Why do people who value time above all else spend years studying the game of Go? Practicing Go gives one a vision and strategic thinking. And with the help of these things you are able to foresee a lot of things, if not everything. The pieces and the board allow one to put one’s events and decisions into perspective before they become irreversible. Not everyone has a naturally good intuition but strategic modeling is something one can learn. And Go has been teaching people that for five thousand years.
The myth about the creation of Go tells about a young heir to the sovereign ruler of Ancient China who was to be trained to govern the country. By his father’s order, a group of wisemen developed an educational program for the young prince which was all in the form a strategy game. Sounds quite modern, doesn’t it? It seems that the game of Go became the first ever MBA program. The tradition goes on, and now a course in strategic Go is available at the Skolkovo Moscow School of Management. But in Asian countries the game of Go has already become a part of an educational standard for political and business elites. But before speaking about the ideas of Go, let me provide a historic example of how strategy works.
Go (also known as wéiqí or baduk) is a logical board game with profound strategic content. It originated in Ancient China 2 to 5 thousand years ago, according to different estimates. It is widely considered to be one of the oldest and most complicated intellectual games in the world. Go is included in the list of the five basic sports of the World Mind Sports Games alongside chess, bridge, draughts (checkers) and xiangqi (Chinese chess).
World War II, Western Front. Bombardment aviation is the key instrument of strategic attacks against Axis Powers for the USA and Great Britain. However, squadrons are suffering big casualties. How do you increase the durability of bomber planes? The easiest way is to enhance their vulnerable spots with armor. But extra armor means extra weight. An integrally armored bomber plane will simply fail to take off. Hence, it is necessary to understand where its main vulnerable spots are.
So the military men address this question to mathematician Abraham Wald, who at that time works with other scientists in the Statistical Research Group of Columbia University. This organization was created by the U.S. government to solve unconventional tasks set by the war. Wald is an expert on statistics. So he asks for data on the damage recorded by military men for bomber planes. Having processed the statistical data, Wald marks the zones that have the most damage.
Pictured above are schematic drawings of two planes. The first one, colored gray, has not been damaged. The second airplane has a zone colored black – it is a zone with shot holes documented by the military experts. The two gray areas are places that never catch any damage. So what do you think was the solution offered by the scientist to the U.S. Air Force? Wald said: “Find the areas that have no damage. These are the most vulnerable spots. The planes only came back because the shots did not hit those spots.” If a plane has returned to its base, it means its damage is not critical. Armoring the fuselage is necessary in the areas where the military men did not detect any holes because bomber planes that got hit in these areas were destroyed by enemy forces. Abraham Wald’s method became known as the Survivorship Bias Method.
Wald’s problem-solving approach is based on the same method that Go masters use to succeed in the game. First of all, examination. Examination does not simply mean collecting information. It is important to see what cannot be seen directly. And it is truly the crucial aspect of the examination. In the case of bomber planes this meant the zones that did not catch any damage and hence were ignored in the military reports. Only after dealing with both the obvious and the hidden sides of the problem the correct decision can be made. Otherwise the analytical insight will not be complete, which can lead to a series of errors.
In the second part of this article you can see by yourself how examination method works when searching for effective solutions in the game of Go. And now let’s talk about the main principles of the game and about how Go helps to think and act in a more effective way.
On the day when I began working on this article famous mathematician, Noble Prize winner in Economic Sciences and big fan of Go John Nash died in a car crash. A prominent American scientist, an intellectual and a true madman, Nash was arguably a typical example of an inquisitive European mind meeting the complex aggregate of knowledge, myths and practices that is Go in China, Japan and Korea.
In the Oscar-winning movie ‘A Beautiful Mind’ Nash was played by Russell Crowe. One of the scenes has the young scientist trying to develop a non-losing strategy while playing Go with a fellow student in Princeton. The culmination of the scene is as follows: Nash, totally disarmed, angrily knocks down the board with the pieces and walks away exclaiming that he does not understand how he lost, because he did everything right!
Indeed, John Nash did develop a passion for Go while studying and then working at Princeton University. Later on he offered a new approach to describing the players’ solutions, but this time in the business sphere. This approach is now known as the Nash equilibrium. According to it, in end game players can use optimal strategies. They allow them to achieve a non-cooperative equilibrium. That is, cooperation of these players has not been conditioned.
Go is one of the most popular games in the world. In 2000 there were about 27 mln. people who played Go, i.e. there was one Go player for every 222 people inhabiting our planet. 22 mln. of these players (i.e. over 80% of their total number) were from East Asia.
