The Monty Hall Problem Is More Than A Game

The Monty Hall Problem was popular on Twitter this week. It’s worth taking a look at it because the Bayesian logic behind the solution is pertinent to reasoning about uncertainty in general.

Let’s start by turning to Wikipedia:

The Monty Hall problem is a brain teaser, in the form of a probability puzzle, loosely based on the American television game show Let’s Make a Deal and named after its original host, Monty Hall. The problem was originally posed (and solved) in a letter by Steve Selvin to the American Statistician in 1975. It became famous as a question from reader Craig F. Whitaker’s letter quoted in Marilyn vos Savant’s “Ask Marilyn” column in Parade magazine in 1990:

Suppose you’re on a game show, and you’re given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what’s behind the doors, opens another door, say No. 3, which has a goat. He then says to you, “Do you want to pick door No. 2?” Is it to your advantage to switch your choice?

I’ll give you space to think about it.

So what happened?

Vos Savant’s response was that the contestant should switch to the other door.

She was right.

Man, if only the “You mad, bro?” was a thing back then. If that answer makes you mad, don’t worry, you are in good company:

Many readers of vos Savant’s column refused to believe switching is beneficial and rejected her explanation. After the problem appeared in Parade, approximately 10,000 readers, including nearly 1,000 with PhDs, wrote to the magazine, most of them calling vos Savant wrong. Even when given explanations, simulations, and formal mathematical proofs, many people still did not accept that switching is the best strategy. Paul Erdős, one of the most prolific mathematicians in history, remained unconvinced until he was shown a computer simulation demonstrating vos Savant’s predicted result.

I’m a dude, and even I got the “mansplain” vibe from the bold-faced section. Wikipedia continues:

The problem is a paradox of the veridical type, because the correct choice (that one should switch doors) is so counterintuitive it can seem absurd, but is nevertheless demonstrably true.

So I chimed in on Twitter with my favorite way to understand why you should switch:

You can find a fuller discussion of the problem and its variants by Professor Jeffrey Rosenthal here.

Rosenthal considers my explanation “shaky” because it fails in some of the variants.

The reason it works in this version is that the host is a “trusted actor”. He is 100% to open an empty door. If he opened a door at random, then your reflex that switching shouldn’t matter would be correct.

Problems like this are Bayesian and can be approached using what Rosenthal calls the “proportionality principle”.

The Proportionality Principle: If various alternatives are equally likely, and then some event is observed, the updated probabilities for the alternatives are proportional to the probabilities that the observed event would have occurred under those alternatives.

That’s a mouthful.

Let me start with an example I made up, then map the proportionality principle’s definition, line by line, to the solution. [don’t crucify me for the made-up shooting percentages]

Paul George and Kawhi Leonard are equally likely to have the ball on the last play of the game down by 2. You discover the Clippers won. Paul George is a 50% 3-pt shooter and Kawhi is a 25% 3-pt shooter.

What’s the probability Paul George took the shot?

So the long way of doing this is to map the paths to victory.

I took the liberty:

What does this tell us?

  1. The Clippers win 3/8 of the time (1/8 + 1/4). This makes sense since 37.5% is the average of their shooting percentages and they each have a 50% probability of getting the ball.
  2. The state of “having won” is 37.5% of the whole set of possibilities. Paul George making the winning shot happens 25% of the time. But since the whole of our restricted “having won” condition is 37.5% we can see that 25% / 37.5% = 2/3

    So Paul George took the shot 2/3 of the time. Going into that last play we expect he wins us the game 1/4 of the time (50% chance of getting the ball x 50% chance of making the shot) but once we “condition” the question on “The Clippers won”, the probability that he took the shot jumps to 2/3!

The proportionality principle allows us to make a shortcut. Follow me step-by- step through the definition laid out above.

  1. If various alternatives are equally likely, [check. Each player is 50% to get the ball]
  2. and then some event is observed, [the event was the Clippers winning]
  3. the updated probabilities for the alternatives are proportional to the probabilities that the observed event would have occurred under those alternatives. [for the Clippers to win one of these players needed to make the shot. Since Paul George is 2x as likely to make the shot (50% shooting % vs 25%), then he is a 2-1 favorite to have taken the shot. 2-1 is the same as 2 out of 3, therefore Paul George was 2/3 to have taken the shot. In case you didn’t notice, the way to convert odds to probabilities is to take a number in the “x to y odds” form and divide it by x+y. So 2-1 odds is the same as 2/(2+1) or 2/3. Just don’t forget which side of that expression is the “favorite” or “underdog”. This is similar how you convert gambling money lines to implied probabilities. All odds are implied probabilities because once you convert them into a fraction they are always less than 1.]

Real-life applications

If you like this kind of probability math, you can look into Bayes Theorem, which is about how we update our “priors”, ie our probability estimates at the get-go, once we get new information (sometimes called “conditioning”, because, well we impose new conditions).

If you don’t like this type of math, perhaps you feel like it’s not relevant. I assure you it is. Just imagine a disease has occurs 1 in 100,000 people but the test for it has a 5% false-positive rate. If you test 100,000 people for it, 5,000 of them will test positive in error yet only 1 person in the population actually has the disease. You’ve got 4,999 doomscrolling WebMD when their odds of actually having the sickness (after the positive test!) are on par with getting struck with lightning at some point in their life.

If this still sounds abstract, then you are my hero for somehow avoiding innumerate covid headlines.

Teach A Math Idea To Internalize It

My 8-year-old Zak is going to be taking the OLSAT soon. It’s a 64-question test that looks an awful lot like an IQ test. The test (or one of its brethren like the CoGat) is administered to all 3rd graders in CA. If you score in the top 2 or 3% you can be eligible for your local ‘gifted and talented’ program. 20% of the questions are considered “very challenging” and that’s where the separation on the high end happens.

I gave Zak a practice test just to familiarize him with it. He’s never taken a test with a time limit before and never filled out Scantron bubbles. Do not underestimate how confusing those sheets are to kids. It took a while for him to register how it worked because he only saw choices A,B,C,D for each of the 64 questions.

Daddy, the answer to question 1 is ‘cat’ not A,B,C, or D

I know, Zak, it’s just that…you know what bud, how about just circle the right answer on the question for now.

Hopefully, some practice breaks the seal so he isn’t scared when he sits for his first test ever. I think a small amount of prep is helpful even though I get the sense that caring about tests is not in style around here. Call me old-fashioned. I’m not bringing out a whip, but having the option to go to the program seems worth putting in a token effort if you think your kid has a shot.

Anyway, he took one test. Poking around a bit, I think his raw score would land him in the 90th percentile. Not good enough but it was his first shot and if he doesn’t improve much, that’s also totally fine too. Plenty of people are content just flipping burgers (I’m kidding, calm down. Also, get your own kid to stuff your insecurities into). One thing did stand out. He got all the math questions (about 1/3 of the test) correct.

Hmm.

It made me think of how I was a decent math student growing up.

I'm Something of a Scientist Myself | Know Your Meme

Not good enough to compete with peers who did math team in HS, but enough to get through Calc BC. Regretfully, I never took another math class after that. I optimized my college courses for A’s not learning. Short-sighted.

