General Tian’s Horse Race and Game Theory

By Math Is Fun, September 28, 2021

Son loves stories. So this episode is a story.

Tian (last name) Ji (first name), a prominent Chinese general 2000 years ago in the Kingdom of Qi, loved horse racing. One day, he raced with the King of Qi. They each had three horses, ranked as “fastest”, “faster” and ”fast”. They held three matches. Whoever won 2 or more matches is the winner. In match 1, General Tian’s fastest horse raced against the king’s fastest horse. Match 2 was between their faster horses. Match 3 was between the fast horses. Of course, the king usually had better horses. In each match, the king beat General Tian by a small margin, and took the race prize money.

Here enters our story’s wise guy: Sun (last name) Bin (first name). He was a descendant of Sun Tzu, the author of The Art of War. Sun Bin himself authored Sun Bin’s Art of War. After he watched Tian Ji’s horse race with the king, he told Tian to invite the king for another race and he had a sure strategy to win the race. Not convinced, Tian followed Sun Bin’s advice, and he did win!

So what did Sun Bin tell Tian? His changed the order of Tian Ji’s horses in the matches:

Match 1: Tian’s fast VS King’s fastest

Match 2: Tian’s fastest VS King’s faster

Match 3: Tian’s faster VS King’s fast

Since Tian Ji’s horses were only slightly slower than the King’s horses of the same rank, they can defeat the King’s horses with the lower rank. Tian Ji lost the first match by a large margin, but won the next 2 matches. 2:1, Tian Ji won the race!

This is a good study example of one of Math’s branches: Game Theory. Put simply, it studies how to win in a game. For example, how to win a chess game. Or how to design a winning strategy in a war.

As a take-home question, for General Tian’s horse race, what if the King also knew he could change the order of his horses, what should Tian Ji do?

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