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Game Theory in Ancient Military Battle Strategy

Few of us have not heard the tale of the Trojan horse, in which a small group of Greek soldiers hid inside a wooden horse to pass through the walls of Troy without so much as raising a single eyebrow – ultimately overthrowing the entire city in the process.  Indeed, the art of deceit in warfare virtually outdates history itself.  In their publication 100 Horsemen and the Empty City: A Game Theoretic Examination of Deception in Chinese Military Legend, Cotton and Liu provide a strikingly new perspective for the two ancient military legends from the vantage point of an economist and financier, respectively.

The first Chinese legend recounts the cunning of Li Guang, who orders his 100 Han warriors to dismount their horses in the midst of several thousand Xiongnu soldiers.  Falsely thinking that the move potentially signaled an ambush, the Xiongnu forces fled.  This payoff matrix is determined primarily by λ, which denotes the strength of Li Guang’s forces.  The valuable of this variable, however, is unknown to the Xiongnu army, which produces three possible equilibria: (1) a strong Li Guang always attacks and a weak Li Guang always retreats; (2) Li Guang prepares for battle despite his strength; and (3) — a mixed-strategy equilibrium — a strong Li Guang attacks while a weak Li Guange decides between retreat and attack.  From these three possible equilibria, and given the historical knowledge that Li Guang’s reputation as a skillful general preceded him, the unique equilibrium will always result in Li Guange preparing for battle – regardless of his strength – and the Xiongnu fleeing.  The key is that Li Guang did not need to make his foes believe he must be strong; rather, he only needed there to be a sufficiently high probability λ that he is strong for this particular point of equilibrium.  However, even when λ is small, the equilibrium will be mixed and will still result primarily in the retreat of the Xiongnu forces.

The second legend – of Zhuge Liang – possesses numerous similarities to the first.  In 228 A.D., during the Three Kingdom period, Zhuge Liang was outnumbered and surrounded by a far more powerful army while in the city of Xicheng.  Instead of fleeing or defending, Xicheng had his men swing the gates wide open, told them to hide from view, and proceeded to play a musical instrument from the watchtower.  Bewildered and confused, the opposing forces interpreted his actions as a decoy for an ambush and decided not to attack the city.

This second example, with a virtually identical payoff matrix to the first, reinforces a crucial finding:  “bluffing” is not as much successful deceit as it is the implanting of uncertainty.  This uncertainty is what prompted both attacking armies to avoid confrontation.  These events, therefore, clearly illustrate game theory, which dictates that what Player A does depends on the action of Player B, and vice versa.  In these scenarios, Player A (the cunning army) has all of the information (army size, location, intent, etc.) while Player B (the attacking army) does not.  We can determine the equilibria by examining the best outcomes/payoffs for each individual player based on the actions of the other players – in other words, their best response.  Incidentally, bluffing was an optimal strategy in many ancient military legends of this type due to its counter intuitively high payoff for the weaker army.

The most interesting facet of these legends is not so much the outcome as it is the fact that these weaker generals chose such optimal strategies without any knowledge of probability theory, specifically game theory.  Both “players” in both situations acted rationally and optimally as well, despite one’s expectation that human emotions (which are more difficult to quantify) will cloud one’s logic and alter his/her payoff so as to change the equilibria.  Overall, the universality of game theory serves as valuable insight into the logical nature of human interactions across cultures and even millennia.

-TJay

Works Consulted

Cotton, and Liu. “100 Horsemen and the Empty City.” 100 Horsemen and the Empty City. Journal of Peace Research, n.d. Web. 03 Oct. 2012. <http://jpr.sagepub.com/content/48/2/217.abstract>.

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