## Zur Theorie der Gesellschaftsspiele. Offprint from Mathematische Annalen, Bd. 100, Heft 1/2, 1928. Bound with 36 other offprints, including five more by von Neumann, three of which deal with the mathematical foundations of quantum mechanics.

Berlin: Julius Springer, 1928.

First edition, extremely rare offprint, of von Neumann’s first and most important work on game theory, containing the first proof of the ‘minimax theorem’ which provides the key to the entire subject. “Of the many areas of mathematics shaped by his genius, none shows more clearly the influence of John von Neumann than the Theory of Games. This modern approach to problems of competition and cooperation was given a broad foundation in his superlative paper of 1928. In scope and youthful vigor this work can be compared only to his papers of the same period on the axioms of set theory and the mathematical foundations of quantum mechanics” (Kuhn & Tucker). “Quantitative mathematical models for games such as poker or bridge at one time appeared impossible, since games like these involve free choices by the players at each move, and each move reacts to moves of other players. However, in the 1920s John von Neumann single-handedly invented game theory, introducing the general mathematical concept of ‘strategy’ in a paper on games of chance [the offered paper]. This contained the proof of his ‘minimax’ theorem that says ‘a strategy exists that guarantees, for each player, a maximum payoff assuming that the adversary acts so as to minimize that payoff.’ The ‘minimax’ principle, a key component of the game-playing computer programs developed in the 1950s and 1960s by Samuel, Newell, Simon, and others, was more fully articulated and explored in *The Theory of Games and Economic Behavior*, co-authored by von Neumann and the Austrian economist Oskar Morgenstern” (Hook & Norman, *Origins of Cyberspace*, p. 473). “In 1928, between papers on mathematical logic and his first papers in physics, he ‘squeezed in’ his first and basic paper on the theory of games. There had been forerunners in the field (E. Zermelo, H. Borel, and H. Steinhaus, among others) by whom most of the concepts like play and counterplay, chance and deception, pure strategy and mixed strategy had been anticipated in one form or another. But von Neumann brought the results of others and his own new ones together, clearly and axiomatically, into twenty-five pages. In the briefest of footnotes he made reference to the analogy with some problems in economics, and in another footnote, added in proof, he made reference to a very recent note of Borel’s in which for the two-person zero-sum game the underlying minimax theorem was proposed, although not proven, as it was in von Neumann's paper” (Bochner, p. 444). “Edward Teller believed that ‘if a mentally superhuman race ever develops, its members will resemble Johnny von Neumann’ … ‘We can all think clearly, more or less, some of the time,’ says fellow Hungarian mathematician Paul Halmos, ‘but von Neumann’s clarity of thought was orders of magnitude greater than that of most of us, all the time … whenever I talked with von Neumann, I always had the impression that only he was fully awake’” (Dyson). Only one copy of this offprint in auction records (Sotheby’s 2000 & 2003). No copy on OCLC or KVK.

*Provenance*: From the collection of Manó (Emanuel) Beke (1862-1946), Hungarian mathematician; two offprints (nos. 31 and 36 in the list below) inscribed to Beke, handwritten index on front endpapers probably in Beke’s hand. Deaccessioned library stamps on the front free endpaper and the first page. Beke is known for his work on differential equations, determinants, and mathematical physics, and particularly for reforming the teaching of mathematics in Hungary. Since 1950 the János Bolyai Mathematical Society has awarded the Máno Beke Commemorative Prize for the teaching and popularization of mathematics.

“The role as founder is even more obvious for the theory of games, which von Neumann, in a 1928 paper [offered here], conjured – so to speak – out of nowhere. To give a quantitative mathematical model for games of chance such as poker or bridge might have seemed a priori impossible, since such games involve free choices by the players at each move, constantly reacting on each other. Yet von Neumann did precisely that, by introducing the general concept of ‘strategy’ (qualitatively considered a few years earlier by E. Borel) and by constructing a model that made this concept amenable to mathematical analysis” (DSB).