I really felt the pain of that decision when I got hired to trade options and was surrounded by a cohort in which 50% of the trainees had an 800 math SAT. (There were 3 people in our office of about 60 that had an SAT verbal > math. I was one of them.) That inferiority exists even to this day. Until Google Translate can decode academic papers, those things are for lining birdcages.

However…

Every now and then, I’ll come across a math topic that seems useful for making estimates about practical things, so I’ll learn it.

And then I’m reminded I have no math gifts because that learning process is uphill in molasses. When I was young I did lots of practice problems (how else are you supposed to become a doctor and please mom) which got me proficient. Today, it’s a similar process. I just power through it.

But there is a difference in how I power through it.

Instead of practice problems, I watch YouTube until I can write the ELI5 version for others. Everyone has heard that if you want to test your knowledge, teach it to others. In that case, it’s a win-win. We all learn.

So that’s what I did this week. I wrote an ELI5 version of a concept called Jensen’s Inequality.

  • Jensen’s Inequality As An Intuition Tool (10 min read)

    You will learn:

    • Why I found Jensen’s Inequality interesting
    • The conditions and statement of the inequality
    • An example that affects us all
    • Spotting Jensen’s in the wild

    If you struggle to understand it after reading it tell me. I am challenging myself to see if I can relay not just the concept but the significance of it with minimal effort on behalf of the reader. If I can get to the point where I’m “putting in the effort so you don’t have to” then I’ll feel like I’m being useful here.

    If you think you got it, test yourself the way I did. Construct an example. (That’s what I did with the “traffic on the way to Sizzler” example.)

  • If you grok Jensen’s Inequality and want to relate it to portfolio construction Corey is your guy. Before I learned of this concept his tweets would have made no sense to me, but now I at least kinda get it.

Jensen’s Inequality As An Intuition Tool

I came across a tool from mathematics called Jensen’s Inequality. I’m going to explain the rule, provide intuitive examples, then end by pointing you to real-world applications.

A warning to math whizzes — I don’t have formal math training so this post is divorced from pedagogical context. Yes, there will be numerical examples. But the real goal is for readers to recognize when the domain they are reasoning about is subject to the surprising predictions of Jensen’s Inequality. For most of us, the value of this tool is how it nudges our intuition to better predictions, not in the direct application of a formula.

Here’s where we’re going:

  1. Why I found Jensen’s Inequality interesting
  2. The conditions and statement of the inequality
  3. An example that affects us all
  4. Spotting Jensen’s in the wild

Why I Found Jensen’s Inequality Interesting

Blindness To Exponents

Exponential phenomena confuse our brains. It has become tiresome to point out that we do not have natural intuition for growth and decay rates. Even finance folk who are apt to appreciate the idea of compounding  seem to not recognize it when the investing skin is pulled off it.

Covid is a timely example. A virus’ R0 (“R naught”) indicates how transmissable it is. Remember “Covid is the flu”. Say the flu has an R0 of 2. So for each person that contracts the flu, they infect 2 more people. Now let’s suppose Covid has an R0 of 3. Here’s how the 2 viruses would spread 1.

R0 is a more complicated function than I’m stylizing here (it should be obvious that behavior, like masks, change it. And if a virus was super effective at replicating itself, well it would find new hosts harder to come by). My point is that even smart people will not hear the “This Is Not A Linear Phenomena” song unless their station is tuned to it. The failure to recognize non-linear domains is serious, because it leads to wildy wrong predictions. And life is prediction. We implicitly predict that the sun will rise tomorrow.

Jensen’s Inequality guides our predictions by forcing us to deliberately consider how the average input maps to the average output. When the function that maps the input to the output is non-linear, Jensen’s Inequality tells us in which direction our predictions will be biased. Stated another way: Jensen’s Inequality informs us when an average occurance is a poor predictor of the average result.

Before we get to any equations, let’s predict the outcome of a simple game.

Dice Payoff

Imagine a game, you stake $1, then roll 2 dice. Whatever the roll returns times your stake amount is how much money you make. That’s the payoff function.

So if you roll a five you receive $5.

Question 1: On average, how much do you expect to get paid?

This is a straightforward expected value problem. You get paid the weighted average of all the outcomes or on average $7.

The average value that you roll will correspond to the average value of the payoff function. If that sounds obvious, that’s the point. So far, so good.

Question 2: If you staked $100, what would you predict the average payoff from playing the game?

A quick way to estimate that would be to ask yourself, “what do we expect to roll on average?” then multiply that by the staked amount. In this case, we roll a 7 on average, and since the staked amount is $100 then on average when we play this game we expect to be paid out $700.

That prediction is correct. We can brute force the expected value of the payoffs.

At this point, things are feeling pretty obvious and redundant, but let me remind you what we did to just answer Question 2. We used a shortcut. We took the expected value of the roll, which was an input, to estimate the expected value of the payoff function or the output. The shortcut worked because the payoff function was linear. We are just scaling the expected input by the staked amount since the function is simply (dice roll x staked amount). This kind of payoff function exists all around us. When you buy a stock, your p/l function is just change in stock price x share quantity. The “staked amount” in our example performs the same scaling role as share quantity.

You can feel the twist coming.

Question 3: Same game but we change the payoff function to (staked amount) x (dice roll)2. What’s the average payoff?

First, what does our shortcut predict? Let’s say we bet $1 again. Since the average value we roll is a 7, then we expect the average payoff to be $72 x $1. So we expect the average payoff of this game to be $49.

As you may have guessed from the unsubtle narrative arc, $49 is the wrong answer. Our shortcut doesn’t work. Brute force method:

It turns out the expected value or average result from the squared game is $54.83, a higher value than what we would predict if we took the average value of the input and simply applied the squared function to it.

It’s intuitive to take the average value of an input, apply a function to it and call that the “expected value of the function”. It turns out that if the function we run the input through is exponential, our estimate will be wrong. So in service of becoming better at making estimates on the fly, we should get better at thinking about what kind of function we are running an input through and if our prediction is likely to be biased higher or lower than the actual expected value of the payoff function.

With that long intro we can now turn to Jensen’s Inequality and its practical applications.

A Look At Jensen’s Inequality

I’ll start with stating the inequality the way I learned it2:

E[f(x)] ≥ f(E[x])

…if f(x) is convex

Let’s try saying this in words several ways, assuming f(x) is convex (a term I will address in a bit):

  • The expected value of a function is greater than or equal to the function applied to the expected value of the input.

  • The average value of a function is greater than or equal to the function applied to the average input.

  • Returning to dice…the [weighted] average of all the squares is greater than the square of the average roll.

In practice this means, you cannot estimate the average value of the function based on the average value of the input IF the function is exponential.

Convex

Let’s address the term “convex”. You know what it is visually.

Mathematically, a convex function:

  • has a first derivative that is greater than 0, meaning that the function has a positive slope everywhere.
  • has a second derivate that is also greater than 0, meaning as X increases the slope itself increases. The steepness of the chart is increasing.

