Commitment Game

• Content map: SMU H3 Game Theory Map

Setup

Definition:

Commitment Game

• Players: Two players, Teacher and Student.
• Strategies: Teacher: Weak or Tough; Student: Punctual or Late; Commitment version: Teacher can commit to Tough before the student chooses.

Rules

• Start with a teacher and student facing a punctuality problem; Players choose Weak or Tough and Punctual or Late, with possible prior commitment to Tough.
• The player who commits credibly or best responds to commitment reaches the preferred outcome.
• Without commitment, teacher and student play the payoff matrix below.
• With commitment, the teacher removes Weak from the later choice set; Payoffs are ordered as (Teacher, Student).

Game Tree

• $C=$ Commit to Tough, $N=$ No Commit, $P=$ Punctual, $L=$ Late.

Payoff Matrix

Derivation (Best Response Analysis)

• Teacher:
  • If Student is Punctual, Teacher prefers Weak since $4>3$.
  • If Student is Late, Teacher prefers Weak since $2>1$.
• Weak is a dominant strategy for Teacher.
• Student:
  • If Teacher is Weak, Student prefers Late since $4>3$.
  • If Teacher is Tough, Student prefers Punctual since $2>1$.

Derivation (Nash Equilibrium)

• In the no-commitment game, Teacher chooses Weak.
• Given Weak, Student chooses Late.
• The Nash equilibrium is therefore (Weak,Late) with payoff $(2,4)$.
• If Teacher commits to Tough, Student chooses Punctual because $2>1$.
• Teacher then receives $3$ instead of the no-commitment payoff $2$.
• Therefore commitment improves Teacher’s payoff.

Nash Equilibrium

Result:

Without commitment:

With commitment:

leading to the Tough-Punctual outcome $(3,2)$.

Social Optimum

• The largest total payoff is at (Weak,Punctual):

• The commitment outcome (Tough,Punctual) gives total payoff $5$.
• Commitment helps Teacher but does not maximise total surplus.

Insights

Insight:

• Commitment can require adopting a dominated action.
• Credibility is key because the teacher must enforce Tough behaviour when the student is late.
• External rules can make the commitment credible.