Utilizing the inclusive relationship between PBE and NE, we sorted out the Nash eqb’ a satisfying (i)<\/mark> and (ii)<\/mark> and identified PBE:<\/p>\n\n\n\n With this new eqbm, we analyze the two most widely applied dynamic Bayesian games:<\/p>\n\n\n\n These two games are often referred to as dynamic games with asymmetric info<\/em> in which<\/p>\n\n\n\n if the informed player moves first \u21d2 signaling One sender<\/mark><\/strong> (player 1) = the player who first receives private information or a type and then chooses an action (sends a message))<\/p>\n\n\n\n One receiver<\/mark><\/strong> (player 2) = the uninformed player who receives the sender’s message and then chooses an action (a response)<\/p>\n\n\n Figure: Example of Signaling Games, Guessing Game<\/p>\n\n\n\n Given receiver’s strategy \\(\\sigma_2\\), the type-t sender’s expected payoff from messages m is<\/p>\n\n\n\n \\(\\sum_{r \\in R}u_1(m, r|t)\\sigma_2^m(r)\\)<\/p>\n\n\n\n – \uc790\uc2e0\uc774 message m\uc744 \ubcf4\ub0c8\uc744 \ub54c receiver\ub294 sender\uac00 m\uc744 \ubcf4\ub0c8\ub358 \uac83\uc744 \uc54c\uace0 \uc788\uc74c If m is sent, the receiver’s expected payoff from response r given beliefs \\(\\mu_2^m\\) is<\/p>\n\n\n\n \\(\\sum_{t \\in T}u_2(m, r|t)\\mu_2^m(t)\\)<\/p>\n\n\n\n – message m\uc774 \ubcf4\ub0b4\uc9c0\uace0 receiver\uac00 response r\uc744 \uc120\ud0dd\ud588\uc744 \ub54c \ub530\ub974\ub294 expected payoff\ub294 belief\uc5d0 \ub530\ub984 EXAMPLE 5.6<\/mark>.<\/p>\n\n\n Suppose \\(\\sigma_2^R\\) = xC + (1-x)F. Then the type-H sender’s expected payoff from sending message R is<\/p>\n\n\n\n \\(u_1\\)(R, C|H)x + \\(u_1\\)(R, F|H)(1-x) = 1+x If R is sent, the receiver’s expected payoff from response C given beliefs \\(\\mu_2^R\\) = p is<\/p>\n\n\n\n \\(\\mu_2\\) = (p, 1-p) -2p + 2(1-p) = 2-4p<\/mark><\/p>\n\n\n\n Assessment \u27e8\\((\\sigma_1, \\sigma_2), \\mu_2\\)\u27e9 is a perfect Bayesian eqbm of a signaling game if<\/p>\n\n\n\n – signaling game\uc5d0\uc11c PBE\ub97c \ub2e4\uc2dc \uc815\uc758 (i) the receiver’s beliefs \\(\\mu_2\\) are Bayesian given (\\(\\sigma_1, \\sigma_2\\)) – Bayesian belief: (\uac01\uac01\uc758 \ub178\ub4dc\uc5d0 \ub3c4\ucc29\ud560 \ud655\ub960)\/(\uc804\uccb4 information set\uc5d0 \ub3c4\ucc29\ud560 \ud655\ub960), \ub3c4\ub2ec\ud558\uba74 Bayesian rule\uc5d0 \ub300\ud574 \uacc4\uc0b0, \ub3c4\ub2ec\ud558\uc9c0 \uc54a\ub294 information set\uc5d0 \ub300\ud574\uc11c\ub294 unrestricted\ub418\uc5b4 \uc544\ubb34\ub7f0 belief\ub098 \uac00\uc9c8 \uc218 \uc788\uc74c<\/mark><\/p>\n\n\n\n Specifically, the second condition (ii) requires that<\/p>\n\n\n\n \\(\\sum_{r \\in R}u_1(\\sigma_1^t, r|t)\\sigma_2^m(r)\\) \u2265 \\(\\sum_{r \\in R}u_1(\\sigma_1^{‘}, r|t)\\sigma_2^m(r)\\) for all \\(\\sigma_1^{‘}\\)<\/mark><\/strong><\/p>\n\n\n\n – sender\uc758 \uc785\uc7a5\uc5d0\uc11c, sender\uc758 eqbm strategy sigma 1\uc740 \ubcf8\uc778\uc774 \uc5b4\ub5a4 type\uc774\ub4e0 \ubaa8\ub4e0 information set\uc5d0\uc11c optimal strategy\uc774\uae30 \ub54c\ubb38\uc5d0, sender\ub294 