Category: Education

Notes From C.Thi Nguyen Interview About Games and Society

Link: https://www.preposterousuniverse.com/podcast/2021/10/18/169-c-thi-nguyen-on-games-art-values-and-agency/

C. Thi Nguyen received his Ph.D. in philosophy from the University of California, Los Angeles. He is currently associate professor of Philosophy at the University of Utah. He has written public philosophy for venues such as Aeon and The New York Times, and is an editor of the aesthetics blog Aesthetics for Birds. He was the recipient of the 2020 Article Prize from the American Philosophical Association. His recent book is Games: Agency as Art.

I only excerpted topics of particular interest to me so you should listen to the podcast for a fuller understanding. Anything bold is my emphasis.

Intro by host Sean Carroll

One of the good things about the artificiality of games, you know when you’ve won or lost, unlike life. But…that clarity of knowing when you’ve won or lost is very seductive outside the context of a formal game. It’s very seductive in life.

Thi has developed this understanding to study things like echo chambers and cult leaders. A cult is in many ways like an echo chamber. In both cases, it’s not just a filter bubble where you prevent information you don’t want from getting in, but it’s like a strategy for preemptively undermining claims from outsiders that the cult leader or the echo chamber doesn’t want you to believe in, right? You give people ways to discount outside information.

One of the reasons why cults and echo chambers are so seductive is that they bring clarity to values and moral reasoning, maybe a little bit too much clarity: They make it too easy, they make things cut and dried in a way that the world itself is often not so cut and dried. So he has a whole understanding of why we’re so seduced by conspiracy theories, by cult leaders, by echo chambers, and how it relates to this seductiveness of clarity that we get from thinking about games and point. We get points, we get likes on our tweets, we get steps on our Fitbit. This engages our brains for interesting evolutionary reasons, and that feature of human psychology can be gamed, if you like, by the leaders of cults or echo chambers.

Games

What is a game?

The notion of a game is really disputed in philosophy. It’s very storied. In philosophy, it’s a particularly famous concept, because when Wittgenstein was like, “No, you can’t define concepts,” his example of an undefinable concept was a game.

There’s an amazing book from Bernard Suits called The Grasshopper, which is an attempt to define games, which actually takes itself as a response of Wittgenstein and also secretly about the meaning of life and the relationship between games and the meaning of life.

The short version is, “To play a game is to voluntarily take on unnecessary obstacles for the sake of making possible the experience of struggling against them.”

  • Part of the idea of the game is that the goal of the game is partially constituted by obedience to certain restrictions:

    If you’re trying to get to the top of a mountain to get some rare drug that’s only there, you’re not playing a game. You’re just trying to get to the top of the mountain. If you’re trying to climb the mountain as a mountain climber, then certain restrictions are part of what you’re doing. So the medical seeker is not a game player, and the mountain climber is. And one way you can tell is if someone goes by in a helicopter and says, “Hey, you want a ride?” The medical climber will say, “Of course, get me the cancer drug.” And the other person is like, “Of course not. What do you think I’m doing?”

  • Autotelic: It’s worth engaging in the activity for the sake of the engagement and the doing rather than the product.

The charm or allure of games

  • Struggle

    The aesthetic experience of struggle can be accessed in a safe way by making the difficulty manageable.

Some struggles are beautiful, some struggles are satisfying. And what games do is, they let you tweak the activity to maximize that satisfaction…Most of life’s challenges are too big or too boring and little for us. In games, we get to modify the world of the game and the abilities we’re allowed in the game until they fit just right.

Example:

I feel like things like chess are kind of tuned to maximize that moment, you get more and more of those moments. In my normal life moving around the world, I get to feel graceful once a week, but rock climbing tunes you into the part of the activity that has that feeling. It’s built to constantly call out of you that incredible experience of delicate, graceful, perfect motion.

Manageable because of clarification

Games are these circumscribed spaces where the actions in space have been often been leaned down and clarified, so your actions can fit. They’re clarified, not just because the actions you can perform have been clarified, it’s because your values have been clarified.

Bounded and limited beings make things that make them feel temporarily okay, like spaces where we don’t feel too little for this vast world. The real world is this existential hell-scape of too many values, and games are like this temporary balm, where the world makes sense for a little bit.

  • Games as art

    Games are sculpted experiences of practicality. I think what’s interesting in games is, if games are sculpted practicality, then the beauty emerges in the practical action. So in other words, when you play a game, it’s not the game that’s beautiful, it’s you that’s beautiful.

    I think a lot of the literature about games has been going around looking for qualities that are in the game, like, Oh, the graphics are beautiful, the sound is beautiful, the story is beautiful. And they’re not looking at how radically different games are. And I think there are other things like this that are also mostly neglected, but the thing that makes games unique is that they’re sculpted action.

