Perfect Bayesian Equilibrium in Extensive- Form Games Ù

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1 Pefect Bayesian Equiibium in Extensive-Fom Games Page Pefect Bayesian Equiibium in Extensive- Fom Games Ù Intoduction Bayes Requiement Bayes Requiement Bayes Requiement 5 Bayes Requiement 4 6 A Refinement 7 Intoduction We have seen that Nash equiibia of extensive-fom games can be undesiabe because they can ey on incedibe theats at off-the-equiibium-path subgames. We wee sometimes abe to efine away such undesiabe equiibia by stengthening ou soution concept demanding subgame pefection, which equies that the estiction of a stategy pofie to any subgame be a Nash equiibium of that subgame. Subgame pefection wi not eiminate a undesiabe equiibia of extensive-fom games, howeve. Conside the extensive-fom game of Figue. Anaysis of its stategic fom quicky shows that this game has two pue-stategy Nash equiibia: (U,) and (A,). This game has ony one subgame, viz. the entie game, so both of these Nash equiibia ae aso subgame pefect. U D A [p] [_p] A U D,,,,,, Figue : Subgame pefection admits undesiabe equiibia. The (A,) equiibium is objectionabe fo the foowing eason. [Note that paye s infomation set is off-the-equiibium path with espect to the (A,) equiibium i.e. it is neve eached when the payes confom to the equiibium specification.] If paye s infomation set wee eve eached, paye woud be uncetain about whethe it was eached via paye having chosen U o via paye having Ù Û 99 by Jim Ratiff, <jim@vituapefection.com>, <

2 Pefect Bayesian Equiibium in Extensive-Fom Games Page chosen D. Howeve, it doesn t matte to paye s decision which move paye had chosen. No matte what paye s beiefs about paye s non-a choice, paye sticty pefes to choose at his infomation set when it is eached. (If paye had chosen U, paye eceives fom and ony fom. If paye had chosen D, paye eceives fom and ony fom.) Because is not a best esponse at paye s infomation set fo any possibe beiefs which paye might have thee, we say that is dominated at paye s infomation set. We can descibe ou dissatisfaction with the (A,) equiibium by objecting to its specification of an action at an infomation set which is dominated at that infomation set. Let s fomaize this easoning. We begin by equiing that at evey one of he infomation sets each paye has some beiefs about the node at which she is ocated conditiona on having eached that infomation set. Fo a paticua stategy pofie ß, we equie that, fo each paye i I, Bayes Requiement and at each of he infomation sets h i H i, paye i has beiefs i ªh i º ÙǪh i º about the node at which she is ocated conditiona upon being infomed that pay has eached the infomation set h i. The beiefs i ªh i º Ǫh i º ae just a pobabiity distibution ove the nodes in the infomation set. Paye i s beiefs in this game, then, ae a specification, fo each paye-i infomation set h i H i, of such conditiona beiefs at that infomation set. The n-tupe =(,, n ) of paye beiefs is a beief pofie. In ode to popey citique aeged equiibia we equie that a candidate equiibium be not just a stategy pofie ß but be a stategy-beief pofie (ß, ). We want to state an equiibium equiement that woud oosey say something ike: Fo evey paye i I and evey paye-i infomation set h i H i, paye i s stategy is a best esponse given he beiefs i ªh i º Ǫh i º at the infomation set h i. Howeve, this is too vague at east to me! so we must be moe pecise in ou statement. Reca that a subgame is fomed by identifying a singeton infomation set and incuding a its successos fom the oigina game. Infomation sets, actions, and payoffs fo the subgame wee deived fom the oigina game by estiction. We now geneaize the concept of a subgame and define a continuation game. A continuation game is an infomation set h i H i fo some paye ifiôªh i º and a of its successo nodes fom the oigina game. Again, infomation sets, actions, and payoffs in the continuation game ae deived fom the oigina game by estiction. If the designated initia infomation set is not a singeton, then this continuation is not a subgame. And with good eason: This continuation game cannot be payed as a game in its own ight, because thee is no initia node. So we incude in the specification of the continuation game the pobabiity distibution i ªh i º ove the nodes of the initia infomation set h i given in the beief pofie. (Think of this continuation game as being peceded by a move of Natue s, whee Natue chooses between the nodes of h i accoding to the pobabiity distibution i ªh i º.) We can estict any stategy ß j and any paye beiefs j to this continuation game We e assuming pefect eca.