If we go back to the dice example and consider the convex payoff function, we can see the average value of the payoff function of $54.76 is greater than the payoff ($49) at the average roll. In other words, the convex function ensured that:

average value of all payoffs > the payoff of the average roll

Concave

For concave functions, like y = sqrt(x), we have a positive slope, but the slope is decreasing as x increases. The second derivative is negative. Let’s look at a concave case for the dice game by making the payoff function = sqrt(roll).

Notice that the average value of the payoff, if you stake $1, is $2.60. But if you tried to predict the expected payoff by using the shortcut of taking the square root of the average roll you’d predict $2.65 which is the sqrt(7).

Wait a minute. The prediction this time overshot the true expected value of the function?!

That’s correct. If you multiply one side of an inequality by -1 you flip the sign…a convex function can be flipped to concave by flipping the sign as well. So a concave function flips the sign of Jensen’s Inequality, making the overshoot the expected result.

Visualizing the concave payoff:

Let’s practice with a highly stylized example I made up, but relates to something we all intuitively feel.

An Intuitive Example That Affects Us All: Traffic!

We are celebrating a big W, so it’s time to take the kids to Sizzler. We’re going to drive. Sizzler is 10 minutes away + some extra time depending on how many cars are on the road. Let’s keep things very simple and assume the number of cars that can be on the road is 10, 20, 30, 40, or 50 and with equal probability. None of these quantities is enough to slow the flow of traffic to a halt, but the impact of the extra cars is not linear.

We’ll create a function called “time to destination” denominated in seconds and make it a function of “cars on the road”:

f(cars on the road) = x2 + 600 

Let’s play “How long will it take to get to Sizzler?”

Before you discovered this post, you likely would have said 25 minutes. Why? Since we can have 10, 20, 30, 40, or 50 cars all with equal probability, then on average we expect to see 30 cars on the road.

302 + 600 = 1500 seconds or 25 minutes.

But because we know about Jensens’s Inequality we:

  1. recognize the traffic output function is convex
  2. realize that the expected value of the traffic function will be greater than sticking the average number of cars in the road into the traffic function

Enlightened, we instead estimate that on average it will take longer than 25 minutes to pounce on that glorious salad bar with the popcorn shrimp.

How much longer? Brute force tells us 28.3 minutes!

Spotting Jensen’s Inequality In The Wild

Here’s a few common applications that abide Jensen’s Inequality

  • Geometric mean ≤ arithmetic mean

    I’ll need to point you to an actual math person. See his beautiful derivation on YouTube. It’s easy to follow along and quite clever. The key to this inequality is recognizing that the geometric mean, which takes the nth root, is concave just like the sqrt(x) function.

    It’s worth noting that the LN(x) function is also concave so when you are in price space we know that the LN(average price) > the average of the LN(prices). Same idea as the geometric means, concavity flips the sign of Jensens.

  • Call options

    Here’s a generic chart3.

A call option is convex payoff function with respect to the stock price. Its first derivative with respect to the price of a stock is delta which is always positive. In other words, as the stock price goes up, all else equal, the call option always goes up (the slope or delta of a way OTM option is 0 so it’s possible for the call to not change in value, but that’s the lower bound). The second derivative with respect to stock price is gamma and it also is always at least worth 0. That means that as a stock price increases the delta or slope itself increases (or hits the zero lower bound). 

In options land, stock prices are assumed to be lognormally distributed. This is a reasonable distribution since a stock is bounded by zero and stretches to infinity. The expected value of a stock is the current stock price (in a no arbitrage framework)4.

Now let’s go back to Jensen’s Inequality:

E[f(x)] ≥ f(E[x])

…if f(x) is convex

Substituting words:

The expected or average value of a call for all possible prices of the stock (of course weighted by their probabilities) will be greater than the value of the call based on the stock being at it’s expected price (which is just the current price in Black Scholes).

In other words, the average value of a call will be higher than the value of a call in the average scenario.

It is easier to see with a binary stock (as opposed to a lognormally distributed stock). Suppose a binary stock is $10. That’s its expected value. Suppose now that the expected value is driven by the fact that it’s 90% to be worth 0 and 10% to be worth $100. The 50 strike call is worthless in the average scenario (since $10 is the weighted avg of the scenarios…again that’s what a stock price is by definition).

 But the weighted average of its value over both scenarios is $5 (90% x 0 + 10% x (100-50))

Again, the average value of a call will be higher than the value of a call in the average scenario.

So next time somebody uses the logic that they don’t buy options because most options expire worthless you can remind them that the typical outcome is not what drives the value of options. Instead, you should care about the average value of the option over all scenarios.

By the way, nothing I said here is revelatory. It’s not like any serious person thinks OTM options are worthless in the first place and just prices options based on the stock’s expected value. 

  • Technology

    Here’s a qualitative one. Suppose teacher skill follows a bell curve. Skill is the independent variable. Our X. Our payoff function is going to be how many people the teacher can effectively educate. A great teacher will impact a higher percentage of the students they actually come into contact with because they are more effective. If we had to predict the payoff, we might be tempted to apply a function to the average teacher. This would be like taking the 7 we rolled and running it through some payoff function.

    But if we consider the average value of the teacher payoff function for all level of teachers we will find that prior estimate to be wildy too low. Sal Khan comes to mind. He literally broke the function despite being just a good or even great teacher but very much on the bell curve. 

    The point is that technology leads to a huge range of possible payoff functions semingly extending to billions. Estimates of output based on the average input will fail to appreciate the convexity of some of the payoff functions in a hyper-connected world network. 

    This might not map perfectly to Jensen’s Inequality but it was a thought I had as I turned these concepts over in my mind.

Takeways

Be careful when trying to make estimates of how a function or process will payoff based upon the average input. If the function has exponential dynamics, Jensen’s Inequality tells us that the weighted average value of the function will not coincide with average input you feed it.

If the function is convex, the average input will underestimate the average output. If the function is concave, it will overestimate the average output.


Progressive Bikeshedding

I’m in Tahoe this long weekend with my extended local family. My in-laws have kids in SF public elementary schools. They are very involved parents in the Richmond district. On Friday night after the kids went to bed we were chatting about school life and the struggles of remote learning in the community.

Deeply regrettable stuff. I learned of the hotline where parents can call in to vent for 60 seconds. Seriously, 1 minute. It’s a desperate outlet. Parents feeling that “they are unfit to be parents”. Totally overwhelmed by the demands of holding down jobs, guiding their kids 24/7, being short with their kids, and their kids becoming distant, troubled, sad or any other strain of negativity you can imagine exacerbating the parents’ dire year even further. A vicious cycle.

Then there’s the clinically tragic. Consider the alarming reason why SF is suing the schools for remaining closed:

UCSF Benioff Children’s Hospital has seen a 66% increase in the number of suicidal children in the emergency room, and a 75% increase in youth who required hospitalization for mental health services, the lawsuit said, quoting pediatricians, child psychiatrists and emergency room doctors.

Last month, UCSF Children’s Emergency Department at Mission Bay reported record high numbers of suicidal children seen and treated, according to the legal filing which did not provide detailed numbers of cases and hospital visits. It also quoted doctors citing an increase in anxiety, depression and eating disorders among children, consistent with national data. (Link)

Now this is all quite bad (after using a thesaurus to find the right word I rejected all the candidates. “Bad” wins.)

