The worse you feel about us, the better you should feel about the world

Epistemic status: attitudinal and naïve

Occasionally I meet someone with a really awful view of human nature. They look at the world, see all the ways we fail to cooperate, and conclude humanity has been doomed from the start. To them, there’s something like an inherent deficit of altruism or foresight that prevents us from solving our most pressing problems. Specifically, they tend to be disappointed with progress on climate change or nuclear weapons, and are pretty dour as a result.

I’m confused by the dour-ness. I’ll grant cooperation is difficult, but if you really hold humans in such low esteem, shouldn’t you be constantly amazed by everything around you?  How can the misanthrope react to the cooperative feat that is you purchasing a pencil with anything but astonishment? Thousands of people had to coordinate their actions to collect the timber, shape the pencil, insert the graphite, and ship it to your locale. This is impressive to almost everybody. To the misanthrope, it should look like a miracle.

And miracles like this happen every day. Food from every corner of the globe is readily available at my grocery store. I can get nearly any book shipped to my door within two days. Humanity’s knowledge is at my fingertips. All of this is due to the coordination of thousands of actors.

There are other, arguably more important miracles. In much (though not all) of the world, I can openly practice any religion and not feel threatened for it. In many of the same places, I can critique those in power without fear of retribution. Throughout much of human history, almost nobody had these privileges.

All of these miracles have some kind of cooperation at their core. Cooperative actors create goods and services and distribute them to those in need. The political freedoms we enjoy are based on tolerance, which is a special kind of cooperative attitude [1].

I stand with the misanthropes. I believe there are dark, myopic parts of humanity that will never go away. But, I believe the natural consequence of this view is amazement, not depression. Things — more or less — work, and such an uncharitable view of human nature by itself could not predict much of the current world. Our faults persist, but in some significant capacity, we’re still able to cooperate. That’s amazing.

[1] “I disagree with their views on X, but these people are still worth cooperating with.”

A Dialogue on Income Inequality

Andrew and Parker are driving on Mulholland. To the south, they peer into the LA basin and glimpse downtown. To the north, the entire San Fernando Valley lies beneath them before it yields to the base of the Santa Susana mountains. Multimillion dollar houses flank them on both sides of the street. Most are gaudy, but a few have good taste. All cost more than the average American will earn over several lifetimes. 

Andrew: Nobody needs that much house! Why should a human on earth have 10,000 square feet, a movie room, wine cellar, infinity pool, vineyard, and 8-car garage?

Parker: The views are nice, though. I wouldn’t mind paying a lot to make sure I see sweeping vistas every day.

Andrew: Yeah, I wouldn’t mind the view either. Still, if you can afford a house like that, it’s a sign you’re making too much money.

Parker: What do you mean?

Andrew: Why is it alright for these people to have so much when others have so little? They live in the lap of luxury while others don’t even have a home. You’ve seen Skid Row; nobody should have to survive in those conditions. People sleeping on the streets and not having enough to eat is unacceptable. Take away 90% of a millionaire’s income and they can still live comfortably. Economic inequality is at an all-time high, and there’s something deeply wrong with that.

Parker: Yeah, I definitely agree. Skid Row should not exist, but I’m confused — what does inequality have to do with this?

Andrew: What’s there to be confused about? Some live in poverty while others spend millions on ugly houses they barely live in. That’s unjust. If we close the gap between the best and worst-off in our society, healthcare, education, housing, as well as other essential goods, will all be more accessible. People will live better lives.

Parker: I still don’t see the connection to inequality. You are —rightfully— upset some can’t afford basic services or have an acceptable standard of living, but what do other people being wealthy have to do with it? The problem, as far as I can tell, is some people don’t have enough resources to live fulfilling lives. I don’t see a clear connection between their deficit and others’ surplus.

There’s a difference between one person having enough of a resource and two people having the same amount. The concepts don’t necessarily follow from each other. Imagine it’s dinnertime and we each only have a bottle of ketchup to eat. This will not do. Ketchup is not nutritious, and eating it alone is pretty gross. We are equal, in the sense both of us have the same resources at our disposal, but it does not follow either of us has enough food. Likewise, it’s possible for both of us to have enough food and be unequal. Suppose we both got groceries and you are having a lobster feast at your house while I settle for a turkey sandwich at mine. There’s a dramatic inequality between us, but this doesn’t imply my turkey sandwich isn’t enough to sate me. I might want to be invited over, but I am still able to live and function on my sandwich. It’s enough.

My point is poverty and inequality are conceptually unrelated. If you care about one you don’t need to care about the other. That’s why I’m having trouble with your position. You talk about reducing poverty but then go on about inequality, which is different.

Andrew: Maybe the concepts of poverty and inequality are distinct, but reducing inequality is still important. Inequality is about making sure the world is fair and everyone gets a chance to lead a fulfilling life. Imagine a single mother who works two jobs supporting her kids and barely makes ends meet. She can’t afford a good education for her children, and she’s sacrificing her long-term health to keep them above the poverty line. Now remember how some of the wealthy earn their money. They just invest, sit back, and live off their returns without having to lift a finger. It’s deeply unfair the single mother has to toil and endanger herself while the millionaires can just park their assets somewhere and live lavishly. When one party works extremely hard and gets little, while the other does almost no work and gets a lot, it’s unfair. We need to remedy that.

Parker: I agree something is wrong there, but it doesn’t have to do with both the millionaire and the single mom. At base, we’re angry the single mom’s efforts go unrewarded. We believe everyone deserves a reasonable standard of living and opportunity for their children, so the fact she works so hard in unsuccessful pursuit of these offends us. Notice the issue fundamentally involves the relationship between her efforts and quality of life.

But the problem ends here. Knowing millionaires are lounging around getting ROI is irrelevant. Even if no millionaires existed, we should be just as angry about the single mom’s situation.

Andrew: But what you’re describing is still unfair. Suppose millionaires exist and the single mom gets a reasonable standard of living, like you say, for herself and her children. The millionaires might still be working half as hard and making twice as much! Think about Jeff Bezos. He might be working incredibly hard, but that cannot possibly justify $130 billion. Some earn a disproportionate amount for their effort, and this should hurt every egalitarian bone in your body.

Parker: Why? Isn’t it alright to be paid differently based on different contributions? LeBron James probably works harder than his fellow Lakers, but let’s imagine he decided to train the exact same number of hours as his teammates from tomorrow on. He shoots the same number of free-throws as them, runs the same distance as them, and lifts the exact same number of weights as they do. Most likely, he’s still going to be a better basketball player, so why shouldn’t he be paid more? He’s still the player putting points on the board, getting rebounds, etc. If anything, it would be unfair if he performed so much better than his colleagues but got paid the same.

Andrew: Your example doesn’t work. The reason why LeBron is a better basketball player is because he practiced much more in the past. Even before he starts training just as much as his teammates, he has still spent more hours on the court than just about anyone else. This is what makes him a better basketball player. If he is paid more like you describe, his income is still commensurate with his efforts.

Parker: But what if Milton Friedman (5’1, ~120 lbs.) and LeBron put in the same effort? Hour-for-hour, Milton spends his life on the court practicing with the same intensity as him. Should they be paid the same? Probably not. It’s safe to assume no matter how much Milton practices he will never be as good as LeBron. He’ll be decent, but don’t expect league MVP or even a triple-double anytime soon. If Milton can’t play at an NBA level, why give him an NBA salary?

LeBron is paid so much because he makes a large contribution. His work ethic, as well as his individual gifts and talents, give him the ability to do so. Milton shouldn’t feel aggrieved he is paid so much less as a basketball player, and we shouldn’t worry about it, either. His contributions are tiny.

Andrew: But you’re still missing something. Being a poor basketball player won’t upset Milton because he can fall back on economics and make his living that way, but not everyone has that option. If basketball is the only way people earn income, and Milton can’t earn enough to escape poverty, then there’s an injustice.

Parker: I agree. Everyone, no matter what kinds of contributions they can make, should be able to have a reasonable standard of living for their efforts. If we were an economy of basketball players and the Milton Freidmans of the world would starve because nobody wants them on their team, we should remedy that. Perhaps a livable basketball minimum wage is in order, or maybe we can change the rules and take Milton off the roster to have him work in the front office where his abilities and inclinations are better suited.

Notice we can solve the problem without talking about LeBron and how much he makes relative to other players. Inequality of income is not the root issue here.

Andrew: You’re not much of an egalitarian, are you.  

Parker: I think I’m an egalitarian in a specific way. Obviously, we need equal political liberties. I’m very enthusiastic about equal voting rights, an equal ability to express ourselves, and fair and equal treatment before the law more generally. I’m pretty sure nothing will shake my confidence in these concepts.

My views on the distribution of wealth and income stem from two principles that also take equality seriously. As mentioned, I believe everyone is entitled to a minimum standard of living in exchange for their efforts, regardless of the size of their contribution. Conscientious people, striving to care for themselves and their loved ones, should not live in poverty. Whether their inclinations or abilities match what the market demands is irrelevant. I should be clear this minimum standard of living is not glamorous, nor would most be comfortable staying there for the long-term. Yet, it is a dramatic improvement from poverty and would protect our ability to pursue meaningful lives. In a slogan, I’m for an equality of minimum condition.

