Games with incomplete information

There are traditional machine learning tasks, where signal is hidden in the data, and once it is found, you're done.

There are games where the situation is well defined, known to everyone, opponents take turns and the outcome is set in stone for each sequence of turns. It's just hard to look through all the possibilities to confirm whether this or that path has a winning ending for you. So you need to have a general idea of how the late stages of the game tend to look like, as well as the ability to see the possibility more steps ahead than your opponent. So the information about opponent becomes irrelevant, instead you just focus on predicting further. Then you are can become good at playing against versions of yourself who are also trying to do so, and that pretty much guarantees you will be good against anyone else.

Now games with incomplete information seem to require you to have some understanding of your opponent. A practical example is poker. Any Nash equilibrium examples are also such. The distinction is kind of blurry though. In chess, you can also try to win by understanding your opponent, it's just not an effective strategy in that context. The distinguishing feature of poker, besides having a lot of freedom with bets and multiple players, is that each player is dealt a hand that is not revealed until the end (if at all). So if in chess there was a clear deterministic outcome that you can calculate depending on actions the players take, in poker at the end the outcome also depends on a hidden variable known to another player but not you. If you just assign a uniform prior to that variable, you will win against completely stupid players, but you will lose pretty quickly to anyone with experience because your actions reveal your hidden variable while you don't even try to learn anything about theirs. In the competition, it becomes very meta: the player who can predict what others think he thinks others think... wins.

However, hidden variable do not even have to be random. If we ask the players to hide the action they choose until both of them chose something, and then reveal - that's exactly the same. Let's start with a game with no hidden info. Consider a game with two players, where they are assigned score based on their choice and the opponents last choice according to the matrix
a b
c d
They keep making choices in turn. One of the players chooses corresponding to max(a,b) and max(c,d), another to max(a,c), max(b,d). Now consider the matrix like
0 -1
-1 1
It has two equilibria - 0 and 1 - where the next choice is always the same for both players. However, swapping between the two incurs a penalty. Clearly, even without communicating players will switch to the second answer. Consider
-1 1
1 -1
This matrix encourages them to answer differently from each other. Consider
-1 1
-1 -1
This one is weird - the only way for any of the players to win is when another answers the first answer. But then the other doesn't get any benefit from doing so, unless he indicates that he will stop giving freebies and wait for the other player to answer the first answer. If they understand each other quickly, they can form a sequence like 0110110110110110.. which will give them only -n/3 value. If one of them submits, he gets -n and the other gets +n.
A matrix like
00
10
is equivalent to the above.
11
00
will result in the game where outcome doesn't depend on your choice locally, but you can be nice to your opponent and let him win. He can then do the same to you


In principle their matrices do not have to be the same.
Also the scoring system may be different: they may want to optimize the difference between their scores, not the individual scores. In that case, even the games above become more complicated! A first step of complication is if the differences are counted locally: that just introduces dependance on your previous step. But then the choice is not so easy anymore!

How about another simplification: yes there are turns, but the first player token shifts once the players made their moves. The score is only counted within a pair of moves. Again there are matrices, in terms of two possible answers of first player and second player:
10
01
everyone just gets zero score. Nothing to see here. Wait what? How is the first player scored? Is only the second player scored? I guess.
01
10,
11
10,
same, but
00
01
is special. Actually only  max in a row matters. In this simple game, you can give something to your opponent in hopes that he will give something to you. And the only way to win is if he gives back more than you gave him by the time the game ends.

The game is actually very simple. You have two choices, 0 and 1. If you press 1, your opponent's score increases by 1. That's all. You take turns pressing the buttons. The game lasts n rounds. Your goal is to get a bigger score than your opponent.

The problem with this game is the presence of a simple solution where the game gets "stuck".