“The crucial innovation of von Neumann, which was to be both the keystone of his Theory of Games and the central theme of his later research in the area, was the assertion and proof of the Minimax Theorem. Ideas of pure and randomized strategies had been introduced earlier, especially by Êmile Borel. However, these efforts were restricted either to individual examples or, at best, to zero-sum two-person games with skew-symmetric payoff matrices. To paraphrase his own opinion, von Neumann did not view

the mere desire to mathematize strategic concepts and the straight formal definition of a pure strategy as the main agenda of an ‘initiator’ in the field, but felt that there was nothing worth publishing until the Minimax Theorem was proved …

“For any finite zero-sum two-person game in a normalized form, [the Minimax Theorem] asserts the existence of a unique numerical value, representing a gain for one player and a loss for the other, such that each can achieve at least this favorable an expectation from his own point of view by using a randomized (or mixed) strategy of his own choosing. Such strategies for the two players are termed optimal strategies and the unique numerical value, the minimax value of the game. This is the starting point of the von

Neumann-Morgenstern solution for cooperative games, where all possible partitions of the players into two coalitions are considered and the reasonable aspirations of the opposing coalitions in each partition measured by the minimax value of the strictly competitive two-party struggle between them. In the area of extensive games, the solution of games with perfect information by means of pure strategies assumes importance only by contrast to the necessity of randomizing in the general case. The Minimax Theorem reappears in a new guise, when von Neumann turned to analyze a linear model of production. Finally, in the hands of von Neumann, it was the source of a broad spectrum of technical results, ranging from his extensions of the Brouwer fixed-point theorem, developed for its proof, to new and unexpected methods for combinatorial problems …

“The impact of von Neumann’s Theory of Games extends far beyond the boundaries of this subject. By his example and through his accomplishments, he opened a broad new channel of two-way communication between mathematics and the social sciences. These sciences were fortunate indeed that one of the most creative mathematicians of the twentieth century concerned himself with some of their fundamental problems and constructed strikingly imaginative and stimulating models with which to attack their problems quantitatively. At the same time, mathematics received a vital infusion of fresh ideas and methods that will continue to be highly productive for many years to come. Von Neumann's interest in ‘problems of organized complexity,’ so important in the social sciences, went hand in hand with his pioneering development of large-scale high-speed computers. There is a great challenge for other mathematicians to follow his lead in grappling with complex systems in many areas of the sciences where mathematics has not yet penetrated deeply” (Kuhn & Tucker).

“That von Neumann turned his attention to the mathematical analysis of games, producing the minimax theorem in 1926, was not so much the result of a detached moment of inspiration, as a reflection of the fact that a number of his peers were doing similar work at the same time. In fact, in the first decades of this century, there existed among Hungarian and German mathematicians something of a ‘conversation’ about the mathematics of games, something which must be seen in the context of the period’s mathematical history. Two features of the latter are worth noting: first, the attempt to establish various areas of mathematics on a secure axiomatic basis, central to the so-called Hilbert program at Gottingen; and, second, an associated imperialistic drive to show how mathematical formalization could constitute a widely applicable tool of explanation, even in areas that were hitherto deemed unapproachable in mathematical terms …

“The second element concerns the success of Hilbert and others in extending the axiomatic approach to a range of areas, the most obvious being physics – the mathematical foundations of the kinetic theory of gases and quantum mechanics – less obvious being such social activities as parlor games. In this context, the first significant step was that taken by Hilbert’s Göttingen colleague and set theorist, Zermelo, who, in 1912, turned his attention to the mathematics of chess. In a lecture, ‘On the Application of Set Theory to the Theory of Chess,’ delivered to the International Congress of Mathematicians at Cambridge, he presented an inductive proof that the outcome of chess is strictly determined, i.e., either white can force a win, or black can force a win, or both sides can force at least a draw. While offering the entire exercise in the context of contemporaneous work on set theory, Zermelo also presents it as part of the attempt to push mathematics into as many realms as possible and to show how other phenomena, be they psychological or physical, may ultimately be ‘explained’ by rendering them accessible to mathematical interpretation. Of the analysis of chess, he says that it is not dealing with the practical method for games, but rather is simply giving an answer to the following question: can the value of a particular feasible position in a game for one of the players be mathematically and objectively decided, or can it at least be defined without resorting to more subjective psychological concepts?

