Oskar Morgenstern’s Review of Samuel Karlin’s Mathematical Methods and Theory in Games, Programming and Economics (1961)

Oskar Morgenstern’s review evaluates Karlin’s two-volume work as both a major mathematical contribution and an instructive example of the distance between formal sophistication and economic realism. His judgment is strongly favorable: Karlin offers lucid proofs, valuable applications, and a substantial synthesis of recent research. Yet Morgenstern questions the economic adequacy of concentrating on two-person zero-sum games and treating standard equilibrium models as faithful representations of actual economic relations. The review’s central distinction is between excellence in developing mathematical methods and the choice of models capable of representing strategically interdependent participants.

Morgenstern begins with the volumes’ independent organization. Each repeats the same introductory chapter on the definition of a game and the min-max theorem, together with identical appendices and bibliography. This arrangement makes either volume usable separately but charges purchasers of both for considerable duplication. His joke about whose welfare this publishing scheme maximizes introduces a recurring concern: the merits of an arrangement depend on whose interests and circumstances it represents. More substantively, he identifies Karlin’s governing orientation:

Karlin writes as a mathematician, more interested in methods than in the realism of the models to which the methods are applied, although the two volumes contain numerous and most valuable applications to concrete, empirical situations and problems.

The qualification matters. Morgenstern does not equate mathematical abstraction with an absence of applications. He praises Karlin’s concrete examples while distinguishing the successful application of a technique from the realism of the model to which it is applied. The mathematical exposition earns especially emphatic approval:

Precision and clarity in formulation and proof are outstanding, a statement that deserves to be made since not all mathematicians or those who employ mathematics write clearly, contrary to what most laymen think.

Karlin uses the tools required for proof without displaying irrelevant mathematical knowledge. Exercises, subsequent solutions, and historical and bibliographical comments give the work a double function: it surveys advanced research, including unpublished or privately circulated results, and provides a textbook for mathematically sophisticated students. Morgenstern situates this achievement against the earlier mathematical economics of Marshall, Edgeworth, Walras, and Pareto. The contrast registers how far the discipline’s mathematical resources have advanced, without implying that greater technical power automatically produces better economic explanations.

Volume I receives the most attention because of its direct relevance to economists. Its first part develops two-person zero-sum games, with applications to bargaining and advertising; its second treats linear and nonlinear programming and economic models. Morgenstern finds the game-theoretic treatment substantial but identifies omissions with conceptual consequences: limited development of the extensive form, inadequate attention to the distinction between perfect and complete information, and almost no account of the utility theory underlying payoff matrices. The restriction to zero-sum games is particularly regrettable because economists may mistake this special case for the whole field. Against that narrowing, he emphasizes both active research and economic significance:

Furthermore, it is the n-person situation which is of decisive importance for economics, especially when the payoff is a nonzero sum, i.e., when all participants together gain or lose.

This objection supplies the bridge to Morgenstern’s criticism of equilibrium theory. He warmly endorses the presentation of programming and its mutually enriching connection with game theory. He also singles out the von Neumann model of an expanding economy as surpassing Walras and Pareto in scope and mathematical conception while retaining unexplored possibilities. His reservation concerns the apparent acceptance of standard equilibrium theories as realistic accounts of economic activity. The Walrasian construction, he argues, is a limiting case rather than an adequate representation of the central economic problem.

The decisive issue is the distribution of control over variables. An economic participant does not simply confront fixed conditions and choose a maximizing response: relevant conditions depend on other participants’ decisions. Nor can variables outside that participant’s control simply be assigned to statistical treatment.

In economic reality no maximum problem is given, since no participant controls all variables and cannot treat those he does not control as subject to statistical procedures.

For Morgenstern, the appropriate conceptual shift is therefore from isolated maximization toward the n-person game. He allows that exploring mathematical methods within the Lausanne framework is justified, but doubts their usefulness when analysis moves to the actual distribution of control. His criticism concerns the starting formulation of the economic problem, not the competence of its mathematical solution.

Volume II receives a shorter, appreciative account. It studies infinite games, chiefly two-person games with infinitely many available strategies, and applies them to situations where timing is decisive, including noisy, silent, and mixed duels. Morgenstern places this research in the context of work at the RAND Corporation, to which Karlin contributed. He especially values the final chapter on poker because that game offers a close model of political and economic negotiations. The review thus closes with an unambiguous recommendation while preserving its central qualification: Karlin’s work powerfully advances mathematical understanding, but economic relevance ultimately depends on representing strategic interaction rather than assuming fixed conditions for maximization.