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Tian Ji (),
courtesy name A courtesy name (), also known as a style name, is a name bestowed upon one at adulthood in addition to one's given name. This practice is a tradition in the East Asian cultural sphere, including China, Japan, Korea, and Vietnam.Ulrich Theobald ...
Qi (齐), was a military general of the Qi state during the early
Warring States period The Warring States period () was an era in History of China#Ancient China, ancient Chinese history characterized by warfare, as well as bureaucratic and military reforms and consolidation. It followed the Spring and Autumn period and concluded ...
(4th century BC) of
Chinese history The earliest known written records of the history of China date from as early as 1250 BC, from the Shang dynasty (c. 1600–1046 BC), during the reign of king Wu Ding. Ancient historical texts such as the ''Book of Documents'' (early chapter ...
. Tian Ji met
Sun Bin Sun Bin (died 316 BC) was a Chinese general, military strategist, and writer who lived during the Warring States period of Chinese history. A supposed descendant of Sun Tzu, Sun was tutored in military strategy by the hermit Guiguzi. He wa ...
and recommended him to
King Wei of Qi King Wei of Qi (), whose personal name was Tian Yinqi (田因齊), was the king of the northern Chinese state of Qi during the Warring States period, when Qi was one of the most powerful states in China. He reigned from 356 to 320 BC. or accordin ...
as a military strategist. Tian Ji commanded the Qi armies in the
Battle of Guiling The Battle of Guìlíng (桂陵之戰) was fought between the states of Qí and Wei in the Warring States period of Chinese history. In 354 BC, an army from Wèi was laying siege to Handan, the capital of the State of Zhao. The next year, Zh ...
and
Battle of Maling The Battle of Maling () took place in Maling, currently Dazhangjia Town (), Shen County (), Henan Province, in 342 BC during the Warring States period (476–221 BC). The combatants were the State of Qi, who fought on behalf of the State of Ha ...
, scoring victories against the Wei state with help and guidance from Sun Bin.


Biography

Tian Ji was descended from an aristocratic clan based in the State of Qi and was later appointed a military general by King Wei of Qi. In 340 BC, Sun Bin arrived in the State of Qi as a refugee from the State of Wei. Tian Ji met Sun Bin and was so impressed with Sun's expertise in
military strategy Military strategy is a set of ideas implemented by military organizations to pursue desired strategic goals. Derived from the Greek word '' strategos'', the term strategy, when it appeared in use during the 18th century, was seen in its narrow s ...
that he kept Sun at his residence as a
retainer Retainer may refer to: * Retainer (orthodontics), devices for teeth * RFA ''Retainer'' (A329), a ship * Retainers in early China, a social group in early China Employment * Retainer agreement, a contract in which an employer pays in advance for ...
(门客). Once, Tian Ji was invited to participate in a horse-racing event hosted by the king and Sun Bin proposed a strategy for Tian Ji to win. Tian used his inferior horse to race with the king's best horse, his average horse to race with the king's inferior horse, and his best horse to race with the king's average horse, winning the latter two out of the three races. The king was impressed by Tian Ji's victory and Tian told the king that he won the race after following Sun Bin's suggestion. Tian recommended Sun Bin to the king and the king appointed Sun as a military strategist and advisor. In 354 BC, Wei attacked the
State of Zhao Zhao () was one of the seven major State (Ancient China), states during the Warring States period of ancient China. It was created from the three-way Partition of Jin, together with Han (state), Han and Wei (state), Wei, in the 5th century BC. ...
, with
Pang Juan Pang Juan (died 342 BC) was an ancient Chinese military general of the Wei state during the Warring States period. Life Early life Pang Juan was a fellow student of Sun Bin and both of them studied military strategy together under the tutela ...
leading the Wei forces. Zhao requested help from Qi whereupon King Wei of Qi ordered Tian Ji and Sun Bin to lead an army to reinforce Zhao. Sun Bin proposed the strategy of "Besieging Wei to rescue Zhao", and Tian Ji followed the plan, ordering an attack on Wei, forcing Pang Juan to turn back to save Wei, and effectively lifting the siege on Zhao. In 341 BC, Wei attacked the
State of Han Han (, Old Chinese: ''*'') was an ancient Chinese state during the Warring States period of ancient China. It is conventionally romanized by scholars as Hann to distinguish it from the later Han dynasty (). It was located in central China ...
with Pang Juan commanding the Wei army. Han was an ally of Qi such that Qi responded by sending an army to reinforce Han. Sun Bin was appointed as commander-in-chief of the Qi army this time, with Tian Ji as his deputy. The Qi army scored a major victory in the ensuing Battle of Maling in 342 BC, which culminated in a crushing defeat for Wei and the death of Pang Juan. Tian Ji was not on good terms with the Qi chancellor Zou Ji who disparaged Tian in front of King Wei. Unable to clear his name, Tian fled from Qi and sought refuge in the
State of Chu Chu, or Ch'u in Wade–Giles romanization, (, Hanyu Pinyin: Chǔ, Old Chinese: ''*s-r̥aʔ'') was a Zhou dynasty vassal state. Their first ruler was King Wu of Chu in the early 8th century BCE. Chu was located in the south of the Zhou hea ...
. He was invited back to Qi and reinstated in his former office after King Xuan ascended the throne of Qi.


