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

[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.