Tuesday, August 9, 2016

How much does losing the first round of a tournament hurt your final result?

This is the second half of a two-part piece about arguments I’ve had with Shengwu Li. Part 1 gives a statistical argument for why in some states it is rational to vote.

Today’s question: can you sleep through the first round of a 9-round debate tournament, like the World or European championships, without hurting your final result? You might think this question is only interesting to debaters, but the mathematical way of answering it is potentially applicable to any tournament with some randomness in pairings in which the winners play the winners and the losers play the losers; this also happens in chess and lots of other competitions.

(Brief note for non-debaters: in each round of the debate world championships, you compete against three other teams, earn up to three points, want to accumulate as many points as possible. In each round, you’re paired against teams with roughly the same number of points, so if you win, you face better teams.)
The argument that the first round doesn’t matter is that, if you lose, you face worse teams in the next eight rounds, an advantage you can capitalize on. Perhaps this evens out.

A first glance at the data would imply this argument is wrong. In the three debate world championships from 2013 - 2015, teams who earned three points in their first round ended up with about six more points by the end of the tournament than teams who earned zero points in their first round. This seems pretty amazing, because you only get three points from winning your first round, and then you have to face better teams; how do you end up six points better? Perhaps winning your first round gives you a confidence boost that improves your performance?

This reasoning is wrong. Winning your first round doesn’t necessarily cause you to do better in later rounds; it’s just a sign you’re a better team. (Similarly, getting in an ambulance doesn’t cause you to die; it’s just a sign that you’re sick.) The teams who win their first rounds would’ve done better whether they won their first rounds or not.

If we want to figure out whether winning your first round causes you to do better overall, we need to control for the fact that teams that win their first rounds are better. In an ideal world, we’d just do an experiment: instead of running the first round, we’d divide teams into four random, equally-sized groups, give one group 3 points, one group 2 points, one group 1 point, and one group 0 points, run the rest of the tournament normally, and see whether the initial advantage ended up mattering. But we can’t run a real random experiment, so we need to find a random factor that affects who wins the first round. We call the random factor an instrumental variable. The math behind how exactly this works is beyond the scope of the post (here’s a less mathy reference, here’s a more mathy one), but the basic recipe is straightforward. If you want to know how cause X affects outcome Y, you need to find a random factor Z which affects X (and is uncorrelated with Y when controlling for X). Then there’s some math that lets you put those three ingredients together to figure out how X affects Y. That’s a lot of symbols, so here are some examples.
How does X...
Affect Y...
Random factor Z which affects X
Serving in the Vietnam War
Future earnings
Draft lottery number
Tea party protests
Election outcomes
Rain on tax day
Iron metabolism
Risk of Parkinson’s
Genetic variants affecting iron metabolism
Height and BMI
Socioeconomic status
Genetic variants affecting height and BMI

To return to our debate problem: our X is how well you do in round 1, our outcome Y is how well you do in the tournament overall, and now we need a random factor Z that affects how well you do in round 1. One random factor that affects a team’s probability of winning round 1 is how good the other teams in the round are, since teams are randomly matched in the first round. We estimate the likelihood that each team will win the round using a number of other factors, including their record in past tournaments, their school’s record in past tournaments, and their EFL / ESL status [1], and use these as our instruments. Then we use these instruments to estimate the true causal effect of winning round 1.  (Another source of randomness you could potentially use is which position you’re assigned to; we discuss this further here [2]).

Here are the results. The red line shows the estimated effect of your round one result on your score after each round (x-axis) using the naive, incorrect method we describe above; the black line shows the estimated effect using the more sophisticated instrumental variable method.
A couple interesting things here. First, the red line and the black line are very different; in particular, the naive approach (red line) suggests that winning round 1 has a much larger effect, and this gets bigger as the tournament progresses, which is wrong for the reasons discussed above. The more sophisticated analysis implies that while winning round 1 affects your performance for a couple rounds, by the end of the tournament (round 9) it doesn’t make much difference. (The 95% confidence interval on the estimate is unfortunately large [3], because the instruments aren’t that strong.)