information set\uc740 \uc790\uc2e0\uc758 type\uc5d0 \ub530\ub77c \ub098\ub268 \\(\\sum_{t \\in T}u_2(m, \\sigma_2|t)\\mu_2^m(t)\\) \u2265 \\(\\sum_{t \\in T}u_2(m, \\sigma_2^{‘}|t)\\mu_2^m(t)\\) for all \\(\\sigma_2^{‘}\\)<\/mark><\/strong><\/p>\n\n\n\n – \uac01 message m\uac00 \uc804\ub2ec\ub418\ub4e0 \ub418\uc9c0 \uc54a\ub4e0 \uc0c1\uad00\uc5c6\uc774 sigma m\uc740 receiver\uc758 expected payoff\ub97c maximize\ud574\uc918\uc57c \ud568 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.<\/p>\n\n\n\n Nevertheless, PBE calls for an optimal behavior at {x, y}.<\/p>\n\n\n 2 is allowed to have any beliefs at {x, y}<\/p>\n\n\n\n \\(\\mu_2^R\\) = (p, 1-p)<\/p>\n\n\n\n 2’s expected payoff from response C given beliefs \\(\\mu_2^R\\) is and from response F is -1<\/p>\n\n\n\n 2 – 4p \u2265 -1 \u21d4 p \u2264 3\/4<\/mark><\/p>\n\n\n\n 2’s BR = \\(\\begin {cases} C & \\text{if } p \\leq \\frac{3}{4} \\\\ F & \\text{if } p > \\frac{3}{4} \\end{cases}\\)<\/p>\n\n\n\n <\/p>\n\n\n\n <\/p>\n\n\n\n\n
Asymmetric Information<\/h3>\n\n\n\n
signaling and screening games.<\/pre>\n\n\n\n
\n
we call that player an informed<\/mark><\/strong> player and the other an uninformed<\/mark><\/strong> player;<\/li>\n\n\n\n
if the uninformed player moves first \u21d2 screening<\/p>\n\n\n\nSignaling Games<\/h3>\n\n\n\n
![\"\"](\"https:\/\/saraheee.com\/wp-content\/uploads\/2023\/07\/image-25-1024x859.png\")
<\/figure><\/div>\n\n\nBasic Elements in Signaling Games<\/h3>\n\n\n\n
Formula<\/td> Description<\/td><\/tr> N = {1, 2}
T = {\\(t_1, …, t_k\\)}
\\(\\pi\\)<\/td>the set of players; 1 = the sender, 2 = the receiver<\/mark><\/strong>
the sender’s type space
– sender’s private information space (w or b)<\/mark>
a common prior on the sender’s type
– common prior: nature\uc758 \uc6c0\uc9c1\uc784(\uc0ac\uc804 distribution)<\/mark><\/td><\/tr>M = {\\(m_1, …, m_s\\)}
\\(\\sigma_1^t(\\cdot)\\)<\/td>the sender’s set of possible messages
a (mixed) strategy of the sender with type t
– sender\ub294 \ud0c0\uc785 \ubcc4\ub85c \uc6c0\uc9c1\uc784(\uc804\ub7b5\uc740 type dependent)<\/mark><\/td><\/tr>R = {\\(r_1, …, r_b\\)}
\\(\\sigma_2^m(\\cdot)\\)
\\(\\mu_1^t(\\cdot)\\)<\/td>the receiver’s set of possible responses
– receiver\uc758 action space<\/mark>
a (mixed) strategy of the receiver who observed m
– receiver\uac00 \ubc1b\uc740 message m \uc5d0 \uc758\ud574 \uc804\ub7b5 \uc120\ud0dd<\/mark>
the receiver’s beliefs about t after observing m
– \ubc1b\uc740 message m\uc5d0 \ub300\ud574 receiver\uc758 update\ub41c belief<\/mark><\/td><\/tr>\\(u_1\\)(m, r|t)
\\(u_2\\)(m, r|t)<\/td>the payoff function of the sender with type t
– \uc790\uc2e0\uc774 \ubcf4\ub0b8 message, receiver\uc758 response, \uc790\uc2e0\uc758 type\uc5d0 \ub300\ud574 dependent<\/mark>
the payoff function of the receiver
– sender\uc758 message, \uc790\uc2e0\uc758 response, sender\uc758 type\uc5d0 \ub300\ud574 dependent<\/mark><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\nExpected Payoffs<\/h3>\n\n\n\n