The danger in having your values clarified

  • Gamification

    Game values are hyper crisped-up, and that’s fine if you put away those values at the end. But when you gamify something like education or communication, then you’re forcing a singular clarified value system on a real-world activity. Think of how Fitbits or Twitter engagement can orient your goals towards local maxima. Twitter’s gamification squashes an individuals’ pluralistic values and gets everyone, insofar as they’re motivated, to be motivated in the same direction.

    • Education examples:
      • Sean Carroll: I used to be at the University of Chicago, which obviously has always been academically super-duper strong, but back in the day, it wasn’t the place you applied if you were interested in Harvard or Stanford or Princeton, it was less well known. So suddenly, there was a strategy that they undertook at the University of Chicago because they were being hurt in the US News rankings, and they were being hurt because the only people who applied to the University of Chicago were the ones who really wanted to go there. And you are rewarded in the US News rankings by having a high selectivity, by rejecting most of the people who apply, so they intentionally encouraged people to apply knowing they would reject them, ’cause it increased their selectivity, and they leapt up in the rankings. That’s an example of maybe the goal perverting the original aspiration.
      • Law School culture
        A study about law school culture when the US News and World Report started ranking them charts a bunch of stuff like what you’re talking about, about people gaming the rankings. One of the things they point out is that different law schools used to follow different missions before the rankings, but if their mission is skewed to the ranking at all, then you drop in the rankings, so it’s forced everyone to pursue the same values.

        Before the rankings, prospective law students used to talk about what different law schools valued and talk about their own values and decide what their values were, to pick which school to go to. Now, they, say 99% of the students just assume their goal is to get into the best school, where the best school is set by the ranking. So they don’t go through the process of value self-deliberation. You end up outsourcing your values, you end up letting somebody else perform value deliberations for you, and what goes into those values are often very much based in what’s in the interest of large-scale institutions and the kind of information management systems at large-scale information.

People worry a lot about games creating violence, and there’s actually a lot of data that mostly they don’t. And I think part of that is that the violence in games is fictional, and we have a lot of information that people are mostly capable of screening off fictions. The thing that I’m really worried about is people becoming used to the idea that the goal is some simple, quantified thing that people share, and what we’re supposed to do is do everything in our power to up that simple measure, and one thing on to note, that’s not fictional.

We shouldn’t worry about games creating serial killers, we should worry about them creating Wall Street bankers.

[obligatory reference to James C Scott’s Seeing Like A State: “Look, what you should think is that large-scale institutions generally see the parts of the world that are processable by large-scale bureaucratic machines, which are quantified data, they can’t register the part.” So, think about this, a large-scale school district or an educational bureaucracy can’t register individual student evaluation data, they can only register aggregatable data like GPA. So, says Scott, large-scale institutions have reason to remake the world along lines that are more regular, so that they can be legible to the institution and actable on by the institution.]

  • The satisfying aspect of games common to conspiracy theories

The satisfaction we crave in the clarity and simplification of games is why we are drawn to conspiracy theories. Conspiracy theories are tuned to give you the exact same pleasure [as games]. Someone has changed the nature of the world, apparently, to make it tractable.

How the argument works:

    1. There’s a source of anxiety: The world is complex

      Scientists are hyper-specialized, no-one understands everything, at some point you realize that you have to just trust tons of stuff that you have no ability to grapple with.

    2. Conspiracy theories circumvent our need to trust

      They say “Don’t be sheep. Don’t trust other people. Here is a vision of the world, where you can contain the world in you. You can explain all of it with this one powerful explanation.” It is a game-like pleasure, but exported to a place where it’s dangerous.

      • Refers to a book by Elijah Millgram, called The Great Endarkenment. It talks about how knowledge must not be individual quest given the fact that the world is so hyper-specialized that no one can know more than a tiny amount of it. [This stands in contrast to the] ideal of intellectual autonomy that drove the Great Enlightenment. But it doomed itself, because it created all the science that made it impossible to be intellectually autonomous. If you still hold to the old ideal of intellectual autonomy, where everyone can understand everything, what you get is being driven to anti-vaxxing and various conspiracy theories in which you reject trust in the sciences.]
    3. Conspiracy theories hack our desire for clarity

      It’s not that clarity is always bad. When you have intellectual success, you do have this feeling of clarity. My worry is the feel of clarity actually comes apart from real understanding, and that outside actors can game it.

      For example, in the course of evolution, it made perfect sense for us to pursue sugar and fat because calories are scarce, it’s hard to get enough fat and so on. Then the world changes and industrial forces figure out that they can maximize the feeling of sugar and fat separate from any nutritive qualities. And then if you’re still stuck on that old heuristic and chasing sugar and fat, then you’re screwed. Clarity can be like cognitive sugar. Someone can aim to max out the feeling of clarity, and the way that looks like is a conspiracy theory.