  1. Don’t freak out too much if you do badly in an early round in a long tournament like Worlds or Euros -- our estimates suggest it doesn’t affect your final performance very much, if at all. On the other hand, in shorter 4 - 6 round tournaments, losing an early round can have a larger effect.
  2. If early rounds have a smaller effect on your final performance, it might be better to run controversial debates in earlier rounds; that way if teams had to opt out, they could do so without taking as large a hit to their final result.
  3. Storing comprehensive data from all Worlds in a single centralized repository, with a standard format, would make these analyses easier and is worth doing.
  4. Naive correlational estimates of effects are often different than those estimated using causal methods. For the love of God, bear this in mind when reading popular social science coverage. Even if the authors claim to have “controlled for other factors”, this phrase does not work magic. It is very difficult to control completely for other factors.

[1] This isn’t, of course, a perfect measure of team strength -- since Oxford A is usually composed of different people from year to year, and many teams do not appear in other Worlds.
[2] In each round, every team will debate the same statement -- for example, “We should ban abortions at all stages of pregnancy” -- but because there are four teams in the same room, two teams will be randomly assigned to argue for the statement, and two teams will be assigned to argue against the statement. It’s hard to come up with statements which are perfectly balanced, so it’s often better to be on one side or the other. We found that a team’s score after the first four rounds was significantly associated with the positions they had been randomly assigned in those rounds, so position in the first four rounds could be used as an instrument. You have to use the first four rounds, not just the first round, to satisfy the exclusion restriction, since most debate teams cycle through all four positions in the first four rounds.
[3] I think you should actually compute standard errors clustering by round, since results within round are highly correlated; when you do this, the errorbars are slightly larger but the conclusions are the same.

Monday, July 25, 2016

Arguments I’ve had with Shengwu Li, part 1: should you vote?

This piece comes from conversations with my housemate, Shengwu, who is leaving us for Harvard. We are all sad about this; I am sad about it in part because Sheng has sharpened my thinking as much as anyone I know. I still remember the first time we talked about math; sitting at a coffee shop I thought, shit, this guy with the Oxbridge accent and the penchant for Latin phrases might actually be as smart as he sounds. You won’t find someone who’s more fluent talking about both math and morality, and you won’t get a better intellectual guarantee than “this argument is Shengwu-proof” . I’m not really sure what we’ll do without him -- less rigorous social science, I guess. Any errors in this analysis remain my own.

Here are two questions we’ve been arguing about recently.

  1. Is it rational to vote in a national US election just because of the remote possibility you could change which candidate wins?
  2. How much does sleeping through the first round of the debate world championships hurt your final result?  

I’m going to talk about the first argument in this post and the second argument in a later post because I’ve been informed that there’s a limit to how much other people want to listen to Sheng and I argue.

There are many arguments for voting -- “what would happen if no one did?” “people died for your right to vote!” -- but let’s consider a crazy one: you should vote because you might be the tie-breaking vote in an election. Obviously, this is incredibly unlikely. On the other hand, if you did, it would be incredibly important. So just how unlikely is it?

Say you live in a state with a million expected voters where one candidate is polling at 52% and the other at 48%. Here’s a first attempt at figuring out how likely you are to be the tying vote: if each voter is like a coin with a 52% chance of landing heads, what is the probability you get exactly 50% heads? The answer is so small -- less than one in 10100 -- that the computer returns 0. If you run a bunch of simulations of the fraction of heads, it looks like this -- pretty much every time you’ll end up with a fraction close to .52.

This implies that voting is a waste of time; you have no chance of influencing the outcome. But this simple analysis turns out to be wrong for a simple reason: polls aren’t perfect. Just because a poll tells you 52% of voters prefer a candidate doesn’t mean that 52% of voters actually do. One source of error is the margin of error in polls: as the Upshot pointed out recently, because pollsters only sample a relatively small group of people, polling results fluctuate just due to random noise. So if a poll tells you that 52% of voters prefer a candidate, the real percentage might be something like:

Which is a lot more uncertainty than we saw before.