Let's consider different scoring mechanisms. Now the first player gets the score. The matrix is again
00
01
so the second player may or may not give him the score if he asks for it. This game is a bit more interesting: the two players ask each other, and then it reduces to the first one.
Now if both players get the score, there are more choices, the score of first and second correspondingly:
00
00,
01
01
second can just increase his score by his choice.
10
10,
01
01
second choses whether to increase himself or the opponent, clearly chooses himself.
10
10,
11
00
 first one chooses to increase second, second chooses to increase first - reduces to the previous game.
10
10,
10
01
first one will always choose the one that leads to his own increase.
Strangely, nothing interesting here.

All of these games can in principle be solved by recursion: the very last step is obvious, regardless of the past configuration. Thus the step before that can be decided by using the knowledge of what your opponent will do on last step for any configuration. Et cetera, et cetera. The only time where interaction with the opponent becomes more personal is when you're "begging" for scores like in the examples above.

Now when the players reveal their answers simultaneously, things get a bit more tricky, and cooperation becomes more important especially if the game is repeated several times. I'd like to give an example of a 3 player game that scores +1 to players who voted together and -1 to the player who was a minority vote, and -1 to all if they all agree.
Interesting thing about this game is that your score does not depend on your actions directly at all, only on whether the two other players agree or disagree. But if the game is repeated several times, they can infer your strategy and your score will start depending on your answers.

Even in the taking turns in sequence case, when each one gets scored according to the previous two, the game is essentially a game of begging. Half of the times the players swap their vote for something completely random, so they can try to give things to other players in hopes of getting something in return.

There's no beating strategy to such game, you can't even win against random opponents. But in the very special cases, if your opponent is consistent, you can hope to win something. Now imagine an algorithm trained to learn this game, to defeat other, previous generation algorithms. What is it gonna be like?

Imagine 3 STEM PhD's paid hundreds of thousands of dollars to compete in this game, with the one losing being fired. Of course they will make really sophisticated algorithms to outperform each other. Or maybe they will immediately form a coalition and kick the odd one out.

So the current state of machine trading is somewhat reminiscent of this game. I'd like to make an estimate of how long it takes to learn about your opponent. Every turn of the game you get 1 bit of information (whether you lose or win). You don't even know how others vote in the hidden variable case, or you do see the pattern of votes (thus know your opponents and get 2 bits every turn). So if you assume the algorithm of size n for your opponents, it will take O(n)*assumed lookback to estimate what algorithm each one of them have, and come up with your own. But that algorithm by itself is much bigger, so the next generation of opponents will have to estimate much bigger parameters space. However it's highly non-generic, and also has a transient where it clearly estimates the opponents behavior. The point is, at fairly low step of "meta" the data sizes become intractable to estimate anything.

Scientific facts about investment

To the first approximation, investment is a roulette. The house (your broker) always wins thanks to fees, while you have 50/50 chances. This was shown pretty clearly in first part of the movie Wolf of Wall St, for instance. If you exchange dollars for euros today, and then exchange them back some other day when you feel like it, then ignoring the fees your expectation value of the difference in the dollar amounts is zero. Even the distribution is fairly uniform. Here by expectation value we mean our knowledge: we know about this number as much as we know about a random variable with zero mean and a normal distribution. In the same way, if you buy Apple stock today, and then sell it some day next year when you feel like it, there's no knowledge in science whether it's gonna be up or down then.

To the second approximation, S&P 500 appears to grow. So you have a small positive expectation value in your scientific knowledge for the purchase of Apple stock, that is essentially drowned in dispersion (random fluctuations). Buying S&P 500 index reduces that dispersion by a bit, so you expect to see that small positive expectation value after 3-5 years. The distribution is also skewed - it grows faster typically, but there are crises where it falls dramatically. Crises happen every 10 years or so. That is what has been observed in US for the last 100 years, and in other countries seemingly independently for some time. So a consensus seems to be that it will continue this way throughout the history of humanity. Here people confuse the more tangible concepts of economic growth, consumption, GDP, quality of life, population growth with the artificial concept of market price of S&P 500 (or any other index) stocks. They are actually not related by anything hard-wired in the system, or any inherent property of human psyche. The relationship between them is more like the relationship between the gold stored in fort Knox and the number of dollar bills in circulation. It used to be kept due to conservatism of the people in charge, but it can be completely abandoned whenever it becomes convenient. The value of dollar bills is actually completely independent of amount of gold in practice.

