Finding Rollback Equilibria

Question 1.

A firm is quite happy monopolizing its industry with profits of $10M. A potential competitor considers entering the industry. If the competitor elects not to enter, it earns profits of $0 and the monopolist maintains its profit of $10M. If the competitor enters, the monopolist must either accommodate the entry or fight. If the monopolist accommodates, both firms earn $5M. If the monopolist fights, both firms lose $5M. The game is represented by the following tree: Game Tree What is the rollback equilibrium of the above game?

Question 2.

Consider the above game (question 1) but suppose that the decision to enter by the competitor is reversible in the following sense: after it has entered, and after the monopolist has chosen to accommodate or fight, the competitor can choose to remain in the industry (and receive either the $5M profits or $5M loss) or to exit. Suppose that exiting at this point results in a loss to the entrant of $1M, and the monopolist regains its $10M profit. The new game is represented by the following tree: Game Tree What is the rollback equilibrium of the above game?

Question 3.

Consider the game represented by the following tree: Game Tree Player 1 is represented by blue circles (and actions in italics). Player 2 is represented by red circles. What is the rollback equilibrium of the above game?

Question 4.

The three person executive committee of the Owen Honor Council is considering a case of cheating on a take home exam. The members decide that, for the sake of consistency, they should first determine if such an offense warrants a strong or a weak punishment, and then determine if the accused should receive that punishment. Hence, the order of the voting will be:

The three members vote by majority rule.

The three members of the committee have the following preferences:
Person 1 prefers:
  strongtono sanctiontoweak
Person 2 prefers:
  no sanctiontoweaktostrong
Person 3 prefers:
  weaktostrongtono sanction

In other words, person one prefers strong sanctions and would rather let the accused walk than lower the reputation of the school by giving him a slap on the wrist. Person two is morally opposed to sanctions, while person three prefers some sanction to none, but prefers the weak sanction.

If the judges vote myopically (i.e. each votes for his or her true favorite option in every round), which alternative will prevail?

Question 5.

The three judges from the previous question decide that prior to voting, they will consult a game theorist to determine how to vote strategically. If, rather than voting myopically, the judges look forward and reason back, voting strategically in the first round, which outcome will prevail?




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