Game Theory 10 – chap19. Signaling games intro

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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

FormulaDescription
N = {1, 2}
T = {\(t_1, …, t_k\)}
\(\pi\)		the set of players; 1 = the sender, 2 = the receiver
the sender’s type space
– sender’s private information space (w or b)
a common prior on the sender’s type
– common prior: nature의 움직임(사전 distribution)
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
– sender는 타입 별로 움직임(전략은 type dependent)
R = {\(r_1, …, r_b\)}
\(\sigma_2^m(\cdot)\)
\(\mu_1^t(\cdot)\)		the receiver’s set of possible responses
– receiver의 action space
a (mixed) strategy of the receiver who observed m
– receiver가 받은 message m 에 의해 전략 선택
the receiver’s beliefs about t after observing m
– 받은 message m에 대해 receiver의 update된 belief
\(u_1\)(m, r|t)
\(u_2\)(m, r|t)		the payoff function of the sender with type t
– 자신이 보낸 message, receiver의 response, 자신의 type에 대해 dependent
the payoff function of the receiver
– sender의 message, 자신의 response, sender의 type에 대해 dependent

Expected Payoffs

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

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

– 자신이 message m을 보냈을 때 receiver는 sender가 m을 보냈던 것을 알고 있음
– m을 받았을 때 response r을 선택할 확률(sigma)에 대한 sender의 payoff
– 이 mixed strategy 하에서 response의 여러 확률을 곱해서 더해준 것이 sender의 expected payoff

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

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

– message m이 보내지고 receiver가 response r을 선택했을 때 따르는 expected payoff는 belief에 따름
– sender의 type이 정확히 H type인지 L type인지 모르기 때문
– m은 받았으니 fixed되어 있지만 receiver는 t에 대해 uncertain하기 때문에 t에 대한 distribution인 자신의 belief를 곱하고 type에 대해 전부 더해준 것이 expected payoff

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

– signaling game에서 PBE를 다시 정의
– sender가 어떻게 움직이고 어떤 메시지를 보낼지, receiver가 어떤 response를 선택할지, sender의 belief는 생각할 필요 없음(각각의 information set이 하나의 decision node (1)로 구성되어 있음), 대신 receiver의 belief를 묘사하여 3가지 element로 구성

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

– Bayesian belief: (각각의 노드에 도착할 확률)/(전체 information set에 도착할 확률), 도달하면 Bayesian rule에 대해 계산, 도달하지 않는 information set에 대해서는 unrestricted되어 아무런 belief나 가질 수 있음

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 \(\sigma_1^{‘}\)

– sender의 입장에서, sender의 eqbm strategy sigma 1은 본인이 어떤 type이든 모든 information set에서 optimal strategy이기 때문에, sender는 information set은 자신의 type에 따라 나뉨
– sigma 1은 자신의 type이 t일 때, sender의 expected payoff를 maximize해야 함, receiver가 sigma 2라는 전략을 선택했을 때
– receiver가 sigma 2 전략을 선택했을 때, 자신이 type t일 때 선택하는 전략이 다른 어떤 것을 선택하는 것(sigma ‘)보다 항상 더 높은 payoff를 줘야 함
– sender가 type별로 선택하는 전략이 optimal strategy이어야 한다는 것

  • 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 \(\sigma_2^{‘}\)

– 각 message m가 전달되든 되지 않든 상관없이 sigma m은 receiver의 expected payoff를 maximize해줘야 함
– 만약 자신이 m이라는 메시지를 받으면 belief가 업데이트되고, expected payoff는 다른 전략에 비해 eqbm strategy는 더 높은 payoff를 안겨다 줌
– receiver는 본인이 어떤 메시지를 받든, 실제 메시지가 도달했든 optimal strategy를 선택해야 한다는 것이 sequential rational하다는 것

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

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