Signalling Game

• Content map: SMU H3 Game Theory Map

Setup

Definition:

Signalling Game

• Players: Two players, Student and Employer.
• Strategies: The student chooses the number $n$ of hard courses on the transcript; The employer offers wages based on the observed transcript.

Rules

• Start with a student type privately known to the student; Players move sequentially: the student chooses hard courses and the employer chooses wages after observing the transcript.
• The player who uses the signal or wage rule optimally maximises payoff.
• There are two student types:
  • type $A$: intrinsically motivated,
  • type $C$: grade oriented.
• Employer values type $A$ at $160{,}000$ and type $C$ at $60{,}000$; Outside options are $125{,}000$ for type $A$ and $30{,}000$ for type $C$.

Payoff Details

• Signalling costs are:

Payoff Matrix

• If the employer interprets the signal as type $A$, the wage is $160{,}000$.
• If the employer interprets the signal as type $C$, the wage is $60{,}000$.
• Student payoffs are wage minus signalling cost.

Derivation (Best Response Analysis)

• For a separating signal $n$, type $A$ must prefer signalling to pooling at the low wage:

• This gives

• Type $C$ must prefer not to imitate type $A$:

• This gives

• Therefore incentive compatibility requires

• Individual rationality for type $A$ requires

so

• Individual rationality for type $C$ is automatic because $60{,}000 > 30{,}000$.

Derivation (Nash Equilibrium)

• Combining incentive compatibility and individual rationality gives the feasible separating range

• The least costly separating signal is therefore

• A pooling outcome with no hard courses gives both types wage $100{,}000p+60{,}000$, where $p$ is the prior probability of type $A$.
• Under intuitive beliefs, type $A$ would deviate to one hard course unless

which requires $p \geq 0.97$.

• Hence pooling is fragile unless the employer already believes the student is almost surely high type.

Nash Equilibrium

Result:

A separating signalling equilibrium is:

• type $A$ chooses $n^\ast=8$,
• type $C$ chooses $0$,
• the employer pays $160{,}000$ after observing $n \geq 8$ and $60{,}000$ after observing $0$.

Social Optimum

• If type were directly observable, no costly signal would be needed.
• Among separating outcomes, the efficient signal is the smallest feasible one, $n=8$, because it minimises wasteful education cost.

Insights

Insight:

• A signal is credible because it is cheaper for the high type than for the low type.
• Signalling can separate types while still wasting resources relative to full information.