Theory of Games Economic Behavior

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<p>CO</p> <p>^</p> <p>66986 [rA-thpnrptip*i the theory-to-be.))) ing</p> <p>Thesepoints</p> <p>1E.g., ewton's N</p> <p>4.1.2.</p> <p>3.7.1.</p> <p>34</p> <p>FORMULATIONOF THE ECONOMIC PROBLEM</p> <p>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.</p> <p>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.</p> <p>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</p> <p>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.</p> <p>i.e.</p> <p>i.e.</p> <p>A SOLUTIONS ND STANDARDSOF BEHAVIOR</p> <p>35</p> <p>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</p> <p>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</p> <p>4.3.2. simplestgamewhere the first requirementis disregardedis a The g three-personame where the sum of all payments is zero. In contrast to</p> <p>positive step.</p> <p>i.e.</p> <p>2 coalitions.</p> <p>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</p> <p>2.1.1.</p> <p>a)))</p> <p>36</p> <p>FORMULATIONOF THE ECONOMIC PROBLEM</p> <p>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</p> <p>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</p> <p>Ch.V.</p> <p>We shall seethat a consistent theory will for solutions which are not single imputations, but</p> <p>situation.</p> <p>4.3.3. last result will The</p> <p>allies.</p> <p>turn</p> <p>out to be the prototype of the general</p> <p>result from looking rather systems of</p> <p>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.)))</p> <p>I.e. 33.1.1.</p> <p>A SOLUTIONS ND STANDARDSOF BEHAVIOR</p> <p>37</p> <p>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 our considerationsin the generalframework of socialheory will t</p> <p>30.1.1.</p> <p>becomelearer. c</p> <p>4.4.The Intransitive Notion of \"Superiority\" or \"Domination\" 4.4.1. us return to a more primitive conceptof the solutionwhich we Let know already must be abandoned. We mean the ideaof a solution as a it singleimputation. If this sort of solution existed would have to be an imputation which in some plausiblesensewas superiorto all otherimputations. This notion of superiority as between imputations ought to be formulated in a way which takesaccountof the physicaland socialstructure of the milieu. That is, one should define that an imputation x is superior to an imputation y whenever this happens:Assume that society, t i.e.he totality of all participants, has to considerhe questionwhether or t not to \"accept\"staticsettlement f all questionsof distribution by the a o imputation y. Assumefurthermore that at this moment the alternative T settlement y the imputation x is also considered. hen this alternative x b will suffice to exclude of acceptance y. By this we mean that a sufficient number of participants prefer in their own interestx to i/, and areconvinced or can be convinced of the possibilityof obtaining the advantages of x. In this comparisonof x to y the participants should not be influenced by w of the consideration any third alternatives (imputations). I.e. e conceive the the relationship of superiority as an elementary one,correlating two imputations x and y only. The further comparisonof three or more ultimately of all imputations is the subject of the theory which must now follow, as a superstructure erected upon the elementary conceptof superiority. Whether the possibilityof obtaining certainadvantagesby relinquishing in y for x as discussed the above definition, can be made convincing to the interestedpartieswill depend upon the physical facts of the situation in the terminology of games, n the rules of the game. o a We prefer to use,instead of \" superior\" ith its manifold associations, w word more in the nature of a terminus technicus. When the above described 1 relationship between two imputations x and y exists,then we shall sayy</p> <p>that x dominates y. If one restates little more carefully what should be expected from a a S solution consistingof a single imputation, this formulation obtains: uch an imputation should dominate all others and be dominated by</p> <p>none.</p> <p>30.1.1.)))</p> <p>as formulated or rather indicated above is clearly in the nature of an ordering, similar to the question of 1 That is, when it holds in the mathematically precise form, which will be given in</p> <p>4.4.2. notion of domination The</p> <p>38</p> <p>FORMULATIONOF THE ECONOMIC PROBLEM</p> <p>or preference, of size in any quantitative theory. The notion of a single w imputation solution1 correspondsto that of the first element ith respect 2 to that ordering. w Thesearch such a first element ould be a plausibleone if the orderfor our notion of domination, possessedthe important ing in question, property of transitivity ; that is, if it were true that whenever x dominates one y and y dominatesz, then alsox dominatesz. In this case might proceed as follows:Starting with an arbitrary x, look for a y which dominatesa:;if such a y exists, hoose and look for a z which dominatesy\\ if such a z one c choose ne and lookfor a u which dominatesz, etc. In most practical o exists, e problemsthere is a fair chancethat this processitherterminates after a o finite number of steps with a w which is undominatedby anything else, r but that these that the sequence y, z, u, , goeson ad infinitum, x, tend to a limiting positionw undominatedby anything else. x, y, z, u, And, due to the transitivity referredto above, the final w will in eithercase dominateall previously obtained x, y, z, w, We shall not go into more elaborate etails which could and should d t be given in an exhaustive discussion.It will probablybe clearo the reader that the progress through the sequence y, z, u, #, correspondsto \" t successive improvements \" culminating in the \" optimum,\" i.e.he \"first\" w element which dominatesall others and is not dominated. All this becomesery different when transitivity does not prevail. v In that caseany attempt to reachan \"optimum\" by successiveimprovements may be futile. It can happen that x is dominatedby y y by z, and</p> <p>i.e.</p> <p>.</p> <p>4.4.3. the notion of domination on which we rely is, indeed,not Now w transitive. In our tentative descriptionof this concept e indicatedthat x dominatesy when there exists group of participants eachone of whom a prefers his individual situation in x to that in y, and who are convinced that they are able as a group i.e. s an alliance to enforce their prefera ences. We shall discuss these matters in detail in 30.2.This group of participantsshall becalledthe \"effectiveset\"for the domination of x over y. Now when x dominatesy and y dominatesz, the effective sets for thesetwo can dominationsmay be entirely disjunct and therefore no conclusions be drawn concerning relationshipbetween z and x. It can even happen the that z dominates x with the help of a third effective set, possiblydisjunct' from both previous ones.1We continu...</p>