff you are talking about is all calculable, but I frankly do not understand your point.

I think the end of my previous post explained it better than the initial post. Because the possibly returns are substantially above investments like the stock market, at some point buying 1 lottery ticket makes sense when the change in average return is very low, and the change in maximum potential return is very high.

But then, how do you really quantify the problem, or can you? There are real-world scenarios that make higher returns quantifiably better (since you lose ~4% of the value of any held money per year, the higher the gain and the faster it comes makes inflation have less of an effect on your current value, which in turn let's you reinvest with greater amounts). So I think the more variables you add, the more quantifiable it becomes, but I don't think you can ever take out the last portion of subjectivity to it... or can you?

 

[Arqane]

Now that I'm thinking about it, the amount that you can possibly invest could be factored into the equation, and might come up with something quantifiable. If you had $1B to invest, then the average returns are more important than a single $1B prize at a very low chance. In that case, the maximum potential return % is less relevant because the total amount acquired through other investments nearly matches the top prize, and is less risky. On the other hand, if you had $10k to invest then you'd still only max out around $1.2 million investing at 110% in the stock market for 50 years, so the prize money obviously makes more of a difference.

So it sounds like you could get a pretty good graph with at least the following variables:
-Initial investment amount
-Average return on each investment
-Prize value in lottery/Maximum potential value of stock (very hard to figure out the max stock value, but you can use economic factors and historical context for this)

I feel like there should be more main variables, so point out any that you think I'm missing. There's lots of minor variables that could be added or included in those calculations (inflation, additional costs/taxes, and possibly one for risk allowance). But there might be critical points from just those 3 variables.

 

[hutchphd]

Whatever makes you happy. It makes no financial sense to me. Yes there is a small possibility you will get rich quick. On average it is foolish IMHO

 

[Arqane]

Whatever makes you happy. It makes no financial sense to me. Yes there is a small possibility you will get rich quick. On average it is foolish IMHO

Heh, well this was really just an example of something that seems like it has some subjective and some objective parts. I just used the lottery as an example of that. Ultimately it was more of a question of whether you can fully turn something subjective into an objective formula. I just got caught up in really finding if it was solvable.

 

[jack action]

Venture capitalists are another good example where they expect many of their investments to fail (some will recoup the initial investments, others will not), but through diversity and a few winners that return very large percentage gains they can thrive.

Venture capitalists must have returns greater than 100% otherwise they wouldn't do the investments.

A 90% return means just that: If you were to buy a lot of tickets (millions of them), you would get 90% of your money. If you buy fewer tickets, your return will drop accordingly.

Venture capitalists do the same calculations: If you do a lot of investments (dozens, maybe hundreds), you will get X% of your money. For sure the value of X is greater than 100. Note also that their investments are not randomly selected, there are some feasibility and market studies done before investing.

 

[Arqane]

Venture capitalists must have returns greater than 100% otherwise they wouldn't do the investments.

A 90% return means just that: If you were to buy a lot of tickets (millions of them), you would get 90% of your money. If you buy fewer tickets, your return will drop accordingly.

Venture capitalists do the same calculations: If you do a lot of investments (dozens, maybe hundreds), you will get X% of your money. For sure the value of X is greater than 100. Note also that their investments are not randomly selected, there are some feasibility and market studies done before investing.

Yes, but again, you're misinterpreting the value of X, and what I'm talking about here because you're assuming an all-or-nothing investment strategy. That's why I pointed out in my last reply that it would be ridiculous to spend 100% of your investment in lottery tickets because the expected return would be <100%.

VCs literally put their expectation of failures into the math. Here's a quote from an article about that: "Sergey Toporov, Principal at LETA Capital venture firm explains “Early stages VCs have to expect at least 10x return to get an average return on their investment in 3-5 years, simply because 8 from 10 projects will fail”." The total X over that set will be over 100%, but they are fully expecting 8 out of them to be under 100%.