The second principle is concerned with compensation. When individuals make contributions, they should be rewarded accordingly. If you invent a vaccine, write a novel, or create an organization, your accomplishment should be recognized in proportion to the good it brings. In the same way paying LeBron the same as his teammates when he’s league MVP is unfair, there’s something wrong with paying a great author the same as a mediocre one. Human achievement should be celebrated, and there is nothing wrong if that involves giving some dramatically more than others.

This idea appears more anti-egalitarian than not, but it embodies equality with respect to accomplishment. Part of rewarding achievement fairly is giving approximately the same level of recognition to contributions of the same import. I acknowledge we need to control for context to see how noteworthy an achievement is, but, in general, we should strive to reward contributions solely based on merit.

As a result, I think income inequality is conceptually unimportant. I’m concerned with an equality of minimum condition and rewarding those who push the needle on human progress and achievement. The bare knowledge two people earn vastly different sums of money does not bother me, and it shouldn’t upset you, either.

Andrew: I see a conflict here. You care about political liberties, but aren’t those inextricably tied to the resources at our disposal? The person with the funds to purchase billboards on Sunset has more political power than you or me. Freedom of speech is much more valuable to them, creating positions of advantage and disadvantage in the political landscape. This seems incompatible with support for equal political liberties.  

Parker: You’re right. I need to clarify my position. There’s a distinction between political liberties and the worth of those liberties to an individual. Liberties grant the ability to undertake certain actions. For instance, having liberty of movement allows us to travel unrestricted between different locales. Freedom of speech allows us to say what we believe (without inciting imminent lawless action, among other restrictions). Having these liberties means we are given the mostly unrestricted option to undertake the actions in question.

However, two individuals having identical liberties may not exercise them in the same way. Suppose we both have freedom of speech, but you’re a celebrity and I’m a recluse. Both of us generally have the ability to say what we believe, but this capacity is much more valuable to you. You have a legion of fans who will listen and react to your opinions, while I don’t have any audience. In fact, taking away my freedom of speech probably wouldn’t have any practical implications, as recluses don’t communicate with anyone in the first place.

This situation seems unobjectionable. If freedom of speech were of equal value to everybody, we would give equivalent coverage and consideration to eccentric hermits as well as established public figures. We care deeply about individuals’ abilities to exercise their liberties, but how much, or if, they benefit from such exercise is beyond our concern.

Andrew: This doesn’t protect against another form of political inequality, though. Even if we all have equal liberties, some might use their wealth and advantage to secure disproportionate political influence. Billionaires can donate to congressional campaigns, hire lobbyists, and purchase political ads to shape the ideological make-up of a government. Instead of having one vote, like the rest of us, the wealthy can in effect unilaterally determine the result of an election. This violates the basic political egalitarianism at the heart of democracy. As soon as some voices count more than others, we become an oligarchy masquerading as something else. It’s difficult to imagine how extreme wealth inequality and a functioning democracy can peacefully coexist.

Parker: This has bothered me, too. Initially, it seems unlikely we can allow radical inequality on one dimension and be committed egalitarians on another when the two seem closely related. Yet, I don’t think this is a knock-down argument against my position. First, notice the problem has to do with a relation between the two dimensions; economic inequality has a negative influence on political egalitarianism. We agree this relation is harmful, but there are at least two ways to address the issue. Acknowledging the relation as fixed and eliminating economic inequality is one method. If there is no inequality, the reasoning goes, then it can’t exercise a negative influence on politics. The other option is combatting the relation. Ensuring economic inequality and political power are independent of each other can also work. If money had no influence on politics, then political equality would be secure. Wealth could vary dramatically between individuals with no adverse consequences.

The objection you raise is valid and important, but we can resolve it without affirming egalitarianism on every dimension. If we were to argue from here, the conversation would then be about which of the two methods described is better. Your point is good, but it only kicks the can down the road.

Andrew: I have another concern about your reasoning. You mentioned if money had no influence on politics, then inequality could run rampant with no cause for concern, yes?

Parker: I would say “fluctuate wildly” rather than “run rampant,” but yes, that’s my view.

Andrew: When you say that, you’re ignoring the psychological effects of inequality. If everyone else is much wealthier than you, you’re bound to feel inferior. This can happen despite achieving a “reasonable standard of living.” You might be ostracized based on the clothes you can afford, the house you live in, or the car you drive if you’re the poorest in a community. The result is real emotional pain that can inhibit normal interaction among peers and your pursuit of a meaningful life.

Parker: But resolving income inequality is not the best solution. Here’s an unconventional example that illustrates my point.

Physically unattractive people are also mistreated. Research has shown unattractive people are presumed guilty more often in court, are less likely to receive help from strangers, friends, and family, and are punished more severely in school than their attractive counterparts. Indeed, men in the bottom third of attractiveness earn 22% less in lifetime earnings than those of average physical attractiveness. This is likely due to discriminatory practices (Minerva).

Presumably, being physically unattractive can also cause psychological distress, but our solution is not the make everyone equally beautiful. If I understand the zeitgeist correctly, we aspire to celebrate these differences and treat one another equally, regardless of physical characteristics. Attraction is unrelated to moral worth or human value, so it should be irrelevant to how we treat one another.

Income inequality is similar. Like the unattractive, low-income people are mistreated because of prejudice. We can either erase the differences to ensure everybody is treated with respect or realize income has no relation to an individual’s moral worth. The latter seems more consistent with the tolerant, accepting people we imagine ourselves to be.

Andrew: But there are two types of psychological harm income inequality can bring. You’ve addressed the damage that comes externally from poor treatment, but what about the internal effects on self-esteem? Even if they’re not mistreated, the least-wealthy in a community can begin to rationalize their relative economic position. Simply seeing others who are better-off can foster feelings of inferiority and lead individuals to conclude they have lesser moral worth.

Parker: But they’re wrong. All humans have equal moral worth, regardless of whether they feel it is true. It could be the case the market values their individual skills differently leading to different levels of income, but wealth does not track moral status. Humans might be treated differently —in a fair and reasonable manner— based on relevant characteristics, but it does not follow they are above or below others in the moral hierarchy. The blind are prohibited from driving, but they are not morally inferior to the rest of us. Celebrities might get preferential treatment at public venues, but their lives are no more valuable than ours.

I acknowledge there are areas where moral equality necessitates equal treatment. Politics is one of them. Our equal rights reflect our equivalent moral worth, regardless of other contingent properties. Wealth, power, sex, gender, or race (ideally) do not give someone more freedom of speech than others, or less of a right to vote than her peers.

The other is living standards. There’s debate about the exact qualities that yield moral status, but our ability to formulate a life plan and follow it is most likely relevant. To respect this, we must ensure each human has reasonable opportunities to pursue her plan. Providing a minimum standard of living in exchange for conscientious behavior does so.  

Large swaths of the economic sphere do not fall under either of these categories. It’s possible to affirm our equal moral worth while still having an unequal distribution of resources.

Andrew: Do you understand how callous this sounds? You’re discounting peoples’ legitimate feelings. You will never be able to comprehend what they’re going through, or how the state of the world makes affects them, yet you dismiss it all without reservation.

Parker: I would be more sympathetic if moral value was the sort of thing where feeling it to be a certain way makes it that way. Moral worth is “real” in the same way trees and rocks are real. It’s a quality imbued in each of us that is independent of what we think or want to believe. Nothing can change it, so it’s unclear why we should accommodate people who believe theirs’ has been diminished or destroyed.

We need a distinction, though. Individuals can believe their moral worth was disrespected without actual disrespect occurring, but it does not follow all instances of perceived offense are illegitimate. We should feel aggrieved when our humanity is ignored or violated, and acting on these feelings brings us closer to justice. I only want to emphasize our unrefined moral intuitions are imperfect guides to whether a transgression actually occurred.

If the theory I’ve described is correct, the internal psychological distress caused by income inequality is unjustified. An unequal distribution of resources is not an affront to anyone’s moral worth. Yes, certain feelings might arise, but they are subtly misguided. That some parties are dissatisfied is no reason to reject what I’ve been saying. Ideally, when people feel disrespected, our first responsibility is to see if disrespect has occurred. If so, we remedy the situation, giving redress when appropriate. Else, we demonstrate why their humanity was unoffended, and how they can come to recognize this. I believe your objection puts us in the latter situation.

Andrew: I’m not convinced, but suppose everything you’re saying is correct. This doesn’t change the fact your theory is politically unworkable. Nobody gets far calling other peoples’ feelings “unjustified” or “misguided.” If we allow income inequality to run rampant, as your theory permits, a new age of political instability will begin. People will feel mistreated, and no amount of philosophizing can change that. Perhaps a new revolution will occur where the wealthy minority is removed from power. Maybe it will be violent. Perhaps the state will dissolve into anarchy — who knows? My point is your theory fails the stability test. No matter how “right” it is, it will never be accepted, and no sane government valuing self-preservation will take it seriously.

Whether or not you think income inequality should be resolved for its own sake, you must recognize its practical implications. Inequality of all sorts foments unrest. Good governance requires a stable government, and constant agitation threatens that. Insofar as you care about good governance, you must also care about income inequality, as attenuating the latter preserves the former. You might not think income inequality is inherently bad, but related commitments can force you to acknowledge it’s a problem.

Parker: You’re right, but I think these distinctions are important. I admit if I was a policymaker, I would keep an eye on inequality and ensure it’s not too severe. Yet, my interest in economic inequality would be derivative in that it stems from an interest in social stability. There is, I maintain, nothing morally repugnant about dramatic differences in wealth. What’s dangerous about inequality are the riots and unrest that might ensue if it gets too dramatic. Social stability is important enough for me to sacrifice some of my ideals, but this doesn’t change my philosophical beliefs. There are good theoretical reasons why economic equality is not inherently valuable, but I don’t want my policymakers dying on that hill. Anarchy doesn’t listen to theoretical reasons.

