## Solve If You Are A Genius — Of Ambiguity

The sun also ariseth, and the sun goeth down, and hasteth to his place where he arose. The wind goeth toward the south, and turneth about unto the north; it whirleth about continually, and the wind returneth again according to his circuits. All the rivers run into the sea; yet the sea is not full; unto the place from whence the rivers come, thither they return again. All things are full of labour; man cannot utter it: the eye is not satisfied with seeing, nor the ear filled with hearing.—Ecclesiastes 1:5-9^{9}The thing that hath been, it is that which shall be; and that which is done is that which shall be done: and there is no new thing under the sun.

Among other things, Ecclesiastes deals with ambiguity, assumptions, and the ways we react emotionally when our assumptions turn out to be inaccurate. Anyone who does the social media/blogosphere thing has seen at least one “Solve If You Are a Genius” meme. They’re great discussion starters, really good for getting folks to take a position on something that shouldn’t be controversial. I found the one I’m about to discuss on LinkedIn. It generated 51 answers, many of them with reasoned explanations, and some folks prompting others to justify their responses.

As far as I can tell, these memes are good at creating discussion for three reasons:

- They appear mathematical. It’s close enough to get everyone thinking mathematically, or at least arithmetically. The notation isn’t quite right, but it’s enough to get our brains to infer that something mathematical is going on.
- There’s just enough information to make the numbers look like they have a linear pattern, and it’s good enough for just about everyone to be able to discern one. Actually, it activates the human instinct to seek patterns, even in random noise. Bud Light’s football season commercials satirize the same tendency.
- Presentation—the words at the beginning take advantage of the math classes of our early school years, when most of us learned that math problems have only one solution.

The story on this meme, however, is that there are many reasonable answers—an infinite number, in fact. But in order to choose an answer, our brain has to make at least one assumption. Some assumptions may be more reasonable or easily explained than others. I’ll run two cases right quick and state my assumptions to show how they impact response to the meme.

**Case One**

**Assumptions:** ***Values on the left side produce the values on the right side via arithmetic operations. ***Not all strategic statements, x…. –> y, are included on the list.

**Corollary:** Completing the list is imperative to answering the question

**Strategy: ** Rewrite the list with gaps included. Number the statements. Relate the right and left sides arithmetic operations. Use the algorithm to fill in the gaps.

At this point, it outta be easy to fill in all the gaps, right? Sure: we can just accept our prior assumptions, input the results, and walk away confident that we got the right answer. Except maybe not: even if the presence of row 3 is reasonable, can the same be said about row 1? And if row 1 doesn’t exist, how do we deal with the non-existent value of 2 on the left side of the equation? Probably the best thing is to treat it as a 0—but maybe 1 is a better choice. On the other hand, maybe we just say it’s undefined. Now there are four reasonable possibilities:

Using the set of assumptions selected in Case One, the solution set is {0, 3, 6, DNE}. Here’s another way to look at it:

** Case Two**

**Assumptions: ** Values on the left side and right side appear correlated, but they are not

**Strategy: ** Fill in the value in question following the pattern on the right side only

By working through these two cases five possible answers have been found—all based on arithmetic patterns and reasonable assumptions. There are probably more answers to be found by following these same lines of reasoning, but for now I’d like to abandon it. I’d like to engage instead in some polynomial interpolation and extrapolate from there. In short, I will cheat the data. (Yes, I really said that!) In this case we just use it to cram together a non-singular matrix using an equation of the form:

p_{n}(x) = α_{n}x^{n} + α_{n-1}x^{n-1} … + α_{0}x^{0 }

For this exercise y = p(x) = right hand values, X = left hand values taken to the appropriate exponent, α-values will be represented by letters, and n starts at the top and ends at the bottom. Therefore the equation will take the form:

p(x) = ax^{3} + bx^{2} + cx + d and p(8) = 56, p(7) = 42, p(6) =30, p(5) = 20 . These assignments yield the following matrix:

This gives a value of p(3) = 6. The same answer pops up when p(x) = ax^{4} + bx^{3} + cx^{2} + dx. Using a standard, perfectly sensible linear interpolation, we find the pattern to be p(x) = x^{2} – x. But if p instead takes the form:

p(x) = ap^{n} + bp^{n-1} + cp^{n-2} + dp^{n-3} where n > 4

values of p(3) other than 6 begin to show up. For example, when n = 5, p(3) = 6.214285714285708…, and when n = 6, p(3) = 1.993112244897960…

Based on this we can safely say that there are many, many answers to the “Solve If You Are a Genius” meme. The set is almost as big as the set of polynomial functions, which just happens to be infinity. But this answer brings in two new questions:

1. How are the real answers bound for different polynomial forms?

2. Are answers limited to real numbers?

I’m not sure how to prove the first one analytically, but I will attempt to address it numerically in a future post. With the second question, my instinct is that under some conditions they are. When I solved the polynomial equation, I assumed the constants a, b, c, and d had only real parts. But it’s not difficult to imagine they take the complex form of q = y + zi. My sense is that under an even set of knowns consisting only of real numbers it is possible to have a set of z-constants that sum to zero for each known in the set. That is to say, the imaginary component is trapped within the null space. When the unknown gives the set an odd number of equations, then I think it’s possible for the imaginary component to show up in the answer. However, if it is an odd set of known consisting only of real numbers, the unknown must be a real number also. Confused yet? Me, too!

Anyway, that’s enough for this post. What do you think of the meme?

**Update (12/15/2013)**

I spent about a week chewing over my last paragraph and another week considering the best format to answer the questions I posed at the end. Here’s a couple comments addressing them….

- First Question–How are real answers bound for different polynomial forms: It’s not necessary to address this numerically because it can be shown analytically that they can be any real number. With polynomial interpolation it was possible to produce four independent equations with which to solve for the four independent variables–a, b, c, and d. I chose to unbind the highest order exponent as a variable j. The rest of the exponents were set to j-1, j-2, and j-3. In doing this, I actually was creating a fifth independent equation. In doing this, the system becomes specifiable to five values in the right hand column. By having five independent equations, any desired value can be specified for 3 = ?. There’s also no rule that says the exponent-values must be bound to the pattern j, j-1, j-2,….j-n. If exponents can be selected at will, then a total of 8 independent equations can be fabricated, allowing one to solve a system of eight question marks from the four known relations.
- Second Question–Is the range of reasonable answers bound only to real numbers:

The answer is no, answers can be real or complex. The imaginary components in the four known relations can be designated as b1 = b2 = b3 = b4 = 0. But if there are five independent equations, b5 is not constrained to zero. The same principle above applies, and it can be any positive or negative number.

The number of ways to consider the Solve If You Are Genius meme are as boundless as imagination, but there’s one more method I want to address here. Using the same polynomial interpolation p(x) can take the form:

p(x) = ap^{n} + bp^{n-1} + cp^{n-2} + dp^{n-3 }+…+ e

In this case the unknown can take the value of any number. What’s really happening is the unknown is being constrained to a one-dimensional line in R^{n}. Personally, I’d rather have a plane. But a line can be just as good. Cheers!

## About Those Exploding SS Disability Benefits

There’s lots of fretting and frowning over the number of folks getting disability payments in the US these days. It seems like everybody’s in on it–established magazines, conservative pundits, educated economists have good theories to explain it, NPR (with a pretty engaging piece), and even WonkBlog has gotten in an explanation (demographics + tough job market, in case you were wondering).

My piece on long-term unemployment got me to wondering–how many long term unemployed, marginally attached, and discouraged workers are leaving the workforce permanently and winding up on disability insurance? In my own work life I’ve known people who really could have qualified for a disability award. But in a solid labor market they were able to find employers who were willing/able to accommodate some special needs. When times are tougher, accommodation above ADA requirements often goes away. So a working person who is technically disabled is pushed out of the workforce by market changes. If I’ve witnessed it myself it can’t be too uncommon. This led me to ask how much it’s going on, and if it’s higher now than in the past.