– m\uc744 \ubc1b\uc558\uc744 \ub54c response r\uc744 \uc120\ud0dd\ud560 \ud655\ub960(sigma)\uc5d0 \ub300\ud55c sender\uc758 payoff
– \uc774 mixed strategy \ud558\uc5d0\uc11c response\uc758 \uc5ec\ub7ec \ud655\ub960\uc744 \uacf1\ud574\uc11c \ub354\ud574\uc900 \uac83\uc774 sender\uc758 expected payoff<\/mark><\/p>\n\n\n\n
– sender\uc758 type\uc774 \uc815\ud655\ud788 H type\uc778\uc9c0 L type\uc778\uc9c0 \ubaa8\ub974\uae30 \ub54c\ubb38
– m\uc740 \ubc1b\uc558\uc73c\ub2c8 fixed\ub418\uc5b4 \uc788\uc9c0\ub9cc receiver\ub294 t\uc5d0 \ub300\ud574 uncertain\ud558\uae30 \ub54c\ubb38\uc5d0 t\uc5d0 \ub300\ud55c distribution\uc778 \uc790\uc2e0\uc758 belief\ub97c \uacf1\ud558\uace0 type\uc5d0 \ub300\ud574 \uc804\ubd80 \ub354\ud574\uc900 \uac83\uc774 expected payoff<\/mark><\/p>\n\n\n\n![\"\"](\"https:\/\/saraheee.com\/wp-content\/uploads\/2023\/07\/image-26.png\")
<\/figure><\/div>\n\n\n
\\(u_1\\)(R, C|H) = 2
\\(u_1\\)(R, F|H) = 1<\/p>\n\n\n\n
p = \\(\\mu_2^R\\)(H)<\/mark><\/p>\n\n\n\nPBE in Signaling Games<\/h3>\n\n\n\n
– sender\uac00 \uc5b4\ub5bb\uac8c \uc6c0\uc9c1\uc774\uace0 \uc5b4\ub5a4 \uba54\uc2dc\uc9c0\ub97c \ubcf4\ub0bc\uc9c0, receiver\uac00 \uc5b4\ub5a4 response\ub97c \uc120\ud0dd\ud560\uc9c0, sender\uc758 belief\ub294 \uc0dd\uac01\ud560 \ud544\uc694 \uc5c6\uc74c(\uac01\uac01\uc758 information set\uc774 \ud558\ub098\uc758 decision node (1)\ub85c \uad6c\uc131\ub418\uc5b4 \uc788\uc74c), \ub300\uc2e0 receiver\uc758 belief\ub97c \ubb18\uc0ac\ud558\uc5ec 3\uac00\uc9c0 element\ub85c \uad6c\uc131<\/mark><\/p>\n\n\n\n
(ii) (\\(\\sigma_1, \\sigma_2\\)) is sequentially rational given \\(\\mu_2\\)<\/p>\n\n\n\n\n
– sigma 1\uc740 \uc790\uc2e0\uc758 type\uc774 t\uc77c \ub54c, sender\uc758 expected payoff\ub97c maximize\ud574\uc57c \ud568, receiver\uac00 sigma 2\ub77c\ub294 \uc804\ub7b5\uc744 \uc120\ud0dd\ud588\uc744 \ub54c
– receiver\uac00 sigma 2 \uc804\ub7b5\uc744 \uc120\ud0dd\ud588\uc744 \ub54c, \uc790\uc2e0\uc774 type t\uc77c \ub54c \uc120\ud0dd\ud558\ub294 \uc804\ub7b5\uc774 \ub2e4\ub978 \uc5b4\ub5a4 \uac83\uc744 \uc120\ud0dd\ud558\ub294 \uac83(sigma ‘)\ubcf4\ub2e4 \ud56d\uc0c1 \ub354 \ub192\uc740 payoff\ub97c \uc918\uc57c \ud568
– sender\uac00 type\ubcc4\ub85c \uc120\ud0dd\ud558\ub294 \uc804\ub7b5\uc774 optimal strategy\uc774\uc5b4\uc57c \ud55c\ub2e4\ub294 \uac83<\/mark><\/p>\n\n\n\n\n
– \ub9cc\uc57d \uc790\uc2e0\uc774 m\uc774\ub77c\ub294 \uba54\uc2dc\uc9c0\ub97c \ubc1b\uc73c\uba74 belief\uac00 \uc5c5\ub370\uc774\ud2b8\ub418\uace0, expected payoff\ub294 \ub2e4\ub978 \uc804\ub7b5\uc5d0 \ube44\ud574 eqbm strategy\ub294 \ub354 \ub192\uc740 payoff\ub97c \uc548\uaca8\ub2e4 \uc90c
– receiver\ub294 \ubcf8\uc778\uc774 \uc5b4\ub5a4 \uba54\uc2dc\uc9c0\ub97c \ubc1b\ub4e0, \uc2e4\uc81c \uba54\uc2dc\uc9c0\uac00 \ub3c4\ub2ec\ud588\ub4e0 optimal strategy\ub97c \uc120\ud0dd\ud574\uc57c \ud55c\ub2e4\ub294 \uac83\uc774 sequential rational\ud558\ub2e4\ub294 \uac83<\/mark><\/p>\n\n\n\n![\"\"](\"https:\/\/saraheee.com\/wp-content\/uploads\/2023\/07\/image-29.png\")
<\/figure><\/div>\n\n\n
2-4p<\/p>\n\n\n\n