Andrew: I’m glad we see eye-to-eye on that. I still think economic equality is independently important, but at least I understand why you believe otherwise. I also think it’s interesting how your relatively extreme philosophical belief is tempered when it confronts certain political facts.

Parker: Yeah, it’s interesting.

Here is the link to the Minverva article. The main ideas covered in the dialogue aren’t particularly original. For instance, Harry Frankfurt’s article on economic inequality has heavily influenced my thinking.

My personal views on economic inequality aren’t Andrew’s and aren’t Parker’s. I think Parker’s onto something deep and important, but I have reservations about some of what she says. If you want to hear my full thoughts, send me an email and we can chat.

Tennis and Probability


Imagine there’s a tennis match in Kyrgyzstan between two players you’ve never heard of. You don’t know their rankings, match histories, or even what they look like, but you’re tasked with predicting who will win. The only piece of information at your disposal is player1 has a p chance of winning any given point against player2. In other words, if player1 and player2 played 100 points, on average player1 would win p*100 of these points.

Given only the information above, who do you think will win? How confident should you be they will win?


I’m sure there’s a mathematical pen-and-paper way to solve this problem. I don’t know it, so we’re going to rely on computer simulations. We can understand how likely it is player1 will win by simulating some large number of matches between both players while factoring in player1’s chance of winning any given point. Our confidence in player1 is represented by the proportion of matches she wins out of the large number simulated.

Here’s an example. Suppose we know player1 has a .51 chance of winning any given point against player2. At 15-0, she has a .51 chance of winning the next point and going up 30-0. At 0-40, she has a .51 chance of getting to 15-40, but a .49 chance of losing the point and the game.

To determine our confidence player1 will win an entire match, we might simulate 1000 such matches between player1 and player2 and count how many player1 won. For instance, she might win 586 of the 1000 matches. As a result, if all we know is player1 has a .51 chance of winning any given point, we might say we are 586/1000 = 58.6% confident she will win the match.

The model we consider does this for a specific type of tennis match. All simulated matches are best-of-3 sets with a 7-point tiebreaker played at 6 games all. What’s more, the deuce is “played out,” meaning players must win by two points to win a game. I chose this format because it’s the most common type played on the ATP and WTA tours.  

We run the simulation and graph the results. 3000 matches were simulated for 50 probabilities on the interval [.4, .6]. The exact code used for all the simulations in this post is here.

Figure 1

Figure 1 plots the probability player1 has of winning any given point against the probability she wins the match. For instance, if she has a .4 chance of winning a point, it’s very unlikely (~0% chance) she will win the match. Likewise, if she has a .60 chance of winning a given point, her probability of victory is ~100%.

As expected, if player1 has a .5 chance of winning a point, she has a 50% chance of winning the match. Based on this, we might expect a .525 chance of winning a point to yield a 52.5% chance of victory. Yet, our intuition fails to grasp the benefits of a slight edge. Just increasing the probability of winning a point from .5 to .525 causes the probability of winning the match to increase from 50% to 75%! A .025 increase in win probability at the point level leads to a 25% increase at the match level. To put this in perspective, we might say the difference between a player that wins 50% of her matches and one that wins 75% of them is the latter wins 2.5 more points out of 100 than the former. To make it even more concrete, we can surmise the difference between Pete Sampras (64 career singles titles) and Carl-Uwe Steeb (3 career singles titles) is only 2.5 points per 100!

Fully explaining Pete Sampras’ dominance and Carl-Uwe Steeb’s adequacy is far beyond the scope of our model, though. At best, we can say a player with a .525 point win rate would have a match record like Sampras’. However, this does not imply Sampras actually won 52.5% of the points he played or a player who does win that proportion will be a 14-time grand-slam champion. We are only noting similarities between the statistics of idealized players considered in the model and actual ones. Resemblance only suggests both players could have other common properties.

Still, the results are striking. Player1 only needs a .55 probability of winning a given point to win ~95% of the matches she plays. Approaching .6 almost guarantees she’ll win every time. Even a .51 probability of winning a point increases the odds of winning to ~.6.

Any tennis player will tell you the margins are slim, but now you’ve seen it computationally demonstrated. Common tennis platitudes like “every point counts” and “make them work for each point” take on additional significance. The practical implication is to focus more on individual points as opposed to games or sets when playing a match. Any competitive tennis player should already do this, but now they have an additional, semi-scientific reason to do so.

Back to Kyrgyzstan. We understand player1 has a p chance of winning a given point. If p > .5, we should be confident she will win. If p > .525, we should be really confident she will win. The same holds on the other side of .5. If p < .475, we should root for an upset. If p < .45, there’s pretty much no hope.

There’s another match happening in Zanzibar. We still know player1 has p chance of winning any given point, but now we know something else: she chokes. Whenever she has game, set, or match point, her probability of winning said point drops from p to pc. As an example, imagine her p = .55 and pc = .35. Suppose she’s also up 40-15 in a game against player2. Although her probability of winning a given point is .55, because it’s a game point, there’s only a .35 chance she wins the next point to secure the game. If she was up 6-4 in a tiebreaker, the same decrease in probability occurs. There’s only a .35 chance she wins the tiebreaker 7-4.

A similar problem presents itself. Given player1’s p and pc, will she win the match? How confident should we be in our prediction?

We run the same model in the previous section with some minor modifications. Whenever player1 is poised to win a game, set, or match, her probability of winning the next point plummets to some pc. As in the previous section, we vary p within [.4, .6] and observe player1’s probability of winning the match. What’s different now is each curve is associated with a constant pc.

Figure 2

The blue “standard” curve represents how player1 would perform sans choking. It is the same curve depicted in Figure 1 where her probability of winning a given point is p regardless of the score. The others record what a player’s match record at a certain p would be like given her pc. If player1’s pc = .45 for instance, her performance suffers heavily. At p = .5, she has only ~40% chance of victory compared to the 50% chance observed with an identical p in the absence of choking.

Past p = .5, a .05 decrease in pc roughly corresponds to a 10% reduction in the chance of winning a match. If player1’s p = .525, and pc = .45, her chances of winning would be 62%. However, if her mental game falters and pc drops to .4, her chance of victory decreases to 53%.

Clearly, any level of choking impedes performance. If we know player’s pc is low, we should expect a rather large p to compensate. Player1’s coach might suggest addressing the root factors of unclutchness. A sports psychologist or deep reflection might increase pc, or eliminate choking altogether.

There’s a third match in Andorra and we’ve learned more about player1. She no longer chokes at this junction in her career, but her play has become streaky. Given she won the previous point in a match, there’s a probability ps she will win the next point as well. If ps = .8, for example, there’s an 80% chance she will win the next point after winning the previous one. However, streakiness goes both ways. Given she lost the previous point, there’s an 80% chance she will lose the next one as well. The first point of every game (and tiebreak) is a fresh start, though. The chance she wins that point is p. From then on, ps reigns.

To get a sense of our credence in player1’s performance, we run another simulation.

Figure 3

This is a heatmap. The color of the block occupying the (probability of winning first point, ps) coordinate corresponds to the probability of winning the match. Light colors indicate high probabilities while darker colors represent low ones. As an example, if player1 has a .42 probability of winning the first point of any given game or tiebreaker and her ps = .9, her probability of winning the match is around .3. Here, we note heatmaps favor concise representation over numerical precision. The bar on the right gives us only a general idea of what probabilities are associated with which colors.

Still, patterns are clear. Variation in ps only has noticeable consequences at the extremes. If p is low, then a high streakiness almost guarantees losing the match. Observe how the probability of victory is only ~20% when p = .4 and ps > .9. However, for any value of ps < .9, the probability of winning the match hovers around 40% if p remains fixed. Similar results occur for higher p’s. Only when ps is large do we see its influence on the chance of victory. At values of p close to .5, ps has no apparent effect. The probability of winning the match stays roughly constant as ps varies.

It might look like high ps, or streakiness, is an advantage when p is sufficiently greater than .5. This is true, but the gain is comparatively small. Recall the Kyrgyzstan model where player1’s probability of winning the next point was constant throughout the match. If p = .525, player1 had 75% chance of victory. Here, for all levels of ps, p = .525’s probability of victory hovers around 60%. Player1 is much better off winning 52.5% of points in general as opposed to embracing streakiness.

We should note streakiness as we’ve defined it “smooths” performance at low values of ps. If ps = .5, this means players have a 50% chance of winning any given point after the first. In other words, we’re saying the players are just about even after the initial point. If p = .4 in the Kyrgyzstan model, a player has almost no chance of winning. In this model, p = .4 gives an ~40% chance. What it captures at these kinds of values is the dynamics of one player frequently winning the first point, and then both players having an equiprobable chance of winning the following ones.

We take these results to Andorra. Streakiness only concerns us at the extremes. If player1 has an exceptional chance of winning the first point and is very streaky, we’re confident. If she’s dismal on first points but just as streaky, we lose faith.


I’m excited about these models for two reasons. First, they suggest other interesting questions to tackle. How do the probabilities change in 5-set matches? What about no-ad? What if a player is brilliant on break points? What happens if we incorporate a serving advantage? Can we combine several of these models? The list goes on.