Fortunately, the Social Security Administration makes figures on disability awards easily available. They also publish the month-to-month number of disability beneficiaries back to 1985, which I copied into Excel and graphed:

I was able to discern four trends in monthly awards in the data:

- An average rate of about 4800 awards per month in the late 1980’s
- That rate to almost 20,000 per month in the early 1990’s
- The rate declined slightly in the mid- and late- nineties and then slightly rose in the aughts
- The average monthly awards fell by 38% to about 12000 awards per month in April of 2012 to present

I attribute the increase in awards in 1990 to a cultural change–Americans began to be more aware and more accepting of individuals with disabilities. Disability accommodation came to be seen as a Civil Rights issue, culminating in the first President Bush signing the ADA. It makes sense that Americans would be more willing to apply for disability benefits, and that the government would be more willing to award them.

Bottom line, folks are bringing up the “explosion” in disability benefits 23 years too late. The rate increase was in 1990. And for the last 15 months, the rate of new awards being added is down, and down by a lot. Sometimes it feels good to go against the conventional wisdom, and this is one of those times.

A piece of Jessica’s art:

## Is Long-Term Unemployment THE Problem

Since the late 1970’s the number of US workers unemployed for 27 weeks or more has generally lingered between one and two million, even during recessions. The ’08-’09 recession and recovery period were different: long-term unemployment exploded to a peak of 6.7 million in April of 2010. It lingered above six million for 17 months before falling in October 2011. Since then the level has declined steadily, but very slowly.

There’s a tendency with noisy data to look for linear trends, but I don’t think it’s the right way to analyze long term unemployment. To begin, looking for a job is a job in itself, and when a person is locked out from formal job–>paycheck ways of earning a living, he or she has to find other ways. That means getting public assistance, asking help from friends, family, and community organizations, doing side work or day labor for cash payments, etc. These all take mucho time, and generally have a very low return. For this reason, I assert that being long-term unemployed can be thought of as an occupation in itself. The corollary is that long-term unemployed folks returning to work is a form of occupational mobility.

In addition, if you’re employed there’s a pretty good chance you know very few (if any) long-term unemployed folks…..Unless you’re very active in your church, community, and social networks like LinkedIn, of course. Even then, you’re probably locked out of regular interaction by circumstance. This isn’t a value judgment–it’s just the nature of human interactions. Long-term unemployed people tend to network with others in the same circumstance and share information. Connections tend to play a gigantic role in finding formal work in a slack labor market, but reducing the pool of the long-term unemployed leads to disconnection for those remaining in it. While it would be nice to think that the long-term unemployed would be able to pull their unemployed friends along, this just isn’t the case. Instead, the rate of job finding in the remaining pool slows as information sharing is reduced.

When examining the long-term unemployment trend post-recession, there are two distinct pieces–the relative steady state beginning in April 2010 and the decline beginning in October 2011. Because of the dynamics I described above I decided the best way to analyze + predict from the measured data was with a deterministic trend; as a first-order ODE specifically. I took the measured data beginning in September 2011 and worked out a long-term predictive trend based on it:

The data smooths out to a disturbing trend where an average of 1.715% of the pool of long-term unemployed individuals find work or exit the labor force in any given month. If we assume the recovery continues as it has, the number of people long-term unemployed will decline to about 2.07 million when the current US President leaves office–a number higher the previous ceiling of 2 million and way above what’s been normal during expansions. In my opinion, this shifts the long-term unemployment problem from being just an economic problem to being a political problem. It will be very troublesome for the next Democratic Presidential nominee to have to defend this kind of economic record, and will be an issue the Republican nominee can seize upon.

Of course, “the recovery will continue as it has” is a big assumption. I’ve run the trend line out longer than the data I’m using, which is pretty gutsy to do. However, data from prior expansions show this is a reasonable expectation, and the trend seems to have reached a steady, if undesirable, equilibrium. Anything can happen in the next couple of years to grow the job market, but if history is an indicator, just growing the job market won’t be enough. What will really be needed are more entry level jobs in every industry. As I mentioned previously, entry level jobs have been in short supply, and are simply needed in general. Can the long term unemployed–with all the stigma that has been unfairly attached to them–be expected to compete successfully with the already working? I don’t think they can. In my opinion this is an economic problem that requires a political solution–one that could come in many forms–from tax credits, to paid placement, to direct hiring. If the current administration doesn’t successfully address it, the next one will have to.