So all that the part of math you're talking about proves is that you must diversify in order for the math to work out. That is correct. A sum where all returns are less than 100% will be less than 100%. It's very easy to say that in that diversity, you should never let your X value go under 100%. In fact you shouldn't really let it go under 104% so you don't lose value to inflation. And even further, as I pointed out, you should probably never buy more than 1 lottery ticket in a lottery they didn't mess up like the rollover, because more than 1 doesn't increase your maximum possible return, it just continues to lower the average return (this may change in two separate lotteries because it refreshes the difference in maximum possible return between the lottery and other investments).

 

[fresh_42]

Venture capitalists must have returns greater than 100% otherwise they wouldn't do the investments.

A 90% return means just that: If you were to buy a lot of tickets (millions of them), you would get 90% of your money. If you buy fewer tickets, your return will drop accordingly.

Venture capitalists do the same calculations: If you do a lot of investments (dozens, maybe hundreds), you will get X% of your money. For sure the value of X is greater than 100. Note also that their investments are not randomly selected, there are some feasibility and market studies done before investing.

And the ROI must be high enough affording to fail in 9/10 cases!

It is still more promising to go to the casino, put whatever you want to risk on red, and see how it goes. The odds are significantly higher than in any lottery.

 

[hutchphd]

The ability to manipulate data having noise or uncertainty is remarkably useful and powerful. The subjective part for investments is the accurate assessment of these uncertainties and an honest appraisal of your personal/financial ability to sustain integrity during the fluctuations. The actual predictive mechanics work very well and in the limit of very large numbers approach the stature of objective fact. Casino owners are not gamblers any more than is the Surgeon General.
My most lucrative work was doing R&D on contract. When asked about the acceptable failure rate for some product or process I always upped my price a lot when the customer who answered "zero". It was tell for future disagreement.

 

[Arqane]

The ability to manipulate data having noise or uncertainty is remarkably useful and powerful. The subjective part for investments is the accurate assessment of these uncertainties and an honest appraisal of your personal/financial ability to sustain integrity during the fluctuations. The actual predictive mechanics work very well and in the limit of very large numbers approach the stature of objective fact. Casino owners are not gamblers any more than is the Surgeon General.
My most lucrative work was doing R&D on contract. When asked about the acceptable failure rate for some product or process I always upped my price a lot when the customer who answered "zero". It was tell for future disagreement.

Yeah, I guess the best you can get it to is an acceptable loss situation. It's easy enough to make a risk factor, so you can at least compare those. On two different choices with equal chance, you may as well take the higher payoff. But in a situation like this, I guess you should compare the average expected gains, the maximum possible gains, and the percentage of investment in each to find a good point along that line. You can do some calculus on it, find any critical points you need to watch, and go from there. There would be some interesting graphs, especially if you included the starting investment amount. In some cases (lower amounts) a lottery win would be out of the possible scope of investment gains through the stock market. In cases with higher starting amounts the risk/reward of a lottery ticket would be ridiculously bad.

 

[Office_Shredder]

Look. Any investment portfolio has a distribution of future outcomes. What distribution is preferable to you is in some sense a personal choice.

I think some things that are generally true are: Having more money on average is better than less.
And also,
Taking risks is bad. If you can make a 0 expected value bet, you would prefer to not do it.

Those two things imply you need to have a positive rate of return that is sufficiently high to overcome any risk of investing out of just money. Betting on a lottery ticket is risk - the risk is small if you only buy one ticket, but you do have a very high probability of losing all that money. So under these principles, you would never want to buy a lottery ticket.

Now, you can decide these two things don't fit your personal preference. Some people like risk, and would pay to gamble (in fact... a lot of people). But there's no mathematical basis for why that's a good idea, it's purely one based on personal preferences for how they want to live their life.

 

[jack action]

That's why I pointed out in my last reply that it would be ridiculous to spend 100% of your investment in lottery tickets because the expected return would be <100%.

If putting 100% of your investments in the lottery ends up in a guaranteed loss, how would putting, say, 1% of your investments get a higher return on that 1% of your investments?

VC may invest in flying cars, robots, social media, AI medicine, or whatever possible future next trend. One of them has to get through because people will necessarily buy something in the future. But they won't invest in buggy whips or typewriters because, who knows, they might just come back.

 

[jack action]

Let's even imagine the following scenario:

You try to convince some VC to invest in your revolutionary invention. You tell them that you need 10 million $ to develop it and that it will probably make 50 million $. Now, to close the deal, tell them that you will take 5 $ out of the 10 million $ to buy a lottery ticket for which the jackpot is 100 million $ because, you know, you never know!

Expect to lose all credibility in the eyes of the VC.

 

[Arqane]

But there's no mathematical basis for why that's a good idea, it's purely one based on personal preferences for how they want to live their life.

I'm kind of curious, though, with the out of scope possibility. How would you explain that mathematically? By that I mean at a certain point with a low enough investment number there are no other valid methods to make $1B after a 100-year limit (I think that's substantially high enough to say the person won't live long enough to care). If you're trying to find the way that the person can make the greatest possible amount of money in their life through investments, then below a certain number the stock investments would never beat a single $1B lottery win (happens to be ~$72,500 at 10%). Though there's obvious variability in that, there is a maximum and I was just using that as an example. Above that number, and it's objectively going to be better on average (not for all possibilities) at all points to put the money in stocks. But below that, mathematically it's technically better to play the lottery since it's the only way to reach that number in the given time-frame. Still doesn't make it a good idea, but the only possible idea.

 

[Arqane]

If putting 100% of your investments in the lottery ends up in a guaranteed loss, how would putting, say, 1% of your investments get a higher return on that 1% of your investments?

Lower average return, much higher possible return.

 

[Office_Shredder]

But below that, mathematically it's technically better to play the lottery since it's the only way to reach that number in the given time-frame. Still doesn't make it a good idea, but the only possible idea.

The only possible idea, is your only goal is to get a billion dollars. Why is that your goal? If your goal is to get 10 trillion dollars, the lottery ticket is worthless. If your goal is to retire by the age of 55, buying lottery tickets probably negatively affects that. If your only goal is to get exactly a billion dollars or bust, then sure, buy a lottery ticket, but that's a dumb goal in my opinion.

 

[fresh_42]

One can attempt to investigate personal risk curves. Most of us would probably have no problem betting one dollar on pair at a roulette table but hesitate to bet a hundred dollars on a fifty-fifty chance. So somewhere is the point where it changes.

Economists consider variances as a measure of risk and the expectation value as yield. One can try to get to a personal risk curve with experiments like the one I just described. It is probably reasonable to consider exclusively normal distributed random variables as long as there is no further information. You can finally compare a certain risky decision with your curve once you know your personal habit on decisions under risk.

 

[Arqane]

The only possible idea, is your only goal is to get a billion dollars. Why is that your goal? If your goal is to get 10 trillion dollars, the lottery ticket is worthless. If your goal is to retire by the age of 55, buying lottery tickets probably negatively affects that. If your only goal is to get exactly a billion dollars or bust, then sure, buy a lottery ticket, but that's a dumb goal in my opinion.

It was an arbitrary number. Just saying that some numbers will be out of the scope of certain investment types. Anything you can possibly reach with a better average is going to be the better average choice (obviously). But how do you deal with comparing two separate slopes that don't intersect? If you were to make a risk calculation, it would probably end up with some asymptotes where the two couldn't be compared. But the funny thing is, if you wanted to maximize your money then after winning the lottery once you'd want to go back to the stock market and stick with it, because it's not only better on average, but usually exponential. Lottery prizes are generally more linear.

 

[pbuk]

I've not studied this in detail, but when there is a head-to-head sports contest (any soccer, tennis or cricket match or boxing match etc.) there are mutiple online betting sites that gives the odds for both sides. No individual site will give odds that add up to more than 1, but what if two different sites favour the different participants? Say online betting site A gives odds of 5/4 on team X and betting site B gives odds of 5/4 on team Y. Then, you could bet the same amount on each team on the different sites and guarantee positive return.

This is called arbitrage.

 

[pbuk]

A big lottery win absolutely dwarfs the amount of the greatest stock market return, so it ends up being high risk, but extremely high reward.