In the same way the price of stocks can in principle become independent from the wellbeing of the companies they represent. We are already in the age where computer algorithms are trading with each other, and the only thing they care about is beating each other, not the concept of value. There are regulators that enforce certain constraints on those algorithms, but the main constraint is against volatility, not against the deconstruction of meaning. So a more pessimistic picture of the future is that traditional investors will slowly become disappointed in value investing, and the prices of stocks will level out in accordance of regulations. The computer algorithms will have a lot of fun trading with each other, but stocks will become just another meaningless electronic currency that fluctuates within prescribed bounds, but is somewhat useless even for the leadership of the company that issued it. This transition is likely to happen around these years, so we might see the last impressive growth in the stock prices around the world. It's probably a good idea to ride this wave, it might be the last one.

To motivate a little bit why stock is decoupled from value, let me just list obvious things:
The company issues stocks, but does not have any obligation to pay dividends on it, or buy or sell it in any way. The company may have ups and downs, the profits may be redistributed as bonuses to the leadership or invested in a company of CEO's wife, either way the stockholders see little of  it. Essentially once the stock is issued, it's just a piece of paper. The voting rights don't matter because someone holds 51% anyways. You can't interact with the company in any significant way using the stock you own. For all intents and purposes, what's happening is just a crowd of people that have in their mind some idea of what price this or that piece of paper has. It's all in their heads. There's no fixed meaning of that piece of paper. If the crowd suddenly decides as a whole to do something else with it, there's no stopping force. Of course real crowd has inertia - there are always conservative people who stick to old ways. But the majority is pretty flexible. And what the majority is doing is going on a wild goose chase of pattern-seeking, sometimes glorified as machine learning or advanced statistics. Of course, those approaches provide good tools to tell whether the pattern is really there, or you're just imagining it. However the malpractice of using these tools in finance introduces several layers of selection biases, after which the original results of those tools become meaningless. First the author of the algorithm or strategy discards a bunch of strategies that didn't work. Then the hedge fund fires the workers whose strategies didn't perform. Then the hedge funds that didn't perform disappear and new ones are created. Finally the investors look at hedge funds, choose the ones that made the best impression using what they call "due diligence". Why is this multi-level scheme of choosing which strategy will the money follow bad? For instance, if at least one of the links in the chain is untrustworthy, the whole thing falls apart, and the huge chunk of money in possession of that given investor is invested according to a terrible, meaningless, essentially random strategy. Moreover, due to constant rotation of algorithms, quants, hedge funds, and investors themselves, there's no memory in the system.  If one tries to ask: how well do hedge funds perform in general? Or: how well do our employees perform in general? One gets an essentially random number that can happen to be positive, which is when it will be used in an advertisement brochure, but it also can happen to be negative, which is when it is not talked about. Selection bias is when we mention a few success stories like Steve Jobs, but never talk about 99% of failure stories. In the same way, the employee may not be 100% honest about his selection of the algorithm and not 100% blind in his crossvalidation. The company's choice of who to fire and whose algorithm to invest money into is usually rushed, and the good statistics are exaggerated (p-value hacking). The successful hedge funds get media and investor attention, while the failed ones disappear quietly, and nobody really counts how many of them were there. Successful investors have a lot of money and a personally confirmed belief that the whole system works so the majority of money in the future is distributed by people whose personal experience was not representative of the system as a whole. Of course locally this industry looks legit, and every quant believe himself to be smarter than the other guy, and his statistical tools to be flawless. So unless you compare his experiences to the experiences of another quant who also believed he was smart but lost, everything seems to be working. Look at people who were fired from hedge funds. Are they really objectively less smart than the ones that work there today? Or were they just unlucky? When there's a hedge fund with flawless 30-year record that you see, how many other hedge funds were there that lost money and lost investors so you don't see them?