Trust’s role in conspiracy theories

1. Trust is tricky because there’s a lot you cannot trust, yet being able to trust is not optional

If you devote your entire life to it, you can understand one one-millionth of the human landscape of knowledge. [Even worse] not only can you not understand everything else, you don’t even have the capacity in yourself to pick the right experts to trust.

I have a PhD in philosophy, I have a lot of education. If you gave me a good, a real statistician, and a fake one, I don’t have the mathematical skills to tell the difference between a good statistical paper and one that gives a bad result. I don’t have that in myself. So what you get is actually this incredibly iterated and very fractal chain of trust.

[An example of why we have no choice but to trust in many cases:

There was one proof of a really famous theorem that literally only one guy understood, and he’s getting old, so a whole bunch of people had to have a project to re-write the proof in a way that other people could understand it. And that whole process of dealing with the fact that you need to trust some things, while you shouldn’t trust everything, is a tricky thing that is probably under-theorized.]

Politically, conspiracy theories tend to be associated with the right, but the left’s appeal to science fall flat because they are not appreciating the role of trust:

      • The false argument they spout:

        “Oh, these fucking anti-vaxxers, anti-maskers. Think for yourself, look at the science, evaluate the science.”

        When the truth is “I can’t evaluate that science.”

      • The obstacle is not whether people trust Science with a capital “S”, it’s the trust of its messengers

        Look at the people that are legitimated in a certain institutional structure, which involves a background trust in those institutions. And I think there’s this vision where for a lot of us, like when you look at anti-vaxxing, anti-masking, the climate change denialist space, what we want to say something like, “Oh, those people are totally irrational.” But I think what you have to think instead is, they have an entirely different basic framework of trust, for a different set of institutions. The degree of rationality there depends on the degree to which we can justify our trust in our institutions. And that’s a really, really complicated matter, and it’s not like the authoritative institutions are always right. There are plenty of historical cases where they are corrupted, right?

        It’s hard to be against conspiracy theories as a blanket statement ’cause sometimes there are conspiracies. We have plenty of historical examples where all the institutions in a particular country have become corrupted, have taken over the news media, are issuing fall statements. That’s a real thing that happens.

2. Echo chambers are not about ignorance, they’re about trust

An echo chamber is a system in which people have been taught to systematically distrust people on the other side. The book Echo Chamber doesn’t quite say the world around Rush Limbaugh is a cult, but they basically almost say it. This book is an empirical analysis of the world around Rush Limbaugh — Rush Limbaugh’s top people just systematically distrust and dismiss people on the other side.

This is is different from not hearing them at all (which is the filter bubble argument). The problem goes back to trust, not irrationality or disbelief of science.

A large segment of the population has had their trust subverted and undermined and directed toward what we think of as like the wrong institutions.

The way back is not to wave the evidence in people’s faces. I think people want to be like, “Oh, climate change denialists, just look at the evidence, here is the evidence,” but of course they’re not showing you the evidence, they’re showing scientists that they trust who process the evidence, ’cause not even a climate change scientist, a particular climate change scientist, has looked at all of the evidence for climate… It’s all processed.

Someone whose trust has been systematically undermined in that set of institutions will not trust evidence from sources they distrust**. And that is rational.

This is a complicated problem without easy solutions. But the first step is recognizing the story is that the other side doesn’t hear us. It’s that their trust has been undermined.**

A lot of public policy figures are fixated on the filter effect thinking we just need to create these public spaces where people can meet each other and talk to each other. That’s a standard view in a lot of political philosophy and public policy. And I think that’s not going to work if trust has already been systematically undermined. It doesn’t matter if you meet and hear the other side, if you already have a prevailing story that says they’re malicious, manipulative, evil people.

The grand takeaway

The need for clarity as evidenced by our love of both games and conspiracy theories can cause us to overoptimize and seek safety in echo chambers.

What do we do about it?

While there are no answers some hints that can point in helpful directions:

  1. Transition between perspectives

    Playfulness is the quality to transition between different world perspectives, easily, lightly, to hold your perspective lightly and slip between different ones. This can be done via literally traveling and playing games where you are forced to slip in between different value perspectives.

    It’s hard to do, it’s hard to put on different worldviews as like different outfits. This is a resonant argument for reading and earning widely:

    This is going to become the world’s oldest chestnut, but sometimes I think like, this is what the fucking humanities are for. Read some art, read some novels, motherfucker, and if you want the background paranoid view, it’s something like the world has very good reasons to get us to onboard to super simple, clear targets. And when I look at universities cutting humanities programs in favor of business schools and STEM, because those are higher-earning jobs or lead to more clearly measurable productivity, I’m like, of course, reading weird subtle art, experiencing weird subtle art, including games, but also including novels and music and all this other stuff, is this stuff that might have clued you in to different value perspectives other than make a lot of money and get a good job. And of course, they’re going to get cut out in a world dominated by hyper-simplified institutionalized values.

