In Chapter Three of Sun Tzu’s the Art of War, there is a sentence which states “故小敵 之堅，大敵之擒也。” Translators like Wu Jiou-lon (吳九龍) explains the situation as a smaller force is outnumbered and that the larger force will always be superior.7 However,
5 Roger A. McCain, Game Theory and Public Policy, (Cheltenham, UK: Edward Elgar, 2009), pp.50-68;
Avinash K. Dixit and Barry J. Nalebuff, The Art of Strategy – A Game Theorist’s Guide to Success in Business
& Life, (New York: W. W. Norton & Company, 1993), pp. 102-138
6 Emerson M. S. Niou and Peter C. Ordeshook, “A Game-Theoretic Interpretation of Sun Tzu's The Art of War”, Journal of Peace Research, Vol. 31, No. 2 (May, 1994), pp. 161-174
7 吳九龍主編，《孫子校釋》(北京: 軍事科學出版社，1991 年)，第 50 頁
other scholars have various perceptions on it:8 Yan Bin-an (楊炳安) believes xiao-di (小敵) depicts troops that are smaller and weaker in terms of strength and the word ‘jien’ (堅) can mean either solid or stubborn. This sentence, therefore, should be appropriately understood as “a troop smaller or weaker in terms of its strength or number should adopt defensive tactics.” Chen Yu (陳宇), another Chinese scholar, believes ‘jien’ means to “fight stubbornly with any reasonable purpose” and that the whole sentence should be read as “therefore, a small force that fights against a big force stubbornly will be captivated by the bigger one.”
His point is that a weak army can only fight weak enemies, but a strong army can fight any enemy. Thus, the general has to know how to manipulate his forces according to the conditions in order to be effective. In the books,9 the author Li Ling states (李零) that both the ‘small force’ and the ‘large force’ refer to the enemy, meaning that ‘small force’ refers to
‘our enemy is relatively small,’ and ‘large force’ signifies that ‘our enemy is large and formidable.’ His idea stems from Xun Tzu (荀子), another ancient classic popular around the time of Sun Tzu, that has a similar expression which can ultimately be translated as, “A small force, when concentrated, can become formidable and have the capability to capture the larger force. (是事小敵毳則偷可用也，事大敵堅則渙焉離耳)”10 In this case, it is possible to interpret this passage as saying that an army can defeat their enemy regardless of their size, so long as they can consolidate their powers. Chinese culture expert, Chu Liang-tsai (褚良才), believes ‘da’ (大) and ‘xiao’ (小) are both verbs and where the former means ‘to help your enemy become larger,’ the latter means ‘to make your enemy become smaller.’ Therefore, the whole statement would be translated as “we become stronger if we are able to make our enemy smaller and we will be captivated if we help to make our enemy become larger.” In other words, an army has to find a way to make a large, intimidating opponent seem like a ‘small army.’ From these exegeses we know when we read Sun Tzu’s the Art of War we are not just reading his words but also the annotative opinions of those who have had a hand in explaining some of his theories. For this reason, we should not necessarily jump to any conclusions as a result of what we read, for only the shrewdest readers will know which ones are most plausible. So for those who embark on the quest to find the most authentic essence of Sun Tzu, they are in a minefield of choices; on the other hand, even the ambitious readers have to decide whether Sun Tzu’s teaching can be applied to solve crises in the real world. In this instance, we need more than exegeses to justify that it is not just the size of the army that has to be accounted for, but also how well one can
8 Keke1969 ，(「故小敵之堅，大敵之擒也」含義辨析)，求知之家， http://keke1969.bokee.com/viewdiary.
13129606.html (20120824)
9 李零，《〈孫子〉十三篇綜合研究》，(北京：中華書局，2006 年)，第 26 頁；李零，《兵以詐立——
我讀〈孫子〉》，(北京：中華書局，2006 年) ，第 138 頁
manage to alter the actual physical perimeters of the enemy itself.
