Celebratio Mathematica — Blackwell

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Works connected to Kenneth J. Arrow

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K. J. Arrow, D. Blackwell, and M. A. Girshick: “Bayes and minimax solutions of sequential decision problems,” Econometrica 17 : 3/4 (July–October 1949), pp. 213–244. MR 0032173 Zbl 0034.07504 article

The present paper deals with the general problem of sequential choice among several actions, where at each stage the options available are to stop and take a definite action or to continue sampling for more information. There are costs attached to taking inappropriate action and to sampling. A characterization of the optimum solution is obtained first under very general assumptions as to the distribution of the successive observations and the costs of sampling; then more detailed results are given for the case where the alternative actions are finite in number, the observations are drawn under conditions of random sampling, and the cost depends only on the number of observations. Explicit solutions are given for the case of two actions, random sampling, and linear cost functions.

Kenneth J. Arrow	Related
Meyer Abraham Girshick	Related

K. J. Arrow, E. W. Barankin, and D. Blackwell: “Admissible points of convex sets,” pp. 87–91 in Contributions to the theory of games, vol. II. Edited by H. W. Kuhn and A. W. Tucker. Annals of Mathematics Studies 28. Princeton University Press, 1953. MR 0054919 Zbl 0050.14203 incollection

A point \( s \) of a closed convex subset \( S \) of \( k \)-space is admissible if there is no \( t\in S \) with \( t_i\leq s_i \) for all \( i=1,\dots \), \( k \), \( t\neq s \). An example is given in which the set \( A \) of admissible points is not closed.

Let \( P \) be the set of vectors \( p=(p_1,\dots \), \( p_k) \) with \( p_i > 0 \) and \( \sum_1^k p_i=1 \), let \( B(p) \) be the set of \( s\in S \) with \[ (p,s)=\min_{t\in S}(p,t) ,\] and let \( B=\sum B(p) \), so that \( B \) consists of exactly those points of \( S \) at which there is a supporting hyperplane whose normal has positive components.

\( B\subset A\subset \overline{B} \). If \( S \) is determined by a finite set, there is a finite set \( p_1,\dots \), \( p_N \), with \( p_j\in P \), such that \( B=\sum_{j=1}^N B(p_j) \), so that, since \( B(p) \) is closed for fixed \( p \), \( B = A = \overline{B} \).

Albert William Tucker	Related
Kenneth J. Arrow	Related
Edward William Barankin	Related
Harold William Kuhn	Related