Ou ast equiibium concet The ast equiibium concet we study afte Nash eqm, Subgame Pefect Nash eqm, and Bayesian Nash eqm is Pefect Bayesian Equiibium. Pefect efes to the fact that the game wi be dynamic, ike the kind we soved using Subgame Pefect Nash Equiibium Bayesian efes to the fact that the game wi incude incomete infomation, so that ayes ae unsue of othe ayes ivate infomation o ast decisions. Thee ae actuay othe imotant soution concets Sequentia Equiibium, Makov Pefect Equiibium that won t be coveed in the couse, but ae commony used. PBE was deveoed in the 1980 s, and is a etty cutting edge idea, given that Nash Equiibium was discoveed in 1950 by Nash, Subgame Pefect Nash Equiibium was discoveed in 1965 by Reinhad Seten, and Bayesian Nash Equiibium was discoveed in 1967 by John Hasanyi. Infomation Sets in the Extensive Fom Definition 1. An infomation set is a set of decision nodes, a beonging to the same aye, ove which that aye cannot distinguish. onside this vesion of the Batte of the Sexes game: The woman W decides whethe to Go Out on a date o Stay in. If W goes out, she goes to a footba game F o the baet B. If W goes out, she eaves a message fo M saying that she s going to meet him, but it doesn t say whee, and he hone goes staight to voice mai. The man M can go to F o B. How can we eesent this game in an extensive fom? Exame The infomation set is eesented by the dashed ine. 1
Subgames Definition 2. A subgame is a of the nodes, banches, and ayoffs that foow afte a singeton infomation set. So in this game: thee ae TWO subgames, not thee. We cannot cut infomation sets u. Beiefs Definition 3. A set of beiefs ae a obabiity distibution ove a the nodes in a given infomation set. Note that beiefs is a technica tem, ike the wod heat o enegy in hysics. We don t mean that beiefs ae feeings (feeings ae not ea in economics) o anything ike that. The wod just suggests the ight idea about the obabiity distibution being a subjective assessment of ikeihoods. Beiefs Fo the exame, we have µ and 1 µ as the beiefs: So µ is (the obabiity that W chose F, given that she went out). 2
Pefect Bayesian Equiibium Definition4. Aefect Bayesian equiibium isasetofstategiesσ = (σ 1,...,σ N ) and beiefs fo evey aye at evey infomation set, so that Bayesian Beiefs: The beiefs ae deived fom the stategies and common io beiefs using Bayes ue, wheeve ossibe! Sequentia Rationaity: The stategies σ ae otima at evey oint in the game, given the ayes beiefs. Note that we use the stategies to deive the beiefs, but the beiefs must be consistent with the stategies. Pefect Bayesian Equiibium Then fo a Pefect Bayesian Equiibium is a set of stategies fo M and W, and beiefs µ fo M, so that no aye has a ofitabe deviation. Equiibium 1 Let s think about this stategy ofie: W goes out. W goes to B. M goes to B. µ = 0 Is this a efect Bayesian equiibium if x = 0? x = 1/2? x = 3/2? 3
Equiibium 2 Let s think about this stategy ofie: W goes out. W goes to F. M goes to F. µ = 1 Is this a efect Bayesian equiibium if x = 0? x = 1/2? x = 3/2? Equiibium 3 Let s think about this stategy ofie: W goes out. W goes to F with obabiity 1/3, and B with obabiity 2/3. M goes to F with obabiity 2/3, and B with obabiity 1/3. µ = 1/3 Is this a efect Bayesian equiibium if x = 0? x = 1/2? x = 3/2? Exame Let s find some PBE s fom scatch: Signaing Games The most inteesting cass of games that ae soved used the efect Bayesian Equiibium concet ae signaing games: (i) Natue assigns a ivatey known tye to the Sende (ii) The sende chooses a message to send to the Receive (iii) The eceive takes an action that detemines both thei ayoffs 4
