The Professional Effect

In a recent discussion with a friend, I talked about why the ability of polls and  markets to predict political outcomes seem to have taken a beating as of late. I described an effect whereas the more faith people place into a certain predictive tool and gain more knowledge of it, the less reliable it becomes. I vaguely remember hearing about it somewhere but when pressed for an academic source, I came up short. I couldn’t even give a name for it, which leads to the disturbing possibility that I simply made it up.

However, weeks later, I found a name for it in a rather unorthodox source: this video on the video game Hearthstone. At around the 11 minute mark, Trump talked about something he named the “professional mage effect”, which I shortened to simply the professional effect, which I believe is one of the main causes in the crumbling of the accuracy of polls and other statistical tools as of late. Trump himself summarized the effect fairly succinctly in the video, but for those of you who don’t know the language of Hearthstone, I will give you an explanation.

In Hearthstone there are nine playable classes. Each class has a distinct set of accessible cards and hero abilities. In the current incarnation of the game, the mage class was given several new cards that gave it a significant advantage over other classes. This meant that every time it was possible to select a mage, people tend to select mage; the difference in win rate between mage and the other 8 classes was staggering. As a consequence, Blizzard decided to introduce some changes to try to balance the game, most notably removing several problematic cards for mages. However, after a month of the changes, it seems that mage still maintained a sizeable lead. Blizzard explained this with the ‘professional mage effect’, where strong players who tend to read the statistics instead of choosing at will or from word of mouth, tend to still prefer mage because its previous high win rate and its current average win rate still gives an above average win rate; and since these players tend to be the strongest, their superior skill allows them to win more, which is conflated with the class being strong.

It appears that a similar phenomenon is at play when it comes to high frequency trading. Whenever too many algorithms use similar methods to choose stocks, a slight advantage is exaggerated and causing many buyers to buy into it, significantly overvaluing the stock. This then creates a lot of uncertainty in the valuation of stocks and makes the market rife for bubbles.

This seems to give a rigorous justification that a diversity of ideas is quantitat

新三国剧《军师联盟》

作为一个铁三国迷,我非常期待吴秀波主演的新三国剧《军师联盟》。看了片花以后,觉得潜力非常大,有可能有让历史焕然一新的作用。

在大众心目中的三国故事,三国演义,其实是元末明初人士罗贯中所写。演义里面的人物正邪鲜明,明显是一个捧蜀汉贬曹魏的著作。最近国内两大三国演义电视剧都没有摆脱这个以刘备为英雄,曹操为枭雄的基本故事结构。最起码《新三国》里面的司马懿戏份还算不少,但仍然是一个负面角色。

然而看完《军师联盟》片花,可以看得出一些与小说不同的说法。其一,故事的主角是司马懿,也就是早就最后三国归晋的政治家和军事家。司马懿是在董卓乱朝,诸侯割据之后才入仕,所以按道理不算是什么汉臣,也就没有跟曹操那样的道德枷锁。再者,司马懿此人的故事很复杂。他辛辛苦苦伺候了三朝大魏郡主(包括曹操在内),其实可以说的上是忠臣。但是到老却无奈曹魏重用曹姓人的秉性,被排挤,甚至有功高震主而有可能招来杀身之祸。最后他只能反击,独揽朝野,造就了晋代魏的局面。虽然他是中国一个朝代的“太祖”,却永远在故事里被描述成曹操的属下或诸葛亮的劲敌,永远不是故事核心。所以这出戏有潜能能把一个杰出历史人物的故事讲好。

另外一个剧情可能没有多少人留意。在三国历史上被黑的最严重的人非蜀后主莫属。“阿斗”已经成为笨蛋庸才的代言词,而每一个跟三国有关的文艺作品都会把刘禅写的一文不值。三国演义为了抬高蜀相诸葛亮的形象,更把后主写的跟猪一样。但是历史上的后主可能不是什么明君圣贤,但是最起码他以一个外来人(刘备是从外地征服蜀地的),年轻人(后主上位时才十七岁)的状况下成功的团结了蜀地原来的官僚和士绅与刘备从荆州带来的部下和军队。从这看来刘禅绝对不是什么笨蛋,而是拥有一定政治能力的。在片花了,后主大怒把走着从案上一扫而下,悲叹“十万大军!这已经第六次了!好一个‘忠臣良相’!”他说的第六次肯定是说诸葛亮六出祁山,而他的悲叹很明显:孔明率领大军北伐,对经济民生造成了极大的负担,也是最后导致蜀国国弱民衰的原因。这种叙述诸葛亮的写法好像从来没在戏里看过,一定很新鲜,很让人值得思考。

希望能尽快看到!