Tian Ji’s Horse Racing Strategy


Background

Tian Ji’s Horse Racing Strategy originated from a well-known ancient Chinese story from The Records of the Grand Historian. Tian Ji was a military general of the Qi state during the early Warring States period of Chinese history. The story is about a horse riding game between Tian Ji and the king of the Qi Kingdom, and how Tian Ji won the horse racing game under the guidance of a military strategist, Sun Bin, with all his horses less competitive than the king. In the original story, Tian Ji was aware of the king’s strategy and changed the sequence of his horse to win the game. However, in the later edition, where the original story was developed into a game theory strategy, players are not aware of other players’ decisions in advance when they are making decisions, which add more randomness and fairness to the game. Our game description and solution are based on the later edition.


The game

Tian Ji and the king of the Qi Kingdom both like horse racing and often make bets. They choose three horses and classify them as good, better, and best. For horses of the same grade, King Qi's horses are better than Tian Ji's horses. The rule of the race is that there are three rounds, and the winner is the one who wins at least two rounds. Both parties cannot know the other party's decision in advance before making a decision, so it can be regarded as choosing a strategy at the same time. Each horse can only participate in one round. The winner of each round gets 1 point, the loser gets -1 point, and the result of the game is the sum of three game scores. * Draw the normal form matrix of this game. Be sure to include all key elements of a strategic game. * Suggest a strategy that the King of Qi should adopt.


Solution

Suppose we use E, B, and G to represent the “fastest,” “fast,” and “slow” horses. Here is the strategy for both players(Please refer to the image): The original story did not mention King of Qi modifying his strategy, so for each of the strategies King of Qi made, Tian Ji can choose the corresponding strategy bringing him the maximum benefit:. * If the King of Qi chose EGB, Tian Ji could choose BEG. * If the King of Qi chose EBG, Tian Ji could choose BGE. * If the King of Qi chose GEB, Tian Ji could choose EBG. * If the King of Qi chose GBE, Tian Ji could choose EGB. * If the King of Qi chose BEG, Tian Ji could choose GBE. * If the King of Qi chose BGE, Tian Ji could choose GEB. From the above analysis, we can find that if Tian Ji knows the strategy that the King of Qi is making, he can choose the corresponding and only winning strategy to win the King of Qi. For example, if the King of Qi chooses EGB and Tian Ji knows that, then he can choose BEG to win. However, for the game, none of these are mutual best responses, so there is no pure Nash equilibrium strategy. But there exists a mixed Nash equilibrium strategy. The hypothesis of the original story is that Tian Ji knew the strategy that the King of Qi would choose. What if Tian Ji did not know that? Then we will discuss how both of them can change their strategies. For the king of Qi, his dominant strategy is to choose randomly. The reason is that no matter which strategy he chooses, the probability for Tian Ji to win is 1/6. For Tian Ji, his dominant strategy is also to choose randomly since the probability for the king of Qi to win is 5/6. Therefore, for both of them, the dominant strategy is to choose randomly.


External links and references

{{DEFAULTSORT:Tian, Ji Zhou dynasty generals Qi (state) Year of death unknown Year of birth unknown 4th-century BC Chinese people