    One of the suspicions I have is that certain domains, especially the domains that science has a lot of success with, are the kinds of domains that admit of extraordinary clarity. And other domains, like the domain of life value and the domains of personal health and fitness and aesthetic joy, are not domains that admit of the same systemic clarity. And when we demand them, we start hitting simplified targets.

    There’s a difference between qualitative and quantitative data. Qualitative data is really rich and nuanced and subtle, but it’s really context-sensitive. It doesn’t aggregate, it doesn’t travel well.

    [insert quote: Not everything valuable can be measured, and not everything we measure is valuable]

  2. Be aware that there are forces that are trying to manipulate us

Like the sugar analogy, is this moral view or worldview too yummy? Is it just too satisfying? Did someone make this just for me and people in my cohort to swallow down?

This is definitely not a blanket. The thing is, you also get clarity and pleasure from really getting at truths. You can’t throw all that stuff away, you just have to realize that the signal has been amenable to perversion and misuse.

Keepsakes From Slatestar’s Fake Graduation Speech

Link to original post: https://slatestarcodex.com/2014/05/23/ssc-gives-a-graduation-speech/

These excerpts capture the gist of the “speech” that I want to retain.

What if education, as you understand it – public or private or charter schooling from age four or five all the way to university as young adults – is, on net, a waste of your time and money?

Addressing The Alleged Benefits

Benefit 1: The philosophical benefits of feeling connected to the beauty of mathematics, the passion of the humanities, the great historical traditions.

Testing the claim: Compare to a control group — the unschooled

“unschooling” movement, a group of parents who think school is oppressive and damaging. They tell the government they’re home-schooling their children but actually just let them do whatever they want. They may teach their kid something if the child wants to be taught, otherwise they will leave them pretty much alone.

And this is really hard to study, because they’re a highly self-selected group and there aren’t very many of them.

What do we know from the small sample?

    1. if you’d stayed out of public school and stayed home and played games and maybe asked your parents some questions, then by the time your friends were graduating twelfth grade, you would have the equivalent of an eleventh-grade education.
    2. Louis Benezet’s experiment: He decreed that in some of the schools in his district, there would be no math instruction until grade six. He found that within a year, these sixth graders had caught up with their peers in traditional schools, and furthermore that they were able to think much more logically about math problems – figure out what was going on rather than desperately trying to multiply and divide all the numbers in the problem by one another.math education before grade six is useless at best. And it’s hard to resist the urge to generalize to other subjects and children even older still.

Why is it so easy for the unschooled to keep up with their better-educated brethren? My guess is that it’s because very little learning goes on at school at all.

      • According to the general survey of knowledge among college students, 3.3% know who Euclid was, 7.6% know who wrote Canterbury, and a full 15% know what city the Parthenon’s in.
      • 36% of high school students know that an atom is bigger than an electron, rather than vice versa. But a full 59% of college students know the same. That’s a whole nine percent better than chance. On one of the most basic facts about the fundamental entities that make up everything in existence.

“But knowledge isn’t about names and dates!” No, but names and dates are the parts that are easy to measure, and it’s a pretty good bet that if you don’t know what city the Parthenon’s in you probably haven’t absorbed the full genius of the Greek architectural tradition. Anyone who’s never heard of Chaucer probably doesn’t have strong opinions on the classics of Middle English literature.

Benefit #2: The practical benefits of being able to get a job and afford nice things like food and shelter.

Testing the claim: Employment

About fifteen percent of you will be some variant of unemployed straight out of college. Another ten percent will find something part-time. And another forty or so percent will be underemployed, working as waiters or clerks or baristas or something else that uses zero percent of the knowledge you’ve worked so hard to accumulate.

As bad as you will have it, everyone who didn’t graduate college still has it much, much worse. All the economic indicators agree with the signs from the desolate wasteland that was once our industrial heartland: they are doomed. Their wages are not stagnating but actively declining

Testing the claim: More education

As bad as the job market is, staying in school looks worse. Economists warn that attending law school is the worst career decision you can make, so much so that newly graduated lawyers have nothing do to but sue law schools for not warning them against attending and established firms offer an Anything But Law School Scholarship to raise awareness of the problem. Doctors are so uniformly unhappy that they are committing suicide in record numbers and nine out of ten would warn young people against going into medicine. Graduate school has always been an iffy bet, but now the ratio of Ph. D applicants to open tenure track positions has hit triple digits, with the vast majority ending up as miserable adjunct professors who juggle multiple part time jobs and end up making as much as a Starbucks barista but without the health insurance.