An ethical question with a concrete answer

Ethics questions often give a scenario (often unrealistic and missing key details) where you have to choose between a set of difficult options. For example, it could be “an old lady who has no family or a young man with a promising career and loved by a lot of people are both admitted to the ER. Who would you prioritize on treating if their probability of survival is the same?”

However, sometimes the answer is quite a bit more straightforward. The question I saw is the following: Suppose that an intelligent machine has the ability to predict, with 99.99% accuracy, whether someone will commit murder. Would it be permissible to arrest people based on the predictions of the machine?

There seems to be no ‘correct’ way to answer this question, but that’s because the person who asked the question doesn’t understand statistics. There is a phenomenon in statistics called the false positive paradox, where even a very accurate test will produce many more false positives than actual positives. This is relevant when the actual probability of a true positive is very low.

Here is an example. You are a unique person, one of say 7 billion. Suppose that there is a machine that can identify someone with 99.9999% accuracy. A person gets scanned by the machine, and the machine says it’s you. What is the probability that the machine is right?

There are two possibilities for the machine to give that reading. Either the person scanned by the machine actually is you and the machine is accurate, or the person scanned by the machine is not you and the machine malfunctioned. The probability that the machine is accurate is 99.9999%, and the probability that the person scanned is you is one in 7 billion. The other possibility is that the machine is wrong and the person being scanned is not you. The probability that the machine is wrong is 0.0001%, and the probability that the person is not you is 6,999,999,999 out of 7 billion. Thus, the probability that the person actually is you, given that the machine says it scanned you, is given by the equation

\dfrac{\frac{999999}{1000000} \times \frac{1}{7000000000}}{\frac{999999}{7000000000} \times \frac{1}{7000000000} + \frac{1}{1000000} \times \frac{6999999999}{7000000000}}.

Evaluating, the probability that the machine is right is less than 0.015%. The paradox is caused by the fact that the machine’s accuracy is very poor compared to the astronomically unlikely phenomenon that you would be picked out of 7 billion people.

Now back to the question. Whether it is ethical or not (that is, whether the machine produces a desirable result an acceptable proportion of the time) will depend on the actual murder rate of a place. The global average is currently 6.2 people out of every 100,000, or 31 out of every half a million. Assuming the machine has accuracy 99.99%, the probability that a person identified by the machine as a murderer is actually a murderer is given by

\dfrac{\frac{9999}{10000} \times \frac{31}{500000}}{\frac{9999}{10000} \times \frac{31}{500000} + \frac{1}{10000} \times \frac{499969}{500000}}

or roughly 38.3%. Therefore, the machine is right way less than half the time, far below what can be considered reasonable. Therefore this question has a concrete answer and is not really debatable.

An elementary statistics problem from Hearthstone

In Hearthstone, a popular online card game, there is a minion called C’Thun which deals damage equal to its attack value, one damage at a time. This prompted a popular streamer named Trump (not the insufferable presidential candidate) to ask the following question once on his channel: when playing a C’Thun with 12 attack (hence, it will deal 12 damage one at a time), what is the probability that it will kill a 6 health minion when played, with no other minions on the board?

To those who don’t know how Hearthstone works, one can think of the question as follows: suppose you flip a coin 12 times in a row. Each time you flip the coin, you record down whether you got heads or tails. As long as you have fewer than 6 heads, the coin is fair and you have equal probability of getting heads or tails on your next flip. However, as soon as you hit 6 heads, then the coin becomes unfair and you can only get tails after. The question that Trump (real name Jeffrey Shih) asks is then equivalent to “what is the probability that you get 6 heads”.

To calculate this probability, we can ask the reverse question: what is the probability that we don’t get six heads? If we don’t get 6 heads, then the probability distribution is the same as if we always had a fair coin, and so the probability is the same. The probability of getting at most five heads when flipping a fair coin 12 times is given by

\displaystyle 2^{-12} \sum_{k=0}^5 \binom{12}{k}.

The well-known binomial theorem tells us that

\displaystyle 2^{-12} \sum_{k=0}^{12} \binom{12}{k} = 1.

This shows that the probability of getting at most 5 heads is equal to

\displaystyle \frac{1}{2} - 2^{-13} \binom{12}{6} = \frac{793}{2048}.

Therefore, the probability of getting 6 heads in the original game is equal to

\displaystyle 1 - \frac{793}{2048} = \frac{1255}{2048},

or roughly 61.28\%. Thus it is far more likely that C’Thun will kill a 6 health minion with only 12 damage than too many shots going to face.