“This paper gave rise to further work on chess by a number of mathematicians close to Zermelo, including the lesser-known Denes Konig and Laszlo Kalmar, and, of course, von Neumann, all of whom were familiar with each other’s work. In short, chess became a well-defined topic in German-speaking mathematical circles, and it was this interest in the relationship between set theory and parlor games which formed the intellectual setting for von Neumann’s work on the minimax theorem. This he first presented to the Göttingen Mathematical Society in December 1926. The paper contains mainly a long and difficult existence proof, based on functional calculus and topology, of the ‘solution’ for all two-person, zero-sum, games with a finite number of strategies. The mathematical concept of a game is completely axiomatized and two examples are offered of simple zero-sum games with solutions only in mixed strategies, ‘matching pennies’ and ‘paper, stone, scissors.’ He also treats the three-person zero-sum game, showing how the possibility of coalition formation introduces into such games a measure of indeterminacy, or ‘struggle.’ In preliminary remarks on games with more than three players, he introduces a ‘system of constants’ describing ‘the sum per play which [each] coalition of the players is able to obtain from the coalition of the other players,’ and conjectures that ‘the complex of valuations and coalitions in a game of strategy is determined by these constants alone. We have seen that this is true for *n* = 2, 3; for *n* > 3 a general proof has yet to be found.’ If this is the case, he concludes, then ‘we have brought all games of strategy into a natural and final normal form’. As in the papers by Zermelo and the others, mathematics is seen as something capable of penetrating the psychology of gaming …

“This is our first hint of the extent to which von Neumann’s view of the application of mathematics to the social domain was conditioned by the philosophy and standards of physical science. We might also note that, at this stage, the prime focus was the analysis of parlor games, not economics. Admittedly, von Neumann suggests that the formalism is tapping the deep structure of more than just simple parlor games, when he says that ‘any event – given the external conditions and the participants in the situation (provided that latter are acting of their own free will) – may be regarded as a game of strategy if one looks at the effect it has on the participants’. And, in a footnote, he even says that ‘[this] is the principal problem of classical economics: how is the absolutely selfish ‘homo economicus’ going to act under given external circumstances?’ But we would be mistaken to overemphasize the suggested economic orientation of this work: it was written for an audience of mathematicians and its purpose was to render abstract games amenable to mathematical treatment. It was not primarily concerned with economics: those links came later. Zermelo had begun with the game of chess; von Neumann was now, with characteristic audacity, trying to do the same for all zero-sum games, two-person, three-person, etc., groping for a completely general theory. It was part of a general effort to push mathematics to the limit, to show how pure, abstract, mathematics of the formal kind could constitute an ‘explanatory’ device, of not only the ‘natural,’ as for example in statistical and quantum mechanics, but also the ‘social’ where bluffing, outguessing, and cooperation among humans were involved. If the Hilbert era had any defining feature, it was this supreme faith in the explanatory powers of mathematical formalism” (Leonard, pp. 732-4).

Three of the other offprints by von Neumann in this collection (nos. 1, 2, 5) relate to his work on the mathematical foundations of quantum mechanics. “When he began work in that field in 1927, the methods used by its founders were hard to formulate in precise mathematical terms; ‘operators’ on ‘functions’ were handled without much consideration of their domain of definition or their topological properties; and it was blithely assumed that such ‘operators,’ when self-adjoint, could always be ‘diagonalized’ (as in the finite dimensional case), at the expense of introducing ‘Dirac functions’ as ‘eigenvectors.’ Von Neumann showed that mathematical rigor could be restored by taking as basic axioms the assumptions that the states of a physical system were points of a Hilbert space and that the measurable quantities were Hermitian (generally unbounded) operators densely defined in that space. This formalism, the practical use of which became available after von Neumann had developed the spectral theory of unbounded Hermitian operators [no. 2], has survived subsequent developments of quantum mechanics and is still the basis of non-relativistic quantum theory; with the introduction of the theory of distributions, it has even become possible to interpret its results in a way similar to Dirac’s original intuition.

