Problems

Chapter 3 (First Edition)

Problems 2 and 4

These problems provide practice for solving sequential games.

Problem 2

Use rollback to find equilibria for the following games:

2a

image 1a

2b

image 1b

2c

image 1c

Problem 4

Consider the rivalry between Airbus and Boeing to develop a new commercial jet aircraft. Suppose Boeing is ahead in the development process, and Airbus is considering whether to enter the competition. If Airbus stays out, it earns zero profit while Boeing enjoys a monopoly and earns a profit of $1 billion. If Airbus decides to enter and develop the rival airplane, then Boeing has to decide whether to accommodate Airbus peaceably or to wage a price war. In the event of peaceful competition, each firm will make $300 million. If there is a price war, each will lose $100 million because the prices of airplanes will fall so low that neither firm will be able to recoup its development costs. Draw the tree for this game. Find the rollback equilibrium.

Solutions

4.2

(a) {S,t} with payoffs of (1,0). Player A's equilibrium strategy is S; B's equilibrium strategy is "t if N." To characterize a rollback equilibrium, one must find the optimal strategy for a player, even if the player is never called upon to use it. In this case, although player B never has to select between "t" and "b," the fact that the player would select "t" is what makes playing "S" an equilibrium for player A.

image 2

(b) { N,N; b; d } with payoffs (2,3,2).

Player A's equilibrium strategy is "N and then N if b follows N or N if d follows N" or "Always N." Player B's equilibrium strategy is "b if N" (or just b). Player C's equilibrium strategy is "d if S" (or just d).

Again, the equilibrium specifies an action for every player at every decision node, even if that decision node is never reached. In the last two decision nodes for player A, player A would choose N. Because of this, player B selects b and player C selects d. Now considering the first period, player A chooses N.

image 2b

(c) { S,S,N ; n,n,s } with payoffs of (4,5).

Fig 2c

3.4

Rollback shows that Boeing chooses peace over war if Airbus enters, so Airbus will enter. Rollback equilibrium entails Airbus playing "Enter" and Boeing playing "Peace if entry"; each firm earns $300 million profit in equilibrium. Simply, we find that Boeing would not fight the entry. Hence, Airbus would enter. image 4