3 Pefect Bayesian Equiibium in Extensive-Fom Games Page just as we esticted a stategy to a subgame: simpy thow out its specifications at infomation sets which don t beong to the smae game. Conside the continuation game defined by some paye-i infomation Bayes Requiement set h i H i and the conditiona beiefs i ªh i º. The estiction of the stategy-beief pofie (ß, ) to this continuation game must be a Nash equiibium of the continuation game. Definition Let (ß, ) be a stategy-beief pofie and et h i H i be an infomation set fo paye i=ôªh i º. Let (ß, ) be the estiction of (ß, ) to the continuation game which begins at the infomation set h i. We say that the paye-i stategy ß i is sticty dominated beginning at the infomation set h i if thee exists anothe paye-i stategy ß i such that, fo a othe deeted stategy pofies ß i fo the opponents, paye i s expected payoff in the continuation game is sticty highe fo (ß i,ß i ) than fo ß. Bayes Requiements and ae sufficient to emove the undesiabe equiibium, viz. (A,), in Figue. To see this we constuct the continuation game beginning at paye s infomation set fo some beiefs paameteized by p [,]. See Figue. The stategic fom of this continuation game is aso shown in Figue, fom which it is cea that is the unique Nash equiibium in the continuation game. Figue : The continuation game beginning at paye s infomation set. Moe geneay, Bayes Requiement ejects a stategy pofies which specify at any infomation set an action which is dominated at that infomation set. Exampe: Resticting a stategy-beief pofie to a continuation game Conside the stategy-beief pofie s=(u,a,d;;p) fo some p [,] in the extensive-fom game in Figue a. Now conside the continuation game beginning at paye s infomation set. Figue b depicts the estiction s of this stategy-beief pofie to the continuation game. The expected payoff vecto to the estiction s is p(,)+(_p)(,4)=(_p,4_p). Let s evauate whethe the stategy pofie s passes Bayes Requiement with espect to the continuation game beginning at paye s infomation set. We can constuct a stategic fom fo the continuation game. Fo exampe, the expected payoff vecto to the esticted stategy pofie (a,c;) is p(,)+(_p)(,)=(p,p+). The expected payoff vecto to (*,*;) is p(,)+(_p)(,6) =

4 Pefect Bayesian Equiibium in Extensive-Fom Games Page 4 (,6_4p). Simia cacuations fo paye s othe stategies yied the payoff matix in Figue. In ode that s passes Bayesian Requiement with espect to the specified continuation game, s=(a,d;;p) must be a Bayesian Nash equiibium of the continuation game. This equies that (a,d) be a best esponse by paye to, i.e. _p maxù{p,,_p}, which is satisfied fo a p [,]. In ode that be a best esponse by paye to (a,d) we must have 4_p 6_4p, which is satisfied if and ony if p [,]. Theefoe s is a Bayesian Nash equiibium of this continuation game if and ony if p [,]. Figue : The estiction of the (U,a,d;;p) of (a) to the continuation game (b). Figue : The stategic-fom matix coesponding to the continuation game of Figue b. We can aso ask whethe is sticty dominated beginning at paye s infomation set. This woud equie that o p<. 6_4p>maxÙ{+p,4_p,,4_p}=4_p,