In Sima Qian’s (司馬遷) biography on Sun Bin (史記‧孫子吳起列傳), there is a little episode showing the fallacy of the Chinese military strategic thinking. According to Sima Qian, Sun Bin serves the minister Tien Ji (田忌). One day, Tien bets heavily against the ruler of Qi, and then he consults Sun Bin for advice. The race works as follows: The ruler has three horses: A (best), B (second best), and C (worst horse); the Minister, Tien, also has three horses: X (best), Y (second best), and Z (worst). One horse competes each round, and the winner is the one who has won at least two of the three races. Sun Bin proposes using his weakest horse (Z) against the opponent’s best horse (A). This leaves Sun Bin’s best horse against his opponent’s weaker horses. In the three races, Tien Ji loses one and wins two.11 However, if we truly study the story, we will see that the reason they won was based on the inaction of his opponent, the ruler of Qi.
In the real world, however, and in the realm of game theory, it is highly unlikely that your opponent will remain still and take no action against you. Although Sima is a great historian, he applies typical Chinese thinking in failing to recognize the realities and unpredictability of war by preparing for what they think war should be like. Let’s take baseball, the Americans’ national pastime, for example: a hitter who is competent from both sides of the plate is an asset for a team and can cause a myriad of challenges for an opposing pitcher.
In the real world, every team manager knows the abilities of his own team as well as their competitors. He must apply this knowledge to all 162 games in the season. The only thing a manager can do is make smart moves in each game with his relief pitchers and pinch hitters in order to get the match-up in his favor. This Major League Baseball scenario overturns the myth of Sun Bing, and to some extent makes us doubt what validity and feasibility of these kinds of military teachings would be. When we are involved in war, every decision we make is based on what we believe is the best strategy at the moment. It is likely that our opponent is doing the exact same thing, and we are both hoping that we prepared adequately. It is fair to say that how we prepare is a reflection of what we think about our enemy and what we think they anticipate we are doing. Therefore, such strategy as “sending our worst horse to eliminate his best horse” (下駟對上駟) does not work in the real world.
Similarly, Sima Qian depicts the Battle of Julu (鉅鹿之戰) in the same way. It was a battle which decided the fate of the dynasty of Qin (秦) when it was defeated by Xiang Yu.
(項羽) Gen. Xiang had all his boats sunken, all his cooking pots broken, and all his
10 王天海，《荀子校釋》，上海：古籍出版社，2005 年）第 611 頁注 18
11 司馬遷，《史記》卷六十五，孫子吳起列傳第五 http://www.novelscape.net/lszl/s/simaqian/sj/065.htm
provisions huts burned when soldiers crossed the Zhang River. Food was issued to the soldiers for three days to demonstrate that his soldiers were ready to die and did not believe that they would survive. Historian Bao Jiashu (鮑家樹) attributes Xiang’s victory to his knowledge gained from Sun Tzu’s the Art of War.12 According to Bao, Sun Tzu reiterates the importance and necessity of a soldier’s will to win in desperate situations. However, according to the game theory, limiting one’s options and destruction of weapon, ships, etc., will increase the chances for victory.13 William the Conqueror burning his own ships is an example of determination to not retreat.14 Bao and other exegetes contribute Xiang’s victory to Sun Tzu, but they did not know that modern game theory may do a better job in explaining Xiang’s victory. In other words, with the help of modern knowledge like game theory we will be able to understand Sun Tzu’s teachings more correctly and holistically.
The Parallel between Sun Tzu and Game Theory
Sun Tzu instructs the general on how he can help strengthen the country. According to the advice of the various annotations on his book, a general with a smaller force can defeat a larger force, if he is better prepared than they are. This might be true on paper, but shouldn’t be taken for granted by small forces. We learn nothing worthwhile if we only learn from exegetes without applying critical thinking to their arguments. If we know that both sides have the same information, as in Sun Bin’s horse example, the other side will do their best to win, so will the odds of Sun Bin’s winning be increased, as stated in Sima Qian’s biography? The answer will definitely be no because there will always be someone who will be caught unprepared and die out, according to game theory.