For the technical and fundamental traders of the 90'ies, the numbers really added up to zero as they should, ignoring the "making money from air" effect (overall S&P 500 growth, and people who grew thanks to market manipulation, fees, bubbles aka pyramid schemes). That's not bad by itself, in every competition there are winners and losers. It's even not surprising to have zero-sum game where the total wealth remains the same. Still you would expect that the person who professionally trades, as in he convinced somebody to pay him fees to trade their money, will perform better than the random number generator. This was not the case - the industry was so deeply rotten that no selection based on the promo materials and no due diligence were able to correlate with the positive performance of the professional investors. They all were in it for the fees after all. Now there's a myth that the machine trading hedge funds are more honest about their performance reports. And some of them definitely are. But the scientific hypotheses is that the algorithmic trading is equally rotten so that from the outside point of view there's no way to tell apart the fakes from the real ones. They themselves don't know if they are fake sometimes, because quants who have doubts about it just get fired, or not get hired in the first place.

Anyways, even if there's a positive returns promise that we can believe from some of the algorithmic trading, it is not accessible to ordinary investors. You need to have a lot of money to even begin talking to some of the hedge funds. There are platforms which allow you to use openly available algorithms and reproduce whatever the hedge funds are doing, at your own risk.  But no statistically significant good strategies really exist. The trick with statistical significance is that for strong enough noise, nothing is statistically significant. Moreover, the more complicated your strategy is, the smaller the noise that will completely ruin it's significance. Suppose your strategy has n parameters that can be 0 or 1. So you have 2^n strategies. The base probability is 1/2^n for each of those to be the best one. Suppose the strategies true expectation value of working at all is e<<1, and dispersion 1. Then for it to be picked, you need a sample of poly(n)/e points. But when you write a strategy, you make a lot of choices, so n is pretty big actually. For any interesting algorithm, the sample size needed is enormous - much bigger than the trading data available. And all the simple algorithms were already been found and washed out. So the robots have to choose algorithms in the pool where there's no statistically significant difference between them. This is just for illustrative purposes and is oversimplified.

I don't think the idea of robots fighting each other in a game of insufficient information is that far-fetched. It's like horse-races. You need to spend a lot of time and money on your horse for it to not lose immediately, but your competitors are doing the same, and in the end there's no way to predict who's gonna win. In the same way, an amateur-made algorithm will immediately lose on the fees, but even the professionally made algorithms are essentially giving back random performance. Of course that performance is properly smooth, so they create an illusion of doing something intelligent (or otherwise they would have been discarded at the selection process), but they can smoothly loose money just as well as they can smoothly win.

How much money will you spend in a lifetime?

A version of this question is often asked: how much money do you need to retire? Let's make a simple yet believable calculation.

The main expenses you incur are housing, transportation and healthcare. The prices change over time: transportation gets cheaper, housing depends on where you want to live (can find cheap options), but price of healthcare seems to be growing, and in fact much faster than the inflation rate. Which makes you wonder why do they define inflation rate as they do. So let's focus on healthcare. If you save money by putting it in the stock market, your wealth grows approximately as fast as the healthcare prices, so you can use the nominal value on your paycheck and add it up over the years you work. Specifically, you add up the amount you manage to save. And then you compare it to the amount Americans spend on healthcare today per person per lifetime (~300k$). This is how much you need to save. So saving up 1k$ a month, you need to work 25 years to provide for yourself and just yourself. Of course, that implies that you take advantage of all the retirement benefits of your work, insurance and social support. And also move to the place in US with the cheapest housing and living expenses. So we get the right ballpark. You're spending about 4 times more than that, so total per lifetime in US you will spend 1.5M$ per lifetime. That is just for an average job. Many people struggle to reach this number, and enter and endless circle of exploitation by companies.

