Games With Incomplete Information Played by Bayesian Players 1

8/2/2019 Games With Incomplete Information Played by Bayesian Players 1

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MANAGEMENT SCIENCEVol U No 3 No vem ber, 1967Printed in V S A

GAMES WITH INCOMPLETE INFORMATION PLAYEDBY "BAYESIAN" PLAYERS, I-IIIPar t I. The Basic Model *f

JOHN C HARSANYIUniversity of California, Berkeley

The paper develops a new theory for the analysis of games with incompleteinformation where the players are uncertain about some important parametersof the game situation, such as the payoff functions, the strategies available tovarious players, the information o ther players have about the game, etc How-ever, each player has a subjective probability distribution over the alternativepossibibtiesIn most of the paper it is assumed that these probability distributions enter-tained by the different p layers are mu tually "co ns isten t", in the sense th at th eycan be regarded as conditional probability distributions derived from a certain"basic probabil i ty distribution" over the parameters unknown to the variousplay ers Bu t later th e theo ry is extended also to cases where the differentplayers' subjective probability distributions fail to satisfy this consistencyassumptionIn cases where the consistencj assumption holds, the original game can bereplaced by a game where nature first conducts a lottery in accordance withthe basic probablity distribution, and the outcome of this lottery will decideTvhich pa rtic ula r sub gam e will be played , l e , what the actual values of the rele-van t par am eters will be in the game Ye t, each player will receive only partialinformation about the outcome of tbe lottery, and about the values of thesepar am eters How ever, every player will know the "basic prob ability d istribu-tio n" go vermng the lotte ry Th us , technically, the resulting game will be agame with com plete information It is called the Bay es-equivalent of theoriginal game P ar t I of the pap er describes the basic model and discusses vari-ous intuit ive interpre tat ions for the lat ter Pa rt II shows tha t the Nash equi-librium points of the Bayes-equivalent game yield "Bayesian equilibriumpo in ts" for the original game Fin ally, Pa rt I II considers the main prop erties ofthe "basic probabli ty distribution"

* Received June 1965, revised June 1966, accepted August 1966, and revised June 1967t P arts II and III of "G ames with Incomplete Information Played by 'Bayesian' P lay ers "will appear in subsequent issues of Management Science Theory' The original version of this paper was read at the Fifth Princeton Conference on GameTheory, in April, 1965 The present revised version has greatly benefitted from personaldiscussions with Professors Michael Maschler and Rob ert J Aum ann, of the Hebrew Uni-versity, Jerusale m, with D r Rein hard S elten, of the Joha nn Wolfgang Goethe U niversity,Frankfurt am M am , and with the other part icipants of the International Game TheoryWorkshop held at the Hebrew Un iversity in Jerusalem , in October-No vem ber 1965 I amindebted to D r Masch ler also for very helpful detailed comm ents on my ma nuscript


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160 JOHN C HARSANYITABLE OF CONTENTSPart I The Basic ModelSection 1 The sequential-expectations model and its disadvantagesSection 2 Different ways in which incomplete information can arise in a gasituationSection 3 The standard form of a game with incomplete mformationSections 4.-5 Bayesian gamesSection 6 The random-vector model, the prior-lottery model, and th e posterilottery model for Bayesian gamesSection 7 The normal form and the semi-normal form of a Bayesian game

Part II Bayesian Equilibrium PointsSection 8 Definition and some basic properties of Bayesian equdibrium poinTheorems I and IISection 9 Two numerical examples for Bayesian equilibrium pointsSection 1 0 How to exploit the opponent's erroneous beliefs (a numerical ample)Section 11 Why the analysis of Bayesian games in general cannot be based their normal form ( a numerical example)

Part III The Basic Probability Distribution of the GameSection 1 2 Decomposition of games with incomplete information The mtheorem (Theorem III) about the basic probability distributionSectvm IS The proof of Theorem IIISection 1 4 The assumption of mutual consistency among the different playesubjective probabihty distributionsSection 1 5 Games with "mconsistent" subjective probability distributionsSection 1 6 The possibility of spurious mconsistencies among the differ

players' subjective probability distributionsSection 17 A suggested change in the formal definition of games with mutuaconsistent probabihty distributionsGlossary of Mathematical Notation

/-game A game with incomplete informationC-game A game with complete informationG The 7-game originally given to usG* The Bayesian game equivalent to G (G* is a C-game )G** Th e Selten game equivalent to G and to G* (G** is hkewise a C-gam3fl(G), 3l(G*), 3l(G**) The normal form of G, G* and G** respectivelyS(G), S(G*) The semi-normal form of G and G* respectively


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GAMES WITH INCOMPLETE INFORMATION 161C = jc) The range space of vector cc' = (ci, , c,_] , c,+i, , Cn) The vector obtamed froTi vector c by

omitting subvector c,C = {c'} The range space of vector c'x, Player I's payoff (expressed in utihty units)X, = U,isi, , Sn) = V,isi, , Sn , Cl, , Cn) Player I's payoff func-

tionP,(ci, , c,_i, c,+i, , Cn) = Pric') = R,ic' I c,) The subjective prob-

ability distribution entertained bj' player iR* = R ici, , Cn) = R ic) The basic probability distribution of the

game7?, = R ici, , c,_i, c+i, , Cn\ Ci) = R*ic' \ c,) The conditional

probability distribution obtained from R* for a given value of vector c,k. The number of different values that player I's attribute vector c, can take

m the game (in cases where this number is finite)K = 2Zr=i kt The number of players m the Selten game G** (when this

number is finite)s* A normalized strategy of player i (It is a function from the range space

C, of player I's attribute vector c,, to his strategj' space ;S. )S* = {s,*j The set of all normalised strategies s,* available to player i6 The expected-value operatorS(x.) = Wiisx*, , Sn*) Player i's normalized payoff function, stating hisunconditional payoff expectation(a;. I c.) = Z,(si*, , s* | c.) Player I's semi-normalized payoff function,

stating his conditional payoff expectation for a given value of his attributevector c.

D A cylmder set, defined by the condition D = Di X X D , v.hereDl Cl ,DnBCn

GiD) For a given decomposable game G or G*, GiD) denotes the componentgame played in all cases where the vector c lies m cylinder D D is called thedefinmg cylinder of the component game

Special Notation in Certain SectionsIn section 3 iPart I)(h r denotes a vector consisting of those parameters of player I's payoff function

U, which (m player j 's opinion) are unknown to all n playersfflt, denotes a vector consisting of those parameters of the function U, which (in

fs opinion) are unknown to some of the players but are known to player ko = (ooi, , aan) i& a vector summarizing all information that (m;'s opinion)

none of the players have about the functions U\ , , Unfli = (a*!, , Oin) IS a vector summarizmg all information that (m j's opinion)


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16 2 JOHN C HARSANYIIn terms of these notations, player I 's information vector (or attribute vectoc, can be defined asc . = (o , , b,)V,* denotes player I's payoff function before vector oo has been integ rated o

After elimin ation of ve ctor ao th e sym bol F , is used to d eno te playe r I's payofunctionIn sections 9-10 iPart II)a' and a' denote the two possible values of player I 's attribute vector ci?/ and b^ denote the two possible values of player 2's at tnbute vector o,ri-r, = R*ici = a'' and C2 = 6") deno tes the pro ba bih ty m ass function co rresping to the basic probability distribution R*

-\- r,n) and q,^ = r,Jir^^ -f r^m) denote the correspondconditional probability mass functionsy^ and y ^ denote player I 's two pure strategiesz^ and z^ denote player 2'fe two pure strategies2/"' = iy"> y ) denotes a normabzed pure strategy for player 1, requirng the of strategy y" li Ci = o\ and requiring the use of strategy j / ' if Ci = a'z"' = (z", z") denotes a normahzed pure strategy for player 2, requiring the uof strategy z" if C2 = 6\ and re qu inn g the use of strategy z" if C2 = b'In section 11 iPait II)a^ and o^ de note th e two possible values th at either player 's at tnbute vector

can taker,,n = R*ici = a"- and d = a")Pirn and q,m have the same meaning as in sections 9-10y* denotes player t 's payoff demandy, denotes player I 's gross payoffX, denotes player t 's net payoffX* denotes player I's net payoff in the case (ci = a , C2 = o )xt* denotes player ?'s net payoff in the case (ci = a^, oi = a)In section IS iPart III)a, P, y , S denote specific values of vector ca. , |S ., 7 . , 5, den ote specific v alue s of vec tor c,', i8'i y \ ^' denote specific values of vector c', etc' " . ( T ' I 7.) = fi.(c' = 7* I c. = 7i) denotes the probabJity mass function respondmg to player I's subjective probabihty distnbution R, (when E, idiscrete distribution)r*iy) = R*ic = y ) denotes the probabihty mass function correspondmg to


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GAMES WITH INCOMPLETE INFORMATION 1 6 3In section 16 iPart III)iS'"' denotes the basic probabihty dis tr ibution R* as assessed by player lii = 1,,n )R*' denotes a given player 's (player j ' s ) revised estimate of the basic probabihtydistr ibution R*c' , = ( c , , di) denotes player fs revised defimtion of player I's at tnbute vectorc, (I t IS m general a larger vector tha n th e vector c, ongm ally assumed byplayer J )R'I denotes player j ' s revised estimate of player i 's subjective probabihty distribu-tion Ri

1.FoUowmg von Ne um an n an d IMorgenstern [7, p 30], we distinguish betweengames with complete mformation, to be sometimes briefly called C-games m thispaper, and games with incomplete information, to be called /-gam es Th e latterdiffer from the fonner in the fact that some or all of the players lack full informa-tion abo ut th e "r ul es " of the gam e, or equiva lently abou t its norm al form (orabou t its extensive form) Fo r exam ple, they ma y lack full mform ation abo utother players' or even their own payoff functions, about the physical facihtiesand strategies available to other players or even to themselves, about the amountof information the other players have about various aspects of the game Situa-tion, etc

In our own view it has been a major analytical deficiency of existing gametheory that i t has been almost completely restricted to C-games, in spite of thefact that in many real-hfe economic, pohtical, mihtary, and other social situa-tions the participants often lack full information about some important aspectsof the "game" they are playmg ^It seems to me that the basic reason why the theory of games with mcompleteinformation has made so httle progress so far hes in the fact that these gamesgive rise, or at least appear to give nse, to an infinite regress m reciprocal expec-tations on the pa rt of th e players, [3, pp 30-32] For exam ple, let us consider anytwo-person ga me m w hich the play ers do no t know each other 's payoff functions(To simplify our discussion I shall assume that each player knows his own payofffunction If we m ade th e opposite assum ption, then we would hav e to introdu ceeven more comphcated sequences of reciprocal expectations )In such a game player I 's strategy choice will depend on what he expects (orbeheves) to be player 2's payoff function U2, as the natuie of the latter will bean im porta nt d ete m un an t of player 2 's behavior m the game This expectation' The dist inct ion between games with complete and incomplete information (between C-games and /-games) must not be confused with that between games with perfect and imper-fect information By common terminological convention, the first distinction always refers


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1 6 4 JOHN C HARSANYIabout U2 may be called player I's first-order expectation Bu t his strategy cwill also depend on w hat he expects to be player 2's first-order expectation abohis own (player I's) payoff fimction Ui This may be called player I's secoorder expectation, as it is an expectation concemmg a first-order expectatIndeed, player I's strategy choice wiU also depend on what he expects to player 2's second-order expectation that is, on what player 1 thinks that pla2 thinks that player 1 thmks about player 2's payoff function U2 This we mcall player I's third-order expectationand so on ad infinitum Likewise, pl2's strategy choice will depend on an mfimte sequence consistmg of his firorder, second-order, third-order, etc , expectations concernmg th e payoff futions f/i and U2 We shall call any model of this kmd a sequential-expectamodel for games with incomplete informationIf we follow the Bayesian approach and represent the players' expectationsbeliefs by subjective probablity distributions, then player I's irst-orderexpection wiU have the natu re of a subjective probabhty distribution P / ( U2) ovealternative payoff functions U th at player 2 may possibly have Likewase, p2's first-order expectation will be a subjective probabhty distribution P2^iUover all alternative payoff functions Ui that player 1 may possibly have On other hand, player I's second-order expectation will be a subjective probadistribution PiiP2^) over all alternative first-order subjective probabLhty disbutions P2' th at player 2 may possibly choose, etc More generally, the fcth-oexpectation (fc > 1) of either player i will be a subjective probab ihty distributP.'(Pj~') over all alternative (A; l)th-order subjective probability distritions Pj~' that the other player j may possibly entertam '

In the case of n-person /-games the situation is, of course, even more compcated Even if we take the simpler case m which the players know at least thown payoff functions, each player m general wiU have to form expectations abthe payoff functions of the other (71 1) players, which means formmg (n different first-o rder expectations He will also have to form expectations abothe (n 1) first-order expectations entertamed by each of the other (n players, which means forming (n 1)^ second-order expectations, etcThe purjwse of this paper is to suggest an alternative approach to the analyof games with m complete mformation This approach will be based on constnimg, for any given /-game G, some C-game G* (or possibly several different games G*) game-theoretically equivalent to G By this means we shall reduceanalysis of /-gam es to the analysis of certam C-games G*, so th at the problem

Proba bility distributions over some space of payoff functions or of probabihty distrtions, and more generally probability distributions over function spaces, involve certechnica l mathematical difficulties [5, pp 355-357] However, as Aumann has shown [1][2], these difficulties can be overcome Bu t even if we succeed in defining the relevant hig


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GAMES WITH INCOMPLETE INFORMATION 1 6 5such sequences of higher and higher-order reciprocal expectations will simply notariseAs we hav e seen, if we use the B ayesian ap proach, then the sequential-expecta-tions model for an y given /- ga m e G will hav e to be analyzed in term s of uifinitesequences of higher a nd higher-ord er subjectiv e p rob abil ity d istribu tions , l e ,subjective pro bab ihty distribution s over subjective proba blity distributions Incontrast, im der our ow n model, it will be possible to analyze any given /-g am e Gm terms of one unique probabihty distribution R* (as well as certain conditionalprobabhty distributions derived from R*)For exam ple, consider a two-person non-zero-sum game G representing pricecompetition between two duopohst competitiors where neither player has precisemformation about the cost functions and the financial resources of the otherplayer This, of course, me ans th at n either player i will know the true payofffunction U, of the other player j , because he will be unable to predict the profit(or the loss) that the other player will make with any given choice of strategies(l e , price and ou tp ut pohces) si and S2 by the two playersTo make this example more reahstic, we may also assume that each player hassome information about the other player's cost functions and financial resources(which may be represented, e g , by a subjective probability distribution over therelevant variables), but that each player i lacks exact information about howmuch the othe r player j actually know s abo ut player I 'B cost structure and finan-cial position

Un der th ese assum ptions th is game G will be obviously an /-gam e, and it iseasy to visuahze the complicated sequences of reciprocal expectations (or of sub-jective pro bab hty distrib ution s) we would have to postu late if we tried to analyzethis game in terms of the sequential-expectations approachIn contrast, the new approach we shall describe m this paper will enable us toreduce this /-g am e G to an equ ivalent C-game G* mvolving four random events(l e , chance moves) ei , e2, / i , and fi, assumed to occur before the two playerschoose their strategies Si and S2 Th e rand om event e,(? = 1, 2) will determm eplayer z's cost functions and the size of his financial resources, and so will com-

pletely determine his payoff function Ui m the game On the other hand, therandom even t / , wall determ m e the am oim t of mform ation t h at player i w ill ob-taui about the cost functions and the financial resources of the other playerli j = 1,2 and 9^ i), and will thereb y determ ine the actua l amo unt of information*that player i will have about player j's payoff function U,Both players will be assumed to know the joint probabihty distnbution^ * ( e i , ez, / i , /2) of these four random events ^ But , e g , playe r 1 will know theactual outcomes of these random events only m the case of ex and / i , whereas' In terms of the ternunology we shall later introduce, the variables determined by therandom events e, a n d /, will constitu te the random vector c, (t = 1, 2), which will be calledplayer t 's information vector or attnbute vector, and which will be assumed to determine


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16 6 JOHN C HARSANYIplayer 2will know the actual outcomes only m the case of e2 and f^ (In our mthis last assumption will represent the facts that each player wall know only own cost functions and financial resources but wiU not know those of his oponent, and that he will, of course, know how much information he himself habout the opponent but wiU not know exactly how much mformation the oponent will have about him )As m this new game G* the players are assumed to know the probability disbution Ttie-x ,ei ,fx,f2), this game G* will be a C-game To be sure, player 1have no information about the outcomes of the chance moves e^ and / 2 , wheplayer 2 will have no mformation about the outcomes of the chance movesand / i But these facts wiU not m ake G* a game with "m com plete" informatbu t will make it only a game with "imperfect" information (cf Foo tnoteabove) Thus, our approach wall basically am ount to replacmg a game G mvolvincomplete information, by a new game G* which mvolves complete but immformation, yet which is, as we shall argue, essentially eqmvalent to G fromgame-theoretical point of view (see section 5 below)As we shall see, this C-game G* which we shall use m the analysis of a gi/-gam e G will also admit of an alternative m tuitive m terpretation Insteadassummg that certam important attributes of the players are determmed by shypothetical random events at the begmnmg of the game, we may rather assutha t the players themselves are drawn at random from certam hypothetical plations contammg a mixture of mdividuals of different "types", characterized different at tnbut e vectors (l e , by different combmations of the relevant tributes) For m stance, m our duopoly example we may assume that each playi( t = 1, 2) IS drawn from some hypothetical population II, containmg individuof different "types," each possible "type" of player t being characterized bydifferent at tr ibute vector Ci, l e , by a different combination of production cosfinancial resources, and sta tes of information Each player i will know his otype or attnbute vector c, but will be, m general, ignorant of his opponenOn the other hand, both players will agam be assumed to know the jomt proabihty distribution fi*(c, , c^) govemmg the selection of players 1 and 2 of difent possible types ci and d from the two hypothetical populations IIi and 112

It may be noted, however, that m analyzmg a given /-game G, constructiof an equivalent C-game G* is only a partial answer to our analjrtical problebecause we are stiU left with the task of definmg a suitable solution concept this C-game G* itself, which may be a matter of some difficulty This is so becam many cases the C-game G* we shall obtain m this way will be a C-game unfamiliar form, for which no solution concept has been suggested yet m tgame-theoretical hterature ^ Yet, smce G* will always be a game with complinformation, its analysis and the problem of definmg a suitable solution concefor it, will be at least amenable to the standard methods of modem game theo


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GAMES WITH INCOMPLETE INFORMATION 16 7

Our analysis of /-games wall be based on the assumption that, in dealing w lt hmcomplete mformation, every player i will use the Bayesian approach T ha t ishe will assign a subjective jomt probabihty distribution P, to all vanables unknf)wnto him-or at least to all unknown independent variables, l e , to all variables notdependm g on th e pla yers ' owm strateg y choices Once this has been done he w illtry to maximize the mathematical expectation of his owai pci3'off a;, m terms ofthis probab ility d istribution P , ^ This a ssumption will be called the Bayesianhypothesis

If m com plete inform ation is m terpreted as lack of full mformation by theplayeis about the normal form of the game, then such incomplete informationcan anse m three main ways1 Th e players ma y not know the physical outcome function Y of the game,which specifies the physical outcome y = r ( s i , , s^) produced b> eachstrategy -tup]e s = ( i i , , s^) available to the players2 Th e players m ay not know their own or some other plaj 'ers ' utility functionsX,, which specify the utihty payoff x, = XJy) that a given player i denvesfrom every possible physical outcome y3 Th e players m ay not know their own or some other players ' strategy spacesS . , I e , the set of all strategies s, (both pure and mixed) available to variousplayers iAll other cases of mcomplete information can be reduced to these three basiccases-m deed som etime s this can be done m two or more different (b ut essen-tially equivalent) ways For example, mcomplete information may arise by someplayers ' ignorance about the amount or the quahty of physical resources (eqmp-ment, raw materials, etc ) available to some other players (or to themselves)This s i tuation can be equally mterpreted either as ignorance about the physicaloutcome function of the gam e (case 1), or as ignorance abou t the strategies avail-able to va no us playe rs (case 3) Wh ich of the two interpretation s we hav e touse will depend on how we choose to define the "strategies" of the players inquestion Fo r m stanc e, suppose th a t in a military engagement our OWTI side does

not know the number of fire arms of a given quahty available to the other bide' A subjective proba bil i ty distr ibution P , entertained by a given player i is defined interms of his own choice beh avio r, cf [6] In con tras t, an objective probability distributionP* IS defined in term s of the long-run frequencies of the relev ant eve nts (presum ably asestablished by an independ ent obse rver, say, the umpire of the game) It is often convenientto regard the subjective probab ilities used by a given player i as being his personal estimatesof the corresponding objective probabilities or frequencies unknown to him' If the physical outcome y is siny)ly a vec tor of money payoffs j/i , ,Vn to the n pla j ersthen we can usually assume that any player I's utility payoff x, = X,(y,) is a (strictly

mcreasing) func tion of his mone y payoff 2/, and th at all play ers will know this How ever,the other players; may not know the specific mathematical form of player t'B utility func-


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168 JOHN C HARSANYIThis can be mterpreted as mabihty on our part to predict the physical outcoiie, the amount of destruction) resultmg from alternative strategies of topponent, where any given "strategy" of his is defined as firmg a given percentaof his fire arms (case 1) But it can also be mterpreted as mabihty to deciwhether certam strategies are available to the opponent at all, where now agiven "strategy" of his is defined as firmg a specified number of fire arms (case 3

Incomplete mformation can also take the form that a given player i does nknow whether another player j does or does not have mformation about thoccurrence or non-occurrence of some specified event e Such a situation will aways come under case 3 This is so because m a situation of this kmd, from a gamtheoretical pomt of view, the crucial fact is player i's mabihty to decide whethplayer j is m a position to use any strategy s/ involving one course of action case event e does occur, and another course of action m case event e does noccur That is, the situation will essentially amount to ignorance by player i abothe availabihty of certam strategies s, to player j

Gomg back to the three mam cases hsted above, cases 1 and 2 are both specicases of ignorance by the players about their own or some other players' payfunctions f/, = ^. ( F ) specifymg the utihty payoff a;, = f7,(si, , Sn) a givplayer i obtams if the n players use alternative strategy n-tuples s = (si, Sn)

Indeed, case 3 can also be reduced to this general case This is so because thassumption that a given strategy s, = s," is not available to player i is equivalefrom a game-theoretical point of view, to the assumption that player i will nevactually itse strategy s," (even though it would be physically available to himbecause by usmg s, he would always obtam some extremely low (z e, highnegative) payoffs x, = U^isx, , s,", , Sn), whatever strategies Si, Sv_i, s,+i , , Sn the other players 1, , i \,i + \, , n may be usin

Accordmgly, let S['^ ij = 1 or j 7^ 1) denote the largest set of strategieswhich m player ^'s opmion may be conceivably mcluded m player I's stratspace )S. Let Si"^ denote player I's "true" strategy space Then, for the purpoof our analysis, we shall define player t's strategy space S, as(21) S. = U:_oSi*'We lose no generahty by assummg that this set S, as defined by (2 1) is knowto aU players because any lack of mformation on the part of some player j abothis set S, can be represented withm our model as lack of information about tnumerical values that player I's payoff function a;, = f7,(si, , s,, , takes for some specific choices of s,, and in particular whether these values aso low as completely to discourage player i from usmg these strategies s.

Accordmgly, we define an /-game G as a game where every player j kno


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GAMES WITH INCOMPLETE INFORMATION 1 6 93.

In terms of this definition, let us consider a given /-game G from the point ofview of a partic ula r pla yer ^ H e can write the payoff function C/, of each player i(including his own payoff function U,iox i = j) m a, more explicit form as(3 1) I , = C/,(si, ,Sn)=V*{si, ,s, ao. , a i . , , o . , , , o,),where V*, unlike [/,, is a function whose m athe m atica l form is (in player j'sopmion) known to all n p lay ers, whereas ao, is a vector con sisting of those para m e-ters of function [7, whieh (in j ' s opm ion) are unknown to all players, and whereeach a*. for fc = 1, , w is a vec tor consisting of those pa jam ete rs of ftinction[7, which (m J 'S opmion) are unknovm to some of the players but are known toplayer A; If a given p ara m ete r a is known hoth to players k and m (without bemgknown to aU play ers), th en this fact can be represented by lntroducm g twovariables at, and a^, with a*, = , = a, and then making at , A component ofvector at, while making a^, a component of vector a^.For each vector a*, {k = 0, 1, , n), we shall assume that its range spaceAu = {at,}, I e , the set of all possible values it ean take, is the whole Euclidianspace of the requ ired n um be r of dimensions Th en T',* will be a function from th eCartesiaji product Si X X iS, X ^o , X X ^n . to player I's utihty lineS , , w hich IS itself a copy of th e real Ime R

Let us define at as the vector combinmg the components of all n vectors a t i ,, atn T hu s we wr ite(3 2) OA = ( a i l , , a*n),for A: = 0, 1, , i, ,71 Cle arly, vec tor ao sum m arizes th e inform ation th a t(in player J 'S opm ion) none of the players has about th e n func tions f/ i, , C/n ,whereas vecto r at(fc = 1, , n ) summ arizes the mform ation th at (m. faopmion) player k has about these functions, except for the information that (mJ 'S opmion) all n players share abo ut them For each vector ak, its range spacewill be the set A^ = {at} = Aki X X A^n

In equation (3 1) we are free to replace each vector at^(k = 0, ,n) by thelarger vector at = ( a ti , , atx, , dkn), even though this will mean that ineach case th e (n 1) sub-v ectors a t i , , at (, -i ), atc-t-i) , , akn will occurvacuously m th e resulting new equation Th us, we can write(3 3) X, = V*(si, , Sn , ao, ai, , a,, , an)

For any given player t th e n vectors ao, a i , , a,_ i, a,+ i, , an m generalwill represent unknown vari able s, and th e same will be tru e for th e (re 1)vectors 6 ,, , h^i, b,+i, , & to be defined below The refore, un der th eBayesian hypothesis, player i will assign a subjective joint probability distribu-tion


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170 JOHN C HARSANYIb = (bi, ,bn) The vectors obtamed from a and b by omittmg the sub-veco, and 6,, respectively, wdl be denoted by a' and b' The corresponditig ranspaces can be wntten as A = Ai X X An , B = = Bi X X Bn , A' Al X X A,_i X A,+i X X An , B' = Bi X X B._i X B.+, XXBn

No v we can wnte equations (3 3) and (3 4) as(3 5) X. = F,*(si, ,Sn,ao,a)(3 6) P. = P,(ao, a', b')where P, is a probabihty distribution over the vector space Ao X A^ X B'

The other (n 1) players in general will not know the subjective probabildistnbution P, used by player i But player j (from ^vhose point of view we analyzing the game) will be able to write P, for each player i (both i = j aI 7^ j) m the form(3 7) P . ( a o , a ' , 6 ' ) = R,(ao, a' ,b'\K),where R , unhke P, , is a function whose mathematical form is (m playeropinion) known to all n players, whereas 6, is a vector consisting of those paramters of function P, which (m j' s opmion) are unknown to some or all of the playk 9^ I Oi course, player j will realize that player z himself wiU know vectorsince 6, consists of parameters of player I's own subjective probabihty distnbtion P,

The vectors bi, , 6._i, fe.+i, , ?) occurrmg in equation (3 4), whichfar have been left tmdefined, are the parameter vectors of the subjective proability distnbutions Pi , , P,_i, P,+i, , Pn , unknown to player i Tvector 6' occurring in equations (3 6) and (3 7) is a combination of all thevectors fei, , b,_i , b,+i , , bn , and sum.marizes the inform.ation that player j's opmion) player i lacks about the other (n 1) players' subjectprobabihty distributions Pi , , P,_i, P.+i, , Pn

Clearly, function R,, is a function yielding, for each specific value of vector a probability distribution over the vector space A' X B'

We now propose to ehmmate the vector oo, unknown to all players, from equtions (3 5) and (3 7) In the case of equation (3 5) this can be done by takie.rpected values with respect to a a m terms of player z's own subjective probabdistribution P.(ao, o', 6') = i2.(ao, a', b' | b,) We define( 3 8 ) V,(si, ,Sn ,a \b , ) = V,isi, , S n , o , 6 . )

f *(J A .. ' '


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GAMES WITH INCOMPLETE INFORMATION 17 1In th e case of equ ation (3 7) we can elunma te ao by tak ing the app ropriatemarginal prob ability distributio ns We define

(310) P.(a*,6') = fand( 3 1 1 ) R.(a\b'\b,) == f d( . . , f l , (ao,a ' ,6* |6 , )Then we wnte(3 12) P.(a\ 6') = fl,(a', b' | 6,)

We now rewrite equation (3 9) as(3 13) I , = F , ( s j , , s , , a, 6 , , 6') = F ,( s i, , s , , a, 6),where vector b' occurs only vacuously Likewise we rewrite equatton (3 12) as( 3 14 ) P ,(a *, 6') = i ? , ( a ' , 6 ' | a . , b . ) ,where on the right-hand side vector a, occurs only vacuouslyFinally, we m trodu ce the defimtions c, = (a, ,b,),c = (a, b), and c' = (a ', &')Moreover, we wnte C, = A, X B, , C == A X B, and C = A' X B' Clearly,\ector c, represents the total information available to player i in the game (if wedisregard the information available to all n players) Th us, we m ay call c, playerI's mformation vector

From another pomt of view, we can regard vector c, as representing certainphysical, social, and psychological attributes of player i himself, m that i t sum-marizes some crucial parameters of player I's own payoff ftinction U, as well asthe mam parameters of his behefs about his social and physical environment(The relevant parameters of player i's payoff function f/, a gain pa rtl y repre sen tparameters of his subjective utility function X^ and part ly represent parametersof his environment, e g , the amounts of various physical or human resourcesavailable to him, etc ) From this pomt of view, vector c, may be called playerz's attribute vector

Thus, under this model, the players ' incomplete lnformatton about the truenature ot the game situation is represented by the assumption that m generalthe actual value of the attribute vector (or information vector) c, of any givenplayer i will be knowTi only to player i himself, but will be unknown to theother {n I) players T ha t is, as far as these other players are concerned,c, could have any one of a ntimberpossibly even of an infinite numberofaltern ative valu es (wh ich tog eth er form th e ran ge space C, = {c,} of vec tor c,)We m ay also express this assum ption b y saying tha t m an /-g am e G, in general,the rules of the game as such allow any given player z to belong to any one of


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17 2 JOHN C HAESANTImight ha ve in the game) Eac h player is always assumed to know his own a ctutype but to be in general ignorant about the other players ' actual typesEq uatio ns (3 13) and (3 14) now can be writ ten as(3 15 ) a;, = V,(si, , Sn , c) = F, ( s i , , s , C i, , c)(3 16) P. (c') = R,(c'\ c.)or(3 17 ) P, (C l , , C_l , C.+1 , ,Cn)=R,{Cl, , C,_l , C.+1 , , Cn| C.)W e shall regard e qua tions (3 15) and (3 17) [or (3 16)] as the s tan da rd formof the equations defining an /-game G, considered from the pomt of view of somparticular player j

Formally we define the standard form of a given 7-game G for some particulplayer j as an ordered set G such that(3 18) G = {Si , , Sn , Cl , , Cn , Vl , , Vn , Rl , , Rn]where for i = 1, , n we ^\Tite S, = {s,\, C, = {c.}, m oreov er, where V,a function from the set Si X X Sn X Ci X X (7 to player I 's utilhne E. (which is itself a copy of the real line R), and where, for any specific valuof the vector c,, the function fl, = R,(c'\ c,) is a probabihty distribution ovthe set C = CiX X C._i X C.+i X X Cn

4.Among C-games the natural analogue of this /-game G wiU be a C-game Gwith th e sam e payoff functions F , and the same stra teg y spaces S, Howevein G* the vectors c, wiU have to be reinterpreted as bemg random vectors (chanmoves) with an objective jomt probabihty dis tnbut ion

( 4 1 ) R* = R*(ci, ,Cn) = R*{c)known to aU n p la ye rs '" (If some players did not know i2*, then G* would nbe a C-game ) To make G* as similar to G as possible, we shall assume theach vector c, will take its values from the same range space C, m either gamMoreover, we shall assume that m game G* , just as m game G, when playerchooses his strategy s,, he will know only the value of his awn random vectc, bu t wiU no t know the rand om vectors Ci , , c,_i, c,+ i, , Cn of the oth(n 1) players Accordmg ly we m ay agam call c, the information vectorplayer i

Alternatively, we may agam mterpret this random vector c, as representincertain physical, social, and psychological attnbutes of player i himself (Bu


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GAMES WITH INCOMPLETE INFORMATION 1 7 3W e shall say t h at a given C-game G* is m standard form if1 th e payoff functions F , of G* hav e the form ind icated by equation (3 15 ),2 the vectors Ci, , C n occurring in equation (3 15) are rand om vectorsR [equation (4 1)] known to all players,3 each player i is assumed to kn ow only his own vector c, , and does n otthe vectors Ci, , c,_i, c,+i, , c^oi the other players when he choosesSometimes we shall agam express these assumptions by saying that the rulesi to belong to any one of a number of alternative(correspondmg to alternative specific values that the random vector c.

Form ally w e define a C-game G* m stan da rd form as an ordered set G* such

G* = {Sx, ,Sn,Cx, ,Cn,Vx, ,Vn,R*]th e ordered set G* differs from t h e ordered set G [defined b y equ ation18)] only m the fact th at the n-tuple Ri, , Rn occurring m G is replacedR*

If we consider the normal form of a game as a special hmiting case of a standardorm (viz as the case where th e rand om vectors Ci, , c are empty vectorst only a C-game G* containing rand om variables (chance m oves) will hav estan dar d form no n-tn vially different from its norm al form

Indeed, if G* contains more than one random vanable, then it will haveal different sta nd ard forms Th is is so because we can alway s ob tain newG**- intermediate between the original stand ard form G*G***-if we suppress some of the random variables occur-all of them (as we would do if we wanted toG*** itself) Th is pro cedu re can be called partial nor-full normalization, which would yield theG*** "

5.Suppose tha t G is an /-ga m e (considered from player j ' s po m t of view) while* IS a C-game, both games being given m s tand ard form To o btain comp lete

, (Sn , th e ra ng e sp aces C i , , Cn , and the payoff functions V i, , V n" Partial normalization involves essentially the same operations as full normalization

(see sectio n 7 below) It invo lves taking th e expected values of the payoff functions V , with


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17 4 JOHN C HAHSANTIof the two games are the sam e I t is necessary also th a t each player t m eithgame should alwaj^s assign the same numerical proababili ty p to any givespecific event E Yet m game G player i will assess all probabilities m terms ohis subjective probabili ty distr ibution R,(c'\ c,), whereas m game G*smvector c, is known to himhe wiU assess all probabilities in terms of the ojective conditional probabihtj ' ' distribution R*(c'\ c.) generated by the basprobabili ty distr ibution R (c) of the game G Therefore, if the two games ato be equ ivalent, th en nu me ncaU y the distribu tions i2 ,(c' | c,) and fi*(c' | cmust be identically equal

T his leads to the followmg defimtion L et G be an 7-gam e (as considered bplayer j) , and let G* be a C-game, both games bemg given m standard formWe shall say that G and G* are Bayes-equwaient for player j if the followinconditions are fulfilled

1 Th e two games m ust have the same strategy spaces Si, , Sn and tsame range spaces Ci, , C2 Th ey m us t hav e the same payoff functions Vi, , Vn3 T he subjective pro bab ihty dis tnb utio n 72, of each player tiaG must satisthe relationship( 5 1 ) RXc\c,) = R*{c'\c,),where R*ic) = R*(c,, c') is the basic probabihty distribution of game G* anwhere(5 2) R*(c\c,) =R*{c,,c

In view of equations (5 1) and (5 2) we can w n te(5 3 ) i i;*(c) = K*(c. ,c*) = E .( c ' |c . ) f d^c',R*ic,,c')In contrast to equ ation (5 2) , which ceases to have a clear m athem atical meaing when the d enom ma tor on i ts ngh t-ha nd side becomes zero, equation (5 3always retains a clear mathematical meanmg

We propose the following postulatePostulate 1 Bayes-equivalence Suppose that some 7-game G and some C-gamG* are Bayes-equivalent for player j Then the two games will be completeequivalent for player j from a game-theoretical standpomt, and, m particulaplayer j 's strategy choice will be governed by the same decision rule (the samsolution concept) m either gameThis postulate follows from the Bayesian hypothesis, which implies thevery player will use his subjective probabJit ies exactly m the same way as h


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GAMES WITH INCOMPLETE INFORMATION 1 7 5Of course, under the assumptions of the postulate, all we can say is thatthe two games are completely equivalent for game-theoretical

We cannot conclude on the basis of the information assumed th at theother players k 9^ j In ordertter conclusion we would have to know th at G and G* wouldir Bayes-equiva lence even if G were analyzed m ter m s of the func-ons F l , , Vn and Ri, , Rn postulated by these other players k, insteadf being analyzed m term s of the functions T^i, , Vn and Ri, , Rn postu-j himself B ut so long as we are intereste d on ly m th e decisionj hunself will follow m game G, all we have to know are theVi, Vn and Rx, Rn th at player j wiU be usmgPo stula te 1 natu rally gives nse to the followong questions Given any /-g am eIS it alw ays possible to con struc t a C-game G* Ba yes-e quiv alent to G? And ,

G* always unique? These questionsability dist ribu tion s i2i(c '| Cj), , /2n(c"| Cn), the re always exists a prob-R ici, , Cn) satisfymg the functional equation (5 3) ,R* is always unique m cases where it does exist

e th at a given /-g am e G wiU ha ve a C-game analogue G* only if Gself satisfies certain consistency re quir em ents On the other hand , if such aat , m cases where two different C-games Gi*, an d G2* are bo th Baye s-equiva -G2*r the analysis of G) In t he rest of the prese nt P ar t / of this pap er, we shallnc t our analysis to /-ga m es G for which a Bayes-eq mv alent C-game analogueAs we shall make considerable use of Bayes-equivalence relationships betweengam es G and certam C-games G* given m stan dar d form, it wiU be

t to hav e a sho rt designation for the latte r Therefore, we shall m tro -Bayesian games as a shorter nam e for C-games G* given m stan d-form De pend mg on the natu re of the /-ga m e G we shall be dealmg with in

6.In view of the important role that Bayesian games will play m our analysis,

So far we have defined a Bayesian game G* as a game where each player's


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17 6 JOHN C HAHSANYIrandom vectors , but that m general the actual value of any given vector cwiU be known only to player i himself whose information vector (or attributvec tor) it represents Th is mod el will be called th e random-vector model foBayesian games

An alte rna tive model for Bay esian gam es can be desc nbe d as follows Thactual mdividuals who wiU play the roles of players 1, , n m game G* oany given occasion, wiU be selected by lot from certam populations II i , , nof pote ntial players E ac h pop ulation II , from w hich a given player i is to bselected will contam mdividuals with a variety of different attributes, so thaevery possible combmation of attnbutes (l e , every possible "type" of player icorrespondm g to any specific value c, = c," th a t th e att rib ut e vector c, can ta km th e game, will be represented m this population II , If m population n , given md ividua l's att rib ut e vector c, ha s the specific value c, = c,, then we shasay that he belongs to the attribute dass c, Thus, each population II, will bpartitioned into that many attribute classes as the number of different valuethat player I's attribute vector c, can take m the game

As to the random process selectmg n players from the n populat ions II i, En , we shall assume that the probabihty of players 1, ,n bemg selectefrom any specific n-tu ple of a tt n b u te classes ci, , Cn" will be governedby the probabihty distribution R*{ci, , c^) We shall also retam the assumptions that this probabihty distribution R* will be known to all n players, andthat each player t wiU also know his own at tn b u te class c, = c, bu t, m g eneralwiU no t know th e oth er play ers' at trib ut e classes Ci = ci , , c,_i = c_ic,+i = c+i, , Cn = Cn" As m this model the lottery by which the playare selected occurs prior to a ny othe r mo ve m th e gam e, it wiU be called thpnw-lottery model for Bayesian games.

Le t G be a real-hfe gam e situation w here the players hav e mcom plete inform ation , and let G* be a Bayesian gam e Bayes-equ ivalent to G (as assessed by agiven player 7) Th en this Bayesian gam e G*, mterp reted m terms of the priorlottery model, can be regarded as a possible representation (of course a highlyschematic representation) of the real-hfe random sociai process which has actually created this game situation G More particularly, the pnor-lottery modepictures this social process as it would be seen by an outside observer havmmformation about some aspects of the situation but lackmg information abousome other aspects H e could not hav e enough information to predict the attribute vectors Ci = Ci, , Cn = Cn" of the n mdividuals to be selected by thsociai process to play th e roles of players 1, , w m game situatio n G B u t hewould have to have enough mformation to predict the jomt probabili ty distnbution iti* of the at tn b u te v ectors Ci, , Cn of these n mdividuals, and, o


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GAMES WITH INCOMPLETE INTORMATION 1 7 7

Vi,, Vn (But he could not have enough information to predict the payoffUi, , Un because this would require knowledge of the attributen players )

In other words, the hypothetical observer must have exactly all the informa-common to the n players, but must not have access to any additional infor-

players) W e shall call such an observer a properly informed observer Thus,

As an example, let us agam consider the price-competition game G with m-G*, discussed mion 1 above He re each player's at tri bu te vector c, will consist of th e va ria-abou t the other player " Th us, th e prior-lottery m odel of G* wiU

fferent m form ation-gatherm g facihties W e ha ve argued th a t such a model can

D r Selten has sugges ted" a thir d model for Bay esian games, which we shallSeUen model or the posterior-lottery model Its basic difference from theafter each potential player

More particularly, suppose that the attr ibute vector c , of player i {i = 1,, n ) can take fc, different valu es m the gam e (W e shall assume th at alls are fimte b u t th e m odel can b e easily ex tended also to th e infinite case )i m the game, we shalli will be played at the same tune by fc, differente set of all fc, md ividuals p laym g t he role of player i m t he gam e wiU be callede role class i Different mdividuals m the same role class t will be distinguishedsubscripts as playe rs zi , i^, U nde r these assum ptions, obviously the

n but rather wiU be the larger


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178 JOHN C HARSANTIIt Will be assumed that each player i^ from a given role class i will choo

some strategy s, from player I'astrategy space S, Different members of the samrole class i may (but need not) choose different strategies s, from this strategspace S,

After all K players have chosen their strategies, one player z^ from each roclass t wiU be randomly selected as an active player Suppose that the attnbuvectors of the n active players so selected will be Ci = Ci, , Cn = Cn",that these players, prior to their selection, have chosen the strategies Si =

, Sn = Sn Then each active player i^ , selected from role class i, will obtaa payoff(6 2) X, = V,iSl\ , Sn\ Cl", , Cn")All other (K n) players not selected as active players will obtam zero payof

It will be assumed that, when the n active players are randomly selected frothe n role classes, the probabihty of selectmg mdividuals with any specifcombmation of attribute vectors ci = ci, , Cn = c" will be governed by probabihty distnbution R*(ci, , Cn) ' '

It IS easy to see that m all three models we have discussedin the randomvector model, the pnor-lottery model, and the posterior-lottery modelthplayers' payoff ftmctions, the mformation available to them, and the probabihof any specific event m the game, are all essentially the same " Consequentl

" In actual fact, we could just as well assume that each player would choose his strategonly after the lot tery, and after being informed whether this lottery has selected him as active player or not (Of course if we made this assumption then players not selectedactive players could simply forget about choosing a stra tegy at all ) From a game-theoretcal poin t of view th is assum ption would make no real difference so long as each active playwould have to choose his strate gy witho ut being told the names of the other players selectas active players, and in part icular without bemg told the attribute classes to which theother active players would belongThus the fundamental theoretical difference between our second and third models is nso much in the actual timing of the postula ted lot tery as such It is not so much in the fatha t in one case the lottery precedes, and in the other case it follows, the players ' s tra tegchoices The fundamental difference rather hes m the fact that our second model (like oufirst) conceives of the game as an n-person game, in which only the n active players arformally "players of the ga m e", whereas our thir d model conceives of the game as a K-peson game, in which both the active and the inactive players are formally regarded as "playe r s " Yet, to make it easier to avoid confusion between the two models, it is convenient tassume also a difference in the actual timing of the assumed lottery" Technically speaking, the players' effective payoff functions under the posteriolottery model are not quite identical with their payoff functions under the other two model

but this difference is lm m a te na l for our purposes Under the posterior-lottery model, ler = r.ic") be the probability (margmal probability) that a given player with a t t nb utvector c, = c," will be selected as the active player from role class i Then player im wi


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GAMES WITH INCOMPLETE INTOEMATION l 7 9thr ee models can be considered to be essentially equiv alen t B ut , of course,

e G* with complete mform ation,K-person game G** withplete inform ation In w ha t follows, unless th e cont rary i=; m dicate d, by th eG correspondingSelten game

In contrast to the other two models, the prior-lottery model formally doesn players arehe formal gam e-theoretical definition of a gam e the iden tity of the players

before any chance move orT hus , we may characterize the situation as follow s The real-hfe social processe /-g am e G we are considering is best represented by t he prior-B ut th e latter does not correspond to a tru e "ga m e" m a game-

mto a tru e "g am e" In both cases this conversion entailsce m th e form of introducing some unreahstic assum ptions In t he case ofr-lottery m odel correspondmg to the Selten game G , the pnceiK n) fictitious players m addition to the n realrs participating m th e game ^'

In the case of the random-vector model corresponding to the Bayesian gamei is determm edafter the begmnmg of the gamewhich seems to imply thati will be m existence for some period of time, however short, durmg whichth e specific va lue c, = c, his at tri bu te v ector c, will take

fference B ut , as we shall see, when w e conve rt G* m to its norm al form

ose his normalized strate gy (l e , his strateg y for th e normal-form versionbefore he learns th e value of his owm at tn b u te vector c. An im por tan tanta ge of the Selten game G** hes m the fact th at it does not require thislar unreahs tic assumption we are free to assume th at every player i^r

In any case, the postenor-lottery model can be made completely eqmvalent to the othero models if we assume tha t each active player % will obta in a payoff corresponding to the


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18 0 JOHN C HARSANYIwiU know his own attnbute vector c, from the very begmnmg of the gamand will always choose his own strategy m hgh t of this information '*

Thus, as analytical tools used in the analysis of a given 7-game G, both thBayesian game G* and the Selten game G** have their own advantages adisadvantages "7 .

