- 3 Colonel Spicces Has Four Companies That He Can Distribute Among Two Locations In Three Different Ways 3 1 2 2 1 (163.72 KiB) Viewed 15 times
3. Colonel Spicces has four companies that he can distribute among two locations in three different ways: (3, 1), (2, 2) or (1,3). His opponent, Count Vicces, has three companies the he can distribute among the same two locations in two different ways: (2, 1) or (1,2). In oder to describe the possible outcomes of the game, let s; denote the companies that Colonel Spicces sends to location i and let v; denote the companies that Count Vicces sends to location i. Note that i can be either 1 or 2, given that there are two locations in this game. If s₂ = = v₁, the result is a standoff, and each commander gets a payoff of zero for location i. If si vi, the larger force overwhelms the smaller force without loss to itself. If si > vi, Colonel Spicces gets a payoff v₁, and Count Vicces gets a payoff of -v; for location i. If si < vi, Colonel Spicces gets a payoff -si, and Count Vicces gets a payoff of s; for location i. Each player's total payoff is the sum of his payoffs at both locations. (a) Find the strategic-form representation of this simultaneous-move game, and show that the game has no Nash equilibrium in pure strategies. 1 point (b) Find the mixed-strategy Nash equilibria of the game. Hint: This is a rather difficult task, because Colonel Spicces has three pure strategies in this game. You have to follow the usual logic in order to find the mixed-strategy Nash equilibrium, but in this game Colonel Spicces will have to assign two prob- abilities: p₁ to (3,1), p2 to (2, 2); therefore (1,3) will be chosen with probability 1- P₁ - P2- 2 points