5. Dynamic Bayesian Games

5.3 Dynamic Bayesian Games: Signaling

– Definition and Equilibrium (Chapter 28)

– Forward Induction and Cho-Kreps’ Intuitive Criterion*

– Application 1: Job-Market Signaling (Chapter 29, Spence (1979))

– Application 2: Pecking Order Theor* (Myers and Majluf (1984))

In the previous section, we studied how to compute a PBE in dynamic games with imperfect information and how it is related to the other equilibria

Recall the two requirements of PBE ⟨\(\sigma, \mu\)⟩

(i) Bayesian beliefs \(\mu\) given \(\sigma\) and (ii) sequential rationality of \(\sigma\) under \(\mu\)

Utilizing the inclusive relationship between PBE and NE, we sorted out the Nash eqb’ a satisfying (i) and (ii) and identified PBE:

- finding players’ Bayesian beliefs \(\mu\) given Nash eqbm play \(\sigma\)
- checking sequential rationality of \(\sigma\) given Bayesian beliefs \(\mu\)

### Asymmetric Information

With this new eqbm, we analyze the two most widely applied dynamic Bayesian games:

signaling and screening games.

These two games are often referred to as dynamic games with *asymmetric info* in which

- there are two players;
- information is asymmetric, i.e., one of the two players has more or better info;

we call that player an**informed**player and the other an**uninformed**player; - each player moves only once;

if the informed player moves first ⇒ signaling

if the uninformed player moves first ⇒ screening

### Signaling Games

One **sender** (player 1) = the player who first receives private information or a type and then chooses an action (sends a message))

One **receiver** (player 2) = the uninformed player who receives the sender’s message and then chooses an action (a response)

Figure: Example of Signaling Games, Guessing Game

### Basic Elements in Signaling Games

Formula | Description |

N = {1, 2} T = {\(t_1, …, t_k\)} \(\pi\) | the set of players; 1 = the sender, 2 = the receiverthe sender’s type space a common prior on the sender’s type |

M = {\(m_1, …, m_s\)} \(\sigma_1^t(\cdot)\) | the sender’s set of possible messages a (mixed) strategy of the sender with type t |

R = {\(r_1, …, r_b\)} \(\sigma_2^m(\cdot)\) \(\mu_1^t(\cdot)\) | the receiver’s set of possible responses a (mixed) strategy of the receiver who observed m the receiver’s beliefs about t after observing m |

\(u_1\)(m, r|t) \(u_2\)(m, r|t) | the payoff function of the sender with type t the payoff function of the receiver |

### Expected Payoffs

Given receiver’s strategy \(\sigma_2\), the type-t sender’s expected payoff from messages m is

\(\sum_{r \in R}u_1(m, r|t)\sigma_2^m(r)\)

If m is sent, the receiver’s expected payoff from response r given beliefs \(\mu_2^m\) is

\(\sum_{t \in T}u_2(m, r|t)\mu_2^m(t)\)

EXAMPLE 5.6.

Suppose \(\sigma_2^R\) = xC + (1-x)F. Then the type-H sender’s expected payoff from sending message R is

\(u_1\)(R, C|H)x + \(u_1\)(R, F|H)(1-x) = 1+x

\(u_1\)(R, C|H) = 2

\(u_1\)(R, F|H) = 1

If R is sent, the receiver’s expected payoff from response C given beliefs \(\mu_2^R\) = p is

\(\mu_2\) = (p, 1-p)

p = \(\mu_2^R\)(H)

-2p + 2(1-p) = 2-4p

### PBE in Signaling Games

Assessment ⟨\((\sigma_1, \sigma_2), \mu_2\)⟩ is a perfect Bayesian eqbm of a signaling game if

(i) the receiver’s beliefs \(\mu_2\) are Bayesian given (\(\sigma_1, \sigma_2\))

(ii) (\(\sigma_1, \sigma_2\)) is sequentially rational given \(\mu_2\)

Specifically, the second condition (ii) requires that

- for each type t ∈ T, player 1’s eqbm strategy \(\sigma_1^t\) maximize her expected payoff given player 2’s eqbm strategy \(\sigma_2\):

\(\sum_{r \in R}u_1(\sigma_1^t, r|t)\sigma_2^m(r)\) ≥ \(\sum_{r \in R}u_1(\sigma_1^{‘}, r|t)\sigma_2^m(r)\) for all **\(\sigma_1^{‘}\)**

- for each message m ∈ M,
*whether messages m was indeed sent*, player 2’s eqbm strategy \(\sigma_2^m\) maximize his expected payoff given \(\mu_2\):

\(\sum_{t \in T}u_2(m, \sigma_2|t)\mu_2^m(t)\) ≥ \(\sum_{t \in T}u_2(m, \sigma_2^{‘}|t)\mu_2^m(t)\) for all **\(\sigma_2^{‘}\)**

To illustrate the second bullet point, let’s revisit Example 5.6 and suppose player 1 sent message “F” regardless of his type. In this case, P2’s info set {x, y} is never reached.

Nevertheless, PBE calls for an optimal behavior at {x, y}.

2 is allowed to have any beliefs at {x, y}

\(\mu_2^R\) = (p, 1-p)

2’s expected payoff from response C given beliefs \(\mu_2^R\) is

2-4p

and from response F is -1

2 – 4p ≥ -1 ⇔ p ≤ 3/4

2’s BR = \(\begin {cases} C & \text{if } p \leq \frac{3}{4} \\ F & \text{if } p > \frac{3}{4} \end{cases}\)

- Reference: Chang-Koo Chi, (38/50) Game Theory and Applications 10 – Signaling games intro, Jul 14, 2020, https://youtu.be/-0yOmhll7XY