Now another important question: when do you actually want to work those 25 years? Do you want to spend your 20ies and 30ies working and then spend your time with kids in the 40ies and 50ies? Would you rather party away the 20ies and see little of your kids in your 40ies? Would you prefer to work 40 years but not too hard or 25 years very hard, and then retire at 40? Would you not stop at 40 and keep working hard to earn a better, more comfortable living conditions, like many Americans do?

EBITDA

Is a good way to value companies. There are several main methods, all producing different value, well, values. This one involves finding similar companies, and averaging over them the ratio of market cap (not sure if that's the right term, but essentially all that company is worth right now not counting the debt) to this EBITDA number of that company. Then one looks at the EBITDA of company of interest, calculate the worth of it using the averaged ratio, and then uses the result -debt/ shares outstanding to get where the stock price should be.

This method does not work for fast-growing tech companies, because for one thing, they don't have anyone to compare to. Sometimes it can be circumvented by splitting a tech company into sectors doing different things, each one of them comparable to existing companies (like Apple can be split into phone company and computer/software company).

This method is more robust than the seemingly more reasonable DCF (Discounted Cash Flow), where the company is valued depending on the cash it generates for the owner. The DCF is very sensitive to internal parameters, for instance the way the discounting is done (without discounting, any company naively generates infinite cash so has infinite worth). It also doesn't work for natural resource- related companies, because then what's important is how much resources is in the land they own, not how much cash they decided to generate this year. Also note that coal mine companies typically have lifespan of about 10 years, whereas "usual" companies effectively stay around forever (except for a few rare cases).

Method #3 is to look up expert opinions. Typically they are either public or available with your broker. Experts are a bit biased in a sense that they prefer to write about companies they think are good to buy, even though the typical company you find on the market is not any good. So if you put together all expert opinions on all companies, there will be 90% of opinions that tell you to buy something, just because the negative opinions do not get written. Nonetheless, when they talk about specific companies they do not lie, and they probably have spent a month of their lives looking into this company's internal workings, so it's good to check with them.

There's another very good method that seems to be more relevant to what's actually going on in the stock market, but unfortunately it's not available unless you're "in the know". It's called LRO or something, I don't seem to find it on the web. The idea is to see what big investors are actually buying. Look at the big transactions, look at how many offers are out there to buy some kind of company. Having that information, one can estimate how much money can be made if the company was to be bought now and resold in a year, or something like this. Maybe in five years. From that, one can figure out what's a good price. But since only big players are involved in such transactions, you really need to know what they are doing to predict like this.

Macroeconomics doesn't make sense

The purpose of this post is to present IS-LM model and also my own made-up model, just to demonstrate how arbitrary are some assumptions made in economics, and how it obscures the truth. The models like IS-LM and the conclusions drawn from them have their criticism within economic community as well, but the criticism is being no less obscure than the models themselves.

So here's a theoretical physicist's take on IS-LM model. We aim to describe a country with a government, a bunch of banks, businesses and people. Their economic activity is exchange of money and goods/services. It is measured by following well-defined quantities:

CPI -  price of a fixed list of goods that are supposedly the most common consumer choices. Is used to adjust for inflation, so all other quantities get the adjective "real": real GDP, real interest rate. We talk about real values without mentioning it from now on.

GDP - every time money is exchanged for goods/services this year, the amount is added to GDP. Buying shares in a company (or stock) is also counted. Taking/paying loan in a bank, and paying taxes are not counted.

r - interest rate. Reflects what kind of loans are available. Is set by either government, or some agreement/competition between banks (depends on a country).

MS - money supply. Counts how much money (how many bills) the government has printed.