But it’s not shocking. If Covid revealed how our economic supply chains were globally optimized to the penny, we should not be surprised to discover that a typical household was already driving on a spare.

What was shocking was what I read about the SF Unified School District, coincidentally, just before the conversation with my in-laws. After they told me their stories, I simply read aloud what I learned in Bob Seawright’s Better Letter earlier that day:
The school board of the San Francisco Unified School District recently voted to move ahead with a plan to change the names of more than 40 schools. The plan called for removing from schools names of those who “engaged in the subjugation and enslavement of human beings,” who “oppressed women,” who committed acts that “led to genocide,” or who “otherwise significantly diminished the opportunities of those amongst us to the right to life, liberty, and the pursuit of happiness.” Among those to be excluded are Abraham Lincoln, George Washington, Thomas Jefferson, Theodore Roosevelt, Franklin Roosevelt, John Muir, Robert Louis Stevenson, Paul Revere, and Dianne Feinstein.

[pause for family to grok the irony of reading this together on President’s Day Weekend]

The board eliminated Lincoln’s name because of his policies toward Native Americans; Washington’s and Jefferson’s names were struck because they held slaves. The Paul Revere Elementary School will be renamed because of Revere’s role in the Penobscot Expedition of 1779, a Revolutionary War naval assault on a British fort from the Penobscot Bay that the committee assumed, bizarrely and wrongly, was intended to colonize the Penobscot people.

Perhaps it will become Robespierre Elementary and the school board will offer instruction in Maoist constructive self-criticism. 

Robert Louis Stevenson, an important area literary figure, is having his name removed because the poem, “Foreign Children,” from his famous collection, A Child’s Garden of Verses, used the rhyming word “Japanee” for “Japanese.” James Russell Lowell was wrongly claimed to have opposed allowing Black people to vote. It was enough to cancel Lowell. The name of James Lick was ordered removed because his legacy foundation funded an allegedly racist art installation nearly two decades after his death.

Clarendon Elementary, named for Clarendon Avenue, on which it sits, will lose its name because, as the Board of Education explained, the name “can be traced to a county in South Carolina, one of the 13 Colonies named for Edward Hyde Earl of Claredon [sic] impeached by the House of Commons for blatant violations of Habeas Corpus.”

[I was reading all this aloud and decided to take a let-that-sink-in pause. They would need a moment to collect themselves before the crazy train picked up speed again]
Gabriela López, the head of the San Francisco Board of Education, defended the overall decision along with the decision not to consult any historians during the process because she doesn’t want to “discredit the work that this group has done” despite their questionable judgment and glaring use of false information. In her view, those pointing out even obvious errors are “trying to undermine the work that has been done through this process.” 

López insisted that people are “up in arms” because they “have a problem with the discussion of racism.”

Oh, and “Lincoln is not someone that I typically tend to admire or see as a hero.”

In general, any breach of political purity precluded a name from fronting a school irrespective of countervailing good works. There was one exception, however. When a member questioned whether Malcolm X Academy should be renamed because Malcolm was once a pimp, and therefore subjugated women, the committee decided that his later deeds redeemed prior errors. Lincoln, Washington, Jefferson, the Roosevelts, and the others did not receive similar forbearance.

In other San Francisco school news, the school board has deemed acronyms racist, and SFUSD’s vice president, Alison Collins, asserted that the concept of merit is also racist. Just this week, after two hours of debate, the board rejected a gay dad of mixed-race children from volunteering for one of several empty seats on a parent advisory group that didn’t have any gay members or men. Their problem was that he’s white and doesn’t bring diversity to the group. Really.

Ok.

Let me stop here for a moment. I don’t especially like bringing attention to the most ridiculous and therefore straw version of progressivism. Doing so undermines progressive ideas that actually deserve attention (Moontower readers might be surprised that I almost agree with “meritocracy is racist”. In fact, the reason I don’t agree, is because that statement is an object level instance of my meta belief that meritocracy is largely “besides the point”. Maybe I’ll discuss this at some point when I’d feel less bad about a flock of unsubscribes. Like right before there’s enough readers to shove me into the paid tier of Mailchimp.) But also, I couldn’t help but share the insanity as it collided with what my family was telling me.

I’ll let Bob’s pragmatic sentiment be the outro…

Meanwhile, the SFUSD has no plan to reopen its schools despite the weight of scientific authority establishing that reopening can be done safely and that remote school is bad for kids. Priorities, people.

While nearly half (48%) of San Francisco’s residents are white, only 15 percent of public-school students are white. It’s hardly a coincidence that San Francisco’s private school are open, all but conclusively establishing that the city’s care for “the least of these” is far more symbolic and performative than real. As the Apostle James made clear, believing the right things without action on them is worthless.

Sparking My Kid’s Interest In Coding

My niece is learning to code using pygame. Pygame is a Python based module for writing games. When she told me, I hopped on to YouTube to watch a tutorial with Zak (7) to see if we can learn together. Hmm. It was quickly clear. He was going to need some basics first.

After he went to bed, I decided to script a simple text-based game that might enjoy. It also had the oblique purpose about teaching him the basics of business math. You’ll recall I planted this seed with Zak and his podmate back in the fall. I described the process in A Socratic Money Lesson for 2nd Graders.

I’ve shared the game with cousins and his friends, and it gives them a quick little competition. It encourages them to read, reason, and do basic arithmetic. After they played it for awhile, I told him I wrote it. Now I can’t really code but it was just enough for the desired effect — he would think “whoa, we have the power to make a game!”. Now he is interested in at least messing with Python and trying to learn some basic syntax.

Share it with your kids and see how much money they can make at the Ice Cream Shoppe. (game)

Here’s the code. It’s not pretty but it should be easy enough to spot how to change it. (code)

The game takes less than 2 min to play. And it will introduce you to Trinket, a great site for writing and running programs in a browser.

The 4-year-old wanted to play a game too. He’s learning basic addition and loves chess so I wrote him an even simpler game. It encourages arithmetic by adding the values of chess pieces. He can’t read but his bro wrote a cheatsheet on a piece of paper so he can match the words on the screen with the word on the paper. Over time, he is figuring out that the word that starts with “p” is “pawn”.

Play it here. (chess piece game)

Notes from Marc Andreessen on Education

Link: https://a16z.com/2020/09/10/education-myths-monopoly-oligopoly-cartel-costs-past-present-change/


The education system is based on model that pre-dated the printing press. It has had little innovation in light of the technological advancements. Yes there are experiments like Lambda School and its ISA alignments. There are MOOCs which offer micro degrees. But in 2020, distance learning as necessitated by Covid, has accelerated the questions we have about a system whose costs were already outpacing inflation. We are left to wonder who our current system is serving and if it is time to examine more efficient possibilities.

Recently Google dropped the requirement that new hires need college degrees and it’s expected other large employers will follow suit. It begs the question, what were degrees good for?

The CEO of Figma, Dylan Field, interviews Marc Andreessen to hear what the cost/benefit of our college system is and how recent developments will test theories about what college is good for and what alternatives may serve those requirements better or more cheaply.