We were counting the benefits of formal education. We did not do so well in trying to prove that it left you more knowledgeable, but it did seem like it had some practical value in getting you a little bit more money.

What of the costs of education?

Musing for society at large

Well, first about twenty thousand hours of your youth. That’s okay. You weren’t using that golden time of perfect health and halcyon memories when you had more true capacity for creativity and imagination and happiness than you ever will again anyway.

the financial side of it. At $11,000 average per pupil spending per year times thirteen years plus various preschool and college subsidies, the government spends $155,000 on the kindergarten-through-college education of the average American.

How about to each of the individual types?

To the one who says:

I am the 3.3%! I know who Euclid was and I understand the sublime beauty of geometry. I don’t think I would have been exposed to it, or had the grit to keep studying it, if I hadn’t been here surrounded by equally curious peers, under the instruction of enthusiastic professors.

  • to you my advice is: if you’ve sacrificed everything for knowledge, don’t forget that. When you are a paralegal in Brooklyn, and you get home from work, and you are very tired, and you want to curl up in front of the TV and watch reality shows until you are numb, remind yourself that you value knowledge above everything else, that you will seek intellectual beauty though the world perish, and read a book or something. Or take a class at a community college.

my education was worth it….because of the friends, I made here

  • my advice is similar: if you’ve sacrificed everything for friendship, don’t forget that. When you are a paralegal in Brooklyn, or a market analyst in Seattle, or God forbid an intern in Michigan, and you get home from work, and you are very tired, and you want to curl up in front of your computer and check Reddit, remind yourself of the friends you made here and give them a call. See how they’re doing. Write them a Christmas card, especially if it is December. Anything other than declaring friendship your supreme value and drifting out of touch.

my education was worth it…because of the connections I made

  • If you’ve sacrificed everything for ambition, be ambitious as hell. When you are a paralegal in Brooklyn or whatever, claw your way to the top, stay there, and use it to do something important. If you’ve sacrificed everything for ambition, don’t you dare stop at middle manager.

my education was worth it… because it helped me learn civic values, become a better person who is better able to help others.

  • If you’ve sacrificed everything to help others, don’t let it all end with donating a tenner to the OXFAM guy on the street now and then. Join Giving What We Can or go volunteer somewhere. If you’ve sacrificed everything for others, make sure others get something good out of the deal!

my education was worth it because formal education in the school system taught me how to think.

  • Sorry, one second, HAHAHAHAHAHHAHAAAHAHAHAHHHAAHAHA…I’m sorry. Ahem. To you my advice is, again, similar. If you’ve sacrificed everything to learn how to think, learn how to think. When someone says something you disagree with, before you dismiss a straw man it and call that person names and slap yourself five for your brilliant rebuttal, take a second to consider it fairly on its own terms. Go learn about biases and heuristics and how to avoid them. Read enough psychology and cognitive science to figure out why your claim might kind of inspire hysterical laughter from people even a little familiar with the field. Just don’t sacrifice everything to learn how to think and end up only rearranging your prejudices.

And To Those Us Who Feel Fleeced

Finally, some of you will say, wait a second, maybe my education wasn’t worth it. Or, maybe it was the best choice to make from within a bad paradigm, but I’m not content with that.

To you, I can offer a small amount of compensation. You have learned a very valuable lesson that you might not have been able to learn any other way. You have learned that the system is Not Your Friend. I use those last three words very consciously. People usually say “not your friend” as an understatement, a way of saying something is actively hostile. I don’t mean that.

The system is not your friend. The system is not your enemy. The system is a retarded giant throwing wads of $100 bills and books of rules in random directions while shouting “LOOK AT ME! I’M HELPING! I’M HELPING!” Sometimes by luck you catch a wad of cash, and you think the system loves you. Other times by misfortune you get hit in the gut with a rulebook, and you think the system hates you. But either one is giving the system too much credit.

To you I don’t have very much advice…if I knew how to fix the system, it’s a pretty good bet other people would know too and the system would already have been fixed. Maybe you, armed with a degree from the University of [mumble], will be the one to help figure it out.

On the other hand, someone a lot smarter than I am did have some advice for you. Poor Kurt Vonnegut never did get to give a real graduation speech, but one of his books has some advice targeted at another major life transition:

Hello babies. Welcome to Earth. It’s hot in the summer and cold in the winter. It’s round and wet and crowded. On the outside, babies, you’ve got a hundred years here. There’s only one rule that I know of, babies-“God damn it, you’ve got to be kind.”

I don’t know how to fix the system, but I am pretty sure that one of the ingredients is kindness.