I - investment. Part of the GDP that involves buying shares (or stock).

C - consumption. The rest of the GDP. For economists in the room, we ignore export and government spendings.

Then economists write several equations to determine the equilibrium values of these quantities, and also see how they change if some of the internal mechanics of the system changes. First equation follows directly from definitions:

GDP = C + I

Second equation follows from a simple consideration about Investment and interest rate. Supposedly for investors there's a "safe" option to put their money in a savings account that has rate of return that follows the interest rate r. So some of the investment opportunities are not interesting (if their rate of return is smaller than one of savings account). So investment depends on the interest rate with a negative coefficient:

I = Io - a*r

Here Io and a are describing how many investment opportunities are there, They may, and in principle should, depend on what's up in the economy. If people tend to spend more money, there should be bigger investment opportunities, and more profitable ones. But for some reason IS-LM model doesn't consider that, and instead fixes Io and a to be constants.

Consumption, on the other hand, is allowed (in IS-LM model) to depend on how much money people and businesses got. What they got this year is GDP. They probably have paid taxes, so the amount decreased by a bit. Then they have decided which fraction of the rest to spend this year (or possibly overspend and reduce their savings):

C= c0 +c1*GDP

Here c_1<1, or else we can't balance the equation for GDP. If we collect our knowledge so far, we get the IS (investment-savings) part of IS-LM model:

(1-c1)*GDP  = c0 + I0  - a*r

it's a line with \ slope on (GDP,r) axis. Note that somewhat in spite its name, IS equation doesn't actually talk about savings. What's going on is that of total amount of money MS some part have been circulating (possibly several times) to count towards GDP, while other part were people's savings. Some people made money, some people lost this year. GDP by itself does not really tell us anything about savings. We can imagine a year when two monkeys are selling a banana to each other ad infinitum for a dollar, while the rest of MS in never used. In this case, GDP may be arbitrary huge, but at the same time MS-1 dollars never left people's savings.

To describe savings, one needs to involve heavier math. Suppose that MS is split between people according to a distribution m(i) - how much money does i'th person/business have. Everyone wants to consume c(i) and invest I(i). Also this year they earn g(i). In case m(i)+g(i)>c(i)+I(i), this particular individual/business can achieve it's goals. In fact, the quantities c(i) and I(i) should probably depend on m(i) +g(i). But we ignore it for now. If there's not enough money, then an individual can take a loan, or decrease its consumption goals. In the end, we get an inequality:
 m(i)+g(i) + l(i)>c(i)+I(i)
In fact some individuals also need to pay for older loans, so:
 m(i)+g(i) + l(i)>c(i)+I(i)+p(i)
If we sum these inequalities, we don't get anything interesting:
MS+GDP + LOANSissued>GDP+LOANSpayed
Any significant MS will allow this to balance always.

But the breakdown into individual agents has taught us an important lesson: investment and consumption depends on their savings. Economy where people have tons of savings will grow until they start spending most of them every year, but or IS-LM model doesn't capture that.

Finally, the LM ("liquidity preference/money supply")  part of the model describes how the interest rate is set. Looking at the above, intuitively more consumption means more loans (because for more people their savings are not enough). As we noted, this cannot be seen in the aggregates, one actually needs to consider a distribution. Roughly half of the agents will not need loans this year. But how big will be LOANSissued really depends on the economic inequality accumulated from the past, this year's salaries and this year's desire to consume. We assume that both salaries and consumption depends linearly on GDP for those who are in debt. So
LOANSissued = a + b*GDP
Here b>0. We then make a simple assumption about bank operation (that is very far from what they actually do). The global interest rate is set by the following procedure: they look how many loans are requested this year, and think: ok, the bigger the demand, the bigger we can set the "price". The "price" of loans is the interest rate:

r = a' + b'*GDP

Here b'>0. This is the LM part of the IS-LM model. Together these equations can be solved, and if we assume that all the used coefficients except one are constant, then we can find the dependencies between different economical indicators. Like, when we change c0, we observe that if the GDP grows, I  decreases. Other parameters would lead to other relations, so for instance GDP and I can both increase.