Given this uncertainty, it turns out the chance that you're the tying vote is not that low -- somewhere between 1 in 100,000 and 1 in 10 million (see [1] if you want modeling details). Which sounds low but is actually astonishingly high. If the difference in value between the two candidates is, say, a trillion dollars -- which is plausible, given the size of the US budget and the very different uses Trump and Clinton would put it to -- then a 1 in 10 million chance of influencing the election outcome means that in expectation, you change the allocation of $100,000. Of course, I’ve only simulated a single state, not the whole election, using a very simple model. But after doing this I discovered a more sophisticated analysis from Andrew Gelman, Nate Silver, and Aaron Edlin which reached a similar conclusion: if you live in a state which is at all close, voting is probably worth your while simply because of the possibility you will be the tie-breaking vote in an election. If you are trying to decide whether your state is at all close, err on the side of “yes”; here are 538's predictions for every state.

This was surprising to both of us; I am definitely going to vote now (in Virginia, a swing state), and you should too!

Addendum: a corollary to this is that you should not vote for a third party candidate on the grounds that "my vote will never matter anyway, so I'll just vote my conscience". Vote for a third party candidate if you want to, but if your state is at all close -- and current polls imply that many states will be -- weigh your desire to vote your conscience against the fact that, mathematically, your vote really could affect the outcome of the election.


[1] Very simple model: draw the true fraction of voters ut preferring a candidate from a normal, ut  ~ N(up, sigma), where up is the fraction preferring a candidate according to polls and sigma is the uncertainty on the polls (due not just to margin of error but to other things as well); then draw the number of voters voting for a candidate, V, from a binomial distribution, V ~ binom(T, ut), where T is the total number of voters. What is the probability that V = T / 2?

Wednesday, June 29, 2016

How Angry to Be

This essay was written over the course of six months and was initially titled “Why I don’t write angry feminist essays”. Then I realized that I actually was pretty angry. The final product is two-part: first I talk about the benefits of staying calm, and then I talk about the benefits of anger.

The TV show Jessica Jones pits a woman with superhuman strength against her rapist, Kilgrave, who can control minds. Many people have commented on how Jones is really fighting the patriarchy: Kilgrave forces women to smile for him and sleep with him, stalks them and threatens the people they love, and Jones confronts many lesser sexists throughout the season. It is deeply satisfying to watch her throw sexist assholes through walls; I watched the entire season three times in three months.  As I did, however, I realized the very thing that made the show satisfying makes it a poor metaphor for fighting sexism.

Kilgrave is an Unequivocally Bad Dude -- a serial rapist and murderer who casually tortures people for no reason:
Figure 1: Seriously, dude?
Few agents of the patriarchy are so one-dimensional. Consider:

Female engineer 1: God, this guy at work asked a coding question today -- and when I answered, he ignored me until a male engineer gave the exact same answer.
Female engineer 2: I hate that. So what did you do?
Female engineer 1: I threw him through a wall. Fractured his spine.
Female engineer 2: You go, girl!

said nobody ever.

This isn’t a complaint about Jessica Jones; it’s a superheroine show, not an advice manual. But it is a complaint about the tactics used by many people who claim to be fighting the patriarchy. Too often, I think, we treat people as Kilgraves when they aren’t.

There are innumerable examples of this. Take the backlash against Scott Aaronson (professor accused of being a sexist, entitled nerd) or Chris Herries (student accused of comparing rape victims to bikes) or any of the targets of viral internet outrage, take this uncharitable response to Stack Overflow’s well-intentioned attempt to get more data on women among their users, take many, many things written by Jezebel. I try not to demonize people in my work for two reasons.