5 Pefect Bayesian Equiibium in Extensive-Fom Games Page 5 Figue 4: Requiements and don t even guaantee Nash equiibium. Bayes Requiements and ae not stong enough howeve to geneay captue even the concept of Nash equiibium. Conside the game in Figue 4. The stategy-beief pofie (U,;p=) satisfies Bayes Requiements and, yet (U,) is not even a Nash equiibium of the game. So we add a thid equiement. It ensues that in equiibium each paye s beiefs ae coect. The beiefs at any on-the-path infomation set must be detemined fom Bayes Requiement the stategy pofie accoding to Bayes Rue. I.e. if h i H i is a paye-i infomation set eached with positive pobabiity when the payes confom to ß, then i ªh i º Ǫh i º must be computed fom ß using Bayes Rue. This equiement eiminates the non Nash-equiibium pofie (U,; p=) fom the game of Figue 4 because, given that paye is choosing U, paye s beief at his infomation set must put a weight on the node eached by U i.e. we must have p= athe than p=. Howeve, the addition of Bayes Requiement does not guaantee that suviving stategy-beief pofies ae even subgame-pefect equiibia of the game. Conside the game of Figue 5. Conside the subgame beginning with paye s singeton infomation set. This subgame has a unique Nash equiibium of (U,). Theefoe the unique subgame-pefect equiibium of the entie game is (B,U,). Evey infomation set is on the path, and Bayes Rue impies p=. This stategy-beief pofie (B,U,;p=) satisfies Bayes Requiements,, and.

6 Pefect Bayesian Equiibium in Extensive-Fom Games Page 6 A B U D [p] [_p] Figue 5: Requiements,, and do not guaantee subgame pefection. But now conside the stategy-beief pofie (A,U,;p=). This is a Nash equiibium and satisfies Bayes Requiements,, and. (Note that paye s infomation is off-the-path, and theefoe Requiement puts no estiction on p.) Yet this pofie is not subgame pefect, because we have aeady seen that subgame pefection equies (U,) by payes and. The pobem with this pofie can be taced to the beiefs at paye s infomation set. The ony way that paye s infomation set coud be eached is if paye chose B. In this case, accoding the stategy pofie, paye woud choose U. Theefoe paye, conditiona on eaching he infomation set, shoud infe that she is ocated at he eft-hand node and theefoe shoud beieve that p= athe than p=. We now add an additiona equiement, which eiminates the non subgame-pefect equiibium just consideed. Bayes Requiement 4 The beiefs at any off-the-path infomation set must be detemined fom the stategy pofie accoding to Bayes Rue wheneve possibe. Bayes Requiements,,, and 4 taken togethe constitute the definition of pefect Bayesian equiibium in extensive-fom games. (Fo the sende-eceive games we studied eaie, Bayes Requiement 4 has no efining powe. Theefoe Bayes Requiements,, and constitute the definition of pefect Bayesian equiibium in sende-eceive games.) As we saw with sende-eceive games, pefect Bayesian equiibium can admit equiibia which ae objectionabe because they ey on off-the-path beiefs which ae in some sense suspect. Conside the game in Figue 6. Thee ae two casses of pue-stategy pefect Bayesian equiibia: (U,;p=) and {(A,;p):p [, ]}.

7 Pefect Bayesian Equiibium in Extensive-Fom Games Page 7 Figue 6: Pefect Bayesian equiibium can be usefuy efined. In cass of these equiibia, paye s infomation set is off-the-path. Paye s beiefs at this infomation set ae that, given that this infomation set was eached, thee is a positive pobabiity that the infomation set was eached via paye choosing D. But note fom the stategic fom that D is dominated fo paye by A. On the othe hand, U is not dominated. If paye obseves that, contay to his expectation, paye did not choose A, what shoud paye beieve about paye s actua choice? Did paye choose U o D? Paye coud neve pofit by paying D instead of the dominating stategy A. Howeve, paye coud conside paying U in hopes that paye woud pay, giving paye a payoff of instead of. Theefoe we shoud attach zeo weight to the event that a deviation by paye was D athe than U. Theefoe we shoud equie, when paye s infomation set is off-the-path, that p=. If possibe, each paye s beiefs off the equiibium path shoud put A Refinement zeo weight on nodes which can ony be eached if anothe paye pays a stategy that is sticty dominated beginning at some infomation set.