I think of kindness not only as the moral virtue of volunteering at a soup kitchen or even of living your life to help as many other people as possible, but also as an epistemic virtue. Epistemic kindness is kind of like humility. Kindness to ideas you disagree with. Kindness to positions you want to dismiss as crazy and dismiss with insults and mockery. Kindness that breaks you out of your own arrogance, makes you realize the truth is more important than your own glorification, especially when there’s a lot at stake.

I Felt Bad For Picking My 3rd Grader Off

In trading, “picking someone off” means trading against a counterparty who would flake on the price they offered you if they knew what you know.

If I lift an offer on a TSLA March call option because I’m bullish, the market-maker on the other side of my trade doesn’t care. They would still sell me the call even if I texted them my rationale. But if I had the divine knowledge that Elon was going to tweet in the next 10 seconds that earnings would be reported in March not February, then I would be knowingly “picking off” the market-maker. As an options trader, you need to defend against pick-offs. You also want alerts if someone else is making a price that is not incorporating material, public info. For example, if an OPEC meeting date was moved there might be a tiny window when you could disguise a calendar spread as a routine roll when really you are trying to pick off the other side before they get the memo.

[In reality, when such news happens, market-makers will “sweep” all the resting customer orders with price limits below or above the option’s new fair value. It’s very difficult to pick off another professional who is consuming a real-time news feed.]

A few categories of pick-offs:

  • Pickoff trades related to changes in dates.

    If earnings are moved from early Feb to early March, then the “earnings volatility” needs to come out of the Feb expiry since you are no longer exposed to it, and the March options which now contain that volatility must appreciate relatively.

  • Pickoffs related to change in carry

    If a stock announces a change in its dividend that will affect the carry embedded in the options. If a dividend is slashed, the calls go up relative to the puts. Market makers need to be on top of how corporate actions affect the inputs into their pricing models.

  • Pickoffs related to changes in baskets

    If an ETF’s constituents change that affects vol of the underlying basket. If an ETF restricts creations, this can lead to the options and ETF becoming mispriced (one day I’ll tell the story of how this personally cost me 6 figures).

Pickoffs In Real Life

The tradition of trading floors is full of insane prop betting stories (NYMEX vets have so many to choose from but my personal favorite was watching one guy successfully pound 20 Coors Lites in an hour in one of the  upstairs offices).

But picking people off can be as easy as making bets with folks who are arithmetically challenged. Most people are. You only need to look up polls by @10kdiver to see that even professional investors, a subset of the population who should be able to compute a return, struggle numerically.

I am overstating the case a bit.

  1. I don’t know how many of those respondents are professional investors. I’d also admit there are times I’m impatient and just take a guess just to see the results. This is probably common behavior.
  2. When confronted with a bet, people’s defenses go up. They are wary of strangers bearing gifts and will assume there’s a catch.

Now that was a long-winded, but hopefully fun, introduction to a story that I will improve your numerical intuition and illuminate a lesson that is central to both investing and engineering problems.

The Proposition

My 3rd-grader came home with a packet of math worksheets from school. Two of the sheets, a total of 10 questions, were incomplete. He said the teacher didn’t require those sheets to be done.

What a bummer.

I thought they were the best questions from the whole packet. I asked him to solve the total of 10 questions but decided to make my request a little spicier. I offered him the following bet:

If you get them all correct, I will give you $5. Otherwise, you owe me $5.

He thought for a moment, then accepted. Before he went off to work on them, I started to wonder if my impulsive proposition was fair.

Is It A Fair Bet?

After eyeballing the questions, I estimated he had a 90% chance of getting any question correct. Another way to say that, is I expect he gets 1 wrong on average. Right off the bat,  I think I’m going to win. It’s not a fair bet.

This led me to compute a couple numbers.

  1. If my estimate of 90% hit rate per question is correct, what’s the chance he gets them all correct?

    .9010 = 35%

    That means he’s almost a 2-1 underdog

  2. What would his hit rate need to be per question to make the bet fair?

    First we need to convert what a fair bet means in math language. This is straightforward. Since it’s an even $5 bet then the fair proposition would not be, on average, he gets 1 wrong, but that he gets all the questions correct 50% of the time. 

    x10 = 50%

    x = .50 1/10

    x = 93.3%

    So if he had a 93.3% chance of getting any single question correct, then he has a 50% chance of winning the bet.

Flipping The Odds In His Favor

I’m not trying to take candy from a 3rd grader, I just wanted to make him more eager to do the questions. So, I tweaked the bet after he returned with the answers. Each of the 2 worksheets had 5 questions each. I decided to batch the bet as follows:

If he gets all the questions correct, he wins.

If he gets all the questions in one batch correct but not the other, it’s a push.

If he gets less than 100% on each batch then he loses the bet. 

How did this tweak alter the proposition?

In the first bet, if he got anything wrong, he lost. With these rules, he only loses if he gets, at least, one wrong in each batch.

