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An early attempt to measure risk was the 1738 paper of Bernoulli in which he describes the well‐known Saint Petersburg paradox. Subsequent writers have considered this game to draw conclusions about the nature of risk or to test newly devised risk models. We analyze the paradox, evaluate various theories which have been advanced to resolve it, and briefly examine the implications of these theories on the wider area of risk analysis.

The scientific work of Oskar Morgenstern -- I: Game Theory -- Vers une théorie générale des jeux positionnels -- Erweitertes Sattelpunktkriterium und ?-Rand-Spiel -- Differential games with active and passive variables -- Stochastic linearisation of indeterminateness -- The existence problem for solutions -- Values of games with a priori unions -- Core, Lindahl equilibria and revelation of preferences -- Remark on the transfer operator and the value-equilibrium equivalence hypothesis -- ?-Gleichgewichtspunkte in n-Personenspielen -- A simple game model of kidnapping -- An n-person bargaining game in the extensive form -- An extreme application of core theory -- Optimality and dynamics in coalitional games -- II: Utility Theory and Related Topics -- Expected utility theories: a review note -- The Bernoulli principle and the Dirichlet problem -- Bemerkungen zur Widerspruchsfreiheit der Axiome in der Theorie der revealed preference -- Dynamic utility functions -- On representations of social preferences — an algebraic approach -- Eine Axiomatisierung des erwarteten Nutzens -- III: Economic Models -- Externalities and interdependence in a von Neumann growth model -- International trade and game theory in a context of economic growth -- Economic equilibrium under technological changes -- Infinite-dimensional von Neumann models -- Some variations on the Mardo? model -- IV: Economic Theory -- On the consumer demand theory under uncertainty -- Anwendung eines verallgemeinerten Periodizitätsbegriffes in makroökonomischen Modellen -- A paradox in consumption theory -- An application of the theorem of Whitney in joint production theory -- Spieltheoretische Behandlung der Preisbildung vor und nach Unternehmenszusammenschlüssen -- A new approach to modelling some economic problems -- Economically efficient and politically sustainable economic contraction -- Dynamic indirect production functions -- Dynamic limit pricing in a mature market -- Coalitions and clubs — aspects of their formation and economic efficiency -- V: Econometrics and Statistics -- An econometric study of copper market dynamics -- Bemerkung zur Abschätzung des Wertes bei Stop-Problemen -- On the identifiability of continuous time economic models -- Eine Bemerkung zur Anwendung der MINIQUE-Methode -- The use of preliminary data in economic time-series prediction -- Minimaxschätzungen bei Vorbewertung -- Statistical aspects of economic aggregation -- On the transition from pattern cognition to model building -- VI: Different Topics -- Determination of risk preferences to faciliate customer-portfolio manager interactions -- When is it advantageous to cast a negative vote? -- A bond-share portfolio maximizing von Neumann-Morgenstern utility of present real worth under inflation -- Utility and morality -- A plea for preordinators -- The cost assignment of the cooperative water resource development — A game theoretical approach — -- Addendum -- The Genesis of Dynamic Systems Governed by Metzler Matrices -- Extremal Properties of Equilibria in von Neumann Models -- Banks, Insurance and Futures Markets of a Trading Economy with Money, Exogenous Uncertainty and an Optimal Bankruptcy Rule -- A New Route to the Turnpike -- A Bibliography of the Work of Oskar Morgenstern.

SummaryThe gambler's ruin.When a single trial has two possible outcomes A and B, with probabilities p and q(p+q= 1), a succession of these trials forms a so‐called Bernoulli chain. The well‐known result for the probability of n times A and m times B isIn this article we consider the ruin problem, in which the initial capitals of the gamblers are a and b, respectively. In stead of a Bernoulli chain we then have a Markoff chain, with coefficients that are less simple than the ordinary binomial coefficients.A more general expression (formula 1) is obtained for the probability distribution of the gambler's profit after a certain number of games, provided none of them became ruined beforehand. The probability for ruin after a certain number of games is a special case, similar to the results of Lagrange, Laplace and others, but appears in a form, more suitable for numerical calculations.Some other results, obtained through the same method as developed in this paper are indicated.

AbstractWhen facing uncertain demand, several firms may consider pooling their inventories leading to the emergence of two key contractual issues. How much should each produce or purchase for inventory purposes? How should inventory be allocated when shortages occur to some of the firms? Previously, if the allocations issue was considered, it was undertaken through evaluation of the consequences of an arbitrary priority scheme. We consider both these issues within a Nash bargaining solution (NBS) cooperative framework. The firms may not be risk neutral, hence a nontransferable utility bargaining game is defined. Thus the physical pooling mechanism itself must benefit the firms, even without any monetary transfers. The firms may be asymmetric in the sense of having different unit production costs and unit revenues. Our assumption with respect to shortage allocation is that a firm not suffering from a shortfall, will not be affected by any of the other firms' shortages. For two risk neutral firms, the NBS is shown to award priority on all inventory produced to the firm with higher ratio of unit revenue to unit production cost. Nevertheless, the arrangement is also beneficial for the other firm contributing to the total production. We provide examples of Uniform and Bernoulli demand distributions, for which the problem can be solved analytically. For firms with constant absolute risk aversion, the agreement may not award priority to any firm. Analytically solvable examples allow additional insights, e.g. that higher risk aversion can, for some problem parameters, cause an increase in the sum of quantities produced, which is not the case in a single newsvendor setting. © 2008 Wiley Periodicals, Inc. Naval Research Logistics, 2008