Purpose of college

Overt purpose: A bundle of actual education/skills acquisition, social/dating service, network building, “attached to a hedge fund” (in the form of an endowment)

Cynical purpose: Outsourced personality and IQ testing (via SAT) as these screens have become either socially undesirable or illegal for employers to perform.

The personality dimension being tested for is known as conscientiousness 1 which has 2 components.

  1. Industriousness: Basically self-starting energy
  2. Orderliness: Attention to detail, time management, organization

The “sheepskin effect”

Somebody who goes to college for seven out of eight semesters does not receive seven eighths of the income of somebody who goes for eight out of eight semesters, they receive half the income of somebody who goes for eight out of eight. So the diploma signals your conscientiousness by evidence of you clearing the 4 year hurdle.

A diploma tells employers you are a smart kid who can get their work done, signaling conscientiousness, rather than being about knowledge acquired.

Testing the purpose of college

  • Covid-19 will tease out how much people are willing to pay for an online education which will hint as to how much of the value proposition derives from the degree, from the social, and from the actual learning (this acting as a constant). International enrollment which is unsubsidized would be an especially useful clue as you would expect the loss of social network effects would impact those students the most.
  • The test of college as an outsourced intelligence test will naturally occur as leading universities shed standardized testing requirements

Understanding the source of the student debt crisis

We need a conversation about value given vs value received of college from an economic lens because it is subsidized by Federal and state government. If the ROI is not there the victims are tax payers and the students who cannot discharge the debt via bankruptcy.

How did we arrive at a mountain of debt that cannot be serviced?

The system is a hostage of a govt sponsored cartel.

  • K-12 education is compulsory and state-run. Captive audience.
  • Hallmark of monopoly: real dollars spent on education have 3x in 40 years and outcomes are unchanged
  • Funding is monopolized
    • Accreditation: Loans are subsidized by the government and are only available to accredited institutions that are certified by the govt. Accreditation or admittance to the cartel is nearly impossible.
    • University research funding comes from the government. Can’t remember the last research university to come into existence
    • Operating a university is taxed as a non-profit
    • Endowments are taxed as non-profit

    Meanwhile between sports programs and endowments these institutions have more in common with for-profit businesses.

The spiraling costs are exactly what you might expect from a monopoly and to be contrasted with perfectly competitive businesses such as manufacturing that have led to goods disinflation.

Basically what the government does to education is just like what they do in health care, it’s just like what they do in housing. A two part strategy for managing these markets. They restrict supply. And then, and then restricting supply causes prices to rise, because there’s more kids that want to go to school than can get in. And then on the other side rising prices create political pressure which they resolve by subsidizing demand.

(This was part of his anti-govt rant. I haven’t fact checked any of this. He also points out that spiraling costs without an improvement in service is also the hallmark of 2 other heavily govt influenced areas: housing and healthcare. The story of the ultra-liberal Cal professor who called for subsidized housing while he votes against development to maintain “historical charm” came to mind.)

The value proposition of university for people in “show your work” fields is changing.

One of the most basic revelations the internet has surfaced is the different nature of professions.

Internet has made the largest difference in “show your work” professions: occupations where it is valid and easy to demonstrate your value online. For example, coding, design, music, art, game dev, animation. Open source projects and writing, democratized, pure examples of “show your work” fields.

From an employer’s point of view conscientiousness is a proxy for being a good employee. But this can be circumvented by just showing your work online. This erases the value of a degree that derives from employer demand.

GitHub has like an internal ranking and rating system for software code, and for programmers. So you can actually build an actual professional reputation as a software developer on GitHub without ever actually being face to face with another human being. People all over the world today who were basically taken advantage of this to be able to basically build these incredible track records as a software developer and make themselves more employable. Employers like my venture firm. We recommend that our employers spend as much time on GitHub looking for good programmers as they do on LinkedIn, or going to college fairs.

YouTube, blogs, Figma for design all play a similar role as GitHub does for software developers. He tells the story of South Park as an early example of a viral video that was able to spread organically through a distributed technology. The show born from Matt Parker and Trey Stone’s irreverent holiday card which made its rounds as a downloadable Quicktime vid!

“If you can go to college, go to college”

  • Even jobs that probably shouldn’t require degrees require them.

I think it’s actually quite dangerous to give somebody, somebody as an individual the advice, don’t go to college, like in the current system that we have that’s basically saying don’t prove that you’re smart don’t prove that you’re industrious, and conscientious and then basically be prepared to settle for fundamentally lower income for the rest of your life.

  • Understand the proposition

Gates and Zuckerberg notwithstanding, if you go to college finish college. Get the piece of paper.

  • The 2×2 matrix of what to study and where to study.

The spread of outcomes for technical degrees is not that wide. If you have a technical degree your choice of school matters less. This is exactly the opposite of what you find with liberal arts degrees. Since the output of a liberal arts degree are more subjective or uneven the school issuing the diploma carries more weight. 

Possible explanation: in absence of concrete skills, the network from a top school is valuable.

Tips for those in college or considering college

Execute on the opportunity — take the hardest course load you can. Get the skills (obviously get good grades but focus more on getting the skills).

If you are at a sub-tier college taking liberal arts, de-risk by acquiring marketable technical skills.

For those considering alt paths

At this point Marc, still recommends college and acquiring technical skills but if you choose an alt path be aware of the trade-offs. For example, if you choose to do open source work recognize it’s better to make major contributions to one project (as opposed to minor contributions to multiple projects) because that really demonstrates what employers are looking for. Put yourself in the mind those who will be evaluating you years down the road.

Consistent work demonstrates conscientiousness and the nature of the work is an embedded intelligence test.

What should a software developer do? Unquestionably the answer is create an open source project or go become a member of an existing open source project and make successful high quality sustained contributions to that project over time. At this point I think that’s clearly a better credential than getting a computer science degree. I’d hire people like that myself and the great thing now is you can do that from all over the world.

So what matters to Andreesen when they hire or fund someone?

The good news:

They do not care about a degree or GPA or test scores and in fact question if too much conscientiousness means you are too much of a rule-follower.

The tough news:

They measure you by what you have actually done. Building companies requires being able to do things so that is the capacity they are looking for. List of things a founder will need to be able to do:

  • Building an actual product that somebody will actually pay for.
  • Figuring out a way to actually sell it to them
  • Actually collect the money
  • Actually service the customer so they actually have a good experience
  • Actually tell their story so that anybody will even know that they exist
  • Run a finance function so that they don’t lose all the money
  • Run a legal function so they don’t get sued all the time
  • Actually get others to work with them.

There are many talented people so the way to stand out is to actually demonstrate the ability to build or create.

Steve Martin best career advice ever: Be so good they can’t ignore you.

Developments to watch

  • New credentials2 to replace bachelor’s degrees (ie Google certification program, coding tests, and math puzzles)

  • Still early innings of “show your work” online as way to qualify yourself



Let Chess Help Kids

Two years ago my wife Yinh started her podcast Growth From Failure. Her second guest was Berkeley Chess School founder Elizabeth Shaughnessy. It is one of my favorite interviews ever. We have referenced wisdom from it on many occasions since. Yinh texts with her from time to time and always comes away so invigorated. This past week I was stoked to meet the 83-years-young chess whiz. My expectations were high.

