Theory of Games Economic Behavior

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66986 [rA-thpnrptip*i the theory-to-be.))) ing


1E.g., ewton's N





we (Of course have not yet formulated theserequirements.Foran exhausc c b tive discussion, f. loc. it. elow.) The structure of the societyunder considerationwould then be extremelysimple: Therewould existan absolute state of equilibrium in which the quantitative shareof every participant would be preciselydetermined. It will be seenhowever that such a solution, possessingall necessary properties, does not exist in general. The notion of a solution will have to be broadenedconsiderably,and it will be seenthat this is closelyconnectedwith certaininherent features of social organization that are well \" known from a \" common sense point of view but thus far have not been viewed in proper perspective. (Cf. 4.6.nd 4.8.1.) a 4.2.3. mathematical analysis of the problem will show that there Our exists, indeed,a not inconsiderable family of gameswhere a solution can be defined and found in the above sense: .e. s one singleimputation. In i a such cases obtains at leastthe amount thus imputed to every participant him by just behaving appropriately, rationally. Indeed, e gets exactly h this amount if the other participants too behave rationally; if they do not, he may get even more. Thesearethe gamesof two participants where the sum of all payments is zero. While these gamesare not exactly typical for majoreconomic processes, contain some universally important traits of all gamesand they the resultsderivedfrom them arethe basis of the generalheory of games. t We shalldiscussthem at length in Chapter III.

obtains by behaving rationally. Considertheseamounts which the several \" participants''obtain. If the solution did nothing more in the quantitative with the well sensethan specify these amounts, 1 then it would coincide of i known concept imputation:t would just statehow the total proceeds 2 areto be distributed among the participants. We emphasize that the problem of imputation must be solved both are when the total proceeds in fact identicallyzero and when they arevariable. This problem,in its general orm, has neither been properlyformuf latednor solvedin economic literature. 4.2.2. can seeno reason why one should not be satisfied with a We a solution of this nature, providing it can be found: i.e. singleimputation which meetsreasonable requirements for optimum (rational) behavior.

1And of course,n the combinatorial sense,as outlined above,the procedure i how to obtain them. *In games as usually understood the total proceedsare always zero; one participant can gain only what the others lose. Thus there is a pure problem of distribution and absolutely none of increasing the total utility, the \"social imputation In all economicquestions the latter problem arisesas well, but the question product.\" of imputation remains. Subsequently we shall broaden the concept f a game by dropo ping the requirement of the total proceedseing zero(cf. Ch. XI).))) b

4.3.The Solution as a Set of Imputations 4.3.1.eitherof the two above restrictionss dropped,the situation is If i altered materially.





The simplest gamewhere the secondrequirement is oversteppedis a two-persongame where the sum of all payments is variable. This corresponds to a social economy with two participants and allows both for 1 and their interdependence for variability of total utility with their behavior. As a matter of fact this is exactly the caseof a bilateral monopoly (cf. The 6L2.-61.6.). well known \"zone of uncertainty \" which is found in current efforts to solve the problemof imputation indicatesthat a broader c conceptof solution must be sought. This casewill be discussedloc. it. above. Forthe moment we want to use it only as an indicatorof the diffiwhich is more suitableas a basis for a first culty and pass to the other case

the above two-persongame, this does not correspondto any fundamental n a economicroblembut it represents evertheless basicpossibilityin human p relations. The essentialfeature is that any two players who combineand against a third can thereby securean advantage. The problem cooperate is how this advantage shouldbe distributed among the two partners in this combination. Any such schemeof imputation will have to take into t accounthat any two partners can combine; while any onecombination of is in the process formation, eachpartner must considerthe fact that his could break away and join the third participant. prospectiveally h of Of coursethe rules of the game will prescribeow the proceeds a discoalition should be divided between the partners. But the detailed shows that this will not be, in general,the cussion to be given in 22.1. a final verdict. Imagine game (of three or more persons) in which two can form a very advantageouscoalition but where the rules participants of the gameprovide that the greatestpart of the gain goesto the first participant. Assume furthermore that the second participant of this coalition can also entera coalition with the third one,which is lesseffective in toto but promises him a greaterindividual gain than the former. In this situation it is obviously reasonablefor the first participant to transfer a part of the gains which he could get from the first coalition to the second participant in order to save this coalition. In other words: One must t expecthat under certainconditionsone participant of a coalition will be willing to pay a compensationto his partner. Thus the apportionment within a coalition depends not only upon the rules of the game but also upon the above principles, under the influence of the alternative

4.3.2. simplestgamewhere the first requirementis disregardedis a The g three-personame where the sum of all payments is zero. In contrast to

positive step.


2 coalitions.

that one cannot expect ny theoretical tateCommon sensesuggests s a ment as to which alliance will be formed3 but only information concerning 1It will be remembered that we make useof a transferable utility, cf. * This doesnot mean that the rules of the game are violated, sincesuch compensatory payments, if made at all, are made freely in pursuance of a rational consideration. 1Obviously three combinations of two partners eachare possible. In the example w to be given in 21., preference ithin the solution for a particular alliance will be any





how the partners in a possiblecombination must divide the spoilsin order to to avoid the contingency that any oneof them deserts form a combination in with the third player. All this will be discussed detailand quantitativelyin

It suffices to state here only the result which the above qualitative make plausibleand which will be establishedmore rigorously considerations o c loc. it. A reasonableoncept f a solution consistsin this case a system of c of three imputations. These correspond to the above-mentionedthree combinationsor alliancesand express division of spoilsbetweenrespecthetive


We shall seethat a consistent theory will for solutions which are not single imputations, but


4.3.3. last result will The



out to be the prototype of the general

result from looking rather systems of

imputations. It is clearthat in the above three-person no single imputation game from the solution is in itself anything like a solution. Any particular alliance describes one particular considerationwhich entersthe minds only of the participants when they plan their behavior. Even if a particular is alliance ultimately formed, the division of the proceeds betweenthe allies will be decisivelyinfluenced by the other alliances which each one might alternatively have entered. Thus only the three alliancesand their imputations togetherform a rational whole which determines all of its a stability of its own. It is, indeed,this whole which detailsand possesses is the really significant entity, more so than its constituent imputations. i Even if one of these is actually applied, i.e.f one particular allianceis actually formed, the others arepresent in a \"virtual\" existence: Although they have not materialized, hey have contributedessentiallyto shapingand t determiningthe actualreality. In conceiving of the general roblem,a socialconomy or equivalently p e a gameof n participants, we shall with an optimismwhich can be justified t only by subsequentsuccessexpecthe same thing:A solution shouldbe a 1 in system of imputations possessing its entirety some kind of balanceand stability the nature of which we shall try to determine.We emphasize that this stability whatever it may turn out to be will be a property of the system as a whole and not of the singleimputations of which it is T composed. hese brief considerationsregarding the three-person game have illustratedthis point. w a system of imputationsas a 4.3.4. exactcriteria hich characterize The solution of our problem are,of course,of a mathematical nature. For a t a precisend exhaustive discussionwe must therefore refer the readero the d mathematical developmentof the theory. Theexact efinition subsequentthe game will be symmetric with respect o all three t limine excludedby symmetry. Of.however participants. 1They may again include compensations between partners in a coalition, asdescribed in 4.3.2.)))

I.e. 33.1.1.



itself is stated in undertaketo give a prelimiWe shall nevertheless nary, qualitative outline. We hopethis will contributeto the understanding the of the ideas on which the quantitative discussionis based. Besides, placeof