The second reason is epistemological. Can models of this type provide ever provide an explanation of real-world phenomenon? I talked about the Kyrgyzstan model being insufficient to explain Pete Sampras’ dominance, yet, there are distinct reasons why people may think this is so. We could say to explain Pete Sampras’ skill, we must address specific aspects of his game. It’s necessary to observe his deft touch at net, booming serve, and flat forehand. Pete Sampras is good, the reasoning goes, because he was able to hit aces and put away volleys. Any explanation of Sampras’ skill has to begin with these factors. Under this perspective, the Kyrgyzstan model fails to give an explanation because it is too abstract. Only tracking the probability of winning any given point obscures the unique advantages Sampras had that contributed to such a high probability. Proponents of this critique believe, in principle, no idealized model can explain why a given tennis player was so successful. Such models are incapable of capturing the unique, individual aspects of a player that contributed to his or her dominance.

A second critique finds no fault with idealized models in general — just the Kyrgyzstan one. It claims idealized models can give explanations of real-world phenomena, but this one in particular is too weak to do so. A model that actually provides an explanation would take parameters like probability of winning break points, probability of winning a point on second serves, probability of winning games/sets when down or up, etc… It would be much more detailed, but it still wouldn’t directly address the unique aspects of Sampras’ tennis the prior camp requires. Explanations from the model would sound like “Sampras was great because he had a high win percentage on second-serve points,” or “Sampras was a champion because he had a high probability of saving break points.” This camp believes idealized models can supply these explanations and they’re sufficient for deep knowledge. To understand Sampras, we only need to understand his propensity to win or lose in general scenarios. We only care about his serve or volleys or movement insofar as they contribute to a high break-point save rate, for instance. The statistical measures provide the base of our understanding; everything else is secondary.

We might be partial to a certain type of explanation. Players and coaches will tend to explain matches in terms of backhands and serves where statisticians will invoke probabilities and win rates. Our intuitions are often with the players. We find it hard to believe understanding Djokovic is possible without seeing a sliding open-stance backhand winner, or a kick-serve that bounces over your head in the case of John Isner.

Are our intuitions correct? Are the modelers correct? Which level of explanation gives us the best understanding of a tennis player? Is one of these levels subordinate to the other?

Inflated statistics, spring admission, and the University of Southern California

A significant number of USC students do not enter as freshmen in the fall. The university offers alternative enrollment options to inflate incoming freshmen statistics and make their student-body appear higher-quality than it actually is.

Something about USC doesn’t add up.

Every fall approximately 3,200 wide-eyed freshmen matriculate. They were admitted from a pool of more than 60,000 applicants and are among the best of what high schools in the 21st-century have to offer. In 2019, the middle 50% range of their SAT scores was 1360-1530. The same stat for GPAs was 3.72 – 3.99. Roughly 20% of enrollees have rèsumès good enough to warrant merit scholarships.

This is impressive. For most of its history, USC was academically underpowered. It was better known for its football team and fraternities than serious study until the university launched a status-raising campaign in the 2000s. Now, some of the brightest high-schoolers in the country become Trojans.

But look closer. For the 2019-2020 academic year, USC claims to have around 20,500 undergraduates enrolled. If approximately 3,200 freshmen enroll every year, and that has been the case for the last 4 years, then USC should only have 3,200 * 4 = 12,800 students. In fact, if you sum the actual number of freshmen that USC reports matriculated in fall 2016, 2017, 2018, and 2019, you only get 3,068 + 3,358 + 3,401 + 3,168 = 12,995 students.

Where is everybody? If we grant that the 20,500 figure is correct, then there are roughly 7,000 students at USC who are unaccounted for on this naive model. In other words, it’s not obvious how almost 35% of the student body got into USC, or how they compare academically to their peers.

In the next section, I want to establish that the freshmen/total enrollment discrepancy at USC is real and significant compared to similar institutions. Afterward, we will discuss possible explanations for the gap.

1. The discrepancy

1.1 Methodology

The guiding thought is we should be able to get reasonably close to a university’s reported undergraduate population by counting all the students they say enroll. For instance, if a university says they enroll 250 freshmen each year, we might expect their total number of undergrads to be around 1,000. Students should graduate in 4 years, so the only people on campus should be those who enrolled 1, 2, 3, or 4 years ago.

If only college was that simple. There are good reasons why the simple calculation I just described will be inaccurate. Not everybody finishes their degree in 4 years, so students who enrolled 5 or 6 years ago might still be on campus. Matriculating class sizes might vary dramatically over that period as well. Adding a couple of numbers together also can’t account for people like dropouts and transfers who might affect the size of a student body.

We can account for all of these concerns. Many universities (including USC) publish their matriculating class sizes on their websites, either part of the Common Data Set or other statistics they distribute. The same schools also frequently publish the number of transfers they receive. As a result, we can get an exact sense of student inflow over the past 4 years.

We can also roughly approximate the number of students who take more than 4 years to graduate. Universities publish 6 and 4 year graduation rates as part of the Student Right-to-Know Act. From these figures, we can estimate the number of students in a freshmen cohort who graduate on time. We assume the rest of the students in the cohort graduate in 5 or 6 years, getting an idea of how many people who enrolled more than 4 years ago contribute to the student population.

(I acknowledge this is an overcount since some students who don’t graduate in 4 years drop out. Yet, note that overcounts are in USC’s favor. Our naive calculations show there are 7,000 missing students, so any fudgery that accounts for more undergrads helps them).

Handling dropouts who enrolled in the past 4 years is tricky but doable. Universities don’t publish the exact number of students who drop out and when, so making any kind of exact calculation is difficult. For instance, a freshman enrolling in 2015 might count towards undergrad enrollment in 2015, 2016, and 2017, but then drop out. If we knew what percent of a freshmen cohort dropped out every year, we could just “thin” their class accordingly and be done with it.

Our strategy is to perform two sets of calculations. The first assumes everybody that enrolls eventually finishes their degree. This is a “best-case scenario” and will give us an upper bound on the student body. Imagine the student who enrolls in 2015 but drops out in 2017. If we’re trying to calculate undergraduate population in 2018 under the assumption of no dropouts, she’s going to be counted even though she’s no longer enrolled. As a result, if we pretend every freshman sticks around and eventually graduates, we’re going to overshoot the number of people actually on campus by tallying people that have already left school.

The second set of calculations gives us a “worst-case scenario.” We assume everyone who will drop out does so the moment they set foot on campus. The proportion of a freshmen cohort that drops out can be approximated via published graduation rates. In our minds, those who will drop out are counted as enrolled freshmen and then disappear. If we’re doing these calculations for 2018, they will not count our hypothetical student who drops out in 2017. In fact, they wouldn’t count her in 2015, 2016, or 2017, even though she was enrolled then. Under this assumption, we get a lower bound on the student body by ignoring attendees who will eventually drop out, even if they are still enrolled.

With rough upper and lower bounds, we can be reasonably confident the number of students we can trace back to freshmen enrolled in the fall or transfers lies somewhere between the two figures. Roughly, the bounds will be computed as follows:

Upper bound = (number of freshmen enrolled over last 4 years) + (transfers from last 2 years) + (students who take 5-6 years to graduate from freshmen cohorts 5 and 6 years ago)

Lower bound = (number of freshmen expected to graduate who enrolled over the last 4 years) + (transfers from last 2 years) + (students who take 5-6 years to graduate from freshmen cohorts 5 and 6 years ago)

What’s left is to compile the data, run the numbers, and see if the bounds on the number of students we can trace back differ significantly from the number of undergrads a university says it has.

1.2  Numbers

By looking through a combination of Common Data Sets, enrollment reports, and university fact pages, I was able to gather the necessary data and compute lower and upper bounds for 8 schools, including USC. In an effort to compare apples to apples, I tried to include schools that are private and similarly sized. Data for those institutions aren’t always available, so the sample also includes public schools of similar size, and smaller, private ones (with UCLA thrown in for good measure).

Below is a table with the 8 schools and their respective reported undergraduate enrollments.

Screen Shot 2020-06-21 at 3.43.25 PM
Table 0

Using the ideas outlined in the previous section, we calculate the number of students traceable to either freshmen or transfer enrollment. The next table displays the upper and lower bounds as a percentage of the undergraduate population.

Screen Shot 2020-06-21 at 3.45.24 PM
Table 1. For details on how upper and lower bounds were calculated, consult Appendix A.

Let’s look at the University of Virginia (UVA) to start. If we add up all the students the University says have enrolled as freshmen in the past 4 years, adjust for students that enrolled 5 and 6 years ago who are taking extended time, and assume nobody drops out, we can account for 102.38% of the students on campus. If we assume all the freshmen that are going to drop out do so immediately, we can account for 97.91% of the students on campus. In this case, our upper and lower bounds contain the number of reported students. This is a sign as our method does a decent job of predicting how many undergrads a university should have given the numbers they publish.

However, this does not always happen. A percentage greater than 100 indicates an overshoot. For instance, if we assumed all the students we expected were going to drop out of Baylor did so immediately, we should expect their undergraduate population to be 114% of their current one.

As Table 1 suggests, our method does not yield pinpoint accuracy. We don’t hit actual undergraduate population numbers exactly, but two things are notable. First, our new model tends to overshoot. Half of the schools in the sample have lower bounds that are greater than stated enrollment, and the lower bounds of two more come within 8% of the actual figure. The upper bounds of 6 schools are well over 100%, or very close. This suggests there may be outflows of students I haven’t considered, leading to systemic overcounts.