This is called a "maximax" strategy.

This branch of mathematics is called "decision theory".

 

[Arqane]

This is called a "maximax" strategy.

This branch of mathematics is called "decision theory".

So yes, I like the example I skimmed through, as well. It's all based on well-researched probabilities, and will never be completely objective, but does help decide between "smarter" and "less smart" outcomes with help from mathematically sound formulas.

 

[mcastillo356]

Nobody mentions Markowitz

https://www.ubs.com/microsites/nobel-perspectives/en/laureates/harry-markowitz.html

Love

 

[BWV]

TL;DR Summary: Though buying a lottery ticket will almost always be a loss on average, the possible percentage return dwarfs any other type of investment. So when is the risk really worth it?

So let's say we have a lottery out there which costs $2 to buy a ticket, and currently has a $1B prize. Let's say the average return is $1.80 on that $2 (90%). Objectively, for every dollar spent, investing long-term in the stock market is mathematically better. But that doesn't take into account the maximum possible returns over a certain amount of time. A big lottery win absolutely dwarfs the amount of the greatest stock market return, so it ends up being high risk, but extremely high reward. The risk someone takes with an investment usually depends on a few circumstances, and often includes subjectivity. If you had $10,000 to invest, put only $9,998 into the stock market and bought a lottery ticket with the extra $2, the math would say that you're likely to lose $0.40 from my lottery example compared to investing the entire $10,000. With the possibility of winning $1B, the 5,500% return for Bitcoin one year looks tiny. So I guess you could do a probability graph, but what variables would be the best to include? And although I think it would still stay subjective even with the probabilities, could you come up with a reasonable rule of thumb to tell people where the probabilities cross (like saying that you should buy one lottery ticket for about every $10,000 invested in the market)? I find it an interesting exercise between the usual objectivity of math and dealing with subjective feelings of risk.

You are asking under what circumstances adding an asset with a negative expected return will add value? you need more info - expected variance and covariance of the assets. The framework for this is the Sharpe Ratio - the return over the available risk-free rate divided by standard deviation.

So to keep things simple set r, the risk free rate to 0%, then say for stocks the standard deviation is 20% (not far from reality) so the Sharpe ratio is 0.5

the formula for portfolio variance is

• The formula for portfolio variance in a two-asset portfolio is as follows:
  • Portfolio variance = w₁²σ₁² + w₂²σ₂² + 2w₁w₂Cov1,2
  • Cov1,2 = ρσ₁σ₂ where ρ is the correlation of the two assets
• 
  Where:
  • w1 = the portfolio weight of the first asset
  • w2 = the portfolio weight of the second asset
  • σ1= the standard deviation of the first asset
  • σ2 = the standard deviation of the second asset
  • Cov1,2 = the covariance of the two assets, which can thus be expressed as p(1,2)σ1σ2, where p(1,2) is the correlation coefficient between the two assets

https://www.investopedia.com/terms/p/portfolio-variance.asp
The expected return is just the weighted average of the expected return of each asset, so you can play around with this. The second asset only adds value if the portfolio Sharpe ratio is better than 0.5 (the SR for 100% stocks). An asset with a negative expected return is only going to add value if the correlation is close to -1, i.e. it provides some sort of cheap hedge. Given that the correlation of lottery payouts to stock returns can be presumed to be zero, it will not add any value to a stock portfolio

 

[Arqane]

@BWV I'll have to translate that a bit to make sure I'm reading it correctly, but I'm not sure it would fully apply here. Here's a quote from the efficient frontier portion of that, which this formula seems to be trying to show:

"The efficient frontier and modern portfolio theory have many assumptions that may not properly represent reality. For example, one of the assumptions is that asset returns follow a normal distribution.In reality, securities may experience returns (also known as tail risk) that are more than three standard deviations away from the mean. Consequently, asset returns are said to follow a leptokurtic distribution or heavy-tailed distribution."

