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Some other differences between "Crusoe" and a participant in a social exchange economy will not concern us either. Such is the non-existence of money as a means of exchange in the first case where there is only a standard of calculation, for which purpose any commodity can serve. This difficulty indeed has been ploughed under by our assuming in 2.1.2. a quantitative and even monetary notion of utility. We emphasize again: Our interest lies in the fact that even after all these drastic simplifications Crusoe is confronted with a formal problem quite different from the one a participant in a social economy faces. 2.2.2. Crusoe is given certain physical data (wants and commodities) and his task is to combine and apply them in such a fashion as to obtain a maximum resulting satisfaction. There can be no doubt that he controls exclusively all the variables upon which this result depends-say the allotting of resources, the determination of the uses of the same commodity for different wants, etc. 2 Thus Crusoe faces an ordinary maximum problem, the difficulties of which are of a purely technical-and not conceptual-nature, as pointed out. 2.2.3. Consider now a participant in a social exchange economy. His problem has, of course, many elements in common with a maximum prob- 1 It is not important for the following to determine whether its theory is complete in all its aspects. 2 Sometimes uncontrollable factors also intervene, e.g. the weather in agriculture. These however are purely statistical phenomena. Consequently they can be eliminated by the known procedures of the calculus of probabilities: i.e., by determining the probabilities of the various alternatives and by introduction of the notion of "mathematical expectation." Cf. however the influence on the notion of utility, discussed in 3.3. THE PROBLEM OF RATIONAL BEHAVIOR 11 !em. But it also contains some, very essential, elements of an entirely different nature. He too tries to obtain an optimum result. But in order to achieve this, he must enter into relations of exchange with others. If two or more persons exchange goods with each other, then the result for each one will depend in general not merely upon his own actions but on those of the others as well. Thus each participant attempts to maximize a function (his above-mentioned "result") of which he does not control all variables. This is certainly no maximum problem, but a peculiar and disconcerting mixture of several conflicting maximum problems. Every participant is guided by another principle and neither determines all variables which affect his interest. This kind of problem is nowhere dealt with in classical mathematics. We emphasize at the risk of being pedantic that this is no conditional maximum problem, no problem of the calculus of variations, of functional analysis, etc. It arises in full clarity, even in the most "elementary" situations, e.g., when all variables can assume only a finite number of values. A particularly st1iking expression of the popular misunderstanding about this pseudo-maximum problem is the famous statement according to which the purpose of social effort is the "greatest possible good for the greatest possible number." A guiding principle cannot be formulated by the requirement of maximii.;ing two (or more) functions at once. Such a principle, taken literally, is self-contradictory. (,n general one function will have no maximum where the other function has one.) It is no better than saying, e.g., that a firm should obtain maximum prices at maximum turnover, or a maximum revenue at minimum outlay. If some order of importance of these principles or some weighted average is meant, this should be stated. However, in the situation of the participants in a social economy nothing of that sort is intended, but all maxima are desired at once-by various participants. One would be mistaken to believe that it can be obviated, like the difficulty in the Cnrnoe case mentioned in footnote 2 on p. 10, by a mere recourse to the devices of the theory of probability. Every participant can determine the variables which describe his own actions but not those of the others. Nevertheless those "alien" variables cannot, from his point of view, be described by statistical assumptions. This is because the others are guided, just as he himself, by rational principles-whatever that may mean -and no mod1ls proccdendi can be correct which does not attempt to understand those principles and the interactions of the conflicting interests of all participants. Sometimes some of these interests run more or less parallel-then we are nearer to a simple maximum problem. But they can just as well be opposed. The general theory must cover all these possibilities, all intermediary stages, and all their combinations. 2.2.4. The difference between Crusoe's perspective and that of a participant in a social economy can also be illustrated in this way: Apart from 12 FORMULATION OF THE ECONOMIC PROBLEM those variables which his will controls, Crusoe is given a number of data which are "dead"; they are the unalterable physical background of the situation. (Even when they are apparently variable, cf. footnote 2 on p. 10, they are really governed by fixed statistical laws.) Not a single datum with which he has to deal reflects another person's will or intention of an economic kind-based on motives of the same nature as his own. A participant in a social exchange economy, on the other hand, faces data of this last type as well: they are the product of other participants' actions and volitions (like prices). His actions will be influenced by his expectation of these, and they in turn reflect the other participants' expectation of his actions. Thus the study of the Crusoe economy and the use of the methods applicable to it, is of much more limited value to economic theory than has been assumed heretofore even by the most radical critics. The grounds for this limitation lie not in the field of those social relationships which we have mentioned before-although we do not question their significancebut rather they arise from the conceptual differences between the original (Crusoe's) maximum problem and the more complex problem sketched above. We hope that the reader will be convinced by the above that we face here and now a really conceptual-and not merely technical-difficulty. And it is this problem which the theory of "games of strategy" is mainly devised to meet. 2.3. The Number of Variables and the Number of Participants 2.3.1. The formal set-up which we used in the preceding paragraphs to indicate the events in a social exchange economy made use of a number of "variables" which described the actions of the participants in this economy. Thus every participant is allotted a set of variables, "his" variables, which together completely describe his actions, i.e. express pre~isely the manifestations of his will. We call these sets the partial sets of variables. The partial sets of all participants constitute together the set of all variables, to be called the total set. So the total number of variables is determined first by the number of participants, i.e. of partial sets, and second by the number of variables in every partial set. From a purely mathematical point of view there would be nothing objectionable in treating all the variables of any one partial set as a single variable, "the" variable of the participant corresponding to this partial set. Indeed, this is a procedure which we are going to use frequently in our mathematical discussions; it makes absolutely' no difference conceptually, and it simplifies notations considerably. For the moment, however, we propose to distinguish from each other the variables within each partial set. The economic models to which one is naturally led suggest that procedure; thus it is desirable to describe for every participant the quantity of every particular good he wishes to acquire by a separate variable, etc. 2.3.2. Now we must emphasize that any increase of the number of variables inside a participant's partial set may complicate our problem technically, but only technically. Thus in a Crusoe economy-where there exists only one participant and only one partial set which then coincides with the total set-this may make the necessary determination of a maximum technically more difficult, but it will not alter the "pure maximum" character of the problem. If, on the other hand, the number of participants-i.e., of the partial sets of variables-is increased, something of a very different nature happens. To use a terminology which will turn out to be significant, that of games, this amounts to an increase in the number of players in the game. However, to take the simplest cases, a three-person game is very fundamentally different from a two-person game, a four-person game from a three-person game, etc. The combinatorial complications of the problem-which is, as we saw, no maximum problem at all-increase tremendously with every increase in the number of players, -as our subsequent discussions will amply show. We have gone into this matter in such detail particularly because in most models of economics a peculiar mixture of these two phenomena occurs. Whenever the number of players, i.e. of participants in a social economy, increases, the complexity of the economic system usually increases too; e.g. the number of commodities and services exchanged, processes of production used, etc. Thus the number of variables in every participant's partial set is likely to increase. But the number of participants, i.e. of partial sets, has increased too. Thus both of the sources which we discussed contribute pari passu to the total increase in the number of variables, It is essential to visualize each source in its proper role.