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Tian Ji’s Horse Racing Strategy – An Ancient Chinese Story based on Game Theory

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Tian Ji’s Horse Racing (田忌赛马) is a story in one of the greatest Chinese literatures Records of the Grand Historian. The story goes like this: This guy Tian Ji and the king of the Qi Kingdom both like horse racing, and often make bets. Of course, the king of Qi has better horses, so Tian Ji loses all the time. Another guy, Sun Bin, says “take me to the race next time and I can help you win”. Sun Bin learns that for every race, Tian Ji and the king both choose three horses, classify them as good, better, and best. The rule of the race is that there are three rounds, and the winner is the one who wins at least two rounds. Right now, both of them are using their “good” horse for the opponent’s “good” horse, “better” horse for the opponent’s “better” one, and “best” for the “best”. The reason that king of Qi is winning is that he has slightly more superior horse in all three levels. Sun Bin then brings up an idea: he uses Tian Ji’s “good” horse for racing the king’s “best” horse, then uses the “best” horse against the king’s “better” one, and the “better” horse against the “good” one. As a result, Tian Ji loses the first round, but wins the second and third round (because his “best” and “better” horse can still beat the king’s “better” and “good” ones respectively), and eventually wins the race.

If we use A, B, C to denote “best”, “better” and “good” horses, here is the strategy for both players. (T is Tian Ji, K is King of Qi)

 King of Qi
 Tian Ji ABC -1, 1 -1, 1 -1, 1 1, -1 -1, 1 -1, 1
ACB -1, 1 -1, 1 1, -1 -1, 1 -1, 1 -1, 1
BAC -1, 1 -1, 1 -1, 1 -1, 1 -1, 1 1, -1
BCA -1, 1 -1, 1 -1, 1 -1, 1 1, -1 -1, 1
CAB 1, -1 -1, 1 -1, 1 -1, 1 -1, 1 -1, 1
CBA -1 ,1 1, -1 -1, 1 -1, 1 -1, 1 -1, 1

In the original story, while we don’t know what order the original race was, it must be one of the six.

  • If it was “ABC” to “ABC”, the new strategy for Tian Ji is “CAB”.
  • If it was “ACB” to “ACB”, the new strategy for Tian Ji is “CBA”.
  • If it was “BAC” to “BAC”, the new strategy for Tian Ji is “ACB”.
  • If it was “BCA” to “BCA”, the new strategy for Tian Ji is “ABC”.
  • If it was “CAB” to “CAB”, the new strategy for Tian Ji is “BCA”.
  • If it was “CBA” to “CBA”, the new strategy for Tian Ji is “BAC”.

The original story did not mention king of Qi modifying his strategy, so knowing which strategy that the king is making, Tian Ji can choose the one and only winning strategy that has that corresponds to king’s strategy.

What happens if both of them are changing the strategy? What is the dominant strategy then?

For both of them, choosing a completely random strategy is the best bet.

For the king, he should choose totally randomly, because there is no strategy of his where Tian Ji would have a higher chance of winning, it is always 1/6. The same goes for Tian Ji, no matter what strategy he chooses, the chance of the King winning is always 5/6. Therefore, choosing randomly is the dominant strategy for both.

Is this a good idea in real life?

Coming from China, I know that this story if very well-known, and praises the wit of Sun Bin. The rule of the race does not forbid Tian Ji from doing this. However, I know in certain sports using such tactic is forbidden: one has to match players by how good they are, and using an inferior player against a knowingly superior one of the opponent team is forbidden. That, of course, is not a concern of game theory.


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