It turns out I still underestimated how special she is.

We went to lunch at Cafenated Coffee in Berkeley and 5 minutes into the conversation I immediately regretted not having a notebook. Elizabeth is bursting with passion for her mission and practical insights for teaching, life, and of course chess.

I did a full write-up that I’d love for you to check out: Lunch With The Amazing Founder Of Berkeley Chess: Elizabeth Shaughnessy (Link)

Here I’ll give a brief version of why it’s so special but the full article gets into ideas you can literally apply today in your life.

The Mission of Berkeley Chess School

BCS is a true Robinhood organization. As a non-profit, they are funded by donations and fees they receive for after-school programs around the Bay Area and private lessons (our son and his friends do group private lessons with BCS instructors). This supports their mission to provide free or low-cost chess instruction to students at poorly sourced Title 1 schools. In the past 40 years, BCS has taught over 250,000 kids.

But when you sit with Elizabeth you realize this is about far more than a game. Today, with Covid decimating enrollment, the school has re-purposed its building to teach disadvantaged kids. These are kids from low-income sections of Oakland, Richmond and Berkeley who are struggling with distance learning. These kids have no internet or computers at home. Without intervention, these kids, already struggling academically before the pandemic hit, may suffer an irreparable learning loss that could affect their health and financial well-being far into their adult lives.

From her experience, Elizabeth is convinced there is hope.

How Does Chess Help?

As a fan of games and games in learning, I like to believe that the skills acquired in play “transfer” to other domains. This is something I’ve wondered aloud about on Twitter. It is rooted in causality. I specifically asked Elizabeth if she thought a joy of chess was simply a symptom of a more general aptitude or if chess was imparting a more generalized skill that could be applied to other fields.

Elizabeth is a big believer that there is transference.

  • Chess asks kids to slow down and be methodical.

    Count how many pieces are threatening your pieces. Do this for every piece, on every turn, to find the strengths and weaknesses on the board.  Then look at all the checks you can deliver, then the captures, then the attacks. When all this is done, then make your move.

  • Consequences matter and compound.

    Chess teaches you that consequences matter. Make a rash move and you get penalized by your opponent.  Mistakes are expensive in chess and life. What scenarios can unfold if you always skip math class? How will this serve your long term objective of being a Wall Street wizard if you’re unable to calculate risk or odds?

  • Chess sharpens your focus.

    She has repeatedly seen firsthand the power of chess to harness kids’ attention. It’s an effective tool to settle kids so they can get into a better headspace for learning. Kids who start out resistant often do not want to go home after school.

Chess can show kids they are smart. It teaches them to believe in their own abilities. Many of the kids BCS teaches face long odds in life but chess can offer lessons in foresight, creativity, problem solving, and self-control.

Helping BCS

Children heatseek that which provides immediate benefit or stimulation. BCS has figured out how to stimulate children that have been written off. Any witness to that transformation will see one thing — the longest lever we have as a society to improve a child’s well-being today and into adulthood. When I listen to Elizabeth, I can feel what she has seen.

If you are looking for high impact ways to give back I encourage you to check out my full post or if you prefer you can simply head over to BCS site to learn more. (Berkeley Chess School)

Tips and Insights

Elizabeth cannot help but spill insights all over the place when talking. Check out the full post to get:

  • practical tips for learning chess today
  • how to play chess with children and why
  • insights into teaching girls specifically
  • the role of genius
  • the pros and cons of being a good loser

And if you are wondering her view on Netflix’s Queen’s Gambit — she thought it was too long but the beginning and end were fantastic. Ultimately, she thought it deserved high marks for making chess so compelling.

Wrapping Up

My 7-yr-old has been taking lessons with BCS intermittently since he was 5. Even our copycat 4-yr-old is into it. It took him all week of multiple games per night to learn how the knight (he’ll correct you if you call it a “horse”) moves. I better start learning more, they are hot on my heels. I’m MoontowerMeta on Lichess.org if you want to add me. I’m a beginner. I’m still beating the 7-yr-old but it’s getting tougher.

This is one of Elizabeth’s sons teaching chess at our pod a few weeks ago.

Lunch With The Amazing Founder Of Berkeley Chess: Elizabeth Shaughnessy

I’m going to tell you about Elizabeth Shaughnessy. Exactly 2 years ago, Yinh contacted to Elizabeth to be one of the first guests on her podcast Growth From Failure. Elizabeth has been an inspiration to our family since that first conversation.  I have been telling anyone interested in chess, children, education, or simply hope to listen to that interview.

This past weekend I was privileged to meet Elizabeth for lunch in Berkeley.

About Elizabeth

Nearly 40 years ago, Elizabeth, 83-years-young, volunteered to teach an after-school chess enrichment class at her children’s school. She expected a handful of people to take interest. Instead a diverse group of 72 children showed up. She was stunned. In the eighties, chess was the underground world of nerds. And if you think back to 80s movies, nerds were people jocks stuffed in lockers. (It probably didn’t help that we were in the Cold War — think about it — if Drago was a Grandmaster and Stallone a genius orphan from Brooklyn, Rocky IV would have swept the Oscars).

How times have changed. Today, Elizabeth oversees a chess school that teaches nearly 7,000 kids per year. She estimates the school has provided instruction and community for nearly 250,000 people in the past 40 years!

Her illustrious history can be found here.

The Berkeley Chess School

The chess school is a true Robinhood organization. As a non-profit, they are funded by donations and fees they receive for after-school programs around the Bay Area and private lessons (our son and his friends do group private lessons with BCS instructors). This supports their mission to provide free or low-cost chess instruction to students at poorly sourced Title 1 schools. While they are in over 120 local schools, Covid-19 has been devastating to enrollment as in-person instruction has cratered.

Always the optimist, Elizabeth has pivoted resources. With generous support from philanthropists and organizations working through the Berkeley Public Schools Fund, the Chess School now administers a program to help our most vulnerable neighbors. Her son, Stephen Shaughnessy, a former California State Scholastic Chess Champion and a teacher with more than twenty years of experience, guides cohorts of students struggling with remote learning in a safe setting in the School’s spacious tournament hall. Steven’s gifts and calling has always been to teach children, but this year this mission has been extra special. And challenging.

Coming from low-income neighborhoods in Berkeley, Oakland, and Richmond, his 5th graders are from extremely disadvantaged backgrounds. Not one to mince words, Elizabeth says that without intervention, these kids, already struggling academically before the pandemic hit, may suffer an irreparable learning loss that could affect their health and financial well-being far into their adult lives. They are at a critical age, steps from a dark a road without an offramp.

From her experience, Elizabeth is convinced there is hope. BCS is determined to help kids believe in themselves and their own ability to be smart. Many of the kids BCS teaches face long odds in life but chess can offer lessons in foresight, creativity, problem solving, and self-control. It can give these youngsters a chance for better futures far from the disadvantages of their childhoods. The hope is they would have kids of their own one day for whom the sky is the limit. If the realism is off-putting, then you can imagine just how important the work is.