“After 1927 von Neumann also devoted much effort to more specific problems of quantum mechanics, such as the problem of measurement and the foundation of quantum statistics and quantum thermodynamics, proving in particular an ergodic theorem for quantum systems [no. 1]” (DSB). The ergodic theorem is a fundamental principle of classical statistical mechanics according to which the time average of a physical quantity is equal to the average of the same quantity taken over ‘phase space’. Phase space is a description of all the possible configurations of the physical system – fixing a point in phase space corresponds to fixing all the positions and momenta of the constituent parts of the system. The main difficulty in obtaining a quantum mechanical version of this principle is that, because of the uncertainty principle, the positions and momenta cannot be fixed simultaneously, so ‘fixing a point in phase space’ no longer makes sense. Von Neumann summarises how he overcame this problem in the abstract of no. 1: “It is shown how to resolve the apparent contradiction between the macroscopic approach of phase space and the validity of the uncertainty relations. The main notions of statistical mechanics are re-interpreted in a quantum-mechanical way, the ergodic theorem and the H-theorem are formulated and proven (without assumptions of disorder), followed by a discussion of the physical meaning of the mathematical conditions characterizing their domain of validity [of the two theorems]” (Translation from Mehra & Rechenberg, p. 451). The problem had been considered earlier by Schrödinger, and he wrote admiringly of Von Neumann’s paper. “Erwin Schrödinger addressed von Neumann’s proof of the H-theorem directly in a letter to the author, dated 25 December 1929, which he opened by stating: ‘Your statistical paper did interest me quite a lot. I am quite happy about it, especially about the wonderful clarity and accuracy of the conceptions and the careful presentation of what has been achieved in all points’ (Schrödinger to von Neumann, 25 December 1929). Schrödinger recognized as the central issue of von Neumann’s considerations the introduction of the concepts ‘operator of a macroscopically measurable quantity’ and the ‘orthogonal system of the macroscopic observation’. ‘This is an extraordinarily happy move, and I believe that it will obtain a significance far beyond the present problems’” (*ibid*., p. 460).

“Von Neumann’s ideas on rings of operators broke entirely new ground. He was well acquainted with the non-commutative algebra beautifully developed by Emmy Noether and Emil Artin in the 1920s and he realized how these concepts simplified and illuminated the theory of matrices. This probably provided the motivation for extending such concepts to algebras consisting of (bounded) operators in a given separable Hilbert space, to which he gave the vague name ‘rings of operators’ and which are now known as ‘von Neumann algebras.’ He introduced their theory [in no. 5] in the same year as his first paper on unbounded operators [no. 2], and from the beginning he had the insight to select the two essential features that would allow him further progress: the algebra must be self-adjoint (that is, for any operator in the algebra, its adjoint must also belong to the algebra) and closed under the strong topology of operators” (DSB). “This last work [no. 5] won for him the admiration of mathematicians” (Bochner, p. 444).

“In 1929 he published the important result [no. 3] that a group of linear transformations in a finite dimensional space, if closed, is a Lie group. Élie Cartan, who was a geometer, immediately recognized that the theorem is a special case of the general proposition that a closed subgroup of a Lie group is again a Lie group, and he proved it by the same procedure, although in an abbreviated form. There is hardly a young mathematician today who realizes that this familiar general proposition was von Neumann’s discovery originally” (*ibid*., p. 441). In 1933 von Neumann used the same ideas to solve Hilbert’s “fifth problem” for compact groups, proving that such a group admits a Lie group structure once it is locally homeomorphic with Euclidean space.

Of the works by authors other than von Neumann, one of the most significant is the triple offprint no. 33. In these papers Lanczos showed how to derive Dirac’s equation for the electron from a more fundamental system that predicted that spin 1/2 particles should come in pairs. Today, these pairs can unambiguously be interpreted as isospin doublets. From the same fundamental equation, Lanczos derived also the correct form of the wave equation of massive spin 1 particles that would be rediscovered in 1936 by Proca.

Bochner, ‘John von Neumann 1903-1957. A Biographical Memoir,’ National Academy of Sciences, 1958. Dyson, *Turing’s Cathedral*, 2012. Kuhn & Tucker, ‘John von Neumann’s work in the theory of games and mathematical economics,’ *Bulletin of the American Mathematical Society* 64 (1958), pp. 100-122. Leonard, ‘From Parlor Games to Social Science: Von Neumann, Morgenstern, and the Creation of Game Theory 1928-1944,’ *Journal of Economic Literature* 33 (1995), pp. 730-761. Mehra & Rechenberg, *The Historical Development of Quantum Theory*, Vol. 6, 2001.

CONTENTS

- VON NEUMANN. Beweis des Ergodensätzes und des H-Theorems in der neuen Mechanik,
*Zeitschrift für Physik*, Bd. 57, Heft 1/2 (1929), pp. [i, blank], 30-70. Original printed front wrapper. - VON NEUMANN. Allgemeine Eigenwerttheorie Hermitescher Funktionaloperatoren,
*Mathematische Annalen*, Bd. 102, Heft 1 (1929), pp. [49], 50-131. Original printed front wrapper. - VON NEUMANN. Über die analytischen Eigenschaften von Gruppen linearer Transformationen und ihrer Darstellungen,
*Mathematische Zeitschrift*, Bd. 30, Heft 1/2 (1929), pp. [3], 4-42 - VON NEUMANN. Zum Beweise des Minkowskischen Satzes über Linearformen.
*Mathematische Zeitschrift*, Bd. 30, Heft 1/2 (1929), pp. [1], 2. Single leaf. - VON NEUMANN. Zur Algebra der Funktionaloperatoren und Theorie der normalen Operatoren,
*Mathematische Annalen*, Bd. 102, Heft 3 (1929), pp. [i, blank], [370], 371-427. Original printed front wrapper. - VON NEUMANN. Zur Theorie der Gesellschaftsspiele,
*Mathematische Annalen*, Bd. 100, Heft 1/2 (1928), pp. [295], 296-320. Original printed wrappers (front and rear). - KALMÁR, Laszló. Über die Abschätzung der Koeffizientensumme Dirichletscher Reihen
*, Acta Litterarum ac Scientiarum Regiae Universitatis Hungaricae Francisco-Josephinae*, Tom. IV, Fasc. III (1929), pp. 155-181. Original front printed wrapper. - KALMÁR, Laszló. Über die mittlere Anzahl der Produktdarstellungen der Zahlen,
*Acta Litterarum ac Scientiarum Regiae Universitatis Hungaricae Francisco-Josephinae*, Tom. V, Fasc. II (1931), pp. 95-107. Original printed front wrapper. - KALMÁR, Laszló. Eine Bemerkung zur Entscheidungstheorie,
*Acta Litterarum ac Scientiarum Regiae Universitatis Hungaricae Francisco-Josephinae*, Tom. IV, Fasc. IV (1929), pp. 248-252, [2, blank]. Original printed front wrapper. - Duplicate of no. 7. One blank leaf at end with pencil calculations on verso.
- KALMÁR, Laszló. Review of: Joszef Kurschak,
*Matematikai versenytételek*. Szeged, 1929. Single leaf printed on recto only. - SZÜCS [SCHLESINGER], Adolf. A valószinüségszámitás néhány fontos tételéröl.
*Középiskolai Matematikai és Fizikai*, vol. VI (1930), pp. [3], 4-11. Original printed front wrapper. - VERESS, Pál. Euler poliédertételéröl,
*Középiskolai Matematikai és Fizikai*, vol. VII (1930), pp. [3], 4-9. Original front printed wrapper. - VERESS, Pál. A munkabérek hatása a szociális biztosítás díjára,
*Középiskolai Matematikai és Fizikai*, vol. VII (1930), pp. [1], 2-5, [2, blank]. Self wrappers. - SZÁSZ, Ottó. Korlátos hatványsorok együtthatóiról.
*Mathematischer und Naturwissenschaftlicher Anzeiger der Ungarischen Akademie der Wissenschaften*, Bd. XLIII (1926), pp. [i, blank], [488], 489-502. Self wrappers. - SZÁSZ, Ottó. Uber die Koeffizienten Beschrankter Potenzreihen,
*Mathematischer und Naturwissenschaftlicher Anzeiger der Ungarischen Akademie der Wissenschaften*, Bd. XLIII (1926). Single sheet printed on recto only. - SZÁSZ, Ottó. Korlátos hatványsorokról,
*Mathematischer und Naturwissenschaftlicher Anzeiger der Ungarischen Akademie der Wissenschaften*, Bd. XLIII (1926), pp. [i, blank], [504], 505-520. Self wrappers. - JORDAN, Charles [Károly JORDAN]. Sur la détermination de la tendence séculaire des grandeurs statistiques par la méthode des moindres carrés,
*Journal de la Société Hongroise de Statistique*, no. 4 (1929), pp. [3], 4-35, [1]. Original front printed wrapper. - JORDAN, Charles. Véletlen, valószínüség és természeti törvény,
*Athenaeum*, nos. 5-6 (1929), pp. [3], 4-31. Original front printed wrapper. - JUVANCZ, Iréneusz. A variatiosorok mellékcsúcsairól,
*Ranchburg Emlékkönyv*(1929), pp. [3], 4-8. Original front printed wrapper. - HUSZÁR, Géza. A kamatlábfeladat s a Waring-formulák,