What one can do to check that the above makes sense, is to try other things. For instance, one can fix r=0, and instead develop a relationship between I and people's savings. Unfortunately, as soon as savings are involved, the simple line-crossing economists do is not applicable anymore, instead one needs to run numerical simulations. I have run a simple one and observed the relationship between the supply of shares of a specific company (people willing to sell their shares) and the GDP. It turns out that the supply is bigger when GDP is bigger, which would imply that the corresponding stock price is anticorrelated with the GDP.

Two levels of understanding of the Market

New traders start approaching the market as an object of scientific method. They think it is a black box that is given to us, like in a problem statement in high school. They absorb all the knowledge they can find about it, believe successful people and their models of efficient market and stock price somehow representing the value of the company. They probably learn to avoid scam, but they are sure that the truth is out there. That the market will be operating forever according to yet undiscovered laws. That if they see contradictions in different known results and theories, they should just ignore it. And such optimism pays off - they do find relations in the historic data and utilize them to trade in the future. If the market were a static black box, they would do just fine.

Yet there is one situation where looking at history and black box approach may lead you in trouble. Quite literally, imagine a trail of candies lying on the ground. The above strategy is like picking up the candy without questioning who left it there. One may easily get into trouble at the end of the trail.

But once one starts asking questions like "who left the candy", it's really easy to stop trading and be overwhelmed by the complexity of what's inside the box. During meetings of our local investment club, I rarely say anything at all. Many other people jump into arguments, but to me none of their arguments are convincing at all. There is absolutely no reason to trust or believe into any principle that somebody tells you about the market.

So who left the candy? In fact, people like you did. Other traders who believed in similar things that you believe in made "mistakes", and you are getting their money now (assuming that you win). That is a simple picture. Once one starts to dig deeper, it is even more disturbing.

It will probably not be too far off to say that 90% of the money in the market is managed by people called "portfolio managers". That means this is their full-time job, they often have business and economy background and they are put in charge of large sums of money. It is generally not completely automated, it's more like a machine-human interface. Human still does the steering, and the machine takes care of the details.

Now these guys do not actually take money from you. You don't even have enough money to feed their greed. Most of their wins is money taken from each other. That is, even though these people have lengthy resumes with tons of accomplishment and expertise in the area, roughly 50% of them end up losing every year. Stock market does not "generate" money by itself. The only way for someone to win is for someone to lose.

It's ironic then that all of them were able to convince their rich employers that their portfolio management skills are above average. In fact, they are not even rational players. If one tries to use game theory to this problem, one sees that many of the portfolio managers never really tried to optimize their game strategy against other portfolio managers, like they should optimally. Instead, they have empirically collected a huge body of knowledge about how to do their job, that was based essentially on the first approach described above (black box) plus some evolutionary dynamics that made them slowly abandon the methods that do not play well against other players. They still probably have tons of methods used daily that make absolutely no sense from the point of view of game theory. And they will keep using them for a while.

The problem is how fanatic they are on this erroneous path. An ideal game theory strategy against them involves studying their thought process, however inappropriate for the problem, then modeling them by a few math equations and coming up with optimal strategy. But their thought process is so sophisticated (and probably not even deterministic) that it defies any simple modeling. In this way, even though they are not getting closer to the better 50% of their crowd (the winning one) by indulging in all those economy studies, they somehow protect themselves from bright people who would want to attack them with correct math tools. And as a bonus, they also manage to charm their rich employers with the obscure language of finance.

I find this field to be not in the realm of science, it more resembles alchemy, where you secretly develop outlandish recipes that do not actually work, and make colorful sparks to awe the king so he does not think of beheading you this time. Over the years, alchemists developed some kind of understanding of nature, but they also had tons of misconceptions that held them back. 

