Special case

Abstract: The assignment game is a model for a two-sided market in which a product that comes in large, indivisible units (e.g., houses, cars, etc.) is exchanged for money, and in which each participant either supplies or demands exactly one unit. The units need not be alike, and the same unit may have different values to different participants. It is shown here that the outcomes in thecore of such a game — i.e., those that cannot be improved upon by any subset of players — are the solutions of a certain linear programming problem dual to the optimal assignment problem, and that these outcomes correspond exactly to the price-lists that competitively balance supply and demand. The geometric structure of the core is then described and interpreted in economic terms, with explicit attention given to the special case (familiar in the classic literature) in which there is no product differentiation — i.e., in which the units are interchangeable. Finally, a critique of the core solution reveals an insensitivity to some of the bargaining possibilities inherent in the situation, and indicates that further analysis would be desirable using other game-theoretic solution concepts.

Abstract: (1.) If measurements be made of the same part or organ in several hundred or thousand specimens of the same type or family, and a curve be constructed of which the abscissa x represents the size of the organ and the ordinate y the number of specimens falling within a definite small range δx of organ, this curve may be termed a frequency-curve . The centre or origin for measurement of the organ may, if we please, be taken at the mean of all the specimens measured. In this case the frequency-curve may be looked upon as one in which the frequency—per thousand or per ten thousand, as the case may be—of a given small range of deviations from the mean, is plotted up to the mean of that range. Such frequency-curves play a large part in the mathematical theory of evolution, and have been dealt with by Mr. F. Galton, Professor Weldon, and others. In most cases, as in the case of errors of observation, they have a fairly definite symmetrical shape and one that approaches with a close degree of approximation to the well-known error or probability-curve. A frequency-curve, which, for practical purposes, can be represented by the error curve, will for the remainder of this paper be termed a normal curve . When a series of measurements gives rise to a normal curve, we may probably assume something approaching a stable condition; there is production and destruction impartially round the mean. In the case of certain biological, sociological, and economic measurements there is, however, a well-marked deviation from this normal shape, and it becomes important to determine the direction and amount of such deviation. The asymmetry may arise from the fact that the units grouped together in the measured material are not really homogeneous. It may happen that we have a mixture of 2, 3, . . . n homogeneous groups, each of which deviates about its own mean symmetrically and in a manner represented with sufficient accuracy by the normal curve. Thus an abnormal frequency-curve may be really built up of normal curves having parallel but not necessarily coincident axes and different parameters. Even where the material is really homogeneous, but gives an abnormal frequency-curve the amount and direction of the abnormality will be indicated if this frequency-curve can be split up into normal curves. The object of the present paper is to discuss the dissection of abnormal frequency-curves into normal curves. The equations for the dissection of a frequency-curve into n normal curves can be written down in the same manner as for the special case of n = 2 treated in this paper; they require us only to calculate higher moments. But the analytical difficulties, even for the case of n = 2, are so considerable, that it may be questioned whether the general theory could ever be applied in practice to any numerical case. There are reasons, indeed, why the resolution into two is of special importance. A family probably breaks up first into two species, rather than three or more, owing to the pressure at a given time of some particular form of natural selection; in attempting to procure an absolutely homogeneous material, we are less likely to have got a mixture of three or more heterogeneous groups than of two only. Lastly, even where the heterogeneity may be threefold or more, the dissection into two is likely to give us, at any rate, an approximation to the two chief groups. In the case of homogeneous material, with an abnormal frequency-curve, dissection into two normal curves will generally give us the amount and direction of the chief abnormality. So much, then, may be said of the value of the special case dealt with here.

Abstract: Regularization Networks and Support Vector Machines are techniques for solving certain problems of learning from examples – in particular, the regression problem of approximating a multivariate function from sparse data. Radial Basis Functions, for example, are a special case of both regularization and Support Vector Machines. We review both formulations in the context of Vapnik's theory of statistical learning which provides a general foundation for the learning problem, combining functional analysis and statistics. The emphasis is on regression: classification is treated as a special case.

Abstract: WE SHALL DEFINE a syndicate to be a group of individual decision makers who must make a common decision under uncertainty, and who, as a result, will receive jointly a payoff to be shared among them. Our concern is to analyze the decision process of a syndicate when the members have diverse risk tolerances and/or diverse probability assessments of the uncertain events affecting the payoff. Of particular interest is the possiblity of constructing a surrogate "group utility function" and a surrogate "group probability assessment." Such constructions potentially have a role in the theory of finance; e.g., for determining the forms of organizational charters and financial instruments, as well as the modes of delegating the group decision process to professional managers. The present treatment, however, is confined to tractable features embodying only a small measure of the complexity of practical situations. Of comparable importance are the ramifications for welfare theory; in particular, we shall be able to specify conditions under which Pareto optimal behavior by the group satisfies the Savage axioms [15] foy consistent decision making under uncertainty, and to isolate the inconsistent characteristics in the contrary case. Arrow's original treatise [1] has been the source of most of the work on group decision theory. Marschak [13], Radner [17], and Bower [6] have considered the case of a team, in which there is a joint utility function for the members. Harsanyi [9] and Theil [16] have considered the criterion that the group decisions satisfy the Von Neumann-Morgenstern axioms, and others. Madansky [12] has imposed the "external Bayes axiom" in the case of a common utility function but differing probability assessments among the members. Christenson [7] has constructed an axiomatic system for the case of an investment banking syndicate that is a special case of the present study, except for certain institutional factors. Borch [3, 4, 5]