Signaing Games The fun at of signaing games is that the message often has nothing to do with the infomation the sende has e se, but the signas simy vay in cost: Educationa signaing: (i) a student finds out whethe he ow o high oductivity, (ii) the student chooses to get a high schoo o coege degee, and (iii) a fim decides what wage to offe. Lawsuits: (i) a cient finds out whethe he has a genuine o suious awsuit against a fim, (ii) the cient chooses to get an exensive o chea awye, and (iii) the fim decides whethe to sette o go to cout. Advetising and Intoductoy Offes: (i) a fim eans whethe its new oduct is good o bad, (ii) the fim chooses to advetise/discount the oduct o not, and (iii) consumes decide whethe to ty it out o not. Limit icing: (i) a fim finds out whethe he is ow- o high-cost, (ii) the fim chooses whethe to chage a ow o a high ice, and (iii) a otentia entant decides whethe to ente the maket o not. Seaating, Pooing, and Hybid Equiibia Thee ae two kinds of efect Bayesian equiibia we might find in signaing games: Seaating equiibia: A sende tyes choose diffeent messages to send to the eceive. Pooing equiibia: Mutie sende tyes send the same message to the eceive. Hybid equiibia: Some tyes mix ove the signa they send, whie othes use ue stategies. (We won t cove these) Seaating equiibia ae we-behaved, whie ooing equiibia can be moe comicated to anayze. Two Exames: Timing Natue gives the sende tye S 1 with obabiity and tye S 2 with obabiity 1 The sende sends message u o message d Knowing the message but not the sende s tue tye the eceive chooses action o action 5
Exame 1 Hee s a signaing game that has two seaating equiibia: Eqm 1: Ou fist seaating equiibium has the stategies: Tye S 1 chooses u, tye S 2 chooses d R ays if he sees u, and if he sees d O 5,2 1,0 0,0 5,2 [a] u S 1 d [b] R N R [1 a] u 1 S 2 [1 b] d 2,3 1,2 1,0 3,3 whee the eceive s beiefs ae given by: a = [S 1 u] 1 a = [S 2 u] b = [S 1 d] 6
1 b = [S 2 d] Now, given these stategies, we deive the beiefs using Bayes ue: a = [S 1 u] = [S 1 u] [u] = = 1 b = [S 1 d] = [S 1 d] = 0 [d] 1 = 0 These beiefs ae Bayesian, so an equiibium with a = 1 and b = 0 wi satisfy that at of the citeia. Let s check sequentia ationaity: Tye S 1 has no ofitabe deviation, because if he sends the message d athe than u, the eceive then beieves he is facing an S 2 tye and ays, and S 1 then gets a ayoff of 1 athe than 5. Tye S 2 has no ofitabe deviation, because if he sends the message u athe than d, the eceive then beieves he is facing an S 1 tye and ays, and S 2 then gets a ayoff of 0 athe than 3. At the bottom infomation set, given R s beiefs, R get a ayoff of 3 fom and 0 fom, so the eceive has no ofitabe deviation at the bottom infomation set. At the to infomation set, given R s beiefs, R gets a ayoff of 2 fom and 0 fom, so the eceive has no ofitabe deviation at the to infomation set. That coves a the cases. Then S 1 ays u, S 2 ays d, R ays if d is ayed and R ays if u is ayed, a = 1, b = 0 is a efect Bayesian equiibium of the game. It haens to be a seaating equiibium. See if you can figue out the othe seaating equiibium. Ask me if you have questions. Exame 2 Hee s a signaing game that otentiay has two ooing equiibia: 7