Low income strategy

So you have that minimal wage 2k$. You live next to your job and can get by without a car. You share a house with a few roommates so your rent is <1k$. You are healthy so you don't need insurance (or you have subsidized insurance). Your family is all fine so you don't have any financial burden from them. It goes without saying that you don't spend on alcohol and have inexpensive hobbies. Then you're good to go - saving money actually makes sense for you.

You get 1k$ net profit every month. You get credit cards from different banks and accumulate promotions and credit line. You get 0-interest first X month loans and pay them back on time. You hold your money in both liquid and illiquid investments. The amount you loan is equal to twice your current worth that is invested in liquid part of your portfolio. Let's do a calculation:
M is the money that you own. 0.5M of it is in illiquid investments with 5% yearly rate. 0.5M +M owed are in liquid investments - stock market and high rate savings accounts. They say Goldman now has a saving account everybody can open with them. The return rate is 2% for savings, random for stock market, but we assume it is roughly 3% expectation value in the current economy. With a more advanced algorithms small amounts of investment can easily get 20% return rates, but the taxes may be an issue for those ones. So if you don't use our algos, you are stuck at 3% expectation value. If you do, it is let's say 10%.
So over a year you get 12k$ of wages, and (0.025+ 0.045)M= 0.07M of investment income. After taxes it gets reduced to 11k$ and 0.05M. So our new M' = 1.05M + 11k$. The amount of money you save living this way is (for the first 10 years):

0	0
1	11
2	22.55
3	34.6775
4	47.411375
5	60.78194375
6	74.82104094
7	89.56209298
8	105.0401976
9	121.2922075
10	138.3568179

One group of people that easily fits the requirements are gradstudents. They stay for 6 years, and then typically get a postdoc for 2, and then forced to leave their field. With this strategy, they can instead retire :) the 5% income from 100k$ is 5k$/year - enough to have a comfortable life in one of those third world countries. They can even keep doing science - their dream job - in their free time.
Seriously, 5k$/ year is not enough. You don't expect to be able to support your family with that. There are also other nice bonuses like promotions from all those credit cards (500$ a year?), and extra 40k$ of postdoc salaries over those 2 years. So you can maybe get to 150k$ by year 8. Or 200k$ if you get a second postdoc. Moving to another country will mess up your loan game, so 5% interest rate will not be available anymore (however, the interest rate in that country may well be comparable). Also if you are a foreigner, you will need to figure out immigration by then.

Now let's consider an idealistic scenario. You use our algorithm and you get 10% yearly returns on it. Then it doesn't really make sense to use illiquid investments - their rate is lower. You may still do it to diversify your portfolio. But let's say you don't, and put all 3M into this algorithm (email us for details, there should be a form on the right). You get 0.3M yearly returns. Lets say you somehow figure out your taxes, so you just pay 1/3 on stock trading income - your returns after taxes are 0.2M. Let's see what the formula M' = 1.2M + 11k$ spits out after 8 years :)))

0	0
1	11
2	24.2
3	40.04
4	59.048
5	81.8576
6	109.22912
7	142.074944
8	181.4899328
9	228.7879194
10	285.5455032

Now you get 181k$ savings (+44k$ extra from postdoc salary and credit card promos). Also your yearly return is much more noticeable: 45k$/year. That is a decent salary! Our only assumptions are that the algorithm will still be working at 10% yearly returns expectation value, and that the banks will still give out those 0-rate loans as a way to attract you as a customer. If you play the credit card game (see website "dr. credit"), your credit line should be pretty big at year 8, and the banks should be willing to loan you sums like 400k$ with no interest rate for a short amount of time (because they expect you to forget to pay on time). Both assumptions are very feeble. But at least they show that financial stability is possible for those who want to stay in Academia. In the same way they are possible for other low-paying jobs.