We need to analyze all the possible outcomes that can occur with 2 batches.

First, we must ask:

What is the probability he gets all the questions right within a batch of 5 questions?

Assume he has a 90% hit rate again. 

.95 = 59%

So for either batch he has a 59% chance of getting a perfect score and a 41% chance of getting at least one wrong (ie a non-perfect score). 

Now we must consider all the possible outcomes of the proposition and their probabilities.

  1. Perfect score in both batches

    59% x 59% = 34.8%

  2. Perfect score in one batch but not the other

    59% x 41% = 24.2%

  3. At least one wrong in both batches

    41% x 41% = 16.8%

If we sum them all up we get 75.8%.

Wait, these don’t add to 100%, what gives??

We need to weigh these outcomes by the number of ways they can happen.

The possibilities with their probabilities are as follows:

  • Win, Win = 34.8%
  • Win, Lose = 24.2%
  • Lose, Win = 24.2%
  • Lose, Lose = 16.8%

Collapsing the individual possibilities into the proposition’s probabilities we get:

  • Win: 34.8%
  • Push: 48.4% (24.2% + 24.2%)
  • Lose: 16.8%

These probabilities sum to 100% and tell us:

  1. The most likely scenario of the bet is a push, no money exchanged
  2. Otherwise, he wins the bet 2x more than he loses the bet

This is a powerful result. Remember, his hit rate on any individual question was still 90%. By batching, we changed the proposition into a bad bet for him, into a good bet because he gets to diversify or quarantine the risk of a wrong answer.

Even More Diversification

With 2 batches we saw the range of possibilties conforms to the binomial distribution for n = 2:

p2 + 2p(1-p) + (1-p)2

where p = probability of perfect score on a batch

In English, the coefficients: 1 way to win both + 2 ways to push + 1 way to lose both

What if we split the proposition into 5 batches of 2 questions each?

These are the bet scenarios:

  1. Win =  5 pairs of questions correct
  2. Lose = at least one wrong in each of the 5 pairs to lose the bet
  3. Push = and any other combination (i.e. 3 perfect batches + 2 imperfect batches)

How do the batch outcomes roll up to the 3 bet scenarios?

  1. We must count how many ways there are to generate each of the scenarios.
  2. We must probability weight each way.

The summary table shows the work.

[Note the coefficients correspond to the coefficients for the binomial expansion (x+y)5 which is also the row of Pascal’s Triangle for N = 5]

Outcome Scenario # of ways to win probability weights Count x probabilities
5 wins (wins the bet scenario)  win combin(5,5) = 1 0.81⁵ = 34.9% 1 x 34.9% = 34.9%
4 wins, 1 loss (push) push combin(5,4)= 5 .81⁴ x .19 = 8.2% 5 x 8.2% = 41%
3 wins, 2 losses (push) push combin(5,3) = 10 .81³ x .19² = 1.9% 10 x 1.9% = 19%
2 wins, 3 losses (push) push combin(5,2) = 10 .81² x .19³ = .45% 10 x .45% = 4.5%
1 win, 4 losses (push) push combin(5,1) = 5 .81 x .19⁴ = .11% 5 x .11% = .55%
5 losses (lose the bet) lose combin(5,0) = 1 .19⁵ = .025% 1 x .025% = .025%
Total: 2⁵ = 32 100%

The net result of quarantining the questions into 5 groups of 2:

35% chance he wins the bet

65% chance he pushes on the bet

Nearly 0% chance he loses on the bet!

If you took this quarantining logic further and treated each question as its own bath then the new phrasing of the bet would be:

If he gets every question correct, he wins

If he gets every question incorrect, he loses

Any other scenario is a push

The corresponding probabilities:

Win = .910 = 35%

Lose = .110 = E-10 or 0%

So the benefits of separating the bets were mostly achieved above when we batched into groups 5 pairs of questions.

Conclusion

The first required my son to get a string of questions correct. Even though I estimated he had a 90% chance of getting any question correct, by making the net outcome dependent on each chain-link the odds of his success became highly contingent on the length of the chain.

With a chain length of 10, the bet was unfair to him. By changing the bet to be the result of success on smaller chains (the batches) I changed the distribution of outcome. It did not increase his chance of winning, but it reduced his chance of losing by creating offsetting scenarios or “pushes”. In other words, the smaller chains reduced his overall risk without sacrificing his odds of winning. It was a free lunch for him.

When you create long chains of dependency (ahem, positive correlations), an impurity in any link threatens your entire proposition. In these examples, we are dealing with binary outcomes. This is a trivial analysis. With investing, the distributions of the bets are not easily known.  It is well-known, however, that diversifying or not putting all your eggs in one basket is a free lunch. Still, if the investments are highly correlated, you may be fooling yourself and depending on a long chain.