onside the stategies: Both S 1 and S 2 send message d; R chooses if d is ayed, and if u is ayed. 3,0 0,0 2,1 [a] u S 1 d [b] R N R [1 a] u 1 S 2 [1 b] d 0,1 2,0 1,0 3,1 Then the eceive s beiefs shoud be b = [S 1 d] = [S 1 d] [d] = a = [S 1 u] = [S 1 u] [u] +(1 ) = = 0 0 =? So the wheeve ossibe! scenaio has finay shown u to the aty. Asking, What s the obabiity that S 1 sends u? hee is ike asking, What s the obabiity a good job maket candidate doesn t wea a suit to an inteview? It neve haens, but we can sti ose the question. So we wi sove fo the ange of beiefs that make sense of the situation, since Bayes ue is no he. Let s check sequentia ationaity (and sove fo a): The S 1 tye has no incentive to deviate, since if he sends u instead of d, the eceive ays, and the S 1 tye gets a ayoff of 1 athe than 2, which is sticty wose. So S 1 has no ofitabe deviation. The S 2 tye has no incentive to deviate, since if he sends u instead of d, the eceive ays, and the S 2 tye gets a ayoff of 0 athe than 3, which is sticty wose. So S 2 has no ofitabe deviation. At the bottom infomation set, fo this to be an equiibium, the exected ayoff to must be geate than the exected ayoff to, o (0)+(1 )1 (1)+(1 )(0) imying that 1/2. At the to infomation set, fo this to be an equiibium, the exected ayoff to must be geate than the exected ayoff to, o a(1)+(1 a)(0) a(0)+(1 a)(1) imying that a 1/2. 8
Then the stategies S 1 and S 2 send message d; the eceive esonds to d with and to u with ; and the eceive s beiefs ae b = 0 and a 1/2; ae a efect Bayesian equiibium if 1/2. (sime, ight?) See ifyou canfindanotheooingbe in this game. Askifyou have questions. Exame: Educationa Signaing Natue assigns a oductivity tye High o Low to the woke. The woke chooses to go to High Schoo ony,, o High Schoo and oege,. The fim obseves the woke s education, but not the woke s tye, and offes eithe a high wage,, o a ow wage,. Educationa Signaing: Extensive Fom Educationa Signaing: Seaating Equiibia (?) Is thee a seaating equiibium of the foowing tye: High tyes send, ow tyes send. The fim ays the high wage to aicants with degee, and the ow wage to aicants with degee. In equiibium, the fim beieves ony high tyes go to coege, and ony ow tyes go to high schoo. Let s ut beiefs in the extensive fom: 9
2,2 2,0 [a] High [b] N [1 a] 1 Low [1 b] 3,2 2, 1 0,1 Then the beiefs ae a = [High ] 1 a = [Low ] b = [High ] 1 b = [Low ] OK... what do we need to do? We, we have the oosed stategies: High tyes go to coege, ow tyes go to high schoo; the fim ays the high wage to coege gads, the ow wage to high schoo gads. If these stategies ae actuay adoted in equiibium, we have 2,2 2,0 [a] High [b] N [1 a] 1 Low [1 b] 3,2 2, 1 0,1 Given these stategies, we can amost ead the beiefs off the above extensive fom: a = [High ] = [High ] [] 1 a = [Low ] = [Low ] [] = 0 1 = 0 = 1 1 = 1 10
b = [High ] = [High ] = [High ] = [] = 1 1 b = [Low ] = [Low ] [] = 0 = 0 Nice. OK, so we have stategies and beiefs. an any tye ofitaby deviate, given the stategies and beiefs? The fim gets an exected ayoffof3byaying to coegegads; switching to gives a ayoff of 1, which is sticty owe. The fim gets an exected ayoff of 1 by aying to high schoo gads; switching to gives a ayoff of 0, which is sticty owe. So given the fim s beiefs, its stategy is otima. The high tye gets a ayoff of 3 fom going to coege. By switching to high schoo, it gets a ayoff of 1 (since the fim infes that it must be a ow tye, and ays ). This is an unofitabe deviation, so the high tyes don t want to deviate. The ow tye gets a ayoff of 1 fom going to high schoo. By switching to coege, it gets a ayoff of 2 (since the fim infes that it must be a high tye, and ays ). This is a ofitabe deviation, so the ow tyes wi neve go aong with the oosed equiibium. Theefoe, this game does not have a seaating equiibium in which the degee is fuy infomative. Educationa Signaing: Pooing Equiibia Is thee a seaating equiibium of the foowing tye: High tyes send, ow tyes send. The fim ays the high wage to eveyone. In equiibium, the fim has beiefs that ae coect. Again, et s ut beiefs in the extensive fom: 2,2 2,0 [a] High [b] N [1 a] 1 Low [1 b] 3,2 2, 1 0,1 11
