Question

General Tian gambled on horse races with King Qi (around 340 BC, see the handout). Each had a team of three horses, whose speeds are respectively: K1, K2, and K3; G1, G2, and G3 (K for King’s horses, and G for General’s horses). These numbers satisfy:

K1>K2>K3, G1>G2>G3; K1> G1, K2> G2, K3> G3; and G1 > K2, G2 > K3.

In the historical event, General Tian and King Qi agreed on a wager of one thousand gold coins on each of three races. The General ended up winning 1000 gold coins from the King. Let the worth of winning a race as 1 (= 1000 gold coins), and that of losing as –1. This is a two–person zero–sum game. Find General Tian’s payoff matrix, and draw the game tree.

Hint: Each has six choices. For example, one choice is: run the high speed horse in the first race, middle speed horse in the second race, and low speed horse in the last race. This can be denoted as 123. Similarly, 312 denotes the choice of running the low speed horse first, the high speed horse second, and the middle speed horse last.

Answer 1

This is old story in China which used for decision making.

If we use 1, 2, 3 to denote "high speed", "middle speed", "low speed".

We will have six orders as follows:

A. If it was “123” to “123”, the new strategy for Tian is “312”.

B. If it was “132” to “132”, the new strategy for Tian is “321”.

C. If it was “213” to “213”, the new strategy for Tian is “132”.

D. If it was “231” to “231”, the new strategy for Tian is “123”.

E. If it was “312” to “312”, the new strategy for Tian is “231”.

F. If it was “321” to “321”, the new strategy for Tian is “213”.

Strategy for both players is as follows:

King Qi
		123	132	213	231	312	321
Tian	123	-1, 1	-1, 1	-1, 1	1, -1	-1, 1	-1, 1
	132	-1, 1	-1, 1	1, -1	-1, 1	-1, 1	-1, 1
	213	-1, 1	-1, 1	-1, 1	-1, 1	-1, 1	1, -1
	231	-1, 1	-1, 1	-1, 1	-1, 1	1, -1	-1, 1
	312	1, -1	-1, 1	-1, 1	-1, 1	-1, 1	-1, 1
	321	-1 ,1	1, -1	-1, 1	-1, 1	-1, 1	-1, 1