  1. It paints a false picture. People are complicated and contradictory. There are rapists whose wives love them [1]; fraternity men who call some women “slampieces” but treat other women with respect; men who campaign for gender equality but talk over women; women who call themselves feminists but attack Bill Clinton’s accusers; fathers who cheer on their daughters but are biased against their female employees; online trolls who feel remorse; philosophy professors who argue for consent but harass their female students; men who can quote Simone de Beauvoir but won’t do the damn dishes. Demonizing people who do sexist things ignores these complexities. Worse, it lets us relax in the comfortable lie that only demons do sexist things. The scarier truth is that the great injustices in history have not been carried out by Kilgraves, but by ordinary people -- perhaps there was a psychopath at the helm, but they needed willing executioners.

  1. Demonizing people alienates potential allies. Take this recent piece criticizing tech CEOs who think “diversity” refers only to gender: “Is it because they’re racist? Sexist? Ignorant? Some sick combination of all three? Probably.” This is a great way to make CEOs afraid to talk about diversity at all. Of course they should be thinking about diversity in more nuanced ways, but there are less vicious ways to convey that.
I know that such rhetoric alienates powerful men because I’ve talked to them. One male tech leader told me that while he cared about diversity, he was reluctant to speak up about it publicly because he worried about incurring backlash. Another told me that he was less likely to hire gender studies majors because he thought they were more likely to sue for discrimination. Perhaps these men should not have harbored such views. But given that they do, we should adapt our rhetoric if we wish to be persuasive.
And if you’re saying screw ‘em, I don’t care if my essays alienate men -- I would submit to you that men rule the world, and if we really want to change it, as opposed to just writing echo-chamber clickbait, we will do so faster if we don’t lose 90% of CEOs, members of Congress, and tech leaders. Which is not to say that you should try to persuade all men -- but if you don’t persuade any, you might want to revise your rhetorical approach. (It is of course sometimes necessary to voice difficult truths which will alienate people; I am not opposed to all radical feminism, but specifically to feminism which divides by demonizing. You are not speaking truth to power when you demonize people; you are simply being inaccurate in a way which also drives away allies.)
I worry too that the more extreme voices in the feminist movement get disproportionate attention (a phenomenon my sister and I also observed when studying campus activists). Anecdotally, men I talk to are more likely to have heard about public shaming campaigns than about the less controversial aspects of feminism. Zeynep Tufekci, a professor at UNC who studies activist campaigns, notes that if a protest movement does not self-regulate and consciously decide its boundaries, it gets defined by its noisy, flamboyant outliers. Always.


And yet. I can make reasoned arguments for seeing the good in sexists, for preaching to the unconverted, for cutting out vitriol and sarcasm. But the truth is that I’m filled with so much anger. People read my statistical pieces about gender and tell me that they like that I can stay so detached and I laugh -- because why would I possibly spend so much time doing math about gender inequality if I were detached?

And I’m getting angrier with age. I am only 25 so this is somewhat concerning. I have lost the ability to laugh at things. I’m writing this in my room after watching Top Gun, an 80s movie about a bunch of military pilots. In one scene, the hero and his copilots surround a woman in a bar, much too close, red-faced and sweating, and drunkenly serenade her; she eventually escapes, so the hero follows her into the woman’s bathroom and tries to block her from leaving. On the one hand it’s dated and you want to laugh at these ridiculous men. But then I think about the rapes in the US military and how this movie was so influential that signups for the naval air force went up by 500% and I wonder how many rapes it helped cause. I read that the actress in the movie, a lesbian, was actually raped in real life, and spent decades thinking the rape was a punishment for her sexuality -- these are the hilarious things that happen when you aggressively push heterosexuality on everyone, haha! I remember the thousands of sexually aggressive comments of the fraternity men I studied and the papers that show how much more often rape occurs in fraternities and the hundreds of stories from survivors I perused and the ones who were brave enough to talk to me directly and the rape cases my mother prosecuted and I have to admit: I can no longer laugh at sexually aggressive bros. I close the door to my room and slam my fist into the wall over and over again, feeling nauseous.

