Chapter 11.3, Problem 24E

In Exercises 13–27, use the graphical method when the payoff matrix is a 2 × 2 matrix or can be reduced to one after removing rows or columns that are dominated. Otherwise, use the simplex method.

Military Science The Colonel Blotto game is a type of military strategy game. Two opposing armies are approaching two posts. Colonel Blotto has 4 regiments under his command, while his opponent, Captain Kije, has 3 regiments. Each commander must decide how many regiments to send to each post. The army that sends more regiments to a post not only captures that post but also captures the losing army's regiments. If both armies send the same number of regiments to a post, there is a stand-off, and neither army wins. The payoff is one point for capturing the post and one point for each regiment captured. Source: Mathematical Methods and Theory in Games, Programming, and Economics.

(a) Set up the payoff matrix for this game, (Hint: Colonel Blotto has five choices, and Captain Kije has four.)

(b) Find the optimum strategy for each commander and the value of the game.

(c) Show that if Colonel Blotto uses the strategy found, in part (b), then any strategy used by Captain Kije results in the same payoff. (Hint: Show that AM = (14/9)R, where R is a row matrix consisting of all 1's, and then use the fact that RB = [1].)

(d) Based on the result of part (c), what can you conclude about the uniqueness of the optimum strategy found by linear programming?