Lessons From A Lifetime Of Teaching Chess

While this lunch was supposed to be nothing more than friends catching up, I found her passion and enthusiasm for her work the only thread I wanted to pull on. She just oozes hard-won insights into children and learning. I immediately regretted not having a notebook. Here is what I can remember from the 90 easiest minutes I ever had of keeping my mouth shut as I tried to absorb the steady stream of wisdom.

Benefits of Chess

As a fan of games and games in learning, I like to believe that the skills acquired in play “transfer” to other domains. This is something I’ve wondered aloud about on Twitter. It is rooted in causality. I specifically asked Elizabeth if she thought a joy of chess was simply a symptom of a more general aptitude or if chess was imparting a more generalized skill that could be applied to other fields.

Elizabeth is a big believer that there is transference.

  • Chess asks kids to slow down and be methodical.

    Count how many pieces are threatening your pieces. Do this for every piece, on every turn, to find the strengths and weaknesses on the board.  Then look at all the checks you can deliver, then the captures, then the attacks. When all this is done, then make your move.

  • Consequences matter and compound

    Chess teaches you that consequences matter. Make a rash move and you get penalized by your opponent.  Mistakes are expensive in chess and life. What scenarios can unfold if you always skip math class? How will this serve your long term objective of being a Wall Street wizard if you’re unable to calculate risk or odds?

  • Chess sharpens your focus.

    She has repeatedly seen firsthand the power of chess to harness kids’ attention. It’s an effective tool to settle kids so they can get into a better headspace for learning. Kids who start out resistant often do not want to go home after school.

Tips For Learning Chess

Yinh and I are starting to learn chess alongside our 7-year-old who has been getting intermittent instruction since he was 5. Our 4-year-old recently learned how to set up the board and how the peices move (ok, he doesn’t really understand how the “horse” jumps). Learning chess can be a bit overwhelming for good reasons. There are tons of amazing resources out there from software, to YouTube, books, and communities. If you are like me, sometimes you just want to be told “Do this” and be handed a basic recipe from which you branch as you learn.

Here’s the simple recipe.

  • At first, focus on tactics.

    You can think of tactics as a series of maneuvers to gain an advantage over your opponent. They have cool names like “forks” and “pins”. They are the “fun” part of chess. Most major software and websites provide ample puzzles to teach and reinforce tactics.

  • There’s plenty of time to worry about openings later

    Don’t worry about studying openings until you have at least a 1200 rating. More has been written about openings than any other aspect of chess and it is a rabbit hole from which the beginning and intermediate chess player might never emerge.

In the meantime, take solace in the idea that beginners’ focus on tactics is not just the best use of time, but conveniently, fun. Strategy including openings and end games come much later (Elizabeth mentioned that endgame chess is especially fascinating to the mathematically inclined). Although I have just started, I find the puzzles extremely engaging. While it’s humbling, the feeling of “seeing” the move is addictingly rewarding.

Chess and Children

  • High standards

    With a good teacher, many kindergartners can visually play without looking at a board. It is not the realm of genius. In fact, one of the most inspiring feelings you get from spending time with Elizabeth is how bullish she is on children’s abilities. She believes we do not give them enough credit. They are capable of so much. We easily forget how we stunt a plant’s growth when it’s in a small pot.

  • Playing chess with children

    Do not let them undo bad moves. Remember, consequences matter. If they make a weak or ill-advised move (a blunder in chess parlance), turn the board around and play the weaker position. You can continue to do this and by the end there is a sense that nobody has truly lost which can be useful to keep kids encouraged.

  • Genius

    They exist. She has seen her share. There are children out there who can recite every move of the last games they have played. You cannot teach geniuses. They are smarter than the teachers. But you can guide them and help them explore the exponential facets of the game. BCS has had the privilege of coaching three Grandmasters, including Olympiad Gold Medalist and 2018 US Chess Champion GM Sam Shankland. BCS offers Master Classes so the best players can learn from and help one another even as they compete.

  • Not pushing too hard

    Other than World Champion Magnus Carlsen, the life of a grandmaster is hard. There are few things in the world in which you can be so close to the top and have so little to show for it. Most grandmasters are scraping by, writing books, and being paid to play in tourneys. She does not push the geniuses in that direction. The application of genius to real world problems results in easier, productive, and more prosperous lives.

Chess and Gender

  • Girls

    On average, girls in chess are more discouraged by losing than boys and this can lead them to giving up. They were not born like that. But if one child is encouraged towards cooperative play, while another child is encouraged to compete, losing will be a more emotionally significant event to the child who is unfamiliar with it. She has found that girls who play sports do not give up easily, reinforcing the idea that this is learned behavior.

    Losing is an important subject. There is a tension in being comfortable with losing. It’s necessary to be able to lose because it’s part of learning. However, Elizabeth has never seen a great player that was not deeply bothered by losing. So we must examine our own values and how they relate to losing. When daughters come off the floor after a chess tournament what does Elizabeth see? Fathers who ask their daughters “Did you have fun?”. To the boys they ask, “Did you win?”.

    Elizabeth has lots of views on women in society based on what she has seen at formative ages and observing thousands of families. She believes there is bias and while we were too short on time to get into the vast subject one thing was obvious. We carry tremendous responsibility for the scripts children grow up believing about themselves. It is the single most empowering lesson I grokked from taking in her wisdom.

    And by the way, BCS teaches girls to play chess aggressively. They are trying to balance out society’s conditioning.

    There are 65 active Grandmasters in the United States. One is female.

  • Women

    When women compete at tournaments they are extremely competitive with one another but away from the table they can become friends. Elizabeth told a story of a tournament she hosted with women coming from all over the world. After the fierce competition ended, the women organized a guided tour of SF Bay and got along like sisters. She noted this was a very different dynamic from the men. There is a balancing energy missing in our world if that story is any indication.

Why we care

Chess has exploded in popularity as nerdiness has become cool and the internet has spread access to high quality chess tools, matches, and education. Elizabeth’s life mission has coincided with a more secular phenomenon. The chess school boasts 3 of the United States’ 65 Grandmasters with the most recent one being just 17 years old (masters are getting younger thanks to online play).

If Elizabeth’s mission were simply to promote the empowering aspects of a beautiful game then she has the right to be satisfied. That baton is securely passed on to wider zeitgeist than she ever imagined. But as the recent pivot to share the school’s resources with our most neglected has shown, the Berkeley Chess School is not just a Kumon For Chess. It is a sustainable model for meaningful impact. It is a model for fostering local community. And through it’s alumni, a model for global community.

It is a place we feel lucky to have discovered and organization we are honored to give to. With enrollment down and the ongoing renovation of the School to improve ADA access, there is a lot of wood to chop. If you are interested in helping, they have several programs that you may make targeted donation to.

You can find the list here.

I’ll conclude by saying, when I met Elizabeth I had high expectations. Yinh talks about her a lot and her interview is one of my absolute favorite all-time pods, not just Yinh’s. When I met her, I was blown away. She is sharp as hell. She cares so much you can feel it. As I listened to her stories, it was clear I was in the the presence of a special individual who has spliced her DNA into the heart of an institution (this is not so figurative…her son Steven manages the day to day operations now).

We are excited for the future of the Berkeley Chess School!

(And if you want to learn she recommends starting with the tutorials on licchess.org. Hope to see you there!)