Let G be an 7-game given m standard form, and let G* be a Bayesian gaBayes-equivalent to G Then we define the normal form 3l(G) of this 7-gamG as bemg the normal form 5)1 (G*) of the Bayesian game G*To obtam this normal form we first have to replace the strategies s, of eacplayer i by normalized strategies s* A normalized strategy s* can be regaras a conditional statement specifymg the strategy s, = s, (c,) that playerwould use if his mformation vector (or attribute vector) c, took any givespecific value Mathem atically, a normalized strategy s, is a function from thrange space C, = {c,} of vector c, to player t's strategy space S, = |s,) Thset of aU possible such functions s,* is called player I's normahzed-strategspace (S,* = {s*\ In contrast to these normalized strategies s,*, the strategies, available to player i m the standard form of the game will be called his ordinastrategiesIf m a given game the mformation vector c, of a certam player i can take onlfc different values (with fc finite) so th at we can write(7 1) c, = c, \ , c,*,then any normalized strategy s,* of this player can be defined simply as a tuple of ordmaiy strategies(7 2) s,* = (s,\ , s . ' ) ,where s," = s,*(c,'"), with m = 1, , fc, denotes the strategy th at playerwould use m the standard form of the game if his information vector c, toothe specific value c, = c," In th is case player t's normalized strategy spacS* = {s,*j will be the set of all suchfc-tupless,*, that is, it will be the fc-timrepeated Cartesian product of player I's ordmary strategy space S, by itselThu s we can write S* = S'x XS! with S,' = = 5.* = S,Under either of these definitions, the normalized strategies s, will not havthe nature of mixed strategies but rather that of behavioral strategies Nev

" Moreover, as Selten has pointed out, his model also has the advantage that it can bextended to the case where the subjective pro babili ty distributions Ri , , R^ of a give7-game G fail to satisfy the required consistency conditions, so that no probability distribtion R* satisfying equa tion (5 3) will exist, and therefore no Baye sian game G* BayeequivaJent to G can be constructed at all In other words, for any /-ga m e G we can alwa


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GAMES WITH INCOMPLETE INFORMATION 181are admissible because any game G* m standard form

a game of perfect recall, and so it will make no difference whether the playersto use behavioral strategies or mixed strategies [4]Equa t ion (3 15) can now be written as

3) X,= V,iSx* iCx) , ,Sn*iCn) , Cx, ,Cn) = V . i S x * , ,*, c)In order to obtam the normal form %iG) = 91 (G*) , all we have to do now is

m equation (7 3) with respect to the whole randomc, m t e rms of the basic probability distribution R * i c ) of the game We

(x.) = F . ( s i * , ,Sn*) = [ V.isx*, , Sn* , c )dJ i * i c ) > chis expected payoff as his effective payoff from the

we can replace 6(a-,) simply by a;, and write5) X. = TF.(si*, , S n * )

W e can now define the normal form of games G and G* as the ordered set6) DZ(G) = ^iG*) = {Sx*, , Sn*, Wx, , W n ]

(3 18) and (4 2) definmg the standard forms oftwo games, m equation (7 6) the ordmary strategy spaces (S, have beenby the normahzed strategy spaces S*, and the ordmary payoff func-V, have been replaced by the normahzed payoff functions W, On the otherthe range spaces C. as well as the probabili ty distnbutions R, or R* have

the normal form TfliG) = 3l (G*) of games G and G* doesot any more mvolve the random vectors cx, , C nThis normal form, however, has the disadvantage tha t it is defined in t e rmsthe players ' uncondit ional payoff expectations (x,) = W^isx*, , Sn*),

m actual fact each player's strategy choice will be governed by hispayoff expectation (x,| c,), because he will always know his ownc, at the t ime of making his strategy choice This conditionalcan be defined as

7 7 ) f i(x .|c .) = Z. (s i* , ,Sn* |c . ) = /" ' c x* , ,s*, c . , c ' ) d ( c . ) f i * ( c ' | c . )To be sure, it can be shown (see Theorem I of Section 8, P a r t II) t h a t if

i maximizes his uncondit ional payoff expectation W,, thenbe maximizmg his condit ional payoff expectation Z,( \ c,) for each

of Cj, with the possible exception of a small set of c, values whichIn this respect our analysis bears out von


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182 JOHN C HARSANYIPrmciple has only res tncted vahdi ty for t hem, and their normal formmus t be used with special care, because solution concepts based on imcnt iuse of the normal form may give countermtuitive results (see Section 11P a r t II of th is paper) In view of this fact, we shall mtroduce the concept osemi-normal form The semi-normal form S(G) = S(G*) of games G and will be defined as a game where the players ' strategies are the normalized stragies s, descnbed above, but where their payoff functions are the conditiopayoff-expectation ftmctions Z^( \ c,) defined by equation (7 7) Formallydefine the semi-norm^al fo rm of the games G and G* as the ordered set(7 8) S(G) = S(G*) = {5i*, ,Sn*,Cl, ,Cn,Zl, ,Zn,R*\As the semi-normal form, unlike the normal form, does mvolve the randovectors ci, , Cn , now the range spaces Ci, , Cn, and the probabihdis tnbut ion R*, which have been omitted from equation (7 6), reappearequation (7 8)

Instead of von Neumann and Morgenstern's Normalization Principle,shall use only the weaker Serm-normahzation Prmciple (Postulate 2 belowwhich IS lmp hed by the Normalization Prmciple but which does not itself impthe la t terPostulate 2 Sufficiency of the Semi-normal Form The solution of any Bayesigame G*, and of the Bayes-eqmvalent / -game G, can be defined m t e rms of t

semi-normal form S(G*) =h(G), without going back to the standard formG* or of GRefe ren ces

1 ROBERT J AUMANN, "On Choosing a Function at Ran d o m," in Fred B Wright (editoSymposium on Ergodic Theory, New Orleans Academ ic Pr ess , 1963, pp 1-202 , "Mixed and Behavior Strategies m Infinite E xtensive Gam es," in M Dresher , LShapley, and A W T ucke r (ed itors), Adronces in Game T Aeorj/, Pr ince ton Pri nce tU niversity Pr ess, 1964, pp 627-6503 JOHN C HAKSANYI, "Bargaining in Ignorance of the Opponent ' s U t i li ty Funct ioJournal of Cm ifhct Resolution, 6 (1962), pp 29-384 H W K uHN, "E xtensive Games and the Problem of Informat ion," in H W K u h n aA W luiiksr {e6.i\.0TB), Contributions to the Th eory of Gam es,\o\ II , Princeton Prton U niversity Pre ss, 1953, pp 193-2165 J C C McKiNSEY, Introduction to the Theory of Gam.es, New York McG raw-Hill, 16 LEONARD J SAVAGE, TAe fou nda ho nso /S^ ahs iics, New York John Wiley and Sona, 19

7 JOHN VON NEU MANN AND OBK ARMORGENST E K N, Theory of Games and Economic BehavPrince ton Prince ton U niversity Press, 1953


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Games With Incomplete Information Played by Bayesian Players 1

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