The second feature is BU and USC are outliers. BU comes reasonably close in the upper bound, but its lower bound is nearly 15% below stated enrollment. USC is even worse. Its upper bound is 83.99%, which is even lower than BU’s lower bound. Remember, 83.99% as an upper bound means we can only account for that percentage of students if we make the rosy assumption nobody drops out. USC’s lower bound also dips into the 70s, which is far worse than any other school considered.

USC, and to a lesser extent, BU, have notable discrepancies between their published freshmen/transfer numbers and total enrollments. Their published student inflows do not come close to accounting for the students they have. The fact USC and BU’s freshmen/transfer enrollment numbers under predict student population — when the model overshoots considerably for other schools — suggests something is afoot.

2. Explanations

There are two general ways to account for USC and BU’s enrollment discrepancies. The first is to claim they’ve made a mistake. For one reason or another, the reasoning goes, both universities don’t have a grip on either their total enrollment or the number of freshmen/transfers that enroll every year. This could be caused by problems with their internal systems, apathy, or communication issues.

I don’t think this is plausible. For one thing, knowing how many freshmen enroll is crucial to universities. Tuition is a large source of revenue, so institutions invest a lot in ensuring incoming cohorts have the correct size and socioeconomic makeup. Unexpected freshmen yields can also lead universities to rescind acceptances, which is bad for students and administrators alike. Along the same lines, it seems unlikely a university also doesn’t know how many total undergraduates they have. Every student is a paying customer, so I’d imagine schools would know the size of a major income source.

Deceit is also a possibility, but if schools are lying about their enrollments, we have a much larger problem than can be discussed in this blog post. For that reason, I will not consider it.

The second general explanation is USC and BU have additional student inflows that aren’t matriculating freshmen or transfers. This sounds strange: doesn’t everyone enroll as either a fall freshman or a transfer? How else are you admitted to a school?

It turns out there’s a third way. USC and BU both admit applicants for the spring semester. This means a high school senior submits their application like everyone else, but instead of being invited to campus in August, the earliest they can enroll is the following January. In USC’s case, we know spring admits aren’t tallied in the freshmen numbers I used to compute the upper and lower bounds. In their matriculating freshmen reports, they are strangely specific in talking about fall admits and fall enrolls.

We can test if spring admits account for the missing students at USC. According to their admissions blog, USC enrolls between 500 and 600 spring freshmen every year. Let’s be conservative and assume the number is 500 while supposing the practice has been active for the last 4 years. We can account for this in our model by adding 500 to every fall matriculating class going back to 2016.

Table 2 demonstrates the new upper and lower bounds with this assumption. Consult Appendix A for details on how the bounds were computed.

Screen Shot 2020-06-18 at 2.26.25 PM
Table 2

There’s improvement. 93.74% as an upper bound approaches respectability, but 87.89% as a lower one is still concerning.

At this point in my investigation, I thought there has to be a table where spring admits show up in USC’s Common Data Set. It records everything from Pell Grant recipients to the number of philosophy degrees conferred, so spring admits need to be recorded somewhere, if not under the name.

That’s when the “Other first-year, degree-seeking” row in Section B of USC’s Common Data Set caught my eye. According to the definitions table in the back of the document, “Other first-year, degree-seeking” undergraduates are students that have completed fewer than 30 semester hours (units) and are seeking to graduate at the university in question. These are contrasted with “Degree-seeking, first-time, freshmen” who are undergraduates in their first year that have not attended a prior postsecondary institution. In other words, “Degree-seeking, first-time, freshmen” are traditional freshmen. “Other first-year, degree-seeking” students are freshmen that have attended a prior institution.

Two pieces of evidence that suggest this row in the Common Data Set counts spring admits. First, USC recommends spring admits to go to community college or study abroad during the fall semester. On their website, they say “most first-year spring admits choose to enroll in community college during the fall.” For the cosmopolitans (or those that can afford it), they even have fall-semester programs in places like Rome and Prague at partner universities exclusively for USC spring admits.

The second piece is the actual values in the “Other first-year, degree-seeking” row. As mentioned, USC says between 500 and 600 spring admits enroll every year. It turns out, for the last 3 years, the number of “Other’s” has been around 600, with a spike 4 years ago. If we update our upper and lower bounds by adding the number of “Other’s” to the freshmen enrollment numbers for the last 4 years, we get the following:

Screen Shot 2020-06-18 at 3.18.41 PM
Table 3

We’re much closer. Our upper bound is nearly 100%, and the lower one is a respectable 93.63%. For USC, I’m more confident than not the “Others” row approximates their spring admits. As a result, I believe USC’s policy of offering spring admission more or less explains the apparent discrepancy between matriculating freshmen numbers and total undergraduate enrollment.

If what I’m saying is correct, we can follow an identical process with BU and create better upper and lower bounds. Yet, there are slight differences that prevent this. BU’s “Other” rows are often in the low teens. This means very few freshmen have attended a prior institution before landing at BU. However, I attribute this to the nature of their spring admission policy. It appears almost all delayed admits arrive at BU as part of the College of General Studies (CGS). This is a 2-year program where students arrive in the winter, take classes, and then study abroad for the summer after their freshman year. On the CGS FAQ, they recommend students spend their gap semester volunteering, working, traveling, or taking a class. The next entry in the FAQ explicitly prohibits CGS students from enrolling in another institution for the fall semester. If students want to take non-degree courses during the fall, BU advises them to consult their CGS academic advisor.

In other words, BU does not present taking classes during fall as an attractive option. For this reason, I am not surprised the “Other” row in their Common Data Set is so low. Yet, we can still update our bounds by taking into account the number of first-year CGS students. From their website, they claim to enroll approximately 600 students annually. If we add 600 to every freshmen class going back 4 years, our new bounds are:

Screen Shot 2020-06-18 at 4.32.37 PM
Table 4

These look similar to bounds created for other institutions. Taking delayed admissions into account, we can resolve BU’s apparent discrepancy between freshmen enrollment and total undergraduate population. As mentioned, I believe the same is true of USC. Every year, both schools enroll around 600 freshmen during the second semester.

3. Motives

I believe USC and BU do this to inflate the statistics of their freshmen classes. The thought is if they can exclude academically weaker students from matriculating in the fall, their student-body will appear better and more selective than it actually is. After all, spring admits aren’t included in the fancy documents USC spins up for their “Class of 202X” promotions, and I doubt CGS statistics are included in BU materials. Based on my calculations, USC could have inflated their admit rate by ~1.3% in 2018. Consult Appendix B for the exact methodology.

Colleges have another incentive to delay admission for weaker students. US News and World Report uses the test scores and high school GPAs of first-time, first-year students who enter in the fall to calculate their rankings. If students are admitted in the spring, their statistics are irrelevant from US News’ perspective. This means spring admission is a way for schools to shield weaker students from prying eyes. Every year, the high-school performance of around 600 USC and BU freshmen is not considered when calculating college rankings. Put differently, 15% of their entering class is invisible to those who want to discern the academic quality of the average Trojan or Terrier. I would be surprised if including that 15% helped USC or BU’s cause.

Deirdre Fernandes, reporting in the Boston Globe, describes the phenomena.

Many of the students targeted for delayed admissions would have traditionally been wait listed or rejected because their test scores or grades may not have been as strong as other applicants. But since these students aren’t counted as part of the entering fall class, their academic histories don’t weigh down the school’s overall average for that particular year.

There is also a wealth dimension.

The freshmen who come in likely wouldn’t have been accepted for the traditional freshman class because their grades weren’t as strong, but they are usually wealthier and can afford to pay for a spot without relying on financial aid from the school.

Put bluntly,

“The college banks on the fact that the student wants to go there,” said Todd Weaver, a vice president with Strategies for College Inc., a Norwood-based private counseling firm. “This student might not be a best fit, but their bank account is.”

In addition to being academically weaker than traditional admits, it appears colleges also target the students that can afford to pay full freight. Observational evidence stands in favor of this point for USC. To be clear, I do not have data on USC spring admit income. Yet, of the three students who gave testimonials about their spring admit experience on USC’s delayed admission page, two went to $40,000 a year college prep high schools. This is far from damning evidence, but it is suggestive.

I acknowledge there are non-deceitful reasons why a university might offer spring admission. For instance, staggering the arrival of students allows institutions to enroll more people. Spring freshmen can replace upperclassmen that are studying abroad for the semester, which leads to efficient use of dormitory space. Personally, I believe more students should take time off between high school and college, and we can see delayed admissions as an embrace of the idea.

Perhaps university administrators have these thoughts, but the consequences for ranking and status are just too convenient. Universities live off their reputations and, notwithstanding the coronavirus, are finding it more difficult to fund themselves; we should be skeptical of professed altruistic motives. Anyone should have a hard time believing colleges engage in policies that hide low-quality admits and allow them to enroll more wealthy students for reasons other than their own advancement.

4. Normative claims

I’m going to pick on USC because it was the original subject of my investigation, though I believe everything below also applies to BU.

There’s an equality argument against USC’s spring admission policy. As described, they advantage rich, under-qualified applicants who otherwise would have little chance of being accepted. We believe wealth should have no role in allocating educational opportunity, so our ideals about merit and social mobility are violated. Hence, spring admissions policies are inconsistent with our values and should be abolished or altered. I believe this is an effective and important argument but will not pursue it at length. We all know its steps and understand how much of an issue social mobility is.