According to Nash’s model, even rival players will try to maintain the already accomplished balance, because any change will not improve the situation for any of the players. Moreover, it is possible to find such equilibrium that will allow not one player but all the players to be in a winning position. In other words, competition does not necessarily mean that you lose if your rival gets ahead. Let others win and you will win too. The Nash equilibrium can be achieved in any types of games – both in mutually advantageous games and zero sum games in which one player’s acquisition means another player’s loss. But which type game we are going to play depends on us.
The idea that in a competitive struggle all participants of the game can have equal advantage seems weird for a Western mindset that is more used to “chess logics”: kill or be killed. However, this idea has been known since the time of Laozi, and it is manifested not only in economy but in the world around us in general. Just take a look around and you will see it.
When Professor of Kyoto University Kinji Imanishi was still a schoolboy, he spent a lot of time collecting herbaria in Japanese mountains, and in the evenings he studied the game of Go. When he grew up he became an environmental expert and developed the theory of Sumi-Wake (living environment segregation). In the vicinity of the Kamo River in Kyoto he discovered four species of one-day butterflies. These insects formed whole colonies organized according to the river flow rate. Based on that observation, he came to the conclusion that species exist and survive by balanced division of the environment and not by struggling for survival. An ecosystem or a society based on the principles of “every man for himself” and “everyone against everyone else” is doomed to die out. On the contrary, if all members of a social system manage to achieve harmony together, then every member gets the exact niche they have enough abilities and resources for.
It is possible that John Nash found the balance of competitive forces idea in Go. For the point of the game is that the black and white stones compete for coexistence by maintaining balance, dividing the space of the game board between themselves.
The size of a Go board is 19x19 lines. In total there are 361 spaces which a black or a white stone can be placed onto during the game. The stones are only different in color, their shape is the same. The goal of the game is dividing territory. The winner is the player who captures the bigger part of the game board. The consequence of the large board size is the impossibility of complete enumeration of options (to compare: a chess board is 8x8 squares). Just like in real life, we cannot calculate all possible consequences of our decisions.
One of the reasons of Go’s difficulty is an extremely large number of possible game combinations that is usually evaluated as ten to the one hundred and seventieth degree (10170). How big is that number? See for yourselves: even the number of atoms the whole observable Universe consists of is incommensurably less – “only” 1081. By the way, the number of possible unique chess games is about 10118.
What is the purpose of the stones and which laws do they operate on? The stones are entered into the game one by one. A stone that has been placed on the board cannot be moved. It can be removed from the board only if it is surrounded or captured by hostile stones.
The “life” of the stones depends on the force that is determined by the number of open links to empty adjacent lines. Each of these links gives an additional degree of liberty and an extra direction for development. Picture 1 shows the black stone having four links with adjacent lines (marked with Xs) while the white one only has three. And more external links mean more possibilities. Only direct links along the lines count in the game. Diagonal links do not count.
Stones of one color are united into a single chain by occupying common “liberties”. In picture 2 we see that the black formation has grown and now occupies two lines. But it is a single body whose connectedness to external environment is equal to six.
When stones of opposite colors contact each other, they take away each other’s degrees of liberty and hence become weaker. Both the black stone and the white stone that stand next to each other in picture 3 have three links to the outside board each. When a stone or a chain of several stones gets surrounded, they begin losing their degrees of liberty. Having lost their last link to the board, the surrounded stones leave the game and become prisoners. How this happens is shown in pictures 4 and 5. The numbers denote the order of positioning of the stones during the game. The black stones here have deliberately surrounded the two white ones. If the whites’ player decides to surrender the stones, then the blacks’ player will surround the whites and thus remove them from the board, spending six of his stones for this. The result of this capture is shown in picture 6. Can the whites save themselves and escape this mousetrap? For that it will be necessary to build up the whites’ chain and thus increase the number of links to the outside space. Whether or not this is viable depends on the situation.
Of course, hunting the other player’s stones is not the goal of the game. It’s merely a way to persuade your opponent that this is your land. While the actual purpose is to organize your stones in such a way that they are able to take hold of more territory than the opponent’s stones. What does ‘take hold of territory’ mean? It means enclose it with your stones in such a way that playing in that area is meaningless for the opponent. Because if the opponent puts any stones there, they are guaranteed to be surrounded and removed from the board. In other words, taking hold of land means establishing your sovereignty.
To successfully play Go at the beginner level it is enough to be able to count to four and to detect weak spots in your own chains and the opponent’s chains (the ones with connectedness of less than 4 links). These weak spots are the key places to manage your own and the opposing stone formations.