The finest GagVid I’ve seen this week….Don’t worry, it’s clean:

## Job Disatisfaction, Quits:Openings (Q:O)

American workers are notoriously dissatisfied with their jobs when compared to their international peers. Part of the problem is compensation–if a fifth of people paid less than $50K a year “hate” their jobs, and median compensation is $34K (*1*), then high dissatisfaction is unavoidable. As Ross and Saturay showed, Dissatisfaction —> Disengagement —> Lower Productivity via net categorical behaviors. Translation: Happy, engaged employees come up with surprising ways to produce, while the dissatisfied ones come up with endless ways to goof off and/or sabotage the organization. This leads to gajillions of B2B services (20 in just one Google search), surveys, and good idea self-help articles on increasing employee engagement. (Personal opinion: The work time is a sunk cost, so getting the most out of it has to be a priority–being engaged is better than the alternative. Of course, that’s sometimes easier said than done.)

From a high altitude perspective America’s work culture has hinged on many dynamics, but I’ve only got time for two–

1. A person at any nearly any stage of life who wants to be self-sufficient can find work within a reasonable period.

2. If at any point you decide you hate your job/life/other you can just move on to other opportunities.

These reach back to the founding of the British and Spanish colonies–they were mainly comprised of people who needed work. It’s woven into our many waves of immigration, the Homestead acts, the post-WWII inclusive military, Pell Grants, student loans, and the community college system–even into abolition of slavery, and the ongoing extension of rights + opportunities to a growing citizenry. Is it possible that high rates of job dissatisfaction have been made viable in the US because markets and managers have been able to convince the truly ticked-off to move on? To greener pastures hopefully, but any pasture will do.

Here’s the problem though–people aren’t moving on these last few years, satisfied or not. Workers, even the dissatisfied workers, seem to have sunk their teeth into any job they’ve got and they’re not letting go. It seems to me that if workers aren’t getting paid particularly well, and many are burnt out on their work, they ought to be willing to quit if there are openings. So I mmmbopped over to JOLTS at the BLS and pulled the openings and quits. In an amazing mathematical feat known as division, I calculated and graphed the ratio of Quits to Job Openings (Q:O ratio). First the monthly data:

The BLS’s treasure chest only goes back to 12/2000, but there’s definitely a trend of declining Q:O. To the extent Q:O is a proxy for occupational mobility it matches with the academic research. Here’s what the annual averages show:

Two notes on the annual average trends: **First**, the ratio increases during recessions. This is because the number of job openings (denominator) declines more than quits (numerator). Example: Job openings in 01/09 were down 38.0% from openings in 01/07. Quits were down by 33.5% in the same period. **Second**, the ratio trended down in the ’02 to ’07 recovery. The downward trend during the ’09 to ’12 recovery accelerated to 1.71 times that. I partly attribute this to worker anxiety about the overall job market. It’s also due to high selectivity in screening and hiring processes. The slight increase in the 2013 data is due to rise in both the number of job openings and the number of quits. In short: an improvement.

Something to keep in mind about job satisfaction–it’s not just about money or attitude. It’s about circumstances and ambition as well. Many workers are bored in their current careers and would like to try other things. For example, I know a few engineers who would like to move into more creative work or teaching. I know people who are in very demanding jobs and would like to have a family, and need to change their careers in order to do it. Life changes, needs change, and people need markets that allow them to adapt.

This brings me to the problem–America is in a very unusual labor market, where there are very few entry level positions in nearly every industry. It’s hurting many workers who are currently employed, whether they are dissatisfied or just having to put up with the dissatisfied. It’s hurting young people very badly, and the long-term unemployed even worse. And yes, it’s hurting employers as well.

But here’s what I’m getting at–this cannot go on forever. It flies in the face American culture, so it will be fixed. It’s started to improve already, and I bet this will accelerate over the next two years. If it doesn’t occur by “natural means”, I’m sure it will come in the form of a political solution. What do you think?

*(1) Assumed total compensation = (net compensation)/0.8*

Krokodil was a Russian satirical magazine that ran from 1922 to 2006. A 1952 cover: