Common p-belief: The General Case

Transcription

1 GAMES AND ECONOMIC BEHAVIOR 8, ARTICLE NO. GA97053 Commo p-belef: The Geeral Case Atsush Kaj* ad Stephe Morrs Departmet of Ecoomcs, Uersty of Pesylaa Receved February, 995 We develop belef operators for formato systems where dvduals have a ucoutable umber of possble sgals, ad we gve a geeral verso of Moderer ad Samet s Ž 989. theorem relatg teratve ad fxed pot otos of commo p-belef. Joural of Ecoomc Lterature Classfcato Numbers: D Academc Press. INTRODUCTION A evet s commo p-belef f everyoe beleves t wth probablty at least p, everyoe beleves wth probablty at least p that everyoe beleves t wth probablty at least p, ad so o ad ftum. Moderer ad Samet Ž 989. hereafter, MS provde a characterzato of commo p-belef whch relates ths terate defto to the followg fxed-pot defto: A evet s sad to be p-evdet f wheever t s true, everyoe beleves t wth probablty at least p. A evet E s commo p-belef at state f ad oly f s a elemet of some p-evdet evet F wth the property that everyoe beleves E wth probablty at least p wheever F s true. Sce commo -belef s essetally equvalet to the usual oto of commo kowledge, ths result s a geeralzato of Auma s classc result gvg a fxed pot characterzato of commo kowledge. Commo p-belef has bee show to be a atural oto of almost commo kowledge : for p suffcetly close to oe, ecoomc outcomes *Facal support from the Ceter for Operatos Research ad Ecoometrcs Ž CORE. fellowshp s gratefully ackowledged. Curretly at the Isttute for Soco-Ecoomc Plag, Uversty of Tsukuba. Facal support from a E.E.C. Huma Captal ad Moblty Program sttutoal fellowshp for a vst to C.O.R.E. s gratefully ackowledged. Stchcombe Ž 988. depedetly troduced a alteratve oto of almost commo kowledge. Auma Ž $5.00 Copyrght 997 by Academc Press All rghts of reproducto ay form reserved.

2 74 KAJII AND MORRIS are smlar to outcomes uder commo kowledge. 3 Ufortuately the MS result requres each formato set of each dvdual to have postve probablty ad thus each dvdual to have at most a coutable umber of possble sgals. These assumptos remove the determacy of codtoal probablty at partcular states ad so make t possble to defe a belef operator whch specfes at whch states a gve evet s beleved wth probablty p. Ths smplfes both the techcal aalyss ad the terpretato, but excludes mportat applcatos. I a aalyss of commo -belef, Nelse Ž 984. detfed sets whch dffer oly by zero measure sets. I ths ote, we follow ths approach ad defe belef operators o classes of equvalet evets Žrather tha o evets drectly.. We prove a straghtforward aalogue of the MS result usg such belef operators. Ths approach has the attractve feature that all deftos ad results are depedet of ay partcular verso of codtoal probablty: there s a uque represetato of belef operators despte the multplcty of codtoal probabltes. However, we emphasze that a alteratve terpretato of our results s that we are cofrmg that t does ot matter Ž almost surely. f a partcular codtoal probablty s fxed. The ote s orgazed as follows. After troducg some measure theoretc propertes Secto, we the defe class based belef operators Ž subsecto 3.. ad use them to defe ad characterze commo p-belef Ž subsecto 3..; ths characterzato cludes a ucoutable sgals verso of Moderer ad Samet s theorem. I Secto 4, we dscuss the applcato of the approach to a smple example.. SETUP Throughout the paper we wll fx a set, a -feld F o, a probablty measure P o, F, a fte collecto of dvduals I, wth sub -felds represetg the formato of each dvdual, F 4 I.We wll detfy evets whch dffer oly by zero probablty evets. We frst summarze some measure theoretc facts, whch are versos of stadard results that ca be foud, for stace, Halmos Ž Although our termology s slghtly dfferet, we are followg Nelse s Ž 984. work o kowledge Ž -belef., who tur attrbutes the dea to Koopma Ž 940. ad Halmos Ž Say two measurable sets E, F are equalet Ž uder P. f PŽ EF. 0, where E F deotes the symmetrc dfferece of E ad F. We shall wrte 3 Ths has bee show for games of complete formato Ž MS., o dsagreemet results Ž MS., o trade results Ž Soso, 995., ad learg Ž Moderer ad Samet, 995..

3 COMMON p-belief 75 E F f E ad F are equvalet. It s easy to show that the relato s a equvalece relato o F. Let E deote the equvalet class of evets that cotas evet E. The E F f ad oly f E F. Deote the set of equvalet classes by F; that s, F F. We wrte E F f there are measurable sets E E, F F such that E F. It s readly verfed that s a reflexve ad trastve relato o F. Notce that the relato s ot the class cluso ad that t depeds o the measure P. Bary relatos o F aalogous to uo ad tersecto are defed as follows: EF EF: EE, FF 4; EF EF: EE, FF 4. It ca be readly verfed that E F ad E F are equvalece classes, cotag E F ad E F, respectvely. The relatve complemet s also defed aalogously: E F E F : E E, F F 4, whch s a equvalece class, cotag E F. The uo ad tersecto of coutably may classes are also well defed: ½ 5 E½E : EE 5. for E,,,..., E E : E E, Ž It s easy to show that E resp. E. s a equvalece class, Ž cotag E resp. E.. To sum up, the relatos,, ad work exactly as ther set theoretc couterparts as log as we are dealg wth coutable operatos o F ad the choce of represetatve elemet each class does ot matter. We say F F s essetally F-measurable f there s F F such that FF. By costructo, f F s essetally F-measurable, the the class F cossts of essetally F-measurable sets, ad we say F s F-measurable. It s easy to see that F-measurable classes are closed uder coutable operatos of,, ad. A real-valued F-measurable fucto f o s sad to be essetally F-measurable f f cocdes wth a F-measurable fucto almost surely. Let E F. A codtoal probablty fucto of evet E F wth respect to F s a essetally F-measurable fucto f such that H f dp H dp Ž PŽ EA.. for ay A F, where s the A A E E dcator fucto of E. Sce fuctos that are detcal almost every-

4 76 KAJII AND MORRIS where have detcal tegrals, a codtoal probablty fucto s essetally uque the sese that f s a codtoal probablty of E f ad oly f ay f that cocdes wth f almost everywhere s a codtoal probablty of E. Codtoal probablty s coutably addtve the followg sese: Fact. Let E, E,...,E,...F be a sequece of dsjot sets. Suppose for each, f s a codtoal probablty fucto of E. The the fucto Ý f s well defed ad t s a codtoal probablty fucto of E. DEFINITION. The posteror probablty of a class E wth respect to F Ždeoted by Ž E.. s the collecto of all codtoal probablty fuctos of all sets E E, wth respect to F. To summarze the above dscusso, we have: Fact 3. f Ž E. f ad oly f f s F-measurable ad H f A dp H A E dp for ay E E ad A F. Therefore, partcular, f Ž E. f ad oly f ay f that cocdes wth fp-almost surely belogs to Ž E.. 3. p-belief OPERATORS: A GENERAL APPROACH 3.. Iddual Belef Operators For ay fucto f: 0,, let F p Ž f. : f Ž. p 4. The from Fact 3, the followg ca be readly verfed: Fact 4. For ay f, f Ž E., F p Ž f. F p Ž f.. If A F satsfes AF p Ž f. for some f Ž E., the there exsts g Ž E. such that F p Ž g. A. The followg descrbes the class of evets where the dvdual p- beleves E. 4 B p E F p f : f E. The followg proposto summarzes basc propertes of the p-belef operator B p, whch are completely aalogous to the fte or the coutable case Ž as studed by MS., except that we are ow dealg wth classes of evets.

5 COMMON p-belief 77 PROPOSITION 5. The followg propertes hold: Ž B0. B p Ž E. s a class; Ž B. B p Ž E. s F-measurable; Ž B. f E s F-measurable, the E B p Ž E. ; Ž B3. f E F, the B p Ž E. B p Ž F. ; p p B4 f E s F-measurable, the B E F E B F ; Ž B5 f F s a decreasg sequece of classes.e., F F for all,,...., the B p Ž F. B p Ž F.. Proof. B0 By fact 4. Ž B. By costructo. Ž B. If E s F-measurable, the there exsts E E wth E F ; thus Ž E. ad the result follows. E Ž B3. Let E F ad f Ž E.. Choose E E ad F F such that EF. Choose a codtoal probabltes Ž wth respect to F. f of E ad g of F. The by defto, g f o E almost surely, thus B pž E. B pž F.. Ž B4. Choose ay E E, F F. Recall that f E s F-measurable, the E s a codtoal probablty of E. Let f be a codtoal probablty fucto of F. The g mž Ž., f. E s a codtoal probablty of E F. Ideed, for ay A, HA g HA E f HA E F H. Sce g p f ad oly f E ad f p, Ž B4. holds. A EF B5 We ca choose a decreasg sequece of measurable sets F F, ad set F F. F F by costructo. Let f, f be codtoal probablty fuctos of F, F, respectvely. Sce F s decreasg, applyg fact to E F F, we have that f coverges mootocally, almost surely to f. I partcular, : f p4 : f p 4; that s, B p Ž F. B p Ž F., as desred. 3.. Commo p-belef Operator We ow wat to cosder belef operators for may dvduals ad formulate the cocept of commo p-belef ths geeral framework. Defe the everyoe beleves operator by the rule B p Ž E. B p Ž E.. I

6 78 KAJII AND MORRIS LEMMA 6. B p ŽB p Ž E.. B p Ž E.. Proof. 4 ½ ½ 55 ž / B B Ž E. B BŽ E. p p p p I B p B p j E, so by B ad B4, I ji ½ 5 B p E B p B p j E, so by B3, I j 4 B p E I B p E. Defe commo p-belef operator C p by the rule Ž. C p E B p E B p B p E B p B p B p E. Because C pž E. s a coutable tersecto of classes, t must tself be a class. I ths framework, t o loger makes sese to ask f a evet or class of evets s commo p-belef at a partcular state, uless the sgleto set 4 occurs wth postve probablty. We rather defe whe a class E s commo p-belef at some other class F. DEFINITION 7. Class E s commo p-belef at F f F C pž E.. Ths terate defto wll be related to the followg fxed-pot characterzato. DEFINITION 8. Class E s p-evdet f E B p Ž E.. The followg result s thus a ucoutable verso of MS s result, whch s tur a geeralzato Ž from -belef to p-belef. of Auma s Ž 976. characterzato of commo kowledge. THEOREM 9. The class E s commo p-belef at A f ad oly f there s a p-edet class F such that A F B p Ž E.. We ca clarfy the theorem by restatg t two parts. Pck ay class E; the Ž. a for ay p-evdet class F such that F B p Ž E., we have F C p Ž E..

7 COMMON p-belief 79 p Coversely, b for ay class A such that A C E, there exsts a p-evdet class F such that A F B p Ž E.. Proof. Now the f statemet follows from Ž. a, ad the oly f part follows from Ž b.. Proof of Ž. a : Suppose a p-evdet class F satsfes F B p Ž E..ByŽ B3,. ad so p p B F B E, for 0,,,... p p C F Ž B. F p 0 p B E B Ž F. sce F s p-evdet, C p Ž E.. By the defto of p-evdet, we also have F B pž F. Ž B3., ad thus, by p p B F B F, for 0,,,..., ad F B p F, for,,... p p p So F B Ž F. C Ž F. C Ž E.. p p p Proof of b : Let F C E. By costructo of C, A F B E. So t s eough to show that F s p-evdet. Let E 0 E, ad defe k E B p Ž E k. teratvely. By costructo, F k E k. By Lemma 6, E k, k,,..., s a decreasg sequece of classes. Therefore, by Ž B5., B p Ž F. B p Ž E k.,so k 4 B p F B p I F, usg the relato above, p I k B E k, ad exchagg the order, 4 B Ž E. p k I k p k B E k. k O the other had, sce E s decreasg, p k B E k B p Ž E k. E k F. Therefore, we have F B p Ž F. k k, so par- tcular F s p-evdet as desred.

8 80 KAJII AND MORRIS 4. DISCUSSION 4.. A Fxed Codtoal Probablty Approach Nelse Ž 984. gave a characterzato of commo -belef for ucoutable state spaces usg the class based approach whch we pursue here. Bradeburger ad Dekel Ž 987. pursued the alteratve approach Žfor commo -belef. of fxg a partcular codtoal probablty, ad cotug to aalyze commo -belef about eets defed at states. We beleve that whle the former approach Ž whch we have pursued here. s mathematcally more elegat, t s ofte useful applcatos to use a atural codtoal probablty. Oe terpretato of the results preseted here s that as log as we terested probablty statemets, there s o loss of geeralty fxg a codtoal probablty. We therefore coclude by brefly summarzg how to do so. For each, fx a regular codtoal probablty P : F : that s, PŽ E. Ž E. for all ad E F, ad P s a probablty measure wth probablty oe such regular codtoal probabltes exst f s a separable metrc space ad F s the Borel feld. For these fxed P, defe evet belef operators B p Ž E. : PŽE. p 4, B p Ž E. I B p Ž E., ad C p Ž E. B p Ž E. B p ŽB p Ž E.. B p ŽB p ŽB p Ž E.... The by Ž B. of Proposto 5 ad costructo, t ca be readly verfed that for ay E F, E B p Ž E. holds f ad oly f B p Ž E. E, ad so E s p-evdet f ad oly f there s E E such that for all, B p Ž E. E s a ull set. So we say that a evet E s p-evdet f PŽE. p holds for almost every E. Also C p Ž E. s the class cotag C p Ž E.. Thus f we say that E s commo p-belef at f C p Ž E., we ca rephrase Theorem 9 as follows: COROLLARY 0. For ay eet E, the followg statemet s true for almost eery : Eet E s commo p-belef at f ad oly f there exsts a p-edet eet F such that F B p Ž E.. I the remader of ths ote, we wll formally preset a example whch llustrates the usefuless of usg fxed codtoal probabltes to characterze commo p-belef. 4.. The Nose Example The followg example was aalyzed by Morrs et al. Ž There are two dvduals ad. Idvdual observes a umber x 0,. ad dvdual observes a umber x.. 0,. So let 0, wth ts Borel sets ad F be the collecto of the sets of the form Ž x, x.: x x 4, for some x 0,.. By coveto, we wll detfy ay umber wth ts decmal part, so that f,,,. 0, ad

9 COMMON p-belief 8 m,ž.4. We cosder probablty measures dexed by. 0, : P s the uform measure o the set Ž x, x.: x x 4 ; 0 partcular, P s uform o the dagoal set D Ž x, x.: x x 4. Thus the margal dstrbuto of x s uform o the terval 0,.. Gve x, dvdual j observes a sgal x j whch s dstrbuted uformly o the terval x, x j j. Let F* be the collecto of evets whch exclude some ope terval o the dagoal,.e., there exsts x 0,. ad 0 F* E F:. ½ such that E Ž x, x. : Ž x, x. Ž x, x. 45 We wll show that for ay p, ay evet E F* s ever commo p-belef for all suffcetly small 0. However, f 0, such evets are p-evdet ad thus commo p-belef wheever they are true, for ay p 0. We wll show how the aalyss of ths paper ca be used to make these clams precse ad depedet of the choce of codtoal probablty. Cosder frst the case wth Ž 0,.. The codtoal desty s, f xjx, x fž xj x., 0, otherwse so, wrtg for the Lebesgue measure o 0,, the atural regular codtoal probablty s j Ž j. j Ž 4. P Ex x : x, x E ad x x. The assocated evet belef operators are deoted by B p Ž E.. Now suppose E F*, wth E Ž x, x.: Ž x, x. Ž x, x. 4. Cosder ay p ad Ž 0, 3. ad let j ; ow f x Ž x, x., dvdual assgs probablty to x beg the terval Ž x, x.. But ' j ' j Ž x, x. Ž x, x. Ž 5. f x Ž x, x. ad x Ž x, x.; so dvdual assgs probablty 0 to Ž x, x. j j beg a elemet of E f x Ž x, x. ad so 4 B p Ž E. Ž x, x.: x Ž x, x. 4 ½ 5 B p Ž E. Ž x, x.: x Ž x, x. for, p x xgž, p,., xgž, p,. B Ž E. Ž x, x. :, for,

10 8 KAJII AND MORRIS where gž, p,. Ž Ž.Ž p... Thus C p Ž E.. Thus for p ay E F*, C Ž E. for all p, for suffcetly small. Now cosder the case where 0, ad thus probablty s dstrbuted uformly o the dagoal. The atural codtoal probablty has dvdual observg x assgg probablty to dvdual observg x ; the correspodg evet belef operator s therefore 4 B p Ž E. Ž x, x. : Ž x, x. E, for all p Ž0,. Thus ay E s p-evdet ad C p Ž E. DE for all E F ad all p Ž0,. The above aalyss Ž lke that of Morrs et al., 993. fxed the codtoal probabltes. But the aalyss of ths ote shows that the choce of codtoal probablty does ot matter. So f 0, we have C pž E. E for all p Ž0, ad all classes E; but f s postve but suffcetly p small, C Ž E. s the ull class for all E F* ad all p Ž,. Ths extreme sestvty of commo p-belef to ose s mportat a umber of applcatos; see, for example, Carlsso ad va Damme Ž REFERENCES Auma, R. Ž Agreeg to Dsagree, A. Statst. 4, Bradeburger, A., ad Dekel, E. Ž Commo Kowledge wth Probablty, J. Math. Eco. 6, Carlsso, H., ad va Damme, E. Ž Global Games ad Equlbrum Selecto, Ecoometrca 6, Halmos, P. Ž The Foudato of Probablty, Amer. Math. Mothly 5, Halmos, P. Ž Measure Theory. Prceto, NJ: Va Nostrad. Koopma, B. Ž The Axoms ad Algebra of a Itutve Probablty, A. Math. 4, 699. Moderer, D., ad Samet, D. Ž Approxmatg Commo Kowledge wth Commo Belefs, Games Eco. Beha., Moderer, D., ad Samet, D. Ž Stochastc Commo Learg, Games Eco. Beha. 9, 670. Morrs, S., Rob, R., ad Sh, H. Ž Rsk Domace ad Stochastc Potetal, CARESS Workg Paper Nelse, L. Ž Commo Kowledge, Commucato ad Covergece of Belefs, Math. Soc. Sc. 8, 4. Soso, D. Ž Impossblty of Speculato Theorems wth Nosy Iformato, Games Eco. Beha. 8, Stchcombe, M. Ž Approxmate Commo Kowledge, Uversty of Calfora at Sa Dego, upublshed workg paper.