Some thoughts after reading an article on Peter Scholze

Recently the following article appeared on my Facebook feed: https://www.quantamagazine.org/20160628-peter-scholze-arithmetic-geometry-profile/Aside from the usual bland mix of singing praises of an archetypal ‘genius’, the article does contain some genuine insights. The most striking of which is Scholze’s description of him learning the proof of Fermat’s Last Theorem by Sir Andrew Wiles: that he “worked backward, figuring out what he needed to learn to make sense of the proof”. Later he also said things like “I never really learned the basic things like linear algebra, actually – I only assimilated it through learning some other stuff.”

If you have experiences learning mathematics at the senior undergraduate level or post-graduate level, you will likely find that these experiences are orthogonal to your own. We spend an inordinate amount of time learning the ‘basics’ for various things, which very much includes linear algebra for example, in order to do research… or so we are told. If you have passed the part of your career where you do more courses than self-learning, then you have likely reached the epiphany that usually it’s not efficient to learn everything there is to know on a subject before actually doing work on the subject.

Some of you have had advisors telling you things like “read these five books (each 300+ pages) before you attempt any research work in the area”. Sometimes this advice comes from highly proficient researchers, which seems odd: if the above ethos is ubiquitous among researchers, why tell your students to do something totally different and entirely more dreadful? I am not sure what the right answer is, but probably part of the reason is a misguided attempt to make research ‘easier’ for students. Perhaps many advisors recall the struggle of trying to understand ‘simple’ phenomena that they encountered in their research careers, that if someone had just told them to read a book or if they were better prepared, would have been trivial to overcome. Perhaps they wish to save their students some time by telling them the shortcut. However, the struggle to understand phenomena on your own is part of what makes research rewarding, and more importantly, it is critical in forging a mind suited to making discoveries.

Of course, I am but a pebble to the avalanche that is Peter Scholze, so my advice may not be worth much. Nevertheless, I feel like I should say this to all prospective and current graduate students: be bold, and give every difficult paper in your field a read. Don’t be intimidated by them. If you don’t understand something, google it until you find what you need to learn the language of the subject. Don’t feel like you need to understand all of Harthshorne before you can read any research papers related to algebraic geometry. Your future self will thank you for this.

A question for students you dislike

At the recent CNTA conference, Professor Joe Silverman gave an explicit homomorphism of \text{GL}_3(\mathbb{R}) into \text{GL}_6(\mathbb{R}) which he jokingly called a great question to ask undergraduates to work out explicitly… if you don’t like them very much. In a similar vein, here is a question that one might ask undergraduates they don’t particularly like:

Let \mathbf{a} = (\alpha, \beta, \gamma) be a triple of co-prime integers which are not all zero and such that \alpha \gamma > 1. Prove that the ternary quadratic form given by:

K_{\mathbf{a}} (F) = \dfrac{1}{8\alpha^3} \bigg( 72 \beta^2 \gamma A^2 + 9 \alpha(\beta^2 + 4 \alpha \gamma) B^2 + 8 \alpha^3 C^2 - 18\beta (\beta^2 + 4 \alpha \gamma)AB
+ 12 \alpha (3 \beta^2 - 4 \alpha \gamma)AC - 24 \alpha^2 \beta BC \bigg)

takes on integer values whenever the triplet (A,B,C) lies in the lattice defined by the congruence conditions

4 \beta \gamma x - (2 \beta^2 + \alpha \gamma) y + 2 \alpha \beta z \equiv 0 \pmod{\alpha^2}

and

\gamma(2 \beta^2 + \alpha \gamma)x - \beta(\beta^2 + \alpha \gamma)y + \alpha \beta^2 z \equiv 0 \pmod{\alpha^3}.

I will reveal the solution in due time (and it does not involve explicit computation of congruences), but if come up with a solution let me know!

 

Manjul Bhargava’s advice to mathematics graduate students

Last night, I had the honour of attending a panel discussion featuring eminent mathematician Manjul Bhargava. During the panel, the moderator, Professor Kumar Murty asked the very productive Fields Medalist to give some advice to graduate students in the audience who may be struggling with their research. Professor Bhargava’s response, paraphrased, is essentially the following:  always work on several problems at a time (at least three), of varying difficulty. There should be a problem which is quite difficult and if you make any progress on it it will be a major breakthrough; there should be one of moderate difficulty, and there should be an ‘easy’ problem that you know you can make progress on eventually. Further, never think too much about a problem at a time and instead rotate between the problems to change your mindset. Sometimes when you approach a problem with a fresh perspective you will gain some insight that would’ve been impossible if you stared at the same problem continuously, since you are subconsciously trying to apply the same techniques.

I thought Professor Bhargava’s advice was very helpful to the graduate students in the audience. It is something I started doing a few years ago, but it wasn’t something that I was aware of consciously. Hopefully heeding this advice will be helpful to your work.