Of course, they're using the tail risk more as the possibility of losing more than 3 standard deviations (hence the risk part, I guess), but lottery curves tend to have insane deviation. And that was part of the point of a few of my posts. It's harder to fit a lottery win into certain formulas then you'd think, and very easy to dismiss. The shortcomings (high risk) would easily be balanced out if stocks or other investments could attain similar numbers. But unless you already have a significant amount to invest, along with substantial time, big lottery wins are out of the scope of being able to compare them directly to other investments. There has to be something in there to compensate to make a fair comparison, I think. What you wrote is pretty similar to a standard possibilities schedule in economics, it looks like. Just in this case, you'll have a pretty normal curve with one attainable point way, way out on the side.

 

[BWV]

@BWV I'll have to translate that a bit to make sure I'm reading it correctly, but I'm not sure it would fully apply here. Here's a quote from the efficient frontier portion of that, which this formula seems to be trying to show:

"The efficient frontier and modern portfolio theory have many assumptions that may not properly represent reality. For example, one of the assumptions is that asset returns follow a normal distribution.In reality, securities may experience returns (also known as tail risk) that are more than three standard deviations away from the mean. Consequently, asset returns are said to follow a leptokurtic distribution or heavy-tailed distribution."

Of course, they're using the tail risk more as the possibility of losing more than 3 standard deviations (hence the risk part, I guess), but lottery curves tend to have insane deviation. And that was part of the point of a few of my posts. It's harder to fit a lottery win into certain formulas then you'd think, and very easy to dismiss. The shortcomings (high risk) would easily be balanced out if stocks or other investments could attain similar numbers. But unless you already have a significant amount to invest, along with substantial time, big lottery wins are out of the scope of being able to compare them directly to other investments. There has to be something in there to compensate to make a fair comparison, I think. What you wrote is pretty similar to a standard possibilities schedule in economics, it looks like. Just in this case, you'll have a pretty normal curve with one attainable point way, way out on the side.

Trying to keep things simple - the same principles apply if you are dealing with lottery type binary payouts, adding fat tails just abscures the key fact that allocating to a negative expected return with zero correlation to stocks adds no value. Allocating any amount reduces the return expectation. If you disagree, then try quantifying your argument

 

[DrClaude]

I've not studied this in detail, but when there is a head-to-head sports contest (any soccer, tennis or cricket match or boxing match etc.) there are mutiple online betting sites that gives the odds for both sides. No individual site will give odds that add up to more than 1, but what if two different sites favour the different participants? Say online betting site A gives odds of 5/4 on team X and betting site B gives odds of 5/4 on team Y. Then, you could bet the same amount on each team on the different sites and guarantee positive return.

I have occasionally checked this out and it does seem that the different sites align with each other in this respect. I.e. the best odds on team X across the Internet plus the best odds on team Y across the internet always adds up to less than 1.

James Holzhauer, Jeopardy! champion and professional gambler, made some good money betting on baseball while exploiting inefficiencies in the odds of gambling houses. There is an informative interview in this episode of the Effectively Wild podcast: https://blogs.fangraphs.com/effectively-wild-episode-1371-what-is-sabermetrics/

 

[hutchphd]

I took a course called "Decision and Choice"as an undergrad. It covered mostly Bayesian Decision Theory. I recommend the formalism. The fact that it was in the psychology department indicates both the answer to the OP initial question and the impetus for me to take it to fulfill distribution requirements for a shiny BA degree. There are objective factors and methods available but the actual process is always subjective in the OP context.

 

[DrJohn]

I do wonder if any of you have ever heard of the gamblers falicy? Google it.

If you don't want to google it, come and play roulette with me as the banker, please! I was talking to a friend in a fun club casino night (profits to the club) and he called me over just before I put my £10 on 18 - no idea why it was 18. As I walked to the table he pulled me back to chat about something, the coupier said rien ne va plus so I just watched. You can guess what happened next!

Anyway.
Suppose in a 50 ball lottery, the winning numbers are 1 ,2, 3, 4, 5, 6 this week.
Or 19, 18, 3, 8, 26 and 7 with these ones winning last week AND they came out in that order?

