“Consider now a participant in a social exchange economy. His problem has, of course, many elements in common with a [Robinson Crusoe] maximum problem. 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 “results”) 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 striking expression of the population misunderstanding about the pseudo-maximum problem is the famous statement according to which the purpose of the social effort is the “greatest possible good for the greatest possible number.” A guiding principle cannot be formulated by the requirement of maximizing two (or more) functions at once.
Such a principle, taken literally, is self-contradictory. (In 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.
[…] 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.”
— Von Neumann and Morgenstern (1944) — prior to Arrow-Debreu — on the “peculiar and disconcerting” properties of general equilibrium, on the motivation of game theory, and on political rhetoric.