Game Theory and Business Strategy

Solutions to Self Test

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?


SOLUTION:

If the competitor enters, the monopolist must select between accommodating, and earning $5M, or fighting and losing $5M. Hence, the monopolist would elect to accommodate. Rolling back to the first period, the competitor may stay out, earning $0, or enter, which results in accommodation and profits of $5M. Hence, the competitor enters.
{In,Accommodate}
{In,Fight}
{Out,Accommodate}
{Out,Fight}


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?


SOLUTION:

The intuition here is that, by not being able to commit to entry, the competitor will be worse off. Start by working at the end:
  • If the monopolist fights entry, the competitor will exit (out3) since losing $1M is better than losing $5M.
  • If the monopolist accommodates, the competitor will stay in the industry (in2), earning $5M.
  • Rolling back to the monopolist's decision: accommodating leads to the competitor staying in, and each earning $5M, while fighting leads to the entrant leaving and the monopolist reclaiming $10M, so the monopolist fights.
  • In the first period, the competitor now recognizes that entry leads to eventual withdrawl and losses of $1M, so the entrant stays out (out1).

{In1,In2,In3; Accommodate}
{Out1,Out2,Out3; Fight}
{Out1,In2,Out3; Fight}
{In1,In2,Out3; Accommodate}


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?


SOLUTION:

  • Period 3: 100>50 so steal better than share
  • Period 2: 10>0 so distrust better than trust
  • Period 1: 1>0 so hold rather than play
So the game ends as soon as player one elects to "hold." However, to fully describe the equilibrium, all of the other potential decisions must be specified. Why? The reason player one holds is because she anticipates that player two will distrust, who in turn anticipates that player one will steal. This must be specified, especially since both players would be happier with the 50,50 outcome.
{ Play, Share; Distrust }
{ Hold }
{ Hold, Distrust }
{ Hold, Distrust, Steal }


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:
  • Round 1. Strong versus weak sanctions
  • Round 2. Winner of round 1 versus no sanction
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?

SOLUTION:

Myopic voting means that they will vote non-strategically, and will not look forward and reason back:
  • Period 1: weak beats strong two votes to one
  • Period 2: no sanction beats weak sanction two votes to one

No sanction
Weak
Strong
It depends


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?

SOLUTION:

Now, we have to start with period 2. Period 2 could see one of two votes: strong versus none, or weak versus none.
  • Strong vs. none --> Strong wins (2-1)
  • Weak vs. none --> None wins (2-1)
Now, the judges should recognize that, no matter how they vote in period one, the weak sanction can never prevail. Hence, "a vote for the weak sanction is a vote for no sanction" quite literally. The true vote in period one is between the strong sanction (by voting for strong) or no sanction (by voting for weak). Two out of three judges prefer strong sanctions to none, so strong will prevail.
No sanction
Weak
Strong
It depends