A more interesting argument concerns honesty.

Suppose you don’t mind whether private institutions lower academic standards for wealthy students. You might not believe these universities deserve tax breaks, but in principle, there’s nothing wrong with private actors imposing an income qualification on applicants. In the same way only the wealthy can buy birken bags, only the rich can get a degree from USC.

In light of this, you can still think spring admission is problematic because it functions to misrepresent the institution. Perhaps it’s within USC’s right to compromise rigor in admitting some students, but their statistics should reflect that. It’s disingenuous to tout 96th percentile SAT scores and falling acceptance rates when 15% of freshmen are not included in those figures. It’s deceptive to submit unrepresentative data to college rankings for status and prestige while slipping in hundreds of students that might be under-qualified.

I think it’s admirable USC is on such a deliberate campaign to improve itself. It’s clearly a much better institution than it was in the past, but only substantive growth should be rewarded. Its spring admissions policies are evidence it wants all the benefits of a high-powered student body while still reserving the right to lower standards for the wealthy. It cannot pick both and remain honest. If USC wants to remain a playground for the rich, so be it, though it should not pretend otherwise.

5. Conclusion

I could be wrong. It’s possible all spring admits are highly qualified, even more so than regular admits. In fact, some of them probably are [1]. It’s also possible most spring admits are low-income first-generation students, on their way up the socioeconomic ladder. Given the evidence, though, I think this is unlikely. The consequences for universities are just too convenient. I invite USC, BU, and all other universities to release statistics on spring admits. Until then, I stand by my critique.

My criticisms are also directed entirely towards institutions, not individuals. Being accepted to any college, during any semester, is a reason to celebrate. When confronted with opportunity, individuals are obliged to take it. Yet, this does not prevent debate at the institutional level about how some opportunities are allocated, or if organizations are being dishonest about who receives them.


[1] Let F(1): (1) is a spring admit and G(1): (1) is academically underqualified.

My position is not:


But rather:

∀x[P(Gx|Fx) > P(Gx)]

(with a little abuse of notation)



Appendix A: Calculations

Let’s begin with an example. Here is the excel sheet I used to calculate UVA’s figures.

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Table A1

The calculations start with the same intuition we had in the introduction: adding up freshmen enrollment numbers over the last 4 years, should reasonably approximate total undergraduate population. This happens in the “Fall matriculating class size” column. The total of 4 years of freshmen enrollment is tallied at the bottom.

The figures in the “students expected to graduate” column are calculated by multiplying the corresponding fall matriculating class size with the university’s 6-year graduation rate. UVA’s 6-year graduation rate is 95%. This means the “students expected to graduate” figure for 2017 is 3788*(.95) = 3598.6.

Transfers per year are often obtained from the latest edition of a university’s Common Data Set. Occasionally, transfer numbers aren’t available for the year 2019-2020, so in that case I assume they took in the same number of transfers as in 2018-2019. Recall that I’m assuming all transfers stay on campus for only 2 years. This means the only transfers on campus are those who arrived this year or the one prior.

I also attempt to approximate the number of students who enrolled as freshmen more than 4 years ago but are still on campus. These are the “students taking non-standard time.” To approximate the number of students from the freshmen class of 2014 that are still at UVA, I multiplied the total freshmen class by (1-[4-year graduation rate]). The rough thought is if you don’t graduate in 4 years, you will in 5 or 6. UVA’s 2014 freshmen enrollment is 3672 and their 4-year graduation rate is 89%. Hence, the number of freshmen from 2014 who are still around is 3672*(1-.89) = 403.92.

The upper bound of accounted-for students is calculated by summing the totals of “fall matriculating class size,” “transfers per year,” and “students taking non-standard time.” This assumes that every freshmen in the 4 prior classes is still around. The lower bound is calculated similarly, but the total of “fall matriculating class size” is replaced by that of “students expecting to graduate.” This calculation assumes all the students that are expected to drop out will immediately, and don’t contribute to the current undergraduate population.

The following table results:

Screen Shot 2020-06-16 at 11.15.02 PM

“Students accounted for (no dropouts)” is the upper bound, and “Students accounted for (all dropout immediately)” is the lower bound.

The full excel sheet I used can be found here. It includes additional notes on where I found enrollment data and graduation rates, and the assumptions I made when those weren’t available.


Appendix B: admit rate

Approximating how much USC inflates their admission rate is straightforward. Let’s use their published data from 2018 as an example, as this is the latest year where good data are available.

In 2018, USC admitted 8,339 students for the fall from a pool of 64,352 applicants. This translates into a 8,339/64,362 = 12.9% acceptance rate. However, in order to get a better sense of their total acceptance rate, we must factor in those they rejected for fall but admitted for spring. To my knowledge, you can’t directly apply for spring admission to USC, so the spring admits must have come from fall applications.

If what I mentioned about the “Other first-year, degree-seeking” row in the Common Data set is correct, USC had 642 spring admits enroll in 2018. Note that fewer students enroll than are admitted. Hence, the total number of spring admits is likely higher than the 642 recorded in the “Other” row. If we knew the “yield rate,” or percentage of students offered spring enrollment who took it, we could divide 642 by the rate and get the number of students accepted for the spring. To my knowledge, USC doesn’t publish that figure.

Yet, we can estimate a yield rate. Suppose it’s true the students offered spring admission to USC are academically under-whelming relative to fall admits. This means they probably would not have been accepted to USC, or schools of similar quality, under regular circumstances. As a result, even though USC offered them a spring position, it’s likely the best option they have. In other words, the choice could be between attending a #22 ranked college in the spring, or a #32 ranked college in the fall. I imagine most students opt for the former, leading to a higher yield rate.

Other factors can also influence the rate. Even though USC might be the best option for many students, it could still be unattractive to start in the spring. Students might want the entire “freshmen experience” that comes with arriving in the fall with other freshmen. Starting in the spring jeopardizes that.

It’s clear USC realizes the concern and took pains to alleviate it. As mentioned, it offers numerous foreign programs with other USC students to make spring admits feel they are not missing out. This increases the appeal of a spring enrollment.

How do we balance these competing concerns? USC’s fall 2018 yield rate was 41%. Harvard’s 2018 yield rate was 82%. I propose we think of a spring admission from USC as more attractive than a fall offer (given the spring admits’ poorer alternatives) but less attractive than Harvard (given the need to delay enrollment on campus). I think a fair figure is 75%. After all, spring admission programs are popular. In 2019, Babson College offered spring admission to 100 students and 86 took the offer. Babson is a small, specialized school, though, so I’m unwilling to read too much into their 86% yield. Still, it indicates rates should be higher rather than lower.


Using admit and application data, we can estimate USC’s actual 2018 admit rate given a 75% spring enrollment yield.

Screen Shot 2020-06-24 at 4.58.55 PM

Our assumptions entail USC is inflating their admit rate by around 1.3%. This might look paltry, but every percentage point counts in the prestige arms race.




Probability and God

Occasionally, rare things happen to us.

You might land a competitive job (3/100), appear on the big screen at a sporting event (1/70,00), or win the lottery (1/12,271,512). It’s also possible for you to get a US green card (1/126), be struck by lightning (1/700,000), or have an idea so good it’s “like getting struck by lightning” (1/???).

Whether it’s good or bad when the improbable becomes actual, there’s always a question lurking in the background: is this evidence of anything? If what seemed impossible is staring us in the face, what can we say about it?

This question is fascinating with respect to life in the universe and God. “God” in this post will not refer to the God of the new testament, the God of the old testament, Allah,  Shiva, Mahavira, Zeus, Ra, Spinoza’s God of substance, or any other popular deity. Formal religion aside, we will be interested in the quite general question of whether a being designed the universe to support life. This designer, whether s/he exists, will be referenced as “God.” I repeat, there is nothing Judeo-Christian, Muslim, Hindu, Wiccan, etc… about my invocation of “God.” I chose the capital-g for ease of reference and because I knew it would grab your attention.


Our existence is an anomaly.  We can get an intuitive feel for this by gazing at the night sky. Billions of stars, millions of planets, and somehow, we’re alone (so far). We have yet to find evidence of even microbes in the vast expanse of the universe, so the fact beings as sophisticated as humans came about represents something uncommon and significant.

The improbability goes deeper. As it turns out, even the laws of the universe that allow life to exist are rare and unlikely to come about by chance. If we were to slightly change the basic rules of force and gravity, for instance, the resulting universe would be hostile to life. Philip Goff has examples. The following three bullets are his words.

  • The strong nuclear force has a value of 0.007. If that value had been 0.006 or less,
    the Universe would have contained nothing but hydrogen. If it had been
    0.008 or higher, the hydrogen would have fused to make heavier elements. In
    either case, any kind of chemical complexity would have been physically
    impossible. And without chemical complexity there can be no life.
  • The physical possibility of chemical complexity is also dependent on the
    masses of the basic components of matter: electrons and quarks. If the mass
    of a down quark had been greater by a factor of 3, the Universe would have
    contained only hydrogen. If the mass of an electron had been greater by a
    factor of 2.5, the Universe would have contained only neutrons: no atoms at
    all, and certainly no chemical reactions.
  • Gravity seems a momentous force but it is actually much weaker than the
    other forces that affect atoms, by about 10^36 . If gravity had been only slightly
    stronger, stars would have formed from smaller amounts of material, and
    consequently would have been smaller, with much shorter lives. A typical
    sun would have lasted around 10,000 years rather than 10 billion years, not
    allowing enough time for the evolutionary processes that produce complex
    life. Conversely, if gravity had been only slightly weaker, stars would have
    been much colder and hence would not have exploded into supernovae. This
    also would have rendered life impossible, as supernovae are the main source
    of many of the heavy elements that form the ingredients of life.