Then the beiefs ae a = [High ] 1 a = [Low ] b = [High ] 1 b = [Low ] Now we have the oosed stategies: High tyes go to coege, ow tyes go to coege; the fim ays the high wage to coege gads, the ow wage to high schoo gads. If these stategies ae actuay adoted in equiibium, we have 2,2 2,0 [a] High [b] N [1 a] 1 Low [1 b] 3,2 2, 1 0,1 Given these stategies, we use Bayes ue to get beiefs: a = [High ] = [High ] [] 1 a = [Low ] = [Low ] [] b = [High ] = [High ] = [High ] [] 1 b = [Low ] = [Low ] [] = 0 0 =? = 0 0 =? = +1 = = 1 +1 = 1 But yikes, 0/0? Bayes ue doesn t wok on events that don t haen in equiibium: No one sends the message, so the obabiity of eceiving the message is zeo, and that asect of beiefs is not we-defined in equiibium. Educationa Signaing: Beiefs on the equiibium ath To check the fim s beiefs, we need to comute [High ] = [High ] [] = (1) (1)+(1 )1 = which uses Bayes ue, and uses the easoning fom the ectue on datboads and boken machine ats. 12
Educationa Signaing: Beiefs off the equiibium ath Now, we need to comute the fim s beiefs fo evey infomation set, incuding the one that is neve eached in equiibium. In aticua, [High ] = [High ] [] = 0 0 =...? Thisiswhythe BayesianBeiefs atofthe definition ofapbesaid wheeve ossibe! Educationa Signaing: Beiefs off the equiibium ath Off the equiibium ath, we sove fo the entie set of beiefs that ae consistent with the equiibium stategies. In aticua, we just need the fim to beieve that offeing in esonse to is bette than offeing : [High ]1 + [Low ]1 [High ]2 + [Low ]0 o 1 2 [High ] So facing the game 2,2 2,0 [a =?] High [] N [1 a =?] 1 Low [1 ] 3,2 2, 1 0,1 We ask(i) onthe equiibiumath, does anytye have a ofitabedeviation, and (ii) off the equiibium ath, what set of beiefs make the oosed equiibium satisfy sequentia ationaity? So, since Bayes ue doesn t give us we-defined beiefs, we then ask, we, ae thee any beiefs that wok with the oosed stategies? What do they ook ike? Ae they easonabe o not? The fim gets an exected ayoff of 2 + (1 )1 by aying to coege gads, and (1)+(1 )(1) = 1 by aying to coege gads. As ong as aying is bette than aying, the fim has no eason to deviate fom the oosed equiibium, so we need 2+(1 )1 1 13
but this is equivaent to 1+ 1 So the fim wi neve want to deviate fom aying to coege gads. Fo the fim to want to ay the ow wage to high schoo gads athe than the high wage, we need the exected ayoff of to be geate than the exected ayoff of, o o a(1)+(1 a)(1) a(2)+(1 a)0 [High ] = a 1 2 Now, do the sende tyes have any incentive to deviate? If the high tye sends instead of, it gets a ayoff of 1 instead of 2; the high tye has no incentive to deviate. If the ow tye sends instead of, it gets a ayoff of 1 instead of 2; the ow tye has no incentive to deviate. Theefoe, if [High ] 1, thee is a (ooing) efect Bayesianequiibium 2 whee the highand owtyes both gotocoege, the fim awaysays the highwage, and [High ] =. Educationa Signaing: Pooing Equiibia The foow stategies and beiefs ae a efect Bayesian equiibium of the signaing game: A sende tyes choose. The eceive ays fo and fo. The eceive s beiefs ae [High ] =, [Low ] = 1, and 1 2 [High ] 14