Fun Ways To Teach Your Kids Encryption

Here are 2 simple ways to introduce the idea of encryption to elementary school kids.

 

Ciphers

I had them try to decode the following code:

4 15 7

(with some prodding they eventually figured out it was a simple letter-number cipher spelling “dog”)

 

Mastermind

You might recognize this game from your own childhood.

 

 

You don’t need to buy it. You can play it with different color marbles, bingo chips, coins or almost any set of things lying around the house.

How to play:

    1. A codemaker constructs a hidden sequence of 4 different colored beads (out of a possible 6 colors).
    2. The codebreaker tries to guess the sequence by arranging 4 colors in order.
    3. The codemaker gives non-verbal feedback:

      a) Identifies how many of the colors used are correct
      b) Identifies how many beads are the right color AND in the right position

    4. Repeat until the code is cracked

      Enrichment questions:

      How can the game be made easier?
      How can the game be made harder?

      And if you have an older onlooker…how many possible codes can be created?

      And if you have an Excel fan in the vicinity, see how you can solve such problems using the hypergeometric distribution. (A Reddit thread targeting game designers)

A Slightly More Advanced Example: Using a “Mask”

Suppose a group of people are sitting around a table and you all want to know how much money everyone makes but of course nobody is willing to share their own salary.

Here’s a way to uncover the average pay at the table without anybody needing to disclose their pay.

Let’s pretend A, B, C, and D are having dinner together at this table.

Just follow these steps:

Masking Phase

  1. Tell “A” to add an arbitrary number to their pay and write the sum on a piece of paper. It’s very important to write just the sum! So if A makes $100k per year and the arbitrary number is 5,000,000 then they would write: $5,100,000

  2. Pass the paper to “B”

  3. “B” notes the sum and adds it to their own salary plus their own arbitrary number. They write this sum on a fresh piece of paper and hand it to “C”. Important: use a new piece of paper, we don’t want anyone to see the history of how the sum was created. 

    Example:

    “B” receives paper with the sum $5,100, 000
    “B” add this to their own salary $50,000 plus an arbitrary number of $1,000,000
    “B” passes a piece of paper with the total $6,150,000 to “D”

  4. Repeat this process until the paper gets back to “A” 

Un-masking Phase

  1. “A” subtracts their arbitrary number only from the total and passes the new total to “B”.

    Example :

    “A” receives a piece of paper with the number $9,000,000 written on it
    “A” subtracts the $1,000,000 arbitrary number and passes the number $8,000,000 to “B”

  2. Repeat until everyone has subtracted their arbitrary number.

The remaining total is the sum of everyone’s pay. If we divide by 4 (the number of people) we have discovered the average pay at the table and nobody needed to reveal their own number!

You have learned a simple way to “mask” data with arbitrary numbers!

Try it yourself. You don’t even need paper — just explain the rules to some friends in your texting group and find out if you are actually under or overpaid! Just don’t kill the messenger.

(The mask example was inspired by this Twitter thread by @theemilyaccount)

A Socratic Money Lesson For 2nd Graders

I’m sharing the money lesson I did with a pair of 2nd graders on my rotation day with the pod. No worksheets involved. I just used the Socratic method to ask questions and let them develop an understanding on their own.

The concepts:

  • hourly income
  • budgets

Techniques applied:

  • addition and multiplication
  • estimating
  • brainstorming

Materials needed:

  • Poker chips or play money
  • Pencil and paper (we actually used markers and small dry-erase boards)

Script

“Let’s suppose you make $25 per hour.

Pick something you want to buy so you have a savings goal.”

[In our example, they answered gumball machine, so I set the price at $1,000]

Question 1:  How long would it take you to save up for it?

What we are looking for in the answers:

    • How many hours it would take to save up $1,000?
    • How many hours you might work per week? (Expect bad estimates. I found myself asking them what time their parents go to and from work)
    • How the time it takes to save varies with how many hours a week you work.
    • Extend discussion to how much you would make in 1 year.

If you were a grown up you would need to pay for many things. These are known as necessities

Question 2:  Can you make a list of necessities?

What we are looking for in the answers:

We want them to identify categories. I had them create a list on their dry-erase boards. Here’s the ones we settled on:

      1. Food/Drink
      2. Shelter
      3. Transportation
      4. Clothes
      5. Utilities/Energy (including gas for the car)
      6. Hygiene(teeth, hair, soap)
      7. Entertainment/Fun
      8. Furniture/phone/computer (things you buy every few years…durable goods)
      9. Then savings!

Question 3: Create a monthly budget per category item.

What we are looking for in the answers:

In our lesson I had the kids estimate their monthly income and had them count out that much money worth of poker chips. You could use play money from Monopoly just as easily. I then had them estimate the amount of money they would need to spend on each category. They have now learned the concept of budget! Be ready to spend a lot of time here helping them walk through reasonable estimates. When one of the boys said $100 a month for food I explained that you could not even by 1 Happy Meal a day for that amount.

I had the kids stack of the appropriate amount of poker chips on each category.

It is interesting so see how much money the kids allocate to each category once reasonable ranges are established. These 2 boys had a big difference in their fun budgets!

Question 4: How long does it take to save up for that $1,000 gumball machine (or whatever they want to save for)?

What we are looking for in the answers:

Bound by the need to pay for necessities, the kids will discover that they can only make extra discretionary purchases if they amass enough savings. Compared to not having obligations, they should notice that it takes much longer to save up for that gumball machine.

Question 5: How can you get the gumball machine faster?

We want to establish the importance of this category. Adults understand it is a stock variable that changes with monthly flows. We want the kids to establish a link between savings and the things they want to buy. What can you do to save faster?

What we are looking for in the answers:

    1. Work more hours
    2. Spend less on budgeted items
    3. Earn more per hour

Going Further

This is where we stopped but next time this is my plan to continue.

Beyond Labor

This is where we will talk about creating a business. We had already started the discussion when the boys realized they could buy a gumball machine, stock it with candy, place it in Target and pay the machine off. Rinse and repeat. This got their little gears turning!

Here is some ideas I have for exploring:

Basic Accounting

  • Revenue
  • Fixed Costs
  • Variable Costs (COGs)
  • Profit

2 kinds of businesses: service and product

Service: Money-for-time businesses. For example a dentist or dogwalker or haircutter or coach. Earnings potential constrained by hours.

Product: Writing a book, or song (I want to talk about writing credit vs performing credit vs lyrics credit for songs. To make it fun I will look up some examples of a songs they like –Hamilton’s My Shot, Lil Nas X’s Old Town Road, or Bruno Mars’ Uptown Funk)

Brainstorming businesses

  •  Brain dump of business ideas
  • Labeling business examples as type 1 or type 2

Application: Lemonade Stand example

  • Working at a lemonade stand is hourly labor and similar to a service business.
  • Owning a lemonade stand is product business.

Demonstrating the difference in risk: the upside and downside of ownership

Contrast the p/l after an hour’s work at a lemonade stand for the worker vs the owner on a

      • Hot day [above expected revenue]
      • Cold day [below expected revenue]
      • An AQI 500 day where people stay indoors (hey these are CA kids!) [no revenue]