When strategists want to change something in an event or a process, they look for a weak link and attack it. On the above example of Abraham Wald it is clear that such a link can be located in some place not visible within standard analysis. A good rule of thumb is that a grown person can often hardly count even to four. While a child cannot even control situations that require counting only to one in order to find stones with one degree of liberty. Attention deficiency is the scourge of a modern person.
By studying applied aspects of the game it is easy to show that any intelligent solution is based on perception. The reason for any mistake is also likely to be perception. Mistakes based on pure logic are not actually made that often by humans. In his book ‘Blink: The Power of Thinking Without Thinking’, Malcolm Gladwell graphically described the way perception can drown out not only any reasons of conscience but even a person’s professional experience. Or, on the contrary, it can give some clues in situations when the conscious mind gives up.
Perception does not merely influence our decisions. Our success also largely depends on it. You cannot believe it? Then please meet Professor Richard Wiseman of the University of Hertfordshire.
For ten years Wiseman followed the lives of four hundred human test subjects of various ages and occupations. He found them using newspaper ads in which he requested for the people who think of themselves either as spoilt children of fortune or complete losers. They kept diaries and took tests, and Wiseman also had them describe their lives in interviews and reports.
In one of his studies Wiseman asked the test subjects to look through a newspaper and count the number of pictures in it. The people who considered themselves to be losers, spent several minutes for this task. The people who thought themselves to be lucky spent a few seconds on average. Here is how Wiseman himself described the results of this experiment:
“The second broadside of the paper had a huge advertisement that occupied half the page and read: “Stop counting: this paper has exactly 43 pictures”. It was impossible to miss such a huge ad, but the “losers” were so carried away by counting that they paid no attention to it. Next time I repeated the task, but this time the ad read: “Stop counting and instead tell the professor you’ve read this and you’ll get 250 dollars.” The outcome was the same. The “lucky guys” got the money while the “losers” ignored the ad and were left with nothing. This experiment showed that people who considered themselves to be “losers” are more uptight and tense than their “fortunate” colleagues, and this tension often prevents them from noticing something unexpected but useful.”
Wiseman claims: what we call luck is actually just a consequence of our behavior that unites our patterns of perception with the way we deal with situations and people that we come across in our lives. “Losers” have a focus that is too narrow, while a “lucky guy” notices everything. As a result this person can often find something other than what he was looking for but what actually suits him better.
What conclusion can be drawn from that? As is in Abraham Wald’s case, the key to finding efficient solutions is our perception, our ability to take a broader view, to see what can’t be seen, to read between the lines and, of course, to think creatively. And now I offer our readers to test these skills on a strategy task from the game of Go. Solving tasks like this is not only a fascinating part of the game but also good training for people who want to develop the qualities of a strategist.
Take a look at pictures 7 and 8. Which of these pictures do you think has five stones arranged more effectively? Before choosing your answer you need to understand the difference between the two structures. The rules of Go, analyzed above, are sufficient to solve this problem.
The main difference between the black and the white stones here is the type of connection between the structural elements. The whites are connected to each other indirectly, while the blacks have direct connections. This means that the five black stones form an inseparable structure in which all elements are merged into one. The white ones have every part standing separate from the others. The black stones cannot be separated because you cannot place a white stone between the black ones, and moving them apart is not permitted by the rules. As for the white stones, they can theoretically be divided.
The key organizational criterion that is available to us now is the stones’ links to the outside space. It is clear that the physical area is equal: in both cases the stones occupy five lines. But there are also free lines next to the stones, and these lines can be further expanded upon. We have already counted such links for pictures 2 and 3. The more external links stones have, the stronger is their formation and the higher are the chances for further development. So, let’s count.
The blacks have eight links (picture 9). These are marked by Xs. I repeat, we only count the lines connected to the spots already occupied by black stones. Diagonal lines don’t count. Which is why I mentioned that the white stones in picture 7 are not connected to each other. They are standing on diagonal lines. Now let’s count the whites’ links.
It is obvious that the whites’ connectedness level is higher – they have twelve links for the same number of stones, which is five (picture 10). So which formation is more effective? The blacks have all their elements merged, like in a corporation with linear structure and tight control. The whites are organized on the basis of the network principle, which is why each of their elements has extra degrees of liberty.
This seemingly simple example is directly related to strategy. A good illustration would be the Prussian war machine that evolved from a unified organization akin to a conglomerate of black stones to an organization with a high level of internal freedom. The famous army of Frederick the Great could only fight as a whole and only under fierce control. That army could not send away soldiers for a mission because they would just disperse and run away. “The most uncertain thing for me”, King Frederick once said to a general within his circle, “Is our safety inside our own war camp.” A nice reference for one’s own army, isn’t it?