Imagine if the 10 questions my son had were the type of word problem where the answer to each question is the input to the next. That’s a portfolio of highly correlated assets. If you are “yield-farming” crypto stablecoins you have probably thought about this problem. Spreading the risk across many coins can offset many idiosyncratic risks to the protocols. But is there a translucent, hard-to-see chain of correlation tying them all together that only reveals itself when the whole background goes black? That’s systemic risk. Ultimately, the only hedge to such a risk is position sizing at the aggregate level where you sum the gross positions. This is why stress-testing a portfolio to that standard is a quant’s “last level of defense”.

Jarvis, what happens to my portfolio when all correlations go to one?


[If you are in the investing world you will see parallels to this lesson in the ergodicity1 problem.]

Oh yea, how’d the bet with my 3rd grader go?

He got a perfect score on one batch of 5, and he got 1 wrong in the second batch. So the overall bet was a push, and his old man didn’t do so bad in estimating he’d have a 90% hit rate.

If you use options to hedge or invest, check out the moontower.ai option trading analytics platform

Financial Statements For A Kid

This week’s Money Angle is more useful to parents or teachers. Let’s go.

Encouraging Saving

We have had several false starts with an allowance/reward/chore system at home. In the past 6 weeks, our current system for our 8-year-old has gotten traction so I thought I’d share it.

Here are the features:

  • 2 sources of income

    a) Chores

    He has a list of chores he’s responsible for such as making his bed (and my bed, muahaha), taking out the trash, folding laundry, cleaning up. On a weekly basis he can be paid 0, $3, $6, or $9 depending on how good a job he does and whether he goes the extra mile either with thoughtfulness or initiative. The slacker has yet to earn $9.

    b) Interest

    He gets paid 1% a week rounded up to the nearest dollar. Sorry DeFi carry-traders, the Moontower yield farm is neither permissionless nor decentralized.

  • 3 account types

    a) Spend

    This is like a checking account. He has access to these funds but they earn no interest.

    b) Save

    This is the savings account. He cannot access this money (although in the future he’ll be able to invest it) but it earns interest.

    c) Share

    This is a charity account. Funds here have to be used for charity or donation. This account also earns interest.

The rules:

  • Chore income can be split however he wants so long as 1/3 goes to the share account. He has been putting 1/3 into each account uniformly.
  • Interest income which is based on the save + share accounts is deposited into the save account.

You can fiddle with the rules to encourage whatever behavior you like. Below is a screenshot of an earlier “statement”. Notice how it maps to an income statement and a balance sheet.

Teaching Kids About Money

There are many finance professionals who want to share their knowledge with youths so they can be more equipped to handle money, understand debt, compounding, investing, and so forth. Count me among them.

The knowledge is not just practical but understanding money is a generally useful lens for understanding the world. Finance is abstraction. It’s symbols. It’s an operating system. It modulates the relationship between current and future states of the world and your life.

If taught well, it can be highly engaging and overlap with how we make sense of things that on the surface might appear to have little to do with money (the sub-genre of “freakonomics” probably over-optimizes for insight-porn but that’s a meta-recursive proof of my point).

I’d like to workshop some introductory lessons locally and have been doing a bit of research to that end. I’m compiling freely available resources to that end so I can mix it with my own thinking to create lessons.

Please share your favorite money education resources for kids and teens!

If some of the newer readers are parents you might appreciate some of these prior posts:

  • A Socratic Money Lesson For 2nd Graders (3 min read)
  • Hands-On Resources to Teach Kids About Business (2 min read)
  • Bohnanza Is A Great Trading & Business Game (3 min read)
  • Thoughts About Monopoly As A Teaching Tool (2 min read)
  • Investing Books For A Teenager (2 min read)

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.

If you use options to hedge or invest, check out the moontower.ai option trading analytics platform

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 2.

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 it3:

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 second derivative that is 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 chart4.

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)5.

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.

  • Investing

    In Convexity in DCFs, Robert Martin uses this intuition to show why choosing an asset that increases cash flow by 30% on average can be worth more than an asset which always grows at 30%. The average of the growth rate is a poor guide to the average output of the function (the function being the total return) because the function is convex with respect to growth rate.

    The math of compounding is non-linear so the impact of a 40% growth rate is greater than 4/3 of a 30% growth rate. Compounding is not intuitive, but if we keep Jensen’s inequality in mind, we can quickly realize that our minds will misdiagnose the impact of the input parameters (in this case 30% vs 40%) on final payoff functions because compounding is a convex phenomena. By remembering that Jensen’s inequality exists, we remember to slow down before estimating the end result of compounded inital inputs even if the inputs don’t seem to differ by large amounts.

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 6 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 credentials7 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