Problem Set 1Other Problems

A Blotto Game

(From Fun and Games by Binmore, problem 6.34) Colonel Blotto has four companies that he can distribute among two locations in three different ways: (3, 1), (2, 2), and (1, 3). His opponent Count Baloney has three companies that he can distribute in two different ways: (2, 1) and (1, 2). Suppose that Blotto sends m1 companies to location 1 and Baloney sends n1 companies to location 1. If m1 = n1, the result is a stand-off and each commander gets a payoff of zero for location 1. If m1 ≠ n1, the larger force overwhelms the smaller force without loss to itself. If m1 > n1, Blotto gets a payoff of n1 and Baloney gets a payoff of -n1. If m1 < n1, Blotto gets a payoff of -m1 and Baloney gets a payoff of m1. Each player’s total payoff is the sum of his payoffs at both locations.

Find the normal-form of this game and determine a mixed-strategy Nash equilibrium.

Answer: The normal-form is

(3, 1) (2, 2) (1, 3)

(2, 1) -2, 2 -1, 1 0, 0

(1, 2) 0, 0 -1, 1 -2, 2

Let p be the probability Blotto plays (2, 1). Blotto must mix to make Baloney indifferent between any of his strategies, so

2 p + 0 (1 - p) = 1 = 0 p + 2 (1 - p)

p = 1/2 satisfies those equations, so Blotto plays (2, 1) half the time.

Let q be the probability Baloney plays (3, 1) and r be the probability that Baloney plays (2, 2). Blotto must be indifferent between his strategies given q and r, or

-2 q - r + 0 (1 - q - r) = 0 q - r - 2(1 - q - r) -2 q - r = -2 + 2 q + r 4 q + 2 r = 2 q = (1 - r) / 2

ECON 440/640 Problem Set 1

1

Baloney plays (2, 2) with probability r and when he doesn’t play (2, 2), mixes half the time on (3, 1) and half the time on (1, 3). For example, Baloney could play (1/3, 1/3, 1/3) or (1/4, 1/2, 1/4).

Bertrand Duopoly

Consider the homogenous goods Bertrand variant we discussed in class (Gibbons’s problem 1.7 is about this model) but with differing marginal costs of production. Firm 1 pays c1 per unit produced and Firm 2 pays c2. Solve for the Nash equilibrium of this game and show that it is unique.

Answer: Assume c1 < c2 < a/3.1 Then the equilibrium is p1 = c2 - ε, p2 = c2.

We ned to check that neither firm wants to raise or lower its price.

Firm 1 lowering prices: in equilibrium

π1 = p1 (a - p1 - c1) d π1 / d p1 = a - c1 - 2 p1 > 0 (from a/3 > c2 > p1 > c1)

Firm 1 makes higher profits the higher price they set as long as p1 < p2.

Firm 1 raising prices: then firm 1 splits the market with firm 2.

Firm 2 lowering prices: then firm 2 is selling at a loss.

Firm 2 raising prices: if firm 2 raised prices, firm 1 would follow suit, and then firm 2 would want to lower prices. Hence no equilibrium exists where p2 > c2.

ECON 440/640 Problem Set 1

2

1 In the case where a is lower, p1 is set between c1 and c2 rather than at c2 - ε.