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A Game-Theoretic Interpretation of Sun Tzu’s The Art of War

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I grew up reading a lot of Chinese historical novels on wars and domination (like Romance of the Three Kingdoms). They are not exactly based on history because it tells the story in more of a literary approach. However, there is one thing in common that is really fascinating. A battle that could change the flow of the war. That battle is usually not about how many soldiers or how good the armors/weapons are, but a strategist who can predict the enemy action and outsmart them in the battle. Sun Tzu, the writer of “Art of War”, wrote this book during his lifetime around 500 BC. Even thought it was written in such an ancient time, the ideas are still being used by people on politics, finance, business, and so on. Being one of the very early books that talks about game theory (not directly because people don’t know about game theory in ancient times), I want to find out how he came up and translated his thoughts into writings that influenced the war history of China (if I may say, forged the ancient Chinese military history). Sun Tzu himself probably wouldn’t know the Game Theory, but what he wrote in the ancient cold weapon era was really close to the interpretation of the Modern Game Theory.

The key idea of “Art of War” is to codify the general strategic character of conflict and, in the process, offer practical advice about how to win military conflicts. Nevertheless, to what extend did Sun Tzu anticipated the implications of the contemporary theory of conflict – the game theory. The essay claimed that he could be credited with having anticipated the concepts of dominant and mixed strategies. However, he failed to intuit the full implications of the notion of equilibrium strategies. An example being his quote “Know thy self, know thy enemy. A thousand battles, a thousand victories.” This advice remains vulnerable to a more complete strategic analysis. If your enemy knows the same theory, what decision shall you make then? But I say this advice is complete enough for the ancient time because people back then lacked the knowledge of probability theory, advanced mathematics and modern technologies. Sun Tzu understood as much as he could for his times and that’s why people call him the God of War in China now (Chinese version of Athena?! LOL). Also, there won’t be a condition where there were no ambiguous information. Spies already existed in the ancient times. So it came to the end that the side with more accurate information would win the battle. But… what is process to victory?

Using a very famous battle example during the Three Kingdoms era called the Battle of Red Cliffs, we can have a better understanding on Sun Tzu’s ideas. This battle involved all three kingdoms where Wu and Shu were allies against Wei. The set up can be summarized into the following seven points:

  • The battle happened on the Yangtze River
  • Wei was attacking from the north-western side and Wu and Shu defending from the south-eastern side
  • If Wei won the battle, it would conquer lands south of the Yangtze River, and probably unity China soon
  • Wei had more soldiers (around 800,000) then both Wu and Shu combined
  • Most of Wei’s soldiers were from the north, thus no sailing experiences
  • Soldiers of Wu and Shu were from the south, thus experts in sailing
  • All three sides had military strategists about the same level and each other’s information

Both Wu and Shu wanted to become the biggest beneficiary in this corporations (more lands, soldiers, and resources). Thus, they would come up with their best responses with the information they knew about the opponent and try to gain the best pay off. So like Prisoner’s Dilemma but with a lot more conditions and of course, life and death situation. I am not going to talk much about this corporation, because there is a big battle and strategies ahead. Sun Tzu stated “what is of supreme importance in war is to attack the enemy’s strategy (III, 4). . . determine the enemy’s plans and you will know which strategy will be successful and which will not (VI, 20).” Sun Tzu’s general intent is clear – to analyze the diversity of interdependent choice situations in warfare and to deduce efficient strategies. And here is how Wu and Shu successfully carried out their strategies that easily won them the battle.

Because Wu and Shu are good at fighting in the water, wei used the strategy of connecting their battle ships altogether with cast iron chains and create a stable battle”ground” to counter (basically made ships into a huge ancient style aircraft carrier that carries horses and people). Since there were seldom South-eastern wind on the river so without the wind blowing towards Wei army, Wu and Shu couldn’t use fire attack to burn all the ships at the same time. Wu and Shu attacked this strategy by delaying the battle till the day of full moon, when the wind changes direction because the moon influenced the tide. It was too late for Wei to untidy their battle ships when Wu and Shu’s suicidal ships with burning materials hit them. Wei lost the battle and flee back north.

Briefly, game theory, which we can view as either a branch of mathematics or of political science and economics, seeks to isolate general, abstract principles of decision-making when the outcomes of people’s choices depend on what others decide and when everyone is aware of their mutual interdependence. Overall, Game theory’s application include not only strategic military planning like “The Art of War” illustrated, but also in different fields like making business decisions, winning elections, and etc. Sun Tzu’s “The Art of War” is just a small part of it. Still, it is very impressive to see an ancient man creating his own idea that could still be deliberated today.



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