It appears that readers intend to learn how a small force could conquer a large force from Sun Tzu, since a large force with agility and cohesiveness usually wins, and it is very pessimistic to not consider the advantages of being a small force. It is our human nature to fight or take flight. When we feel we are in a disadvantageous position, for instance, we need to follow self-preservation as our guide and flee if necessary. Conversely, in an advantageous position, we will fight. Thus, the duality of a small and large force materializes the spirit of ‘fight-or-flight.’ Furthermore, there are two dynamics at work in the ‘xiao di zi jien (小敵之堅)’ situation Although a small army can overwhelm a large force, the large force itself has to be broken down into manageable operations first. In other words, it is up to the leader of a small force to manage the enemy even in a disadvantageous
12 鲍家樹，(鉅鹿之戰中項羽勝利的主要原因再探——以《孫子兵法》為視角)，《常熟理工學院學報》，
2012 年 03 期，第 107-109 頁
13 Joe Podwol, “What Can Roman Generals Teach us About the Anti-Sweatshop Movement?”
http://joeythepea.blogspot.com/2006/02/what-can-roman-generals-teach-us-about.html
14 Avinash K. Dixit and Barry J. Nalebuff, The Art of Strategy – A Game Theorist’s Guide to Success in Business & Life, p. 215
situation.
These scenarios can be understood in terms of hawk dove game theory and in nature where it is considered the calculation of benefits and costs. Suppose there is a population of hawks and doves competing for food. When two birds want the same piece of food they have two options: bluff or fight for food. If a bird bluffs he uses up a certain amount of energy to look big and mean, but will not get injured because he'll run away without food if things get too rough. If a bird fights and loses, he is injured, which imposes a greater energy loss. In the case in which the cost is greater than the prize, they do not end up with 100%
passive (doves). What they end up with is a mixture of hawks and doves based on the calculation of benefits and costs.15 This most basic hawk-dove game resembles what Sun Tzu states, “…and if equally matched, stand up to him. If we are fewer than the enemy in number, retreat. If we are no match for him, try to elude him. (敵則能戰之，少則能逃之，
不若則能避之)”16 That is to say, in the strategic thinking shown in the game theory, it matters to know the cost of fighting and its prize.
Sun Tzu mentions the word “benevolence” (仁) only twice in the Art of War.17 Confucian scholars accused him of being a cold blooded murderer and warmonger.18 However, Sung scholars might not have been completely correct. Sun Tzu did not set out to randomly murder people. He set out to win, and if he could not win, he wanted to survive.
We can explain his strategy in terms of game theory.19 Imagine that we are on a road that is only one lane wide. Two cars on the road are in the center and driving towards each other and the loser is the person who swerves first. The car that swerves first is considered the benevolent driver, while the car that does nothing is the aggressive driver. If both drivers were passive or aggressive, they would both most likely die.20Our priority is winning, but in order to win, one must survive first. Our survival is highly dependent on how we are
15 John Maynard Smith, Evolution and the Theory of Games, (Cambridge: Cambridge University Press, 1982);
Ross Gressman, The Stability Concept of Evolutionary Game Theory – A Dynamic Approach, (Berlin:
Springer-Valag 1992 ), pp. 95-97, also see Day of the New Dan, “Hawk Dove Game Theory”
http://www.dayofthenewdan.com/projects/evolution-simulation; Philip Hingston and Luigi Barone, “Hawks, Doves and Lifetime Reproductive Fitness,” http://www.wfg.csse.uwa.edu.au/publications/WFG2004e.pdf
16 見《孫子兵法》第三章，謀攻
17 見《孫子兵法》第一章，計篇以及第十三章用間
18 見朔雪寒，《孫子兵法論正》(網路版)，第三章〈相關謬論綜駁〉之蘇洵，〈嘉佑集‧權書‧孫武〉，
http://www.cos.url.tw/sunzi/Sun-7-03.htm 另見第四章〈疑古謬論綜駁〉之葉適〈習學記言‧卷四十六‧
孫子〉http://www.cos.url.tw/sunzi/Sun-6-01.htm
19 Peter Hammerstein, “What Is Evolutionary Game Theory?” in Game Theory & Animal Behavior edited by Lee Alan Dugatkin and Hudson Kern Reeve, (New York: Oxford University Press, 1998), PP. 1-15
20 Ross Gressman, The Stability Concept of Evolutionary Game Theory – A Dynamic Approach, pp. 7-9;
William Poundstone, “The Game of Chicken as a Metaphor for Human Conflict” http://www.heretical.com/
pound/chicken.html; Elmer G. Wiens, “Game of Chicken – Rebel without a Cause”: http://www.egwald.ca/
operationsresearch/chickengame.php
hardwired to survive.
