5. Dynamic Bayesian Games

5.3 Dynamic Bayesian Games: Signaling

Application I : Job-Market Signaling (Chapter 29, Spence (1973))

### Job-Market Signaling, Michael Spence (1973)

In most job markets, a job hunter’s productive capability on the job is *not perfectly observable* when an employer makes a hiring decision

Instead, the employer is able to observe a plethora of personal data and certain attributes such as:

education, previous work experiences, race, age, sex, etc

Michael Spence – Nobel laureate in 2001

**MICHAEL SPENCE (1973)** classifies the set of attributes into *indices* and *signals* according to its alterability

In his pioneering work, a job hunter is represented by a pair of his productivity and one signal,

(\(\theta\),*e*),

where \(\theta\) his unobservable innate talent and *e* is the level of education

The textbook presents a simple version of his job-market signaling model:

- The worker’s type 0 is either high or low
- The common prior is Pr(\(\theta\) = H) =1/3
- After observing \(\theta\), the worker decides whether to obtain an education or not
**Education is costly**, 4 to the high type and 7 to the low type- After observing the worker’s choice, the firm decides whether to appoint him as a manager or a clerk
- The firm earns a profit of 10 (0) by putting the high (low) type in the management position
- When appointed as a clerk, both types yield a profit of 4 to the firm
- The manager receives a wage of 10, whereas the clerk receives 4

The dynamic game can be put in the extensive form:

separate equilibrium: High Type E, Low Type N’

(Since there is no intensive debated, because of M for q and C’ for p)

pooling equilibrium: High Type N, Low Type N’

expected payoff: 10/3 < 4 so C’ is Best Response

<(NN’, C’C), p = 1/3, q <= 2/5>

This signaling game has a unique PBE in *separating strategies* satisfying the Cho-Kreps criterion:

⟨(\(E^H N^L, C^N M^E\)), p = 0, q = 1⟩

This highlights the **signaling role** of education in job markets

- Although it does not improve productivity, education could still have economic value to the worker
- High-type workers are willing to obtain more education in order to signal their productivity, and this costly signaling is compensated with high pay in the market

“*the sheepskin* (diploma) effect”

Spence’s work sparked subsequent empirical research investigating whether this effect indeed exists in the real world

We study how to compute a PBE by going through the next classic example

EXAMPLE 5.7 (THE BEER-QUICHE GAME, CHO AND KREPS (1987)).

- Player 1 can be weak (w) or strong (s), and let Pr(\(t_1\) = w) = 0.1. The strong type likes beer for breakfast, while the weak likes quiche.
- Player 1 is ordering his breakfast, and player 2 is watching and contemplating whether to pick a fight with player 1. Player 2 wants to fight with the weak type but walk away from the strong type.
- Player 1 likes to avoid a fight: he gets a payoff of one from the preferred breakfast, and a payoff of two from avoiding the fight.

- Reference: Chang-Koo Chi, (40/50) Game Theory and Applications 11 – Job-market signaling, Jul 15, 2020, https://youtu.be/O19madhZ2rE