A military organization founded in the 19th century by two Prussian military officials Moltke (the Elder and the Younger), on the contrary, was based on the principle of delegating responsibility to officers. This rid the Prussian war machine of excessive unity, made it flexible and really one-of-a-kind, which in the end made Germany one of the strongest military powers of Europe.
If you look at the business society at this angle, you can see that opaque and monolithic structures with vague responsibility and total control lose to companies that are based on the principles of internal freedom, delegation of responsibilities and transparency. This regularity is enunciated in Tao Te Ching in the formula “The hard and strong will fall. The soft and weak will overcome.”
But here is one more method of research: to try and optimize the structure. Let us remove some of the stones and see what influences the connectedness of the whole system. It is very simple for the white stones. If we remove any of the four external stones, the connectedness will go down to 10 lines (picture 11). If the central element is removed (picture 12) the connectedness will not change but the structure will lose one property that seems unremarkable at a first glance.
If a black stone happens to be between two white ones (picture 13) it can be easily surrounded and removed. Picture 14 shows the result of that. The same will happen if the blacks attempt to take lines A, B and C. This means that these lines in fact belong to the whites, even if they are not occupied by their stones. It is pointless for the blacks to go there. But remove the central white stone and you will see that capturing the black one will not be that easy. In other words, the whites have no out-of-place stones in their structure. Now let us see if we can optimize the blacks. Take a look at pictures 15 and 16. Removing any of the four external links of the black chain does not reduce their connectedness. It is still equal to eight. Which is interesting in and of itself. It turns out that these stones do not influence the number of connections, regardless of whether they exist or do not exist. But they do, which means that these stones are dead weight, they are non-productive expenses for the player. It is a logical consequence of a structure organized by this sort of linear method.
But if we remove the central element, the connectedness will increase up to nine! It is easy to assume that the structure has grown exactly around that central stone by consecutively expanding in four directions. However, an inefficient model of management led to this central element becoming a burden for the blacks instead of being their key link as in the whites’ case. And this could have been foreseen!
And now imagine that you need to scale up your structure to occupy line A (pictures 17 and 18). For the whites doing that is very easy. But what about the blacks?
The blacks will have to add five stones to reach the desired place (picture 19). An attempt to optimize the process (picture 20) will lead to a breach in the structure: a break between the black stones will appear, which is unacceptable for their linear structure.
While for the whites, with a little bit of smart management, the two aforementioned stones will be sufficient (picture 21). Their work will be enough to maintain the connection between all parts of the whites’ structure. By saving resources and competently distributing them along the advantageous parts of the board, the white player can easily beat the black one if the latter constructs linear management chains as opposed to network-based ones.
What advice can be given to a player who has chosen the blacks’ strategy? In Go there is an elegant solution: don’t overload! That is, avoid overconcentration of efforts, control and management. However, the article format does not allow me to elaborate upon this or any other elegant solutions. Nine of them are described in my book “The Martial Art of Strategy. The Russian Style. Nine Elegant Solutions” co-written with Igor Grishin, and I recommend it to the readers who are interested.
Instead of a conclusion I will quote remarkable Russian entrepreneur Sergei Andreev, the President of the ABBYY Group: “Go can teach you to see what most people overlook. To win in Go, you need to see the balance of forces, see the weaknesses – both your own and your opponent’s, see the mistakes. The nature of weaknesses and mistakes and the overall balance in Go is extraordinary because it can be, and needs to be, extrapolated to our lives and activities. Having recognized your mistakes on the board, you learn to notice your mistakes in your life and work, too. As you learn strategy in Go, you learn strategy in life.”
There are several systems of measuring the players’ strength in Go. One of the most popular ones is the Japanese ranking system that is similar to the one used in martial arts. Beginner players get the ranking of 25-30 kyu. 30 kyu corresponds to the level of a player who has learned all the rules but has not played a single game yet. As the player becomes stronger, his ranking goes down. Usually at the early stages players learn fast and most of them achieve 8-12 kyu within a few months. The players that surpass the ranking of 1 kyu get the ranking of 1 dan. With the player’s further development, his dan ranking will grow, contrary to the kyu ranking. The traditional limit is 9 dan, which is awarded only to true masters of the game. There is also the notion of “10 dan”, but it is not a ranking that shows the player’s skills but rather an honorary title of sorts. By definition, the difference between adjacent rankings is equal to a one stone head start, i.e. theoretically an 8 kyu player will have equal chances for a win with a 5 kyu player if he has a 3 stone head start.