A true Sun Tzu’s student would always take the passive strategy even when he knows it is not the smartest or safest strategy. We do not just make it up as we go along, rather there are things that are taught to us the most effective and efficient ways to survive. The animal kingdom has many examples of Sun Tzu’s teachings that allow for certain species to continue thriving. It helps to have basic understanding of game theory to understand Sun Tzu’s the Art of War, although little knowledge of game theory existed when it was written.
For example, in the animal kingdom, a small male elephant seal is able to find a mate if it is able to deceive his enemy, the alpha male.21 There are two mating strategies for the male elephant seal. One successful mating strategy is to be the dominant elephant male with many female elephant seals to mate with. In this case, size does matter.
However, this is not the most efficient means of evolution for the smaller seals in the population. The alternative strategy for them, therefore, is for a smaller male to look like a female elephant seal, and sneak into the population and mate with a few different ones. John Maynard Smith’s study suggests that dimorphic male phenotypes evolve via sexual selection through female choice. Typically this leads to alternative reproductive strategies which are often attributed to phenotypic differences among males or their differential ability to compete for females. It means males often use more than one tactic to gain access to females.22 Evolutionary game theory initially provided two major explanations for the maintenance of alternative mating tactics: a mixed evolutionary stable strategy (ESS),23 with alternative tactics maintained by frequency-dependent selection, and a conditional strategy in which the tactic an individual adopts depends upon some aspect of its (environmental or physiological) state.24 As explained in the example of elephant seals, benevolence (仁) in Sun Tzu’s the Art of War means not sacrificing our own troops if there is another way to reach the goal, even if it will be a longer and more difficult process.25
We can imagine that if Sun Tzu lived today he would be able to successfully compete for a mate like the small elephant seal does since he says, “The difficulty of maneuvering is to deceive our enemy by disguising our short cut to win as a long route, and make ourselves
21 Elephant Seals: Population Ecology, Behavior, and Physiology, edited by Burney J. Le Boeuf and Richard M.
Laws, (Berkeley: University of California Press, 1994); Charles J. Deutsch, Daniel E. Crocker, Daniel P.
Costa, Burney J. Le Boeuf and John Morrissey and James L. Sumich, Introduction to the Biology of Marine Life, Tenth Edition, (Sudbury, MA: Jones & Barlett Learning, 2012), p. 320
22 Samir Okasha, Evolution and the Levels of Selection, (New York: Clarendon Press, 2006), pp. 13-14
23 John Maynard Smith, Did Darwin get it Right? Essays on Games, Sex and Evolution, (New York: Chapman and Hall, 1988), p. pp. 194-197,
24 John Maynard Smith, Evolutionary Genetics, Second Edition, (Oxford: Oxford University Press, 1998), pp.
125-136; also see “Yale Open Course: 11. Evolutionary Stability: Cooperation, Mutation, and Equilibrium”, http://www.youtube.com/watch?v=er9KvYn4ldk
25 孫開泰，(孫武與齊文化) ，《石油大學學報》，1991 年第一期，第 7-8 頁
appear to be suffering. (故迂其途，而誘之以利，後人發，先人至，此知迂直之計者也)”26 Your enemy will find it difficult to attack you because they will not know where you are. In the reverse scenario, the enemy may think that you are very unthreatening, when in fact you hare able to sneak up on them. Taking the shortcut is always preferable, especially if the enemy doesn’t know what you are doing. When we apply knowledge of game theory we learn from the small elephant seals and apply this knowledge to this classic, it does indeed become more understandable to modern readers.
Sun Tzu teaches us that finding a shortcut can result in victory. If a general knows a short cut, he will have an upper hand in battle. However, is it always good for a small elephant seal to find a short cut and get himself into the group of harems? Not necessarily.
The best strategy is the one the alpha male does not suspect. When the enemy fails to react, we know that we are on the right track. By quoting the disciplines from the course of game theory, we may develop a more meaningful understanding of Sun Tzu’s teaching. The beauty of this game-theoretic approach is that it allows us to rid ourselves of useless exegeses who would otherwise hinder our understanding of the text. In order for leaders to make a right decision in the imperial court, Sun Tzu teaches us he will choose an option based on whether one will provide a better result than the other.
In order to know which one will provide the better result, he calculates that given a
In order to know which one will provide the better result, he calculates that given a