*Különlenyomat a Kereskedelmi Szakoktatás*, no. 3 (1928), pp. [119], 120-132. Self wrappers. - HUSZÁR, Géza. A normáljáradék kamatlábfeladatának racionális megoldásai,
*Különlenyomat a Kereskedelmi Szakoktatás*, nos. 1-2 (1929), pp. [67], 68-86 [Parts 1 & 2], [i, blank], [166], 167-181 [Parts 3-6]. Self wrappers. Inscribed by the author (inscription slightly cropped). - HUSZÁR, Géza. Az
*x*^{n}^{+1 }–*x*= 0 egyenlet gyökeiröl,^{n}+ p*Különlenyomat a Matematikai és Fizikai Lapok*, vol. XXXVII (1930), pp. [25], 26-35. Original stiff printed front wrapper. Inscribed by the author. - Duplicate of no. 21.
- REVESZ, Margit. A kutató lélektana,
*Tehetség-problémák*(1930), pp. [95], 96-109, [1]. Self wrappers. Inscribed by the author. - JORDAN, Charles. A matematikai reménységröl,
*Középiskolai Matematikai és Fizikai*, vol. VI (1929), pp. [3], 4-9. Original printed front wrapper. - JORDAN, Charles. Sur une formule d’interpolation dérivée de la formule d’Everett,
*Metron*, vol. 7, no. 3 (1928), pp. [1], 2-6, [2, blank]. Self wrappers. - JORDAN, Charles. Sur des polynomes analogues aux polynomes de Bernoulli et sur des formules de summation analogues a cette de Maclaurin-Euler, Acta Scientiarum Mathematicarum (Szeged), vol. 4 (1928-29), pp. [i, blank], 130-150. Self wrappers.
- NEUGEBAUER, Tibor. A Kerr effektus elmélete a hullámmechanika alapján. Doctoral Dissertation. Budapest, 1931, pp. [1-3], 4-31.
- KÖRÖSY, Franz von. Durchgang langsamer Elektronen durch Edelgase,
*Zeitschrift für Physik*, Bd. 51, Heft 5/6 (1928), pp. [i, blank], 420-428. Original printed wrappers (front and rear). Inscribed by the author (inscription cropped). - HÁMOS, L. von. Bemerkungen zur Wirkungsweise des elektrooptischen Momentverschlusses von I. W. Beams und Deutung einiger damit ausgeführter Versuche,
*Zeitschrift für Physik*, Bd. 52, Heft 7/8 (1928), pp. 549-554. Original printed wrappers (front and rear). Inscribed by the author (inscription cropped). - BRODY, Imry. Elektrische Polarisation in Isolatoren, hervorgerufen durch Beschleunigung,
*Zeitschrift für Physik*, Bd. 52, Heft 11/12 (1929), pp. [1, blank], 884-889. Original printed wrappers (front and rear). - LANCZOS, Cornelius. Die tensoranalytischen Beziehung der Diracsche Gleichung; Zur kovarianten Formulierung der Diracsche Gleichung; Die Erhaltungsätze in der feldmässigen Darstellung der Diracsche Gleichung,
*Zeitschrift für Physik*, Bd. 52, Heft 11/12 (1929), pp. 447-493. Original printed wrappers (front and rear). Inscribed by the author (inscription cropped). - PATAI, Emerich. Eine methode zur Bestimmung von Kontaktpotentialen,
*Zeitschrift für Physik*, Bd. 59, Heft 9/10 (1930), pp. 697-699. Stamped ‘Ubereicht vom Verfasser’ in red ink. Original printed wrappers (front and rear). - FORRÓ, Magdalene & PATAI, Emerich. Messungen von Kontaktpotentialen einiger Metalle.
*Zeitschrift für Physik*, Bd. 63, Heft 7/8 (1930), pp. [1, blank], 444-457. Stamped ‘Ubereicht vom Verfasser’ in red ink. Original printed wrappers (front and rear). - Duplicate of no. 34. Stamped ‘Ubereicht vom Verfasser’ in red ink and inscribed by the author.
- GOLDHIZER, Karl. Methodische untersuchungen zu den bevölkerungsstatischen Grundlagen der schweizerischen Alters- und Hinterlassenen-versicherung,
*Zeitschrift für schweizerische Statistik und Volkswirtschaft,*Jahrg. 66, Heft 4 (1930), pp. [501], 502-509. Self wrappers.

Thirty-seven offprints bound in one vol., 8vo (224 x 152mm), a few light creases, probably for posting, nos. 15-17 strengthened in gutter. Contemporary black half-cloth with original printed wrappers bound in (front joint cracked but firm, minor wear to extremities).

Item #5327

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Price:
$28,500.00
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