On a day to day basis it isn’t the rapists who bother me; it is the subtle inequalities even among my progressive, thoughtful social circle. (I can only imagine how women who face more overt discrimination feel.) It’s the men who sit when the women clean after dinner; the women who don’t speak up when they’re uncomfortable because they want to be accommodating; the men who ramble at me even when I’m supposed to the one giving a talk or giving advice; the dates and friends who can lecture but can’t listen; the professional meetings where comments are addressed only to my male colleague. And if I can see these inequalities already, what will happen if we have children, when our incomes further diverge? I am afraid for my friends and afraid for myself.

Part of this increased awareness is the accumulation of slights that are each insignificant, the chafing of a blister rubbed raw. Part of it is that I spend so much of my research time looking for discrimination. Part of it is dealing with online responses to my writing. My writing voice is mild, but in response, I’ve had people call me a lying cunt; insult my body; speculate that I’m bitter because no one will sleep with me and I’m too prudish to get invited to parties; advise my boyfriend to break up with me; call me a token admit to the schools I’ve attended or the conferences I’ve been invited to. These commenters, of course, are the warts on the long statistical tail of readers, but you don’t forget them. On the advice of a male colleague, I now pay $100 a year to keep my personal information off the internet.  

Part of it is that I’m now single. I could make a joke here about how attempting to date Silicon Valley men would make anyone a feminist, but that would contravene what I said above about not unnecessarily being an ass to men and isn’t the point I’m trying to make anyway. When I was in a relationship and I got angry about something, I’d come home to a boyfriend who, our breakup notwithstanding, was one of the most gentle and decent people I’ve met, and he would both calm me down and by his mere existence remind me that #NotAllMen are Satan.

These days I sleep alone. I wake at four in the morning and brood for hours; there’s no one to break up my thoughts, no one to vent to except the empty page. And I’m beginning to believe in the value of anger untempered.

A few months ago I went to dinner with a man who wouldn’t let me get a word in and I came home to an empty house; soon my anger got the better of me and I wrote an essay about mansplainers so quickly it was as if it was torn out of me. It was one of the angriest essays I’ve written but also, I think, one of the better ones. Another night, I woke up so irritated about a male collaborator who was not pulling his weight that I couldn’t go back to sleep. Perhaps men were simply better at getting credit for projects they hadn’t led, I thought, while women’s contributions were ignored [2]. I grabbed my laptop and wrote a computer program to scrape a database of scientific papers and look for gender disparities. A few months later I published a piece based on that 3 AM analysis. So there are times when I am glad I’ve had to reach for a laptop rather than a hug; when I’ve had to put my anger fresh onto the page because there was no one to dull it.

(My friend advised me to cut out the last three paragraphs because they invite nasty comments about how I’m just bitter because I’m single, but I want to trust you to read what I wrote and not skew my words.)

So as I said at the beginning, I’m conflicted about how angry to be. Often when I work I’m not angry at all -- I’m looking at the world through twin lenses which both help me stay detached. The warm lens comes from years of counseling training: to listen without judging, to want to understand how someone who seems despicable can be the hero of their own story. The cold lens comes from years of quantitative training: rapists, like cancer cells, are simply high-dimensional mathematical structures to be parsed.

At times I can master this difficult balance: to humanize without exculpating; to be both furious and curious; to be motivated by anger but not overwhelmed by it.

And then I read, say, the Palo Alto woman’s statement to her assaulter and my careful detachment shatters and I’m overwhelmed again.


[1] You might say, I don’t really care if a rapist had a golden childhood -- but I think you actually should care, not because it exculpates them but because the whole problem is to understand how people grow up to commit rape. If only devil spawn did evil things, we would have a much easier problem. Obviously, we need to put an end to media pieces which portray rapists as athletes first and criminals second. But we also need to analyze rapists as a complex human beings, not monsters, while still giving full weight to the seriousness of their crimes. David Lisak’s seminal research on repeat rapists, which humanizes them without ever forgetting that they’re serial predators, is an example of this kind of work.
[2] There turns out to be some evidence for this: see here and here.