Would you bet on 1, 2, 3, 4, 5, 6 next week? Or on 19, 18, 3, 8, 26, 7 which won last week.
Are you more or less likely to win if you do the same numbers next week?
I'll give you a clue - dice or lottery results are totally random. And the balls and dice can't remember what happened last week. And the balls don't know if these numbers mean anything special to you. They don't know if they are a pattern in your life. For me, they are not a pattern, but for some people they might be (children and parents birthdays, perhaps).

If you bet on a six and throw a dice and it wins five times in a row as a six, should you bet on six again? I did this once but the school bully had no intention of paying after my first or fifth win in a row so I gave up and just kept going as he said double or quits every time I won. Eventually I lost on the sixth go. He was an evil sod! And a lot bigger than me.

I'll give you a clue, as I said above, the dice don't remember what happened last week. They are not sentient dice or lottery balls.

PS I was the schools illegal bookmaker a long time ago. I won, they lost. Regularly.

PPS UK lotteries don't have a fixed prize, they give a percentage of the take to the winners in each section of the lottery. So you will always loose in the long run.

The probability of winning with six balls in a 50 ball lottery is n!/r!(n-r!)
n = equals number of possible ball values, r = number of balls chosen. ! means factorial
You calculate it - answer is 1 : 45,057,474, so in the words of Dirty Harry " Do you feel lucky, PUNK?"

 

[russ_watters]

Trying to keep things simple - the same principles apply if you are dealing with lottery type binary payouts, adding fat tails just abscures the key fact that allocating to a negative expected return with zero correlation to stocks adds no value. Allocating any amount reduces the return expectation. If you disagree, then try quantifying your argument

I agree with that of course, but my question is, regarding the progressive* mega-jackpots, is there a point where the expected return becomes positive and therefore there is a viable strategy to profit form it?

The odds of winning the mega-millions are 1:302 million and the cost of a ticket is $2. So doesn't that mean that any jackpot over $239 million (tax adjusted, annuity option) is positive expected return?

I'm sure someone's thought this through and I have two potential ways that could fail:
1. Foiled by other players reducing the odds by also winning and splitting the jackpot.
2. Foiled by technical issues that make buying a large fraction of the tickets cumbersome.

*The point of these is that the value grows over time if nobody wins, so most people sit out until the jackpot grows big enough to be interesting, which improves the return. This could also apply to a progressive jackpot slot machine. A similar (reverse) concept would be scratch-and-win games where you can track the payout and buy tickets late if the high payout jackpots haven't been won yet.

 

[BWV]

I agree with that of course, but my question is, regarding the progressive* mega-jackpots, is there a point where the expected return becomes positive and therefore there is a viable strategy to profit form it?

The odds of winning the mega-millions are 1:302 million and the cost of a ticket is $2. So doesn't that mean that any jackpot over $239 million (tax adjusted, annuity option) is positive expected return?

I'm sure someone's thought this through and I have two potential ways that could fail:
1. Foiled by other players reducing the odds by also winning and splitting the jackpot.
2. Foiled by technical issues that make buying a large fraction of the tickets cumbersome.

*The point of these is that the value grows over time if nobody wins, so most people sit out until the jackpot grows big enough to be interesting, which improves the return. This could also apply to a progressive jackpot slot machine. A similar (reverse) concept would be scratch-and-win games where you can track the payout and buy tickets late if the high payout jackpots haven't been won yet.

The TX lottery, to my memory, had something like 1:20!million odds when it started back in the 80s, and the payout kept rolling over so it sometimes resulted in a payout over $20M - so a positive expected value, but the odds are still way low - there was an investor group that tried to buy one of every number in a VA lottery in the 90s as the expected value became significant positive, they weren’t successful getting one of each number, but did manage to win

 

[pbuk]

I agree with that of course, but my question is, regarding the progressive* mega-jackpots, is there a point where the expected return becomes positive and therefore there is a viable strategy to profit form it?

Usually, yes, although some now incorporate ways to avoid that happening.

I'm sure someone's thought this through

Yes, see for instance Stefan Mandel, Jerry and Marge Selbee etc.