This is the cosmological equivalent of tweaking the rules of your favorite game and then finding out it is unplayable. If the laws of physics differed slightly from what they are now, life as we know it wouldn’t stand a chance. It appears every law was formulated to lie just inside the narrow range that allows complex organisms like us to exist.

When we consider the fact life is highly uncommon in our current universe, and the second-order fact that it was incredibly unlikely the fundamental structure of said universe could be compatible with even the potential for life, our existence looks even more astounding. Roger Penrose —winner of a Nobel prize in physics with Stephen Hawking— calculated the odds of a universe such as ours being created by chance as one in 10^1,200. Lee Smolin, another physicist, calculates the probability of life arising in the universe as 10^229. These estimates differ by about a thousand orders of magnitude, but their point is clear. If left to chance, nature conspires against us.

It’s natural to find these odds unsettling. “But,” someone might say, “we exist! The odds say it’s nearly impossible for us to be around, yet here we are. If something so improbable happens there has to be some explanation for it that doesn’t appeal to pure chance.” Here, we reach for God. If the universe wasn’t the result of a random process but the product of a creator with life in mind, it’s much easier to believe we exist despite the astronomical odds against us. God is a much more satisfying, and, in a certain sense, more simple, explanation than blind luck. As the reasoning goes, a low probability of life existing in the universe, coupled with the fact life actually exists, constitutes evidence of a creator.


There’s an alternative perspective to probability and God. Life being necessary, in some sense, should be evidence of a creator. If God exists, we assume she wants life to come about and will not tolerate the possibility it could be otherwise. Such a God would make it impossible for a universe to exist that cannot support life, like us.

For instance, if we discovered there was a 99.99% chance any given universe could support life, wouldn’t this mean that possible universes were optimized for our presence? What better evidence of God could there be than odds stacked in our favor? If anything, a low probability of life originating in the universe might be an indication our existence was somehow left to chance. It’s possible we would not have existed, and that is incompatible with there being a God.

These two camps, those that stress the improbability of life, and those that stress its necessity, are at odds. One claims a low probability of life arising in the universe is evidence of God, while the other asserts high probabilities are. In other words, they use opposite premises to draw identical. At first pass, one of the camps has to be wrong. After all, how can low and high probabilities both be evidence for God? Does this mean we get the evidence either way? Did I just prove the existence of God? Definitely not. Probabilistic inference appears to have gone awry, and we’re going to get to the bottom of it.

Interpreting probability (can skip if frequency-type/objective and belief-type/subjective probabilities are familiar)

There are two ways to interpret probability. The first is characterized by statements like:

“There is a 60% chance you draw a red marble out of the urn.”

“The odds of getting pocket aces is 6/1326.”

“You will only win the lottery 1 out of 12 million times.”

Here, we use probability to talk about the outcomes of repeatable chance events. In this sense, a probability tells you, on average, how frequently an outcome occurs per some number of events. “60%” in the first sentence tells us, on average, if we draw 10 marbles out of the urn 6 of them will be red. Likewise, “6/1326” in the second sentence tells us that if we play 1326 hands of poker we should expect 6 pocket aces. In each case, the probability tells us about the distribution of a certain occurrence over a number of trials. We learn something about how often a chance event yields a specific outcome. This is the frequency-type interpretation of probability.

The second interpretation of probability is characterized by statements like:

“I am 90% sure I left my keys at home.”

“The odds of getting Thai food tonight are 1/10.”

“What’s the probability Trump wins the election? I say 28.6%”

These statements are similar to the others in that they use fractions and percents to express probabilities. The similarities end there. Rather than describe the outcomes of chance events, they express subjective levels of confidence in a belief. This is called the belief-type interpretation of probability. Higher probabilities correspond to more certainty in a belief, while lower ones express doubt. For instance, saying there is a “1/10” chance of getting Thai food means you are very unsure it will happen. Saying there’s a “90%” chance you left your keys elsewhere means you’re very confident you don’t have your keys.

It’s important to note that the frequency and belief-type interpretations apply to different things. We formulate frequency-type probabilities about the outcomes of chance events, like poker hands and lottery-drawings. Belief-type probabilities do not apply to chance events. They’re used to describe our subjective degree of confidence in statements about the world, like who is going to win an election or what we are going to eat tonight.

Reconciling the two camps

In the arguments given for God, which interpretation of probability is operative?

Frequency-type looks unlikely. The creation of a universe does not appear to be the outcome of a repeatable chance event, like drawing a marble from an urn. By most scientific accounts, there was a single “Big Bang” that yielded our universe, and there will never be a similar moment of creation. Because the event is unique, it makes no sense to talk about the frequency of a certain occurrence over a number of trials. We cannot say whether a universe containing life will arise 1 out of 10^10 times as it’s impossible to create 10^10 universes and observe what happens.

Belief-type probabilities don’t run into these difficulties. It’s coherent to say you have more or less confidence God created the universe, though a bit unnatural to express the sentiment as a probability. However, philosophers wrestle with how to further interpret belief-type probabilities and discover complications. Many take belief-type probabilities as the odds an individual would set for an event to avoid being dutch-booked. This view has the advantages of being intuitive and mathematical, but the betting analogy breaks under select circumstances. Individuals might refrain from setting odds (and if we compelled them, would those odds be accurate?), and it’s not clear there’s a single, precise number that we would set the odds at to express our confidence in a proposition.

While belief-type probabilities appear to be the best choice, I’m going to ignore them. This is because my “solution” to this issue relies on a frequency-type interpretation of probability so I’m going to shamelessly ignore the alternative. We will assume the creation of our universe is the outcome of a repeatable chance event. It’s also true belief-type probabilities have been critiqued in the context of reasoning about religious hypotheses, but I will not discuss such objections.

Using frequency-type probabilities can also be somewhat legitimate. We can circumvent the objections to using it in reference to the creation of the universe with a multiverse theory. If you believe multiple universes have been created — perhaps there are 10^10 parallel universes, for example — it’s perfectly acceptable to use a frequency-type probability to describe the odds of life arising. Your statement simply expresses the odds of picking a universe with life at random out of all the created universes. Personally, I have no idea whether multiverse theories are actually plausible, but this is a potential way to justify a frequency-type interpretation.

Given the above, I don’t think the two camps are incompatible with each other. It’s possible for low and high probabilities of life to serve as evidence for God. Both parties are making valid inferences from the probabilistic evidence they have. The caveat is that I believe each can only argue for a certain type of God.

Consistent with our assumption the creation of the universe is the outcome of a repeatable chance event, imagine it is determined by the spin of a roulette wheel. Each slot in the wheel represents a possible universe, and wherever the ball lands, the universe corresponding to that slot is created. One slot might represent a universe with physical laws like our own, and compatible with life. Another slot might represent a universe so different from ours that life could never originate.

Those who think low probabilities of life are evidence of God might imagine the roulette wheel of possible universes to be enormous. There are trillions of possible slots, and only a handful of them will correspond to universes that contain life. The probability the ball lands in a slot that creates a universe containing life will be minuscule. Yet, our universe exists and contains life. Since it defied nearly-impossible odds, the reasoning goes, it must have had some assistance beyond pure chance. The “assistance” in roulette wheel terms might be thought of as God picking up the ball and deliberately placing it in a slot corresponding to life. God intervenes in the chance event, making the highly improbable, actual.

The high-probability camp’s perspective can also be thought of in terms of the roulette wheel. In this case, a high probability of life would translate into every slot in the wheel corresponding to a universe where life exists. No matter where the ball lands, a hospitable universe will be created. Chance can select any slot it desires, but the outcome will be the same. God enters the picture when we ask ourselves who made the roulette wheel and dictated the nature of possible universes. The wheel being as it is constitutes evidence of an intention to bring life about. In this instance, God creates living things by stacking the odds in our favor.

When we consider how both camps might imagine God, the tensions between them fade. High and low probabilities of life can both constitute evidence of a creator because they support different versions of her. Low probabilities imply the existence of a God that chose our universe out of innumerable alternatives. High probabilities suggest a God that creates life by making a desolate universe metaphysically impossible.

This doesn’t guarantee either type of God exists, though. Individuals may use high or low probabilities of life arising in the universe as evidence for certain types of Gods, but how effective these arguments are is an open question. At any rate, neither line of reasoning abuses probability.



A dialogue in defense of business majors

Sophia Booth and Taylor Hutchins are both freshmen in college. Sophia has just told Taylor she wants to switch her major to business.

Taylor: Why would you do that? How can you learn anything about business in the classroom? They’re only going to teach you theory, and we all know that’s worse than useless when you go out into the real world.

Sophia: What makes you qualified to piss on business majors like that? Just because you’re mechanical engineering doesn’t give you a license to demean an entire subject you haven’t even studied.

Taylor: Are you kidding me? The business model canvas? The Boston matrix? You’re going to spend your undergraduate years filling out forms and pointing out farm animals as opposed to learning anything useful. Even if that stuff was the cutting edge of business knowledge at one point it’s surely going to be outdated by the time you graduate. You think you’re learning something applicable but it’s all empty theory.

Sophia: Woah, Taylor, you have it all wrong. I can see how you think we’re at college to learn eternal truths about how things operate and then apply them, but that’s just not how higher education works.

Taylor: What do you mean? Are you saying my education is as worthless as a SWOT analysis?

Sophia: No. You’re probably going to apply much of what you learn here in the future, but you need to understand you’re in a minority. The rest of us come to college to signal.

Taylor: You’re making less and less sense. What the hell is a “signal?” We’re all here to learn. That’s why our university exists in the first place.

Sophia: The people that are here to learn things for their jobs are future engineers, programmers, scientists, doctors, professors, and researchers. If I work outside any of those fields I will most likely never draw upon anything I learned in undergrad. Firms like Deloitte hire junior consultants from any major and train them on the job. The way most people get employed is not by learning whatever skills might be necessary to actually do their job, but by sending strong signals to the labor market.

Employers want to know I’m sharp, have a good work ethic, and am enthusiastic about working for their company. It’s possible for me to demonstrate these things by acing my classes, maxing out on credit-hours, and radiating excitement during my job interview. These are the signals I’m talking about, and I’ll get employed by sending them.

Taylor: Wait, so you’re on my side now? I hear you saying that what most people learn in college is useless. That gives you a much stronger reason to do mechanical engineering or computer science rather than business.

Sophia: My point is that even if you’re right and nothing in my business degree is applicable to my career, it doesn’t matter. A business degree sends a strong signal to the job market so getting one is not a total waste of time. I can still show employers I’m sharp, (by acing my classes) hard-working, (by taking a lot of credits) and enthusiastic about being their employee. Remembering any of the stuff in class just doesn’t matter.

Taylor: So you only care about your signals, right? The actual substance of what you learn doesn’t matter, yes? That sounds pretty cynical.

Sophia: I’m just not willing to delude myself into thinking everyone learns applicable things, including in business. It could be the case what I’m learning is useful, but it really doesn’t matter. Academically, college is only a big obstacle course with employers waiting at the other end to see who gets through first.

Taylor: You said employers want to know you’re sharp, right?

Sophia: Yeah.

Taylor: So you should still switch to computer science. It’s much harder than business, so if you do well you’ll be sending a much stronger “signal” to the job market, as you say. Straight A’s in computer science say much more about your ability than A’s in business, and employers know that. There is no reason to get a business degree.

Sophia: Sure, you’re right that computer science sends a better signal in the “sharpness” category, but there are still two other types of signals I’m trying to send. If I do business, I can still take a lot of credits and show employers I’m hard-working. We’ll say business and computer science are approximately equal on that front. Yet, business sends a much better enthusiasm signal. A business degree tells employers I’ve been thinking about business-related things for four years. Who cares if those things are applicable. My willingness to do that demonstrates a deep commitment to private industry that doesn’t come across in a computer science degree. Employers understand I wanted a job in business when I was 18-19 and had enough conviction to stick with it. Sure, doing well in computer science would show I’m sharp, but business says something deeper about my attitude and commitment, two vital things about any potential employee. Plus, I can still ace my management classes and check the sharpness box.

Taylor: You know you still won’t learn anything about doing business.

Sophia: Maybe I will, maybe I won’t, but who cares? I’m sending a good signal and will be able to get a job in business in the end. Still, we both know I’ll probably learn at least one applicable thing. I might have to take accounting or finance, and everybody agrees those are useful.

Taylor: I still think you’re making a mistake. You can send a good signal in a different major. Just switch to engineering and we can do problem sets together.

Sophia: Too late — I have a meeting with my academic counselor now. See you later!


Is Sophia convincing? I think so. She has given a strong argument as to why you should pursue a business degree, but it’s crucial to distinguish between what she is and is not saying.

Sophia is not arguing everything people learn in a business major is unapplicable.

Her argument is agnostic on this point. Maybe she’ll learn applicable things, and maybe she won’t, but it does not matter. To her, there’s no use arguing about applicability. Education is signaling, and all she’s claiming is that doing business sends a good signal regardless of whether applicable learning happens.

This is a strong and important point. It allows someone to say something like this:

“Ok business major skeptic. Let’s assume I learn nothing applicable in the business major for the sake of argument. I’m still making a good decision because my signal to the labour market will be strong and I will be hired.”

That’s it. You don’t need to say anything about how management 101 is highly applicable or how accounting is useful. Signaling will justify your decision regardless.

A business major can certainly strengthen her case by saying, “oh by the way, management 101 is great and finance is applicable,” but these are independent points. What Sophia has shown is that you can theoretically concede a lot of ground and have a strong position.


Jewish occupational selection

I came across this paper while researching a forthcoming post on Medieval Jews and the Black Death. The abstract:

This paper documents the major features of Jewish economic history in the first millennium to explain the distinctive occupational selection of the Jewish people into urban, skilled occupations. We show that many Jews entered urban occupations in the eighth-ninth centuries in the Muslim Empire when there were no restrictions on their economic activities, most of them were farmers, and they were a minority in all locations. Therefore, arguments based on restrictions or minority status cannot explain the occupational transition of the Jews at that time. Our thesis is that the occupational selection of the Jews was the outcome of the widespread literacy prompted by a religious and educational reform in the first century ce, which was implemented in the third to the eighth century. We present detailed information on the implementation of this religious and educational reform in Judaism based on the Talmud, archeological evidence on synagogues, the Cairo Geniza documents, and the Responsa literature. We also provide evidence of the economic returns to Jewish religious literacy.

This reminds me of Protestant advantages that accrued due to increased literacy. What would the 21st century equivalent of this be? A religion that mandated all adults teach their children algebra? C++?

Questions to ask people

I like to have a couple of these in my pocket so when I meet someone new I can ask something that hopefully yields an interesting answer. My go-to for years has been “what’s the worst advice you’ve ever received?”


-What’s a story your parents like to tell about you? (credit to Anshul Aggarwal for this. I have only heard this one in a formal interview setting, though).

-What’s the best book you’ve read that you disagree entirely with?

-Who did you worship when you were younger?

-Do you think the rate of technological progress has slowed over the last 50 years? (then you proceed to convince them that it has).

-What’s the worst decision you made that turned out well? (and vice-versa: best decision that went terribly)

-Do you know of any fun online blogs?

-What do you do to remain weird?

-What’s something only people from your hometown do?

-Why do you think people buy expensive things they don’t need?

-What’s something you take seriously?

-What’s your opinion of Los Angeles? (works in any locale)

-What’s the taboo thing to do in [insert person’s hobby]?

-What’s the weirdest joke you find funny?

-What do you think is the most underrated personality trait?

-I don’t think computer science is really a science. Do you? (only works for CS people).

-If you could write an op-ed for the NYTimes, what would it be about?

-Do you trust economists?

-What’s the best city that’s not your hometown?

Educational Signaling

I recently finished Bryan Caplan’s “The Case Against Education” which is a rollercoaster of a book. Caplan basically makes two claims: education is much more effective at signaling worker productivity than imparting practical/employable skills, and as a result of this we should cut state and federal funding for it entirely.

It’s natural to approach these assertions with a healthy dose of skepticism. I’ll withhold judgment on the second claim for now, but I admit I am moved by his arguments for educational signaling. In short, he demonstrates we learn nothing in school beyond basic literacy and numeracy. Take science. Caplan supplies the following table with data from the General Social Survey and his own corrections for guessing. IMG_0057He has similar tables for basic political and historical knowledge. Clearly, we retain very little in the form of pure information.

What about the old adage that education is supposed to “teach you how to think”? Caplan has an answer for that as well. He cites studies demonstrating that the entirety of one’s undergraduate education increases general reasoning ability marginally, and only specific areas depending on the choice of major. “Learning transfer,” or the ability to apply principles learned in one situation to another, is also rare, especially when the initial principles were learned in a formal context. Self-reflection confirms this. How many times have you explicitly used information/a pattern of reasoning developed in class to solve a problem outside of a test?

In fact, my own decision to study philosophy, and expect employment post-graduation, presupposes a signaling theory of education. I do not plan on becoming a philosophy professor, but that is the only occupation where what I learn in class will be relevant. Nobody uses Kant on the job, and I knew that ex-ante. Instead, I have to rely on the fact choosing philosophy signals something about me that employers value. In Caplan’s terms, I’m hoping it demonstrates my ability, conscientiousness, and/or conformity, as these are the primary signals a college degree functions to send.

I’m a convert. Signaling explains why students cheer when class is canceled, but are reluctant to skip, why MOOCs can provide an objectively better educational experience than many brick and mortar institutions but pose no threat to the establishment, why STEM graduates do not work STEM jobs, and why the years of college that see the most payoff are graduation years. The personal and academic evidence for signaling is gigantic. Why ignore it?

The individual implications for this conclusion are Excellent Sheepy. Because your degree + extracurriculars are the only measurements employers have of your productivity, maximize those. Do the minimum amount of work for each class. Cheat on tests. Join as many clubs as possible and try to get a leadership position in each. You’re not going to remember course material, and it’s surely not going to be relevant to your job, so who cares? Even if you’re overcredentialed for your ability and turn out to be a poor employee, you’ll stick around as long as your incompetence isn’t egregious. Firms will keep a marginal employee for years to delay finding a replacement and upsetting other employees.

The societal implications are also gigantic. If education is just signaling, should there be less of it? (Yes, but I’m not a full Caplanian). If education is just signaling, should it not be a human right? If education is just signaling, should this be an indicator e-learning